3.646 \(\int \frac {\sqrt {d+e x}}{(a+c x^2)^3} \, dx\)

Optimal. Leaf size=849 \[ \frac {\sqrt {d+e x} x}{4 a \left (c x^2+a\right )^2}+\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{32 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{32 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt {c} (d+e x)-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{64 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt {c} (d+e x)+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{64 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\sqrt {d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (c x^2+a\right )} \]

[Out]

1/4*x*(e*x+d)^(1/2)/a/(c*x^2+a)^2+1/16*(a*d*e+(5*a*e^2+6*c*d^2)*x)*(e*x+d)^(1/2)/a^2/(a*e^2+c*d^2)/(c*x^2+a)+1
/64*e*arctanh((-c^(1/4)*2^(1/2)*(e*x+d)^(1/2)+(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))/(d*c^(1/2)-(a*e^2+c*d^2)^
(1/2))^(1/2))*(6*c^(3/2)*d^3+8*a*d*e^2*c^(1/2)+(5*a*e^2+6*c*d^2)*(a*e^2+c*d^2)^(1/2))/a^2/c^(3/4)/(a*e^2+c*d^2
)^(3/2)*2^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2)-1/64*e*arctanh((c^(1/4)*2^(1/2)*(e*x+d)^(1/2)+(d*c^(1/2)
+(a*e^2+c*d^2)^(1/2))^(1/2))/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2))*(6*c^(3/2)*d^3+8*a*d*e^2*c^(1/2)+(5*a*e^2+
6*c*d^2)*(a*e^2+c*d^2)^(1/2))/a^2/c^(3/4)/(a*e^2+c*d^2)^(3/2)*2^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2)-1/
128*e*ln((e*x+d)*c^(1/2)+(a*e^2+c*d^2)^(1/2)-c^(1/4)*2^(1/2)*(e*x+d)^(1/2)*(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/
2))*(6*c^(3/2)*d^3+8*a*d*e^2*c^(1/2)-(5*a*e^2+6*c*d^2)*(a*e^2+c*d^2)^(1/2))/a^2/c^(3/4)/(a*e^2+c*d^2)^(3/2)*2^
(1/2)/(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2)+1/128*e*ln((e*x+d)*c^(1/2)+(a*e^2+c*d^2)^(1/2)+c^(1/4)*2^(1/2)*(e*
x+d)^(1/2)*(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))*(6*c^(3/2)*d^3+8*a*d*e^2*c^(1/2)-(5*a*e^2+6*c*d^2)*(a*e^2+c*
d^2)^(1/2))/a^2/c^(3/4)/(a*e^2+c*d^2)^(3/2)*2^(1/2)/(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 2.88, antiderivative size = 849, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {737, 823, 827, 1169, 634, 618, 206, 628} \[ \frac {\sqrt {d+e x} x}{4 a \left (c x^2+a\right )^2}+\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{32 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{32 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt {c} (d+e x)-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{64 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt {c} (d+e x)+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{64 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\sqrt {d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (c x^2+a\right )} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + e*x]/(a + c*x^2)^3,x]

[Out]

(x*Sqrt[d + e*x])/(4*a*(a + c*x^2)^2) + (Sqrt[d + e*x]*(a*d*e + (6*c*d^2 + 5*a*e^2)*x))/(16*a^2*(c*d^2 + a*e^2
)*(a + c*x^2)) + (e*(6*c^(3/2)*d^3 + 8*a*Sqrt[c]*d*e^2 + Sqrt[c*d^2 + a*e^2]*(6*c*d^2 + 5*a*e^2))*ArcTanh[(Sqr
t[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] - Sqrt[2]*c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(3
2*Sqrt[2]*a^2*c^(3/4)*(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(6*c^(3/2)*d^3 + 8*a*S
qrt[c]*d*e^2 + Sqrt[c*d^2 + a*e^2]*(6*c*d^2 + 5*a*e^2))*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] + Sqrt[
2]*c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(32*Sqrt[2]*a^2*c^(3/4)*(c*d^2 + a*e^2)^(3/2
)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(6*c^(3/2)*d^3 + 8*a*Sqrt[c]*d*e^2 - Sqrt[c*d^2 + a*e^2]*(6*c*d^
2 + 5*a*e^2))*Log[Sqrt[c*d^2 + a*e^2] - Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] +
Sqrt[c]*(d + e*x)])/(64*Sqrt[2]*a^2*c^(3/4)*(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (e*
(6*c^(3/2)*d^3 + 8*a*Sqrt[c]*d*e^2 - Sqrt[c*d^2 + a*e^2]*(6*c*d^2 + 5*a*e^2))*Log[Sqrt[c*d^2 + a*e^2] + Sqrt[2
]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/(64*Sqrt[2]*a^2*c^(3/4)*(c
*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 737

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(x*(d + e*x)^m*(a + c*x^2)^(p + 1)
)/(2*a*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(d + e*x)^(m - 1)*(d*(2*p + 3) + e*(m + 2*p + 3)*x)*(a + c*x^2
)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && (LtQ[m, 1]
|| (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 827

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]

