3.7.44 \(\int \frac {(d+e x)^{5/2}}{(a+c x^2)^3} \, dx\) [644]

Optimal. Leaf size=846 \[ -\frac {(a e-c d x) (d+e x)^{3/2}}{4 a c \left (a+c x^2\right )^2}-\frac {3 \sqrt {d+e x} \left (a d e-\left (2 c d^2+a e^2\right ) x\right )}{16 a^2 c \left (a+c x^2\right )}+\frac {3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (2 c d^2+a e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{32 \sqrt {2} a^2 c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (2 c d^2+a e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{32 \sqrt {2} a^2 c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (2 c d^2+a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{64 \sqrt {2} a^2 c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (2 c d^2+a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{64 \sqrt {2} a^2 c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]

[Out]

-1/4*(-c*d*x+a*e)*(e*x+d)^(3/2)/a/c/(c*x^2+a)^2-3/16*(a*d*e-(a*e^2+2*c*d^2)*x)*(e*x+d)^(1/2)/a^2/c/(c*x^2+a)+3
/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))*(2*c^(3/2)*d^3+2*a*d*e^2*c^(1/2)+(a*e^2+2*c*d^2)*(a*e^2+c*d^2)^(1/2))/a^2/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)-3/64*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))*(2*c^(3/2)*d^3+2*a*d*e^2*c^(1/2)+(a*e^2+2*c*
d^2)*(a*e^2+c*d^2)^(1/2))/a^2/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)-3/128*
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))*
(2*c^(3/2)*d^3+2*a*d*e^2*c^(1/2)-(a*e^2+2*c*d^2)*(a*e^2+c*d^2)^(1/2))/a^2/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)+3/128*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))*(2*c^(3/2)*d^3+2*a*d*e^2*c^(1/2)-(a*e^2+2*c*d^2)*(a*e^2+c*d^2)^(1/
2))/a^2/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 = 2.18, antiderivative size = 846, 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, 835, 841, 1183, 648, 632, 212, 642} \begin {gather*} -\frac {(a e-c d x) (d+e x)^{3/2}}{4 a c \left (c x^2+a\right )^2}-\frac {3 \left (a d e-\left (2 c d^2+a e^2\right ) x\right ) \sqrt {d+e x}}{16 a^2 c \left (c x^2+a\right )}+\frac {3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (2 c d^2+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^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (2 c d^2+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^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} e^2 d-\sqrt {c d^2+a e^2} \left (2 c d^2+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^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} e^2 d-\sqrt {c d^2+a e^2} \left (2 c d^2+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^{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)^3,x]

[Out]

-1/4*((a*e - c*d*x)*(d + e*x)^(3/2))/(a*c*(a + c*x^2)^2) - (3*Sqrt[d + e*x]*(a*d*e - (2*c*d^2 + a*e^2)*x))/(16
*a^2*c*(a + c*x^2)) + (3*e*(2*c^(3/2)*d^3 + 2*a*Sqrt[c]*d*e^2 + Sqrt[c*d^2 + a*e^2]*(2*c*d^2 + 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^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (3*e*(2*c^(3/2)*d^3 +
2*a*Sqrt[c]*d*e^2 + Sqrt[c*d^2 + a*e^2]*(2*c*d^2 + a*e^2))*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] + Sq
rt[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^(7/4)*Sqrt[c*d^2 + a*e^
2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (3*e*(2*c^(3/2)*d^3 + 2*a*Sqrt[c]*d*e^2 - Sqrt[c*d^2 + a*e^2]*(2*c
*d^2 + 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^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (3*e
*(2*c^(3/2)*d^3 + 2*a*Sqrt[c]*d*e^2 - Sqrt[c*d^2 + a*e^2]*(2*c*d^2 + 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^(7/4)*Sqr
t[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 835

