3.7.32 \(\int \frac {(d+e x)^{5/2}}{(a+c x^2)^2} \, dx\) [632]

Optimal. Leaf size=811 \[ -\frac {d e \sqrt {d+e x}}{2 a c}-\frac {(a e-c d x) (d+e x)^{3/2}}{2 a c \left (a+c x^2\right )}+\frac {e \left (c^{3/2} d^3+a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c^{3/2} d^3+a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c^{3/2} d^3+a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (c^{3/2} d^3+a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]

[Out]

-1/2*(-c*d*x+a*e)*(e*x+d)^(3/2)/a/c/(c*x^2+a)-1/2*d*e*(e*x+d)^(1/2)/a/c+1/8*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))*(c^(3/2)*d^3+a*d*e^2*c^(
1/2)+(3*a*e^2+c*d^2)*(a*e^2+c*d^2)^(1/2))/a/c^(7/4)*2^(1/2)/(a*e^2+c*d^2)^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2)
)^(1/2)-1/8*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))*(c^(3/2)*d^3+a*d*e^2*c^(1/2)+(3*a*e^2+c*d^2)*(a*e^2+c*d^2)^(1/2))/a/c^(7/4)*2^(1/2)/(a*e^
2+c*d^2)^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2)-1/16*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))*(c^(3/2)*d^3+a*d*e^2*c^(1/2)-(3*a*e^2+c*d^2)*(a*e^2+
c*d^2)^(1/2))/a/c^(7/4)*2^(1/2)/(a*e^2+c*d^2)^(1/2)/(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2)+1/16*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))*(c^(3/2)*d^3+a*
d*e^2*c^(1/2)-(3*a*e^2+c*d^2)*(a*e^2+c*d^2)^(1/2))/a/c^(7/4)*2^(1/2)/(a*e^2+c*d^2)^(1/2)/(d*c^(1/2)+(a*e^2+c*d
^2)^(1/2))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 1.93, antiderivative size = 811, 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 = {753, 839, 841, 1183, 648, 632, 212, 642} \begin {gather*} -\frac {(a e-c d x) (d+e x)^{3/2}}{2 a c \left (c x^2+a\right )}-\frac {d e \sqrt {d+e x}}{2 a c}+\frac {e \left (c^{3/2} d^3+a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (c d^2+3 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 )}{4 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c^{3/2} d^3+a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (c d^2+3 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 )}{4 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c^{3/2} d^3+a \sqrt {c} e^2 d-\sqrt {c d^2+a e^2} \left (c d^2+3 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 )}{8 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (c^{3/2} d^3+a \sqrt {c} e^2 d-\sqrt {c d^2+a e^2} \left (c d^2+3 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 )}{8 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(5/2)/(a + c*x^2)^2,x]

[Out]

-1/2*(d*e*Sqrt[d + e*x])/(a*c) - ((a*e - c*d*x)*(d + e*x)^(3/2))/(2*a*c*(a + c*x^2)) + (e*(c^(3/2)*d^3 + a*Sqr
t[c]*d*e^2 + Sqrt[c*d^2 + a*e^2]*(c*d^2 + 3*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]]])/(4*Sqrt[2]*a*c^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sq
rt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(c^(3/2)*d^3 + a*Sqrt[c]*d*e^2 + Sqrt[c*d^2 + a*e^2]*(c*d^2 + 3*a*e^2))*A
rcTanh[(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]]])/(4*Sqrt[2]*a*c^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(c^(3/2)*d^3 + a
*Sqrt[c]*d*e^2 - Sqrt[c*d^2 + a*e^2]*(c*d^2 + 3*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)])/(8*Sqrt[2]*a*c^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sq
rt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (e*(c^(3/2)*d^3 + a*Sqrt[c]*d*e^2 - Sqrt[c*d^2 + a*e^2]*(c*d^2 + 3*a*e^2))*L
og[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)])/(8*Sqrt[2]*a*c^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

Rule 212

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

Rule 632

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 642

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 648

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 753

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

Rule 839

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[g*((d + e*x)^m/(
c*m)), x] + Dist[1/c, Int[(d + e*x)^(m - 1)*(Simp[c*d*f - a*e*g + (g*c*d + c*e*f)*x, x]/(a + c*x^2)), x], x] /
; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && FractionQ[m] && GtQ[m, 0]