Rule 1169

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {\sqrt {d+e x}}{\left (a+c x^2\right )^3} \, dx &=\frac {x \sqrt {d+e x}}{4 a \left (a+c x^2\right )^2}-\frac {\int \frac {-3 d-\frac {5 e x}{2}}{\sqrt {d+e x} \left (a+c x^2\right )^2} \, dx}{4 a}\\ &=\frac {x \sqrt {d+e x}}{4 a \left (a+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {\int \frac {\frac {1}{4} c d \left (12 c d^2+13 a e^2\right )+\frac {1}{4} c e \left (6 c d^2+5 a e^2\right ) x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx}{8 a^2 c \left (c d^2+a e^2\right )}\\ &=\frac {x \sqrt {d+e x}}{4 a \left (a+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {1}{4} c d e \left (6 c d^2+5 a e^2\right )+\frac {1}{4} c d e \left (12 c d^2+13 a e^2\right )+\frac {1}{4} c e \left (6 c d^2+5 a e^2\right ) x^2}{c d^2+a e^2-2 c d x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{4 a^2 c \left (c d^2+a e^2\right )}\\ &=\frac {x \sqrt {d+e x}}{4 a \left (a+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (-\frac {1}{4} c d e \left (6 c d^2+5 a e^2\right )+\frac {1}{4} c d e \left (12 c d^2+13 a e^2\right )\right )}{\sqrt [4]{c}}-\left (-\frac {1}{4} c d e \left (6 c d^2+5 a e^2\right )-\frac {1}{4} \sqrt {c} e \sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )+\frac {1}{4} c d e \left (12 c d^2+13 a e^2\right )\right ) x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{8 \sqrt {2} a^2 c^{5/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (-\frac {1}{4} c d e \left (6 c d^2+5 a e^2\right )+\frac {1}{4} c d e \left (12 c d^2+13 a e^2\right )\right )}{\sqrt [4]{c}}+\left (-\frac {1}{4} c d e \left (6 c d^2+5 a e^2\right )-\frac {1}{4} \sqrt {c} e \sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )+\frac {1}{4} c d e \left (12 c d^2+13 a e^2\right )\right ) x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{8 \sqrt {2} a^2 c^{5/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ &=\frac {x \sqrt {d+e x}}{4 a \left (a+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {\left (e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{64 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{64 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{64 a^2 c \left (c d^2+a e^2\right )^{3/2}}+\frac {\left (e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{64 a^2 c \left (c d^2+a e^2\right )^{3/2}}\\ &=\frac {x \sqrt {d+e x}}{4 a \left (a+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{64 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{64 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\left (e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-x^2} \, dx,x,-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{32 a^2 c \left (c d^2+a e^2\right )^{3/2}}-\frac {\left (e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-x^2} \, dx,x,\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{32 a^2 c \left (c d^2+a e^2\right )^{3/2}}\\ &=\frac {x \sqrt {d+e x}}{4 a \left (a+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}-\sqrt {2} \sqrt {d+e x}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{32 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+\sqrt {2} \sqrt {d+e x}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{32 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{64 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{64 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ \end {align*}

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Mathematica [A]  time = 0.80, size = 412, normalized size = 0.49 \[ \frac {\frac {2 (d+e x)^{3/2} \left (5 a^2 e^3+a c d e (3 d+8 e x)+6 c^2 d^3 x\right )}{a+c x^2}+\frac {\sqrt {\sqrt {c} d-\sqrt {-a} e} \left (5 a^2 e^4+6 \sqrt {-a} c^{3/2} d^3 e+19 a c d^2 e^2+8 \sqrt {-a} a \sqrt {c} d e^3+12 c^2 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )-\sqrt {\sqrt {-a} e+\sqrt {c} d} \left (5 a^2 e^4-6 \sqrt {-a} c^{3/2} d^3 e+19 a c d^2 e^2+8 (-a)^{3/2} \sqrt {c} d e^3+12 c^2 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )-4 \sqrt {-a} c^{3/4} d e \sqrt {d+e x} \left (4 a e^2+3 c d^2\right )}{\sqrt {-a} c^{3/4}}+\frac {8 a (d+e x)^{3/2} \left (a e^2+c d^2\right ) (a e+c d x)}{\left (a+c x^2\right )^2}}{32 a^2 \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d + e*x]/(a + c*x^2)^3,x]

[Out]

((8*a*(c*d^2 + a*e^2)*(a*e + c*d*x)*(d + e*x)^(3/2))/(a + c*x^2)^2 + (2*(d + e*x)^(3/2)*(5*a^2*e^3 + 6*c^2*d^3
*x + a*c*d*e*(3*d + 8*e*x)))/(a + c*x^2) + (-4*Sqrt[-a]*c^(3/4)*d*e*(3*c*d^2 + 4*a*e^2)*Sqrt[d + e*x] + Sqrt[S
qrt[c]*d - Sqrt[-a]*e]*(12*c^2*d^4 + 6*Sqrt[-a]*c^(3/2)*d^3*e + 19*a*c*d^2*e^2 + 8*Sqrt[-a]*a*Sqrt[c]*d*e^3 +
5*a^2*e^4)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[-a]*e]] - Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*(12*c^
2*d^4 - 6*Sqrt[-a]*c^(3/2)*d^3*e + 19*a*c*d^2*e^2 + 8*(-a)^(3/2)*Sqrt[c]*d*e^3 + 5*a^2*e^4)*ArcTanh[(c^(1/4)*S
qrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e]])/(Sqrt[-a]*c^(3/4)))/(32*a^2*(c*d^2 + a*e^2)^2)