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

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 )^3} \, dx &=-\frac {(a e-c d x) (d+e x)^{3/2}}{4 a c \left (a+c x^2\right )^2}+\frac {\int \frac {\sqrt {d+e x} \left (\frac {3}{2} \left (2 c d^2+a e^2\right )+\frac {3}{2} c d e x\right )}{\left (a+c x^2\right )^2} \, dx}{4 a c}\\ &=-\frac {(a e-c d x) (d+e x)^{3/2}}{4 a c \left (a+c x^2\right )^2}-\frac {3 \sqrt {d+e x} \left (a d e-\left (2 c d^2+a e^2\right ) x\right )}{16 a^2 c \left (a+c x^2\right )}+\frac {\int \frac {\frac {3}{4} c d \left (4 c d^2+3 a e^2\right )+\frac {3}{4} c e \left (2 c d^2+a e^2\right ) x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx}{8 a^2 c^2}\\ &=-\frac {(a e-c d x) (d+e x)^{3/2}}{4 a c \left (a+c x^2\right )^2}-\frac {3 \sqrt {d+e x} \left (a d e-\left (2 c d^2+a e^2\right ) x\right )}{16 a^2 c \left (a+c x^2\right )}+\frac {\text {Subst}\left (\int \frac {-\frac {3}{4} c d e \left (2 c d^2+a e^2\right )+\frac {3}{4} c d e \left (4 c d^2+3 a e^2\right )+\frac {3}{4} c e \left (2 c d^2+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^2}\\ &=-\frac {(a e-c d x) (d+e x)^{3/2}}{4 a c \left (a+c x^2\right )^2}-\frac {3 \sqrt {d+e x} \left (a d e-\left (2 c d^2+a e^2\right ) x\right )}{16 a^2 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 (-\frac {3}{4} c d e \left (2 c d^2+a e^2\right )+\frac {3}{4} c d e \left (4 c d^2+3 a e^2\right )\right )}{\sqrt [4]{c}}-\left (-\frac {3}{4} c d e \left (2 c d^2+a e^2\right )-\frac {3}{4} \sqrt {c} e \sqrt {c d^2+a e^2} \left (2 c d^2+a e^2\right )+\frac {3}{4} c d e \left (4 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 )}{8 \sqrt {2} a^2 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 (-\frac {3}{4} c d e \left (2 c d^2+a e^2\right )+\frac {3}{4} c d e \left (4 c d^2+3 a e^2\right )\right )}{\sqrt [4]{c}}+\left (-\frac {3}{4} c d e \left (2 c d^2+a e^2\right )-\frac {3}{4} \sqrt {c} e \sqrt {c d^2+a e^2} \left (2 c d^2+a e^2\right )+\frac {3}{4} c d e \left (4 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 )}{8 \sqrt {2} a^2 c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ &=-\frac {(a e-c d x) (d+e x)^{3/2}}{4 a c \left (a+c x^2\right )^2}-\frac {3 \sqrt {d+e x} \left (a d e-\left (2 c d^2+a e^2\right ) x\right )}{16 a^2 c \left (a+c x^2\right )}-\frac {\left (3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (2 c d^2+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 )}{64 \sqrt {2} a^2 c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (2 c d^2+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 )}{64 \sqrt {2} a^2 c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (2 c d^2+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 )}{64 a^2 c^2 \sqrt {c d^2+a e^2}}+\frac {\left (3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (2 c d^2+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 )}{64 a^2 c^2 \sqrt {c d^2+a e^2}}\\ &=-\frac {(a e-c d x) (d+e x)^{3/2}}{4 a c \left (a+c x^2\right )^2}-\frac {3 \sqrt {d+e x} \left (a d e-\left (2 c d^2+a e^2\right ) x\right )}{16 a^2 c \left (a+c x^2\right )}-\frac {3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (2 c d^2+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^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (2 c d^2+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^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\left (3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (2 c d^2+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 )}{32 a^2 c^2 \sqrt {c d^2+a e^2}}-\frac {\left (3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (2 c d^2+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 )}{32 a^2 c^2 \sqrt {c d^2+a e^2}}\\ &=-\frac {(a e-c d x) (d+e x)^{3/2}}{4 a c \left (a+c x^2\right )^2}-\frac {3 \sqrt {d+e x} \left (a d e-\left (2 c d^2+a e^2\right ) x\right )}{16 a^2 c \left (a+c x^2\right )}+\frac {3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (2 c d^2+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^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (2 c d^2+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^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (2 c d^2+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^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {3 e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (2 c d^2+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^{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 = 1.66, size = 308, normalized size = 0.36 \begin {gather*} \frac {\frac {2 \sqrt {a} c \sqrt {d+e x} \left (6 c^2 d^2 x^3-a^2 e (7 d+e x)+a c x \left (10 d^2+d e x+3 e^2 x^2\right )\right )}{\left (a+c x^2\right )^2}-3 i \sqrt {-c d-i \sqrt {a} \sqrt {c} e} \left (4 c d^2-2 i \sqrt {a} \sqrt {c} d e+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 )+3 i \sqrt {-c d+i \sqrt {a} \sqrt {c} e} \left (4 c d^2+2 i \sqrt {a} \sqrt {c} d e+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 )}{32 a^{5/2} c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(5/2)/(a + c*x^2)^3,x]