Rule 841

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 1183

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 {(d+e x)^{5/2}}{\left (a+c x^2\right )^2} \, dx &=-\frac {(a e-c d x) (d+e x)^{3/2}}{2 a c \left (a+c x^2\right )}+\frac {\int \frac {\sqrt {d+e x} \left (\frac {1}{2} \left (2 c d^2+3 a e^2\right )-\frac {1}{2} c d e x\right )}{a+c x^2} \, dx}{2 a c}\\ &=-\frac {d e \sqrt {d+e x}}{2 a c}-\frac {(a e-c d x) (d+e x)^{3/2}}{2 a c \left (a+c x^2\right )}+\frac {\int \frac {c d \left (c d^2+2 a e^2\right )+\frac {1}{2} c e \left (c d^2+3 a e^2\right ) x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx}{2 a c^2}\\ &=-\frac {d e \sqrt {d+e x}}{2 a c}-\frac {(a e-c d x) (d+e x)^{3/2}}{2 a c \left (a+c x^2\right )}+\frac {\text {Subst}\left (\int \frac {c d e \left (c d^2+2 a e^2\right )-\frac {1}{2} c d e \left (c d^2+3 a e^2\right )+\frac {1}{2} c e \left (c d^2+3 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 )}{a c^2}\\ &=-\frac {d e \sqrt {d+e x}}{2 a c}-\frac {(a e-c d x) (d+e x)^{3/2}}{2 a c \left (a+c x^2\right )}+\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c d e \left (c d^2+2 a e^2\right )-\frac {1}{2} c d e \left (c d^2+3 a e^2\right )\right )}{\sqrt [4]{c}}-\left (c d e \left (c d^2+2 a e^2\right )-\frac {1}{2} c d e \left (c d^2+3 a e^2\right )-\frac {1}{2} \sqrt {c} e \sqrt {c d^2+a e^2} \left (c d^2+3 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 )}{2 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c d e \left (c d^2+2 a e^2\right )-\frac {1}{2} c d e \left (c d^2+3 a e^2\right )\right )}{\sqrt [4]{c}}+\left (c d e \left (c d^2+2 a e^2\right )-\frac {1}{2} c d e \left (c d^2+3 a e^2\right )-\frac {1}{2} \sqrt {c} e \sqrt {c d^2+a e^2} \left (c d^2+3 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 )}{2 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ &=-\frac {d e \sqrt {d+e x}}{2 a c}-\frac {(a e-c d x) (d+e x)^{3/2}}{2 a c \left (a+c x^2\right )}-\frac {\left (e \left (c^{3/2} d^3+a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right )\right ) \text {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 )}{8 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (e \left (c^{3/2} d^3+a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right )\right ) \text {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 )}{8 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (e \left (c^{3/2} d^3+a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right )\right ) \text {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 )}{8 a c^2 \sqrt {c d^2+a e^2}}+\frac {\left (e \left (c^{3/2} d^3+a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right )\right ) \text {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 )}{8 a c^2 \sqrt {c d^2+a e^2}}\\ &=-\frac {d e \sqrt {d+e x}}{2 a c}-\frac {(a e-c d x) (d+e x)^{3/2}}{2 a c \left (a+c x^2\right )}-\frac {e \left (c^{3/2} d^3+a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (c d^2+3 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 )}{8 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (c^{3/2} d^3+a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (c d^2+3 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 )}{8 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\left (e \left (c^{3/2} d^3+a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right )\right ) \text {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 )}{4 a c^2 \sqrt {c d^2+a e^2}}-\frac {\left (e \left (c^{3/2} d^3+a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (c d^2+3 a e^2\right )\right )\right ) \text {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 )}{4 a c^2 \sqrt {c d^2+a e^2}}\\ &=-\frac {d e \sqrt {d+e x}}{2 a c}-\frac {(a e-c d x) (d+e x)^{3/2}}{2 a c \left (a+c x^2\right )}+\frac {e \left (c^{3/2} d^3+a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (c d^2+3 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 )}{4 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c^{3/2} d^3+a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (c d^2+3 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 )}{4 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c^{3/2} d^3+a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (c d^2+3 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 )}{8 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (c^{3/2} d^3+a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (c d^2+3 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 )}{8 \sqrt {2} a c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.14, size = 281, normalized size = 0.35 \begin {gather*} \frac {\frac {2 \sqrt {a} c \sqrt {d+e x} \left (c d^2 x-a e (2 d+e x)\right )}{a+c x^2}-i \sqrt {-c d-i \sqrt {a} \sqrt {c} e} \left (2 c d^2-i \sqrt {a} \sqrt {c} d e+3 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )+i \sqrt {-c d+i \sqrt {a} \sqrt {c} e} \left (2 c d^2+i \sqrt {a} \sqrt {c} d e+3 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{4 a^{3/2} c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(5/2)/(a + c*x^2)^2,x]

[Out]

((2*Sqrt[a]*c*Sqrt[d + e*x]*(c*d^2*x - a*e*(2*d + e*x)))/(a + c*x^2) - I*Sqrt[-(c*d) - I*Sqrt[a]*Sqrt[c]*e]*(2
*c*d^2 - I*Sqrt[a]*Sqrt[c]*d*e + 3*a*e^2)*ArcTan[(Sqrt[-(c*d) - I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d
+ I*Sqrt[a]*e)] + I*Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]*(2*c*d^2 + I*Sqrt[a]*Sqrt[c]*d*e + 3*a*e^2)*ArcTan[(Sq
rt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - I*Sqrt[a]*e)])/(4*a^(3/2)*c^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1672\) vs. \(2(655)=1310\).
time = 0.46, size = 1673, normalized size = 2.06

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1673\)
default	\(\text {Expression too large to display}\)	\(1673\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(5/2)/(c*x^2+a)^2,x,method=_RETURNVERBOSE)

[Out]