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fricas [B]  time = 1.15, size = 3770, normalized size = 4.44 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)/(c*x^2+a)^3,x, algorithm="fricas")

[Out]

1/64*((a^4*c*d^2 + a^5*e^2 + (a^2*c^3*d^2 + a^3*c^2*e^2)*x^4 + 2*(a^3*c^2*d^2 + a^4*c*e^2)*x^2)*sqrt(-(144*c^3
*d^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 + (a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^
2*e^4 + a^8*c*e^6)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 + 6*a^6*c^8*d^10*
e^2 + 15*a^7*c^7*d^8*e^4 + 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))/(a
^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6))*log((3024*c^3*d^6*e^5 + 7884*a*c^2*d^4*e^7 +
5625*a^2*c*d^2*e^9 + 625*a^3*e^11)*sqrt(e*x + d) + (126*a^3*c^3*d^5*e^6 + 318*a^4*c^2*d^3*e^8 + 200*a^5*c*d*e^
10 - (12*a^5*c^7*d^10 + 55*a^6*c^6*d^8*e^2 + 98*a^7*c^5*d^6*e^4 + 84*a^8*c^4*d^4*e^6 + 34*a^9*c^3*d^2*e^8 + 5*
a^10*c^2*e^10)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 + 6*a^6*c^8*d^10*e^2
+ 15*a^7*c^7*d^8*e^4 + 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))*sqrt(-
(144*c^3*d^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 + (a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^
7*c^2*d^2*e^4 + a^8*c*e^6)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 + 6*a^6*c
^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 + 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^
12)))/(a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6))) - (a^4*c*d^2 + a^5*e^2 + (a^2*c^3*d^
2 + a^3*c^2*e^2)*x^4 + 2*(a^3*c^2*d^2 + a^4*c*e^2)*x^2)*sqrt(-(144*c^3*d^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3
*e^4 + 105*a^3*d*e^6 + (a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6)*sqrt(-(441*c^2*d^4*e^
10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 + 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 + 20*a^8*c^6*d^
6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))/(a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c
^2*d^2*e^4 + a^8*c*e^6))*log((3024*c^3*d^6*e^5 + 7884*a*c^2*d^4*e^7 + 5625*a^2*c*d^2*e^9 + 625*a^3*e^11)*sqrt(
e*x + d) - (126*a^3*c^3*d^5*e^6 + 318*a^4*c^2*d^3*e^8 + 200*a^5*c*d*e^10 - (12*a^5*c^7*d^10 + 55*a^6*c^6*d^8*e
^2 + 98*a^7*c^5*d^6*e^4 + 84*a^8*c^4*d^4*e^6 + 34*a^9*c^3*d^2*e^8 + 5*a^10*c^2*e^10)*sqrt(-(441*c^2*d^4*e^10 +
1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 + 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 + 20*a^8*c^6*d^6*e^
6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))*sqrt(-(144*c^3*d^7 + 420*a*c^2*d^5*e^2 + 385*a
^2*c*d^3*e^4 + 105*a^3*d*e^6 + (a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6)*sqrt(-(441*c^
2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 + 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 + 20*a^
8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))/(a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 +
3*a^7*c^2*d^2*e^4 + a^8*c*e^6))) + (a^4*c*d^2 + a^5*e^2 + (a^2*c^3*d^2 + a^3*c^2*e^2)*x^4 + 2*(a^3*c^2*d^2 +
a^4*c*e^2)*x^2)*sqrt(-(144*c^3*d^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 - (a^5*c^4*d^6 + 3*
a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(
a^5*c^9*d^12 + 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 + 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*
d^2*e^10 + a^11*c^3*e^12)))/(a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6))*log((3024*c^3*d
^6*e^5 + 7884*a*c^2*d^4*e^7 + 5625*a^2*c*d^2*e^9 + 625*a^3*e^11)*sqrt(e*x + d) + (126*a^3*c^3*d^5*e^6 + 318*a^
4*c^2*d^3*e^8 + 200*a^5*c*d*e^10 + (12*a^5*c^7*d^10 + 55*a^6*c^6*d^8*e^2 + 98*a^7*c^5*d^6*e^4 + 84*a^8*c^4*d^4
*e^6 + 34*a^9*c^3*d^2*e^8 + 5*a^10*c^2*e^10)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*
c^9*d^12 + 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 + 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*d^2*
e^10 + a^11*c^3*e^12)))*sqrt(-(144*c^3*d^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 - (a^5*c^4*
d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2
*e^14)/(a^5*c^9*d^12 + 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 + 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a
^10*c^4*d^2*e^10 + a^11*c^3*e^12)))/(a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6))) - (a^4
*c*d^2 + a^5*e^2 + (a^2*c^3*d^2 + a^3*c^2*e^2)*x^4 + 2*(a^3*c^2*d^2 + a^4*c*e^2)*x^2)*sqrt(-(144*c^3*d^7 + 420
*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 - (a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^
8*c*e^6)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 + 6*a^6*c^8*d^10*e^2 + 15*a
^7*c^7*d^8*e^4 + 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))/(a^5*c^4*d^6
+ 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6))*log((3024*c^3*d^6*e^5 + 7884*a*c^2*d^4*e^7 + 5625*a^2*c
*d^2*e^9 + 625*a^3*e^11)*sqrt(e*x + d) - (126*a^3*c^3*d^5*e^6 + 318*a^4*c^2*d^3*e^8 + 200*a^5*c*d*e^10 + (12*a
^5*c^7*d^10 + 55*a^6*c^6*d^8*e^2 + 98*a^7*c^5*d^6*e^4 + 84*a^8*c^4*d^4*e^6 + 34*a^9*c^3*d^2*e^8 + 5*a^10*c^2*e
^10)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 + 6*a^6*c^8*d^10*e^2 + 15*a^7*c
^7*d^8*e^4 + 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))*sqrt(-(144*c^3*d
^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 - (a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*
e^4 + a^8*c*e^6)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 + 6*a^6*c^8*d^10*e^
2 + 15*a^7*c^7*d^8*e^4 + 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))/(a^5
*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6))) + 4*(a*c*d*e*x^2 + a^2*d*e + (6*c^2*d^2 + 5*a*
c*e^2)*x^3 + (10*a*c*d^2 + 9*a^2*e^2)*x)*sqrt(e*x + d))/(a^4*c*d^2 + a^5*e^2 + (a^2*c^3*d^2 + a^3*c^2*e^2)*x^4
+ 2*(a^3*c^2*d^2 + a^4*c*e^2)*x^2)