[Out]

((2*Sqrt[a]*c*Sqrt[d + e*x]*(6*c^2*d^2*x^3 - a^2*e*(7*d + e*x) + a*c*x*(10*d^2 + d*e*x + 3*e^2*x^2)))/(a + c*x
^2)^2 - (3*I)*Sqrt[-(c*d) - I*Sqrt[a]*Sqrt[c]*e]*(4*c*d^2 - (2*I)*Sqrt[a]*Sqrt[c]*d*e + a*e^2)*ArcTan[(Sqrt[-(
c*d) - I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + I*Sqrt[a]*e)] + (3*I)*Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*
e]*(4*c*d^2 + (2*I)*Sqrt[a]*Sqrt[c]*d*e + a*e^2)*ArcTan[(Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sq
rt[c]*d - I*Sqrt[a]*e)])/(32*a^(5/2)*c^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1748\) vs. \(2(690)=1380\).
time = 0.64, size = 1749, normalized size = 2.07

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1749\)
default	\(\text {Expression too large to display}\)	\(1749\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(5/2)/(c*x^2+a)^3,x,method=_RETURNVERBOSE)

[Out]

2*e^5*((3/32*(a*e^2+2*c*d^2)/a^2/e^4*(e*x+d)^(7/2)-1/16*d*(4*a*e^2+9*c*d^2)/a^2/e^4*(e*x+d)^(5/2)-1/32*(a^2*e^
4-17*a*c*d^2*e^2-18*c^2*d^4)/a^2/e^4/c*(e*x+d)^(3/2)-3/16*d*(a*e^2+c*d^2)^2/a^2/e^4/c*(e*x+d)^(1/2))/(c*(e*x+d
)^2-2*c*d*(e*x+d)+e^2*a+c*d^2)^2+3/32/a^2/e^4/c*(1/4/a/c/e^2*(1/2*(-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+(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a*c*d*e^2+2*(2*(a*c*e^2+c^2*d^2)^(1/2)
+2*c*d)^(1/2)*c^2*d^3+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^(1/2)*(a*e^2+c*d^2)^
(1/2)*d-(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^2-2*(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*(4*(a*e^2+c*d^2)^(1/2)*d*a*c*e^2-1/2*(-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+(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a*c*d*e^2+2*(2*(a*c*e^2+c^2*d^2)
^(1/2)+2*c*d)^(1/2)*c^2*d^3+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^(1/2)*(a*e^2+c
*d^2)^(1/2)*d-(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^2-2*(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*(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+(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(
1/2)*a*c*d*e^2+2*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^2*d^3+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^(1/2)*(a*e^2+c*d^2)^(1/2)*d-(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^2-2*(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*(-4*(a*e^2+c*d^2)^(1/2)*d*a*c*e
^2+1/2*(-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+(2*(a*c*e^2+c^2*d^2)^(1/2)+
2*c*d)^(1/2)*a*c*d*e^2+2*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^2*d^3+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^(1/2)*(a*e^2+c*d^2)^(1/2)*d-(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^2-2*(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)^3,x, algorithm="maxima")