2*e^3*((-1/4*(a*e^2-c*d^2)/a/c/e^2*(e*x+d)^(3/2)-1/4*d*(a*e^2+c*d^2)/a/c/e^2*(e*x+d)^(1/2))/(c*(e*x+d)^2-2*c*d
*(e*x+d)+e^2*a+c*d^2)+1/4/a/c/e^2*(1/4/a/c/e^2*(1/2*(-(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(3/2)*(a*e^2+c
*d^2)^(1/2)*d^2+3*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a*c*d*e^2+(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^
2*d^3+(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(1/2)*(a*e^2+c*d^2)^(1/2)*d-3*(a*c*e^2
+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a*e^2-(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/
2)+2*c*d)^(1/2)*c*d^2)/c^(1/2)*ln((e*x+d)*c^(1/2)+(e*x+d)^(1/2)*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2)+(a*e^2
+c*d^2)^(1/2))+2*(2*(a*e^2+c*d^2)^(1/2)*d*a*c*e^2-1/2*(-(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(3/2)*(a*e^2
+c*d^2)^(1/2)*d^2+3*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a*c*d*e^2+(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*
c^2*d^3+(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(1/2)*(a*e^2+c*d^2)^(1/2)*d-3*(a*c*e
^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a*e^2-(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(
1/2)+2*c*d)^(1/2)*c*d^2)*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2)/c^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a
*e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)+(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2))/(4*
(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2)))+1/4/a/c/e^2*(1/2*((2*(a*c*e^2+c^2*d^2)^(1
/2)+2*c*d)^(1/2)*c^(3/2)*(a*e^2+c*d^2)^(1/2)*d^2-3*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a*c*d*e^2-(2*(a*c*e
^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^2*d^3-(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(1/2)
*(a*e^2+c*d^2)^(1/2)*d+3*(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a*e^2+(a*c*e^2+c^2*d^
2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c*d^2)/c^(1/2)*ln((e*x+d)*c^(1/2)-(e*x+d)^(1/2)*(2*((a*e^2+c*
d^2)*c)^(1/2)+2*c*d)^(1/2)+(a*e^2+c*d^2)^(1/2))+2*(2*(a*e^2+c*d^2)^(1/2)*d*a*c*e^2+1/2*((2*(a*c*e^2+c^2*d^2)^(
1/2)+2*c*d)^(1/2)*c^(3/2)*(a*e^2+c*d^2)^(1/2)*d^2-3*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a*c*d*e^2-(2*(a*c*
e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^2*d^3-(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(1/2
)*(a*e^2+c*d^2)^(1/2)*d+3*(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a*e^2+(a*c*e^2+c^2*d
^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c*d^2)*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2)/c^(1/2))/(4*(
a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)-(2*((a*e^2+c
*d^2)*c)^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2)))))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)/(c*x^2+a)^2,x, algorithm="maxima")

[Out]

integrate((x*e + d)^(5/2)/(c*x^2 + a)^2, x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1306 vs. \(2 (643) = 1286\).
time = 1.83, size = 1306, normalized size = 1.61 \begin {gather*} \frac {{\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt {-\frac {4 \, c^{2} d^{5} + a^{3} c^{3} \sqrt {-\frac {25 \, c^{2} d^{4} e^{6} + 90 \, a c d^{2} e^{8} + 81 \, a^{2} e^{10}}{a^{3} c^{7}}} + 15 \, a c d^{3} e^{2} + 15 \, a^{2} d e^{4}}{a^{3} c^{3}}} \log \left ({\left (20 \, c^{3} d^{6} e^{3} + 101 \, a c^{2} d^{4} e^{5} + 162 \, a^{2} c d^{2} e^{7} + 81 \, a^{3} e^{9}\right )} \sqrt {x e + d} + {\left (5 \, a^{2} c^{3} d^{3} e^{4} + 9 \, a^{3} c^{2} d e^{6} - {\left (2 \, a^{3} c^{6} d^{2} + 3 \, a^{4} c^{5} e^{2}\right )} \sqrt {-\frac {25 \, c^{2} d^{4} e^{6} + 90 \, a c d^{2} e^{8} + 81 \, a^{2} e^{10}}{a^{3} c^{7}}}\right )} \sqrt {-\frac {4 \, c^{2} d^{5} + a^{3} c^{3} \sqrt {-\frac {25 \, c^{2} d^{4} e^{6} + 90 \, a c d^{2} e^{8} + 81 \, a^{2} e^{10}}{a^{3} c^{7}}} + 15 \, a c d^{3} e^{2} + 15 \, a^{2} d e^{4}}{a^{3} c^{3}}}\right ) - {\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt {-\frac {4 \, c^{2} d^{5} + a^{3} c^{3} \sqrt {-\frac {25 \, c^{2} d^{4} e^{6} + 90 \, a c d^{2} e^{8} + 81 \, a^{2} e^{10}}{a^{3} c^{7}}} + 15 \, a c d^{3} e^{2} + 15 \, a^{2} d e^{4}}{a^{3} c^{3}}} \log \left ({\left (20 \, c^{3} d^{6} e^{3} + 101 \, a c^{2} d^{4} e^{5} + 162 \, a^{2} c d^{2} e^{7} + 81 \, a^{3} e^{9}\right )} \sqrt {x e + d} - {\left (5 \, a^{2} c^{3} d^{3} e^{4} + 9 \, a^{3} c^{2} d e^{6} - {\left (2 \, a^{3} c^{6} d^{2} + 3 \, a^{4} c^{5} e^{2}\right )} \sqrt {-\frac {25 \, c^{2} d^{4} e^{6} + 90 \, a c d^{2} e^{8} + 81 \, a^{2} e^{10}}{a^{3} c^{7}}}\right )} \sqrt {-\frac {4 \, c^{2} d^{5} + a^{3} c^{3} \sqrt {-\frac {25 \, c^{2} d^{4} e^{6} + 90 \, a c d^{2} e^{8} + 81 \, a^{2} e^{10}}{a^{3} c^{7}}} + 15 \, a c d^{3} e^{2} + 15 \, a^{2} d e^{4}}{a^{3} c^{3}}}\right ) + {\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt {-\frac {4 \, c^{2} d^{5} - a^{3} c^{3} \sqrt {-\frac {25 \, c^{2} d^{4} e^{6} + 90 \, a c d^{2} e^{8} + 81 \, a^{2} e^{10}}{a^{3} c^{7}}} + 15 \, a c d^{3} e^{2} + 15 \, a^{2} d e^{4}}{a^{3} c^{3}}} \log \left ({\left (20 \, c^{3} d^{6} e^{3} + 101 \, a c^{2} d^{4} e^{5} + 162 \, a^{2} c d^{2} e^{7} + 81 \, a^{3} e^{9}\right )} \sqrt {x e + d} + {\left (5 \, a^{2} c^{3} d^{3} e^{4} + 9 \, a^{3} c^{2} d e^{6} + {\left (2 \, a^{3} c^{6} d^{2} + 3 \, a^{4} c^{5} e^{2}\right )} \sqrt {-\frac {25 \, c^{2} d^{4} e^{6} + 90 \, a c d^{2} e^{8} + 81 \, a^{2} e^{10}}{a^{3} c^{7}}}\right )} \sqrt {-\frac {4 \, c^{2} d^{5} - a^{3} c^{3} \sqrt {-\frac {25 \, c^{2} d^{4} e^{6} + 90 \, a c d^{2} e^{8} + 81 \, a^{2} e^{10}}{a^{3} c^{7}}} + 15 \, a c d^{3} e^{2} + 15 \, a^{2} d e^{4}}{a^{3} c^{3}}}\right ) - {\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt {-\frac {4 \, c^{2} d^{5} - a^{3} c^{3} \sqrt {-\frac {25 \, c^{2} d^{4} e^{6} + 90 \, a c d^{2} e^{8} + 81 \, a^{2} e^{10}}{a^{3} c^{7}}} + 15 \, a c d^{3} e^{2} + 15 \, a^{2} d e^{4}}{a^{3} c^{3}}} \log \left ({\left (20 \, c^{3} d^{6} e^{3} + 101 \, a c^{2} d^{4} e^{5} + 162 \, a^{2} c d^{2} e^{7} + 81 \, a^{3} e^{9}\right )} \sqrt {x e + d} - {\left (5 \, a^{2} c^{3} d^{3} e^{4} + 9 \, a^{3} c^{2} d e^{6} + {\left (2 \, a^{3} c^{6} d^{2} + 3 \, a^{4} c^{5} e^{2}\right )} \sqrt {-\frac {25 \, c^{2} d^{4} e^{6} + 90 \, a c d^{2} e^{8} + 81 \, a^{2} e^{10}}{a^{3} c^{7}}}\right )} \sqrt {-\frac {4 \, c^{2} d^{5} - a^{3} c^{3} \sqrt {-\frac {25 \, c^{2} d^{4} e^{6} + 90 \, a c d^{2} e^{8} + 81 \, a^{2} e^{10}}{a^{3} c^{7}}} + 15 \, a c d^{3} e^{2} + 15 \, a^{2} d e^{4}}{a^{3} c^{3}}}\right ) + 4 \, {\left (c d^{2} x - a x e^{2} - 2 \, a d e\right )} \sqrt {x e + d}}{8 \, {\left (a c^{2} x^{2} + a^{2} c\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)/(c*x^2+a)^2,x, algorithm="fricas")