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giac [A]  time = 0.84, size = 1056, normalized size = 1.24 \[ -\frac {{\left ({\left (a^{2} c d^{2} e + a^{3} e^{3}\right )}^{2} {\left (6 \, c d^{2} e + 5 \, a e^{3}\right )} {\left | c \right |} - 2 \, {\left (3 \, \sqrt {-a c} a c^{2} d^{5} e + 7 \, \sqrt {-a c} a^{2} c d^{3} e^{3} + 4 \, \sqrt {-a c} a^{3} d e^{5}\right )} {\left | -a^{2} c d^{2} e - a^{3} e^{3} \right |} {\left | c \right |} + {\left (12 \, a^{3} c^{4} d^{8} e + 37 \, a^{4} c^{3} d^{6} e^{3} + 38 \, a^{5} c^{2} d^{4} e^{5} + 13 \, a^{6} c d^{2} e^{7}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a^{2} c^{2} d^{3} + a^{3} c d e^{2} + \sqrt {{\left (a^{2} c^{2} d^{3} + a^{3} c d e^{2}\right )}^{2} - {\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )} {\left (a^{2} c^{2} d^{2} + a^{3} c e^{2}\right )}}}{a^{2} c^{2} d^{2} + a^{3} c e^{2}}}}\right )}{32 \, {\left (a^{4} c^{3} d^{4} e - \sqrt {-a c} a^{3} c^{3} d^{5} - 2 \, \sqrt {-a c} a^{4} c^{2} d^{3} e^{2} + 2 \, a^{5} c^{2} d^{2} e^{3} - \sqrt {-a c} a^{5} c d e^{4} + a^{6} c e^{5}\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e} {\left | -a^{2} c d^{2} e - a^{3} e^{3} \right |}} - \frac {{\left ({\left (a^{2} c d^{2} e + a^{3} e^{3}\right )}^{2} {\left (6 \, c d^{2} e + 5 \, a e^{3}\right )} {\left | c \right |} + 2 \, {\left (3 \, \sqrt {-a c} a c^{2} d^{5} e + 7 \, \sqrt {-a c} a^{2} c d^{3} e^{3} + 4 \, \sqrt {-a c} a^{3} d e^{5}\right )} {\left | -a^{2} c d^{2} e - a^{3} e^{3} \right |} {\left | c \right |} + {\left (12 \, a^{3} c^{4} d^{8} e + 37 \, a^{4} c^{3} d^{6} e^{3} + 38 \, a^{5} c^{2} d^{4} e^{5} + 13 \, a^{6} c d^{2} e^{7}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a^{2} c^{2} d^{3} + a^{3} c d e^{2} - \sqrt {{\left (a^{2} c^{2} d^{3} + a^{3} c d e^{2}\right )}^{2} - {\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )} {\left (a^{2} c^{2} d^{2} + a^{3} c e^{2}\right )}}}{a^{2} c^{2} d^{2} + a^{3} c e^{2}}}}\right )}{32 \, {\left (a^{4} c^{3} d^{4} e + \sqrt {-a c} a^{3} c^{3} d^{5} + 2 \, \sqrt {-a c} a^{4} c^{2} d^{3} e^{2} + 2 \, a^{5} c^{2} d^{2} e^{3} + \sqrt {-a c} a^{5} c d e^{4} + a^{6} c e^{5}\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e} {\left | -a^{2} c d^{2} e - a^{3} e^{3} \right |}} + \frac {6 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{2} d^{2} e - 18 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{2} d^{3} e + 18 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{2} d^{4} e - 6 \, \sqrt {x e + d} c^{2} d^{5} e + 5 \, {\left (x e + d\right )}^{\frac {7}{2}} a c e^{3} - 14 \, {\left (x e + d\right )}^{\frac {5}{2}} a c d e^{3} + 23 \, {\left (x e + d\right )}^{\frac {3}{2}} a c d^{2} e^{3} - 14 \, \sqrt {x e + d} a c d^{3} e^{3} + 9 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} e^{5} - 8 \, \sqrt {x e + d} a^{2} d e^{5}}{16 \, {\left (a^{2} c d^{2} + a^{3} e^{2}\right )} {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + a e^{2}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)/(c*x^2+a)^3,x, algorithm="giac")