[Out]

integrate((x*e + d)^(5/2)/(c*x^2 + a)^3, x)

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Fricas [A]
time = 3.16, size = 994, normalized size = 1.17 \begin {gather*} \frac {3 \, {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt {-\frac {a^{5} c^{3} \sqrt {-\frac {e^{10}}{a^{5} c^{7}}} + 16 \, c^{2} d^{5} + 20 \, a c d^{3} e^{2} + 5 \, a^{2} d e^{4}}{a^{5} c^{3}}} \log \left (27 \, {\left (16 \, c^{2} d^{4} e^{5} + 12 \, a c d^{2} e^{7} + a^{2} e^{9}\right )} \sqrt {x e + d} + 27 \, {\left (2 \, a^{3} c^{2} d e^{6} - {\left (4 \, a^{5} c^{6} d^{2} + a^{6} c^{5} e^{2}\right )} \sqrt {-\frac {e^{10}}{a^{5} c^{7}}}\right )} \sqrt {-\frac {a^{5} c^{3} \sqrt {-\frac {e^{10}}{a^{5} c^{7}}} + 16 \, c^{2} d^{5} + 20 \, a c d^{3} e^{2} + 5 \, a^{2} d e^{4}}{a^{5} c^{3}}}\right ) - 3 \, {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt {-\frac {a^{5} c^{3} \sqrt {-\frac {e^{10}}{a^{5} c^{7}}} + 16 \, c^{2} d^{5} + 20 \, a c d^{3} e^{2} + 5 \, a^{2} d e^{4}}{a^{5} c^{3}}} \log \left (27 \, {\left (16 \, c^{2} d^{4} e^{5} + 12 \, a c d^{2} e^{7} + a^{2} e^{9}\right )} \sqrt {x e + d} - 27 \, {\left (2 \, a^{3} c^{2} d e^{6} - {\left (4 \, a^{5} c^{6} d^{2} + a^{6} c^{5} e^{2}\right )} \sqrt {-\frac {e^{10}}{a^{5} c^{7}}}\right )} \sqrt {-\frac {a^{5} c^{3} \sqrt {-\frac {e^{10}}{a^{5} c^{7}}} + 16 \, c^{2} d^{5} + 20 \, a c d^{3} e^{2} + 5 \, a^{2} d e^{4}}{a^{5} c^{3}}}\right ) + 3 \, {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt {\frac {a^{5} c^{3} \sqrt {-\frac {e^{10}}{a^{5} c^{7}}} - 16 \, c^{2} d^{5} - 20 \, a c d^{3} e^{2} - 5 \, a^{2} d e^{4}}{a^{5} c^{3}}} \log \left (27 \, {\left (16 \, c^{2} d^{4} e^{5} + 12 \, a c d^{2} e^{7} + a^{2} e^{9}\right )} \sqrt {x e + d} + 27 \, {\left (2 \, a^{3} c^{2} d e^{6} + {\left (4 \, a^{5} c^{6} d^{2} + a^{6} c^{5} e^{2}\right )} \sqrt {-\frac {e^{10}}{a^{5} c^{7}}}\right )} \sqrt {\frac {a^{5} c^{3} \sqrt {-\frac {e^{10}}{a^{5} c^{7}}} - 16 \, c^{2} d^{5} - 20 \, a c d^{3} e^{2} - 5 \, a^{2} d e^{4}}{a^{5} c^{3}}}\right ) - 3 \, {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt {\frac {a^{5} c^{3} \sqrt {-\frac {e^{10}}{a^{5} c^{7}}} - 16 \, c^{2} d^{5} - 20 \, a c d^{3} e^{2} - 5 \, a^{2} d e^{4}}{a^{5} c^{3}}} \log \left (27 \, {\left (16 \, c^{2} d^{4} e^{5} + 12 \, a c d^{2} e^{7} + a^{2} e^{9}\right )} \sqrt {x e + d} - 27 \, {\left (2 \, a^{3} c^{2} d e^{6} + {\left (4 \, a^{5} c^{6} d^{2} + a^{6} c^{5} e^{2}\right )} \sqrt {-\frac {e^{10}}{a^{5} c^{7}}}\right )} \sqrt {\frac {a^{5} c^{3} \sqrt {-\frac {e^{10}}{a^{5} c^{7}}} - 16 \, c^{2} d^{5} - 20 \, a c d^{3} e^{2} - 5 \, a^{2} d e^{4}}{a^{5} c^{3}}}\right ) + 4 \, {\left (6 \, c^{2} d^{2} x^{3} + 10 \, a c d^{2} x + {\left (3 \, a c x^{3} - a^{2} x\right )} e^{2} + {\left (a c d x^{2} - 7 \, a^{2} d\right )} e\right )} \sqrt {x e + d}}{64 \, {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} 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)^3,x, algorithm="fricas")