[Out]

1/8*((a*c^2*x^2 + a^2*c)*sqrt(-(4*c^2*d^5 + a^3*c^3*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3
*c^7)) + 15*a*c*d^3*e^2 + 15*a^2*d*e^4)/(a^3*c^3))*log((20*c^3*d^6*e^3 + 101*a*c^2*d^4*e^5 + 162*a^2*c*d^2*e^7
+ 81*a^3*e^9)*sqrt(x*e + d) + (5*a^2*c^3*d^3*e^4 + 9*a^3*c^2*d*e^6 - (2*a^3*c^6*d^2 + 3*a^4*c^5*e^2)*sqrt(-(2
5*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))*sqrt(-(4*c^2*d^5 + a^3*c^3*sqrt(-(25*c^2*d^4*e^6 + 9
0*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)) + 15*a*c*d^3*e^2 + 15*a^2*d*e^4)/(a^3*c^3))) - (a*c^2*x^2 + a^2*c)*sqr
t(-(4*c^2*d^5 + a^3*c^3*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)) + 15*a*c*d^3*e^2 + 15
*a^2*d*e^4)/(a^3*c^3))*log((20*c^3*d^6*e^3 + 101*a*c^2*d^4*e^5 + 162*a^2*c*d^2*e^7 + 81*a^3*e^9)*sqrt(x*e + d)
- (5*a^2*c^3*d^3*e^4 + 9*a^3*c^2*d*e^6 - (2*a^3*c^6*d^2 + 3*a^4*c^5*e^2)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e
^8 + 81*a^2*e^10)/(a^3*c^7)))*sqrt(-(4*c^2*d^5 + a^3*c^3*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)
/(a^3*c^7)) + 15*a*c*d^3*e^2 + 15*a^2*d*e^4)/(a^3*c^3))) + (a*c^2*x^2 + a^2*c)*sqrt(-(4*c^2*d^5 - a^3*c^3*sqrt
(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)) + 15*a*c*d^3*e^2 + 15*a^2*d*e^4)/(a^3*c^3))*log((
20*c^3*d^6*e^3 + 101*a*c^2*d^4*e^5 + 162*a^2*c*d^2*e^7 + 81*a^3*e^9)*sqrt(x*e + d) + (5*a^2*c^3*d^3*e^4 + 9*a^
3*c^2*d*e^6 + (2*a^3*c^6*d^2 + 3*a^4*c^5*e^2)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7))
)*sqrt(-(4*c^2*d^5 - a^3*c^3*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)) + 15*a*c*d^3*e^2
+ 15*a^2*d*e^4)/(a^3*c^3))) - (a*c^2*x^2 + a^2*c)*sqrt(-(4*c^2*d^5 - a^3*c^3*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d
^2*e^8 + 81*a^2*e^10)/(a^3*c^7)) + 15*a*c*d^3*e^2 + 15*a^2*d*e^4)/(a^3*c^3))*log((20*c^3*d^6*e^3 + 101*a*c^2*d
^4*e^5 + 162*a^2*c*d^2*e^7 + 81*a^3*e^9)*sqrt(x*e + d) - (5*a^2*c^3*d^3*e^4 + 9*a^3*c^2*d*e^6 + (2*a^3*c^6*d^2
+ 3*a^4*c^5*e^2)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))*sqrt(-(4*c^2*d^5 - a^3*c^3
*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)) + 15*a*c*d^3*e^2 + 15*a^2*d*e^4)/(a^3*c^3)))
+ 4*(c*d^2*x - a*x*e^2 - 2*a*d*e)*sqrt(x*e + d))/(a*c^2*x^2 + a^2*c)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(5/2)/(c*x**2+a)**2,x)