[Out]

-1/32*((a^2*c*d^2*e + a^3*e^3)^2*(6*c*d^2*e + 5*a*e^3)*abs(c) - 2*(3*sqrt(-a*c)*a*c^2*d^5*e + 7*sqrt(-a*c)*a^2
*c*d^3*e^3 + 4*sqrt(-a*c)*a^3*d*e^5)*abs(-a^2*c*d^2*e - a^3*e^3)*abs(c) + (12*a^3*c^4*d^8*e + 37*a^4*c^3*d^6*e
^3 + 38*a^5*c^2*d^4*e^5 + 13*a^6*c*d^2*e^7)*abs(c))*arctan(sqrt(x*e + d)/sqrt(-(a^2*c^2*d^3 + a^3*c*d*e^2 + sq
rt((a^2*c^2*d^3 + a^3*c*d*e^2)^2 - (a^2*c^2*d^4 + 2*a^3*c*d^2*e^2 + a^4*e^4)*(a^2*c^2*d^2 + a^3*c*e^2)))/(a^2*
c^2*d^2 + a^3*c*e^2)))/((a^4*c^3*d^4*e - sqrt(-a*c)*a^3*c^3*d^5 - 2*sqrt(-a*c)*a^4*c^2*d^3*e^2 + 2*a^5*c^2*d^2
*e^3 - sqrt(-a*c)*a^5*c*d*e^4 + a^6*c*e^5)*sqrt(-c^2*d + sqrt(-a*c)*c*e)*abs(-a^2*c*d^2*e - a^3*e^3)) - 1/32*(
(a^2*c*d^2*e + a^3*e^3)^2*(6*c*d^2*e + 5*a*e^3)*abs(c) + 2*(3*sqrt(-a*c)*a*c^2*d^5*e + 7*sqrt(-a*c)*a^2*c*d^3*
e^3 + 4*sqrt(-a*c)*a^3*d*e^5)*abs(-a^2*c*d^2*e - a^3*e^3)*abs(c) + (12*a^3*c^4*d^8*e + 37*a^4*c^3*d^6*e^3 + 38
*a^5*c^2*d^4*e^5 + 13*a^6*c*d^2*e^7)*abs(c))*arctan(sqrt(x*e + d)/sqrt(-(a^2*c^2*d^3 + a^3*c*d*e^2 - sqrt((a^2
*c^2*d^3 + a^3*c*d*e^2)^2 - (a^2*c^2*d^4 + 2*a^3*c*d^2*e^2 + a^4*e^4)*(a^2*c^2*d^2 + a^3*c*e^2)))/(a^2*c^2*d^2
+ a^3*c*e^2)))/((a^4*c^3*d^4*e + sqrt(-a*c)*a^3*c^3*d^5 + 2*sqrt(-a*c)*a^4*c^2*d^3*e^2 + 2*a^5*c^2*d^2*e^3 +
sqrt(-a*c)*a^5*c*d*e^4 + a^6*c*e^5)*sqrt(-c^2*d - sqrt(-a*c)*c*e)*abs(-a^2*c*d^2*e - a^3*e^3)) + 1/16*(6*(x*e
+ d)^(7/2)*c^2*d^2*e - 18*(x*e + d)^(5/2)*c^2*d^3*e + 18*(x*e + d)^(3/2)*c^2*d^4*e - 6*sqrt(x*e + d)*c^2*d^5*e
+ 5*(x*e + d)^(7/2)*a*c*e^3 - 14*(x*e + d)^(5/2)*a*c*d*e^3 + 23*(x*e + d)^(3/2)*a*c*d^2*e^3 - 14*sqrt(x*e + d
)*a*c*d^3*e^3 + 9*(x*e + d)^(3/2)*a^2*e^5 - 8*sqrt(x*e + d)*a^2*d*e^5)/((a^2*c*d^2 + a^3*e^2)*((x*e + d)^2*c -
2*(x*e + d)*c*d + c*d^2 + a*e^2)^2)

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maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e x +d}}{\left (c \,x^{2}+a \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(1/2)/(c*x^2+a)^3,x)

[Out]

int((e*x+d)^(1/2)/(c*x^2+a)^3,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e x + d}}{{\left (c x^{2} + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)/(c*x^2+a)^3,x, algorithm="maxima")

[Out]

integrate(sqrt(e*x + d)/(c*x^2 + a)^3, x)

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mupad [B]  time = 3.22, size = 6238, normalized size = 7.35 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(1/2)/(a + c*x^2)^3,x)

[Out]

atan(((((32768*a^7*c^3*d*e^7 + 24576*a^5*c^5*d^5*e^3 + 57344*a^6*c^4*d^3*e^5)/(4096*(a^8*e^4 + a^6*c^2*d^4 + 2
*a^7*c*d^2*e^2)) - ((d + e*x)^(1/2)*(4096*a^7*c^4*d*e^6 + 4096*a^5*c^6*d^5*e^2 + 8192*a^6*c^5*d^3*e^4)*(-(144*
a^5*c^5*d^7 - 25*a*e^7*(-a^15*c^3)^(1/2) + 105*a^8*c^2*d*e^6 + 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 - 21*
c*d^2*e^5*(-a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 + a^13*c^3*e^6 + 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(
1/2))/(64*(a^6*e^4 + a^4*c^2*d^4 + 2*a^5*c*d^2*e^2)))*(-(144*a^5*c^5*d^7 - 25*a*e^7*(-a^15*c^3)^(1/2) + 105*a^
8*c^2*d*e^6 + 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 - 21*c*d^2*e^5*(-a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6
+ a^13*c^3*e^6 + 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2) - ((d + e*x)^(1/2)*(144*c^5*d^6*e^2 - 25*a^3
*c^2*e^8 + 276*a*c^4*d^4*e^4 + 109*a^2*c^3*d^2*e^6))/(64*(a^6*e^4 + a^4*c^2*d^4 + 2*a^5*c*d^2*e^2)))*(-(144*a^
5*c^5*d^7 - 25*a*e^7*(-a^15*c^3)^(1/2) + 105*a^8*c^2*d*e^6 + 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 - 21*c*
d^2*e^5*(-a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 + a^13*c^3*e^6 + 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/
2)*1i - (((32768*a^7*c^3*d*e^7 + 24576*a^5*c^5*d^5*e^3 + 57344*a^6*c^4*d^3*e^5)/(4096*(a^8*e^4 + a^6*c^2*d^4 +
2*a^7*c*d^2*e^2)) + ((d + e*x)^(1/2)*(4096*a^7*c^4*d*e^6 + 4096*a^5*c^6*d^5*e^2 + 8192*a^6*c^5*d^3*e^4)*(-(14
4*a^5*c^5*d^7 - 25*a*e^7*(-a^15*c^3)^(1/2) + 105*a^8*c^2*d*e^6 + 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 - 2
1*c*d^2*e^5*(-a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 + a^13*c^3*e^6 + 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))
^(1/2))/(64*(a^6*e^4 + a^4*c^2*d^4 + 2*a^5*c*d^2*e^2)))*(-(144*a^5*c^5*d^7 - 25*a*e^7*(-a^15*c^3)^(1/2) + 105*
a^8*c^2*d*e^6 + 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 - 21*c*d^2*e^5*(-a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^
6 + a^13*c^3*e^6 + 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2) + ((d + e*x)^(1/2)*(144*c^5*d^6*e^2 - 25*a
^3*c^2*e^8 + 276*a*c^4*d^4*e^4 + 109*a^2*c^3*d^2*e^6))/(64*(a^6*e^4 + a^4*c^2*d^4 + 2*a^5*c*d^2*e^2)))*(-(144*
a^5*c^5*d^7 - 25*a*e^7*(-a^15*c^3)^(1/2) + 105*a^8*c^2*d*e^6 + 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 - 21*
c*d^2*e^5*(-a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 + a^13*c^3*e^6 + 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(
1/2)*1i)/((((32768*a^7*c^3*d*e^7 + 24576*a^5*c^5*d^5*e^3 + 57344*a^6*c^4*d^3*e^5)/(4096*(a^8*e^4 + a^6*c^2*d^4
+ 2*a^7*c*d^2*e^2)) - ((d + e*x)^(1/2)*(4096*a^7*c^4*d*e^6 + 4096*a^5*c^6*d^5*e^2 + 8192*a^6*c^5*d^3*e^4)*(-(
144*a^5*c^5*d^7 - 25*a*e^7*(-a^15*c^3)^(1/2) + 105*a^8*c^2*d*e^6 + 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 -
21*c*d^2*e^5*(-a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 + a^13*c^3*e^6 + 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)
))^(1/2))/(64*(a^6*e^4 + a^4*c^2*d^4 + 2*a^5*c*d^2*e^2)))*(-(144*a^5*c^5*d^7 - 25*a*e^7*(-a^15*c^3)^(1/2) + 10
5*a^8*c^2*d*e^6 + 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 - 21*c*d^2*e^5*(-a^15*c^3)^(1/2))/(4096*(a^10*c^6*
d^6 + a^13*c^3*e^6 + 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2) - ((d + e*x)^(1/2)*(144*c^5*d^6*e^2 - 25
*a^3*c^2*e^8 + 276*a*c^4*d^4*e^4 + 109*a^2*c^3*d^2*e^6))/(64*(a^6*e^4 + a^4*c^2*d^4 + 2*a^5*c*d^2*e^2)))*(-(14
4*a^5*c^5*d^7 - 25*a*e^7*(-a^15*c^3)^(1/2) + 105*a^8*c^2*d*e^6 + 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 - 2
1*c*d^2*e^5*(-a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 + a^13*c^3*e^6 + 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))