[Out]

Timed out

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Giac [A]
time = 2.36, size = 479, normalized size = 0.59 \begin {gather*} -\frac {{\left (2 \, a c^{4} d^{4} + 4 \, a^{2} c^{3} d^{2} e^{2} + {\left (c d^{2} e^{2} + 3 \, a e^{4}\right )} a^{2} c^{2} - {\left (\sqrt {-a c} c^{2} d^{3} e + \sqrt {-a c} a c d e^{3}\right )} {\left | a \right |} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a c^{2} d + \sqrt {a^{2} c^{4} d^{2} - {\left (a c^{2} d^{2} + a^{2} c e^{2}\right )} a c^{2}}}{a c^{2}}}}\right )}{4 \, {\left (a^{2} c^{3} e - \sqrt {-a c} a c^{3} d\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e} {\left | a \right |}} - \frac {{\left (2 \, a c^{4} d^{4} + 4 \, a^{2} c^{3} d^{2} e^{2} + {\left (c d^{2} e^{2} + 3 \, a e^{4}\right )} a^{2} c^{2} + {\left (\sqrt {-a c} c^{2} d^{3} e + \sqrt {-a c} a c d e^{3}\right )} {\left | a \right |} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a c^{2} d - \sqrt {a^{2} c^{4} d^{2} - {\left (a c^{2} d^{2} + a^{2} c e^{2}\right )} a c^{2}}}{a c^{2}}}}\right )}{4 \, {\left (a^{2} c^{3} e + \sqrt {-a c} a c^{3} d\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e} {\left | a \right |}} + \frac {{\left (x e + d\right )}^{\frac {3}{2}} c d^{2} e - \sqrt {x e + d} c d^{3} e - {\left (x e + d\right )}^{\frac {3}{2}} a e^{3} - \sqrt {x e + d} a d e^{3}}{2 \, {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + a e^{2}\right )} a c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)/(c*x^2+a)^2,x, algorithm="giac")

[Out]

-1/4*(2*a*c^4*d^4 + 4*a^2*c^3*d^2*e^2 + (c*d^2*e^2 + 3*a*e^4)*a^2*c^2 - (sqrt(-a*c)*c^2*d^3*e + sqrt(-a*c)*a*c
*d*e^3)*abs(a)*abs(c))*arctan(sqrt(x*e + d)/sqrt(-(a*c^2*d + sqrt(a^2*c^4*d^2 - (a*c^2*d^2 + a^2*c*e^2)*a*c^2)
)/(a*c^2)))/((a^2*c^3*e - sqrt(-a*c)*a*c^3*d)*sqrt(-c^2*d + sqrt(-a*c)*c*e)*abs(a)) - 1/4*(2*a*c^4*d^4 + 4*a^2
*c^3*d^2*e^2 + (c*d^2*e^2 + 3*a*e^4)*a^2*c^2 + (sqrt(-a*c)*c^2*d^3*e + sqrt(-a*c)*a*c*d*e^3)*abs(a)*abs(c))*ar
ctan(sqrt(x*e + d)/sqrt(-(a*c^2*d - sqrt(a^2*c^4*d^2 - (a*c^2*d^2 + a^2*c*e^2)*a*c^2))/(a*c^2)))/((a^2*c^3*e +
sqrt(-a*c)*a*c^3*d)*sqrt(-c^2*d - sqrt(-a*c)*c*e)*abs(a)) + 1/2*((x*e + d)^(3/2)*c*d^2*e - sqrt(x*e + d)*c*d^
3*e - (x*e + d)^(3/2)*a*e^3 - sqrt(x*e + d)*a*d*e^3)/(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + a*e^2)*a*c)