^(1/2) + (((32768*a^7*c^3*d*e^7 + 24576*a^5*c^5*d^5*e^3 + 57344*a^6*c^4*d^3*e^5)/(4096*(a^8*e^4 + a^6*c^2*d^4
+ 2*a^7*c*d^2*e^2)) + ((d + e*x)^(1/2)*(4096*a^7*c^4*d*e^6 + 4096*a^5*c^6*d^5*e^2 + 8192*a^6*c^5*d^3*e^4)*(-(1
44*a^5*c^5*d^7 - 25*a*e^7*(-a^15*c^3)^(1/2) + 105*a^8*c^2*d*e^6 + 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 -
21*c*d^2*e^5*(-a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 + a^13*c^3*e^6 + 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4))
)^(1/2))/(64*(a^6*e^4 + a^4*c^2*d^4 + 2*a^5*c*d^2*e^2)))*(-(144*a^5*c^5*d^7 - 25*a*e^7*(-a^15*c^3)^(1/2) + 105
*a^8*c^2*d*e^6 + 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 - 21*c*d^2*e^5*(-a^15*c^3)^(1/2))/(4096*(a^10*c^6*d
^6 + a^13*c^3*e^6 + 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2) + ((d + e*x)^(1/2)*(144*c^5*d^6*e^2 - 25*
a^3*c^2*e^8 + 276*a*c^4*d^4*e^4 + 109*a^2*c^3*d^2*e^6))/(64*(a^6*e^4 + a^4*c^2*d^4 + 2*a^5*c*d^2*e^2)))*(-(144
*a^5*c^5*d^7 - 25*a*e^7*(-a^15*c^3)^(1/2) + 105*a^8*c^2*d*e^6 + 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 - 21
*c*d^2*e^5*(-a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 + a^13*c^3*e^6 + 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^
(1/2) + (125*a^3*c*e^9 + 864*c^4*d^6*e^3 + 1944*a*c^3*d^4*e^5 + 1170*a^2*c^2*d^2*e^7)/(2048*(a^8*e^4 + a^6*c^2
*d^4 + 2*a^7*c*d^2*e^2))))*(-(144*a^5*c^5*d^7 - 25*a*e^7*(-a^15*c^3)^(1/2) + 105*a^8*c^2*d*e^6 + 420*a^6*c^4*d
^5*e^2 + 385*a^7*c^3*d^3*e^4 - 21*c*d^2*e^5*(-a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 + a^13*c^3*e^6 + 3*a^11*c^5
*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2)*2i + atan(((((32768*a^7*c^3*d*e^7 + 24576*a^5*c^5*d^5*e^3 + 57344*a^6*c
^4*d^3*e^5)/(4096*(a^8*e^4 + a^6*c^2*d^4 + 2*a^7*c*d^2*e^2)) - ((d + e*x)^(1/2)*(4096*a^7*c^4*d*e^6 + 4096*a^5
*c^6*d^5*e^2 + 8192*a^6*c^5*d^3*e^4)*(-(144*a^5*c^5*d^7 + 25*a*e^7*(-a^15*c^3)^(1/2) + 105*a^8*c^2*d*e^6 + 420
*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 + 21*c*d^2*e^5*(-a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 + a^13*c^3*e^6 +
3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2))/(64*(a^6*e^4 + a^4*c^2*d^4 + 2*a^5*c*d^2*e^2)))*(-(144*a^5*c
^5*d^7 + 25*a*e^7*(-a^15*c^3)^(1/2) + 105*a^8*c^2*d*e^6 + 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 + 21*c*d^2
*e^5*(-a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 + a^13*c^3*e^6 + 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2)
- ((d + e*x)^(1/2)*(144*c^5*d^6*e^2 - 25*a^3*c^2*e^8 + 276*a*c^4*d^4*e^4 + 109*a^2*c^3*d^2*e^6))/(64*(a^6*e^4
+ a^4*c^2*d^4 + 2*a^5*c*d^2*e^2)))*(-(144*a^5*c^5*d^7 + 25*a*e^7*(-a^15*c^3)^(1/2) + 105*a^8*c^2*d*e^6 + 420*a
^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 + 21*c*d^2*e^5*(-a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 + a^13*c^3*e^6 + 3*
a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2)*1i - (((32768*a^7*c^3*d*e^7 + 24576*a^5*c^5*d^5*e^3 + 57344*a^6
*c^4*d^3*e^5)/(4096*(a^8*e^4 + a^6*c^2*d^4 + 2*a^7*c*d^2*e^2)) + ((d + e*x)^(1/2)*(4096*a^7*c^4*d*e^6 + 4096*a
^5*c^6*d^5*e^2 + 8192*a^6*c^5*d^3*e^4)*(-(144*a^5*c^5*d^7 + 25*a*e^7*(-a^15*c^3)^(1/2) + 105*a^8*c^2*d*e^6 + 4
20*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 + 21*c*d^2*e^5*(-a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 + a^13*c^3*e^6
+ 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2))/(64*(a^6*e^4 + a^4*c^2*d^4 + 2*a^5*c*d^2*e^2)))*(-(144*a^5
*c^5*d^7 + 25*a*e^7*(-a^15*c^3)^(1/2) + 105*a^8*c^2*d*e^6 + 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 + 21*c*d
^2*e^5*(-a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 + a^13*c^3*e^6 + 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2
) + ((d + e*x)^(1/2)*(144*c^5*d^6*e^2 - 25*a^3*c^2*e^8 + 276*a*c^4*d^4*e^4 + 109*a^2*c^3*d^2*e^6))/(64*(a^6*e^