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Mupad [B]
time = 0.83, size = 2031, normalized size = 2.50 \begin {gather*} -\frac {\frac {\left (a\,e^3-c\,d^2\,e\right )\,{\left (d+e\,x\right )}^{3/2}}{2\,a\,c}+\frac {\left (c\,d^3\,e+a\,d\,e^3\right )\,\sqrt {d+e\,x}}{2\,a\,c}}{c\,{\left (d+e\,x\right )}^2+a\,e^2+c\,d^2-2\,c\,d\,\left (d+e\,x\right )}-2\,\mathrm {atanh}\left (\frac {18\,a\,e^8\,\sqrt {d+e\,x}\,\sqrt {-\frac {d^5}{16\,a^3\,c}-\frac {15\,d\,e^4}{64\,a\,c^3}-\frac {15\,d^3\,e^2}{64\,a^2\,c^2}-\frac {9\,e^5\,\sqrt {-a^9\,c^7}}{64\,a^5\,c^7}-\frac {5\,d^2\,e^3\,\sqrt {-a^9\,c^7}}{64\,a^6\,c^6}}}{\frac {27\,a\,e^{11}}{4\,c^2}+\frac {43\,d^4\,e^7}{4\,a}+\frac {15\,d^2\,e^9}{c}+\frac {5\,c\,d^6\,e^5}{2\,a^2}-\frac {9\,d\,e^{10}\,\sqrt {-a^9\,c^7}}{4\,a^4\,c^5}-\frac {7\,d^3\,e^8\,\sqrt {-a^9\,c^7}}{2\,a^5\,c^4}-\frac {5\,d^5\,e^6\,\sqrt {-a^9\,c^7}}{4\,a^6\,c^3}}+\frac {10\,c\,d^2\,e^6\,\sqrt {d+e\,x}\,\sqrt {-\frac {d^5}{16\,a^3\,c}-\frac {15\,d\,e^4}{64\,a\,c^3}-\frac {15\,d^3\,e^2}{64\,a^2\,c^2}-\frac {9\,e^5\,\sqrt {-a^9\,c^7}}{64\,a^5\,c^7}-\frac {5\,d^2\,e^3\,\sqrt {-a^9\,c^7}}{64\,a^6\,c^6}}}{\frac {27\,a\,e^{11}}{4\,c^2}+\frac {43\,d^4\,e^7}{4\,a}+\frac {15\,d^2\,e^9}{c}+\frac {5\,c\,d^6\,e^5}{2\,a^2}-\frac {9\,d\,e^{10}\,\sqrt {-a^9\,c^7}}{4\,a^4\,c^5}-\frac {7\,d^3\,e^8\,\sqrt {-a^9\,c^7}}{2\,a^5\,c^4}-\frac {5\,d^5\,e^6\,\sqrt {-a^9\,c^7}}{4\,a^6\,c^3}}+\frac {18\,d\,e^7\,\sqrt {-a^9\,c^7}\,\sqrt {d+e\,x}\,\sqrt {-\frac {d^5}{16\,a^3\,c}-\frac {15\,d\,e^4}{64\,a\,c^3}-\frac {15\,d^3\,e^2}{64\,a^2\,c^2}-\frac {9\,e^5\,\sqrt {-a^9\,c^7}}{64\,a^5\,c^7}-\frac {5\,d^2\,e^3\,\sqrt {-a^9\,c^7}}{64\,a^6\,c^6}}}{\frac {27\,a^5\,c\,e^{11}}{4}+\frac {5\,a^2\,c^4\,d^6\,e^5}{2}+\frac {43\,a^3\,c^3\,d^4\,e^7}{4}+15\,a^4\,c^2\,d^2\,e^9-\frac {9\,d\,e^{10}\,\sqrt {-a^9\,c^7}}{4\,c^2}-\frac {5\,d^5\,e^6\,\sqrt {-a^9\,c^7}}{4\,a^2}-\frac {7\,d^3\,e^8\,\sqrt {-a^9\,c^7}}{2\,a\,c}}+\frac {10\,d^3\,e^5\,\sqrt {-a^9\,c^7}\,\sqrt {d+e\,x}\,\sqrt {-\frac {d^5}{16\,a^3\,c}-\frac {15\,d\,e^4}{64\,a\,c^3}-\frac {15\,d^3\,e^2}{64\,a^2\,c^2}-\frac {9\,e^5\,\sqrt {-a^9\,c^7}}{64\,a^5\,c^7}-\frac {5\,d^2\,e^3\,\sqrt {-a^9\,c^7}}{64\,a^6\,c^6}}}{\frac {27\,a^6\,e^{11}}{4}+15\,a^5\,c\,d^2\,e^9+\frac {5\,a^3\,c^3\,d^6\,e^5}{2}+\frac {43\,a^4\,c^2\,d^4\,e^7}{4}-\frac {7\,d^3\,e^8\,\sqrt {-a^9\,c^7}}{2\,c^2}-\frac {5\,d^5\,e^6\,\sqrt {-a^9\,c^7}}{4\,a\,c}-\frac {9\,a\,d\,e^{10}\,\sqrt {-a^9\,c^7}}{4\,c^3}}\right )\,\sqrt {-\frac {4\,a^3\,c^6\,d^5+9\,a\,e^5\,\sqrt {-a^9\,c^7}+15\,a^5\,c^4\,d\,e^4+15\,a^4\,c^5\,d^3\,e^2+5\,c\,d^2\,e^3\,\sqrt {-a^9\,c^7}}{64\,a^6\,c^7}}-2\,\mathrm {atanh}\left (\frac {18\,a\,e^8\,\sqrt {d+e\,x}\,\sqrt {\frac {9\,e^5\,\sqrt {-a^9\,c^7}}{64\,a^5\,c^7}-\frac {15\,d\,e^4}{64\,a\,c^3}-\frac {15\,d^3\,e^2}{64\,a^2\,c^2}-\frac {d^5}{16\,a^3\,c}+\frac {5\,d^2\,e^3\,\sqrt {-a^9\,c^7}}{64\,a^6\,c^6}}}{\frac {27\,a\,e^{11}}{4\,c^2}+\frac {43\,d^4\,e^7}{4\,a}+\frac {15\,d^2\,e^9}{c}+\frac {5\,c\,d^6\,e^5}{2\,a^2}+\frac {9\,d\,e^{10}\,\sqrt {-a^9\,c^7}}{4\,a^4\,c^5}+\frac {7\,d^3\,e^8\,\sqrt {-a^9\,c^7}}{2\,a^5\,c^4}+\frac {5\,d^5\,e^6\,\sqrt {-a^9\,c^7}}{4\,a^6\,c^3}}+\frac {10\,c\,d^2\,e^6\,\sqrt {d+e\,x}\,\sqrt {\frac {9\,e^5\,\sqrt {-a^9\,c^7}}{64\,a^5\,c^7}-\frac {15\,d\,e^4}{64\,a\,c^3}-\frac {15\,d^3\,e^2}{64\,a^2\,c^2}-\frac {d^5}{16\,a^3\,c}+\frac {5\,d^2\,e^3\,\sqrt {-a^9\,c^7}}{64\,a^6\,c^6}}}{\frac {27\,a\,e^{11}}{4\,c^2}+\frac {43\,d^4\,e^7}{4\,a}+\frac {15\,d^2\,e^9}{c}+\frac {5\,c\,d^6\,e^5}{2\,a^2}+\frac {9\,d\,e^{10}\,\sqrt {-a^9\,c^7}}{4\,a^4\,c^5}+\frac {7\,d^3\,e^8\,\sqrt {-a^9\,c^7}}{2\,a^5\,c^4}+\frac {5\,d^5\,e^6\,\sqrt {-a^9\,c^7}}{4\,a^6\,c^3}}-\frac {18\,d\,e^7\,\sqrt {-a^9\,c^7}\,\sqrt {d+e\,x}\,\sqrt {\frac {9\,e^5\,\sqrt {-a^9\,c^7}}{64\,a^5\,c^7}-\frac {15\,d\,e^4}{64\,a\,c^3}-\frac {15\,d^3\,e^2}{64\,a^2\,c^2}-\frac {d^5}{16\,a^3\,c}+\frac {5\,d^2\,e^3\,\sqrt {-a^9\,c^7}}{64\,a^6\,c^6}}}{\frac {27\,a^5\,c\,e^{11}}{4}+\frac {5\,a^2\,c^4\,d^6\,e^5}{2}+\frac {43\,a^3\,c^3\,d^4\,e^7}{4}+15\,a^4\,c^2\,d^2\,e^9+\frac {9\,d\,e^{10}\,\sqrt {-a^9\,c^7}}{4\,c^2}+\frac {5\,d^5\,e^6\,\sqrt {-a^9\,c^7}}{4\,a^2}+\frac {7\,d^3\,e^8\,\sqrt {-a^9\,c^7}}{2\,a\,c}}-\frac {10\,d^3\,e^5\,\sqrt {-a^9\,c^7}\,\sqrt {d+e\,x}\,\sqrt {\frac {9\,e^5\,\sqrt {-a^9\,c^7}}{64\,a^5\,c^7}-\frac {15\,d\,e^4}{64\,a\,c^3}-\frac {15\,d^3\,e^2}{64\,a^2\,c^2}-\frac {d^5}{16\,a^3\,c}+\frac {5\,d^2\,e^3\,\sqrt {-a^9\,c^7}}{64\,a^6\,c^6}}}{\frac {27\,a^6\,e^{11}}{4}+15\,a^5\,c\,d^2\,e^9+\frac {5\,a^3\,c^3\,d^6\,e^5}{2}+\frac {43\,a^4\,c^2\,d^4\,e^7}{4}+\frac {7\,d^3\,e^8\,\sqrt {-a^9\,c^7}}{2\,c^2}+\frac {5\,d^5\,e^6\,\sqrt {-a^9\,c^7}}{4\,a\,c}+\frac {9\,a\,d\,e^{10}\,\sqrt {-a^9\,c^7}}{4\,c^3}}\right )\,\sqrt {-\frac {4\,a^3\,c^6\,d^5-9\,a\,e^5\,\sqrt {-a^9\,c^7}+15\,a^5\,c^4\,d\,e^4+15\,a^4\,c^5\,d^3\,e^2-5\,c\,d^2\,e^3\,\sqrt {-a^9\,c^7}}{64\,a^6\,c^7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(5/2)/(a + c*x^2)^2,x)