4 + a^4*c^2*d^4 + 2*a^5*c*d^2*e^2)))*(-(144*a^5*c^5*d^7 + 25*a*e^7*(-a^15*c^3)^(1/2) + 105*a^8*c^2*d*e^6 + 420
*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 + 21*c*d^2*e^5*(-a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 + a^13*c^3*e^6 +
3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2)*1i)/((((32768*a^7*c^3*d*e^7 + 24576*a^5*c^5*d^5*e^3 + 57344*a
^6*c^4*d^3*e^5)/(4096*(a^8*e^4 + a^6*c^2*d^4 + 2*a^7*c*d^2*e^2)) - ((d + e*x)^(1/2)*(4096*a^7*c^4*d*e^6 + 4096
*a^5*c^6*d^5*e^2 + 8192*a^6*c^5*d^3*e^4)*(-(144*a^5*c^5*d^7 + 25*a*e^7*(-a^15*c^3)^(1/2) + 105*a^8*c^2*d*e^6 +
420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 + 21*c*d^2*e^5*(-a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 + a^13*c^3*e^
6 + 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2))/(64*(a^6*e^4 + a^4*c^2*d^4 + 2*a^5*c*d^2*e^2)))*(-(144*a
^5*c^5*d^7 + 25*a*e^7*(-a^15*c^3)^(1/2) + 105*a^8*c^2*d*e^6 + 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 + 21*c
*d^2*e^5*(-a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 + a^13*c^3*e^6 + 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1
/2) - ((d + e*x)^(1/2)*(144*c^5*d^6*e^2 - 25*a^3*c^2*e^8 + 276*a*c^4*d^4*e^4 + 109*a^2*c^3*d^2*e^6))/(64*(a^6*
e^4 + a^4*c^2*d^4 + 2*a^5*c*d^2*e^2)))*(-(144*a^5*c^5*d^7 + 25*a*e^7*(-a^15*c^3)^(1/2) + 105*a^8*c^2*d*e^6 + 4
20*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 + 21*c*d^2*e^5*(-a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 + a^13*c^3*e^6
+ 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2) + (((32768*a^7*c^3*d*e^7 + 24576*a^5*c^5*d^5*e^3 + 57344*a^
6*c^4*d^3*e^5)/(4096*(a^8*e^4 + a^6*c^2*d^4 + 2*a^7*c*d^2*e^2)) + ((d + e*x)^(1/2)*(4096*a^7*c^4*d*e^6 + 4096*
a^5*c^6*d^5*e^2 + 8192*a^6*c^5*d^3*e^4)*(-(144*a^5*c^5*d^7 + 25*a*e^7*(-a^15*c^3)^(1/2) + 105*a^8*c^2*d*e^6 +
420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 + 21*c*d^2*e^5*(-a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 + a^13*c^3*e^6
+ 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2))/(64*(a^6*e^4 + a^4*c^2*d^4 + 2*a^5*c*d^2*e^2)))*(-(144*a^
5*c^5*d^7 + 25*a*e^7*(-a^15*c^3)^(1/2) + 105*a^8*c^2*d*e^6 + 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 + 21*c*
d^2*e^5*(-a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 + a^13*c^3*e^6 + 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/
2) + ((d + e*x)^(1/2)*(144*c^5*d^6*e^2 - 25*a^3*c^2*e^8 + 276*a*c^4*d^4*e^4 + 109*a^2*c^3*d^2*e^6))/(64*(a^6*e
^4 + a^4*c^2*d^4 + 2*a^5*c*d^2*e^2)))*(-(144*a^5*c^5*d^7 + 25*a*e^7*(-a^15*c^3)^(1/2) + 105*a^8*c^2*d*e^6 + 42
0*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 + 21*c*d^2*e^5*(-a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 + a^13*c^3*e^6 +
3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2) + (125*a^3*c*e^9 + 864*c^4*d^6*e^3 + 1944*a*c^3*d^4*e^5 + 11
70*a^2*c^2*d^2*e^7)/(2048*(a^8*e^4 + a^6*c^2*d^4 + 2*a^7*c*d^2*e^2))))*(-(144*a^5*c^5*d^7 + 25*a*e^7*(-a^15*c^
3)^(1/2) + 105*a^8*c^2*d*e^6 + 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 + 21*c*d^2*e^5*(-a^15*c^3)^(1/2))/(40
96*(a^10*c^6*d^6 + a^13*c^3*e^6 + 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2)*2i - (((4*a*d*e^3 + 3*c*d^3
*e)*(d + e*x)^(1/2))/(8*a^2) - ((d + e*x)^(3/2)*(9*a^2*e^5 + 18*c^2*d^4*e + 23*a*c*d^2*e^3))/(16*a^2*(a*e^2 +
c*d^2)) - (c*e*(5*a*e^2 + 6*c*d^2)*(d + e*x)^(7/2))/(16*a^2*(a*e^2 + c*d^2)) + (c*d*(7*a*e^3 + 9*c*d^2*e)*(d +
e*x)^(5/2))/(8*a^2*(a*e^2 + c*d^2)))/(c^2*(d + e*x)^4 + a^2*e^4 + c^2*d^4 + (6*c^2*d^2 + 2*a*c*e^2)*(d + e*x)
^2 - (4*c^2*d^3 + 4*a*c*d*e^2)*(d + e*x) - 4*c^2*d*(d + e*x)^3 + 2*a*c*d^2*e^2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(1/2)/(c*x**2+a)**3,x)

[Out]

Timed out

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