[Out]

- (((a*e^3 - c*d^2*e)*(d + e*x)^(3/2))/(2*a*c) + ((a*d*e^3 + c*d^3*e)*(d + e*x)^(1/2))/(2*a*c))/(c*(d + e*x)^2
+ a*e^2 + c*d^2 - 2*c*d*(d + e*x)) - 2*atanh((18*a*e^8*(d + e*x)^(1/2)*(- d^5/(16*a^3*c) - (15*d*e^4)/(64*a*c
^3) - (15*d^3*e^2)/(64*a^2*c^2) - (9*e^5*(-a^9*c^7)^(1/2))/(64*a^5*c^7) - (5*d^2*e^3*(-a^9*c^7)^(1/2))/(64*a^6
*c^6))^(1/2))/((27*a*e^11)/(4*c^2) + (43*d^4*e^7)/(4*a) + (15*d^2*e^9)/c + (5*c*d^6*e^5)/(2*a^2) - (9*d*e^10*(
-a^9*c^7)^(1/2))/(4*a^4*c^5) - (7*d^3*e^8*(-a^9*c^7)^(1/2))/(2*a^5*c^4) - (5*d^5*e^6*(-a^9*c^7)^(1/2))/(4*a^6*
c^3)) + (10*c*d^2*e^6*(d + e*x)^(1/2)*(- d^5/(16*a^3*c) - (15*d*e^4)/(64*a*c^3) - (15*d^3*e^2)/(64*a^2*c^2) -
(9*e^5*(-a^9*c^7)^(1/2))/(64*a^5*c^7) - (5*d^2*e^3*(-a^9*c^7)^(1/2))/(64*a^6*c^6))^(1/2))/((27*a*e^11)/(4*c^2)
+ (43*d^4*e^7)/(4*a) + (15*d^2*e^9)/c + (5*c*d^6*e^5)/(2*a^2) - (9*d*e^10*(-a^9*c^7)^(1/2))/(4*a^4*c^5) - (7*
d^3*e^8*(-a^9*c^7)^(1/2))/(2*a^5*c^4) - (5*d^5*e^6*(-a^9*c^7)^(1/2))/(4*a^6*c^3)) + (18*d*e^7*(-a^9*c^7)^(1/2)
*(d + e*x)^(1/2)*(- d^5/(16*a^3*c) - (15*d*e^4)/(64*a*c^3) - (15*d^3*e^2)/(64*a^2*c^2) - (9*e^5*(-a^9*c^7)^(1/
2))/(64*a^5*c^7) - (5*d^2*e^3*(-a^9*c^7)^(1/2))/(64*a^6*c^6))^(1/2))/((27*a^5*c*e^11)/4 + (5*a^2*c^4*d^6*e^5)/
2 + (43*a^3*c^3*d^4*e^7)/4 + 15*a^4*c^2*d^2*e^9 - (9*d*e^10*(-a^9*c^7)^(1/2))/(4*c^2) - (5*d^5*e^6*(-a^9*c^7)^
(1/2))/(4*a^2) - (7*d^3*e^8*(-a^9*c^7)^(1/2))/(2*a*c)) + (10*d^3*e^5*(-a^9*c^7)^(1/2)*(d + e*x)^(1/2)*(- d^5/(
16*a^3*c) - (15*d*e^4)/(64*a*c^3) - (15*d^3*e^2)/(64*a^2*c^2) - (9*e^5*(-a^9*c^7)^(1/2))/(64*a^5*c^7) - (5*d^2
*e^3*(-a^9*c^7)^(1/2))/(64*a^6*c^6))^(1/2))/((27*a^6*e^11)/4 + 15*a^5*c*d^2*e^9 + (5*a^3*c^3*d^6*e^5)/2 + (43*
a^4*c^2*d^4*e^7)/4 - (7*d^3*e^8*(-a^9*c^7)^(1/2))/(2*c^2) - (5*d^5*e^6*(-a^9*c^7)^(1/2))/(4*a*c) - (9*a*d*e^10
*(-a^9*c^7)^(1/2))/(4*c^3)))*(-(4*a^3*c^6*d^5 + 9*a*e^5*(-a^9*c^7)^(1/2) + 15*a^5*c^4*d*e^4 + 15*a^4*c^5*d^3*e
^2 + 5*c*d^2*e^3*(-a^9*c^7)^(1/2))/(64*a^6*c^7))^(1/2) - 2*atanh((18*a*e^8*(d + e*x)^(1/2)*((9*e^5*(-a^9*c^7)^
(1/2))/(64*a^5*c^7) - (15*d*e^4)/(64*a*c^3) - (15*d^3*e^2)/(64*a^2*c^2) - d^5/(16*a^3*c) + (5*d^2*e^3*(-a^9*c^
7)^(1/2))/(64*a^6*c^6))^(1/2))/((27*a*e^11)/(4*c^2) + (43*d^4*e^7)/(4*a) + (15*d^2*e^9)/c + (5*c*d^6*e^5)/(2*a
^2) + (9*d*e^10*(-a^9*c^7)^(1/2))/(4*a^4*c^5) + (7*d^3*e^8*(-a^9*c^7)^(1/2))/(2*a^5*c^4) + (5*d^5*e^6*(-a^9*c^
7)^(1/2))/(4*a^6*c^3)) + (10*c*d^2*e^6*(d + e*x)^(1/2)*((9*e^5*(-a^9*c^7)^(1/2))/(64*a^5*c^7) - (15*d*e^4)/(64
*a*c^3) - (15*d^3*e^2)/(64*a^2*c^2) - d^5/(16*a^3*c) + (5*d^2*e^3*(-a^9*c^7)^(1/2))/(64*a^6*c^6))^(1/2))/((27*
a*e^11)/(4*c^2) + (43*d^4*e^7)/(4*a) + (15*d^2*e^9)/c + (5*c*d^6*e^5)/(2*a^2) + (9*d*e^10*(-a^9*c^7)^(1/2))/(4
*a^4*c^5) + (7*d^3*e^8*(-a^9*c^7)^(1/2))/(2*a^5*c^4) + (5*d^5*e^6*(-a^9*c^7)^(1/2))/(4*a^6*c^3)) - (18*d*e^7*(
-a^9*c^7)^(1/2)*(d + e*x)^(1/2)*((9*e^5*(-a^9*c^7)^(1/2))/(64*a^5*c^7) - (15*d*e^4)/(64*a*c^3) - (15*d^3*e^2)/
(64*a^2*c^2) - d^5/(16*a^3*c) + (5*d^2*e^3*(-a^9*c^7)^(1/2))/(64*a^6*c^6))^(1/2))/((27*a^5*c*e^11)/4 + (5*a^2*
c^4*d^6*e^5)/2 + (43*a^3*c^3*d^4*e^7)/4 + 15*a^4*c^2*d^2*e^9 + (9*d*e^10*(-a^9*c^7)^(1/2))/(4*c^2) + (5*d^5*e^
6*(-a^9*c^7)^(1/2))/(4*a^2) + (7*d^3*e^8*(-a^9*c^7)^(1/2))/(2*a*c)) - (10*d^3*e^5*(-a^9*c^7)^(1/2)*(d + e*x)^(
1/2)*((9*e^5*(-a^9*c^7)^(1/2))/(64*a^5*c^7) - (15*d*e^4)/(64*a*c^3) - (15*d^3*e^2)/(64*a^2*c^2) - d^5/(16*a^3*
c) + (5*d^2*e^3*(-a^9*c^7)^(1/2))/(64*a^6*c^6))^(1/2))/((27*a^6*e^11)/4 + 15*a^5*c*d^2*e^9 + (5*a^3*c^3*d^6*e^
5)/2 + (43*a^4*c^2*d^4*e^7)/4 + (7*d^3*e^8*(-a^9*c^7)^(1/2))/(2*c^2) + (5*d^5*e^6*(-a^9*c^7)^(1/2))/(4*a*c) +
(9*a*d*e^10*(-a^9*c^7)^(1/2))/(4*c^3)))*(-(4*a^3*c^6*d^5 - 9*a*e^5*(-a^9*c^7)^(1/2) + 15*a^5*c^4*d*e^4 + 15*a^
4*c^5*d^3*e^2 - 5*c*d^2*e^3*(-a^9*c^7)^(1/2))/(64*a^6*c^7))^(1/2)

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