[Out]

1/64*(3*(a^2*c^3*x^4 + 2*a^3*c^2*x^2 + a^4*c)*sqrt(-(a^5*c^3*sqrt(-e^10/(a^5*c^7)) + 16*c^2*d^5 + 20*a*c*d^3*e
^2 + 5*a^2*d*e^4)/(a^5*c^3))*log(27*(16*c^2*d^4*e^5 + 12*a*c*d^2*e^7 + a^2*e^9)*sqrt(x*e + d) + 27*(2*a^3*c^2*
d*e^6 - (4*a^5*c^6*d^2 + a^6*c^5*e^2)*sqrt(-e^10/(a^5*c^7)))*sqrt(-(a^5*c^3*sqrt(-e^10/(a^5*c^7)) + 16*c^2*d^5
+ 20*a*c*d^3*e^2 + 5*a^2*d*e^4)/(a^5*c^3))) - 3*(a^2*c^3*x^4 + 2*a^3*c^2*x^2 + a^4*c)*sqrt(-(a^5*c^3*sqrt(-e^
10/(a^5*c^7)) + 16*c^2*d^5 + 20*a*c*d^3*e^2 + 5*a^2*d*e^4)/(a^5*c^3))*log(27*(16*c^2*d^4*e^5 + 12*a*c*d^2*e^7
+ a^2*e^9)*sqrt(x*e + d) - 27*(2*a^3*c^2*d*e^6 - (4*a^5*c^6*d^2 + a^6*c^5*e^2)*sqrt(-e^10/(a^5*c^7)))*sqrt(-(a
^5*c^3*sqrt(-e^10/(a^5*c^7)) + 16*c^2*d^5 + 20*a*c*d^3*e^2 + 5*a^2*d*e^4)/(a^5*c^3))) + 3*(a^2*c^3*x^4 + 2*a^3
*c^2*x^2 + a^4*c)*sqrt((a^5*c^3*sqrt(-e^10/(a^5*c^7)) - 16*c^2*d^5 - 20*a*c*d^3*e^2 - 5*a^2*d*e^4)/(a^5*c^3))*
log(27*(16*c^2*d^4*e^5 + 12*a*c*d^2*e^7 + a^2*e^9)*sqrt(x*e + d) + 27*(2*a^3*c^2*d*e^6 + (4*a^5*c^6*d^2 + a^6*
c^5*e^2)*sqrt(-e^10/(a^5*c^7)))*sqrt((a^5*c^3*sqrt(-e^10/(a^5*c^7)) - 16*c^2*d^5 - 20*a*c*d^3*e^2 - 5*a^2*d*e^
4)/(a^5*c^3))) - 3*(a^2*c^3*x^4 + 2*a^3*c^2*x^2 + a^4*c)*sqrt((a^5*c^3*sqrt(-e^10/(a^5*c^7)) - 16*c^2*d^5 - 20
*a*c*d^3*e^2 - 5*a^2*d*e^4)/(a^5*c^3))*log(27*(16*c^2*d^4*e^5 + 12*a*c*d^2*e^7 + a^2*e^9)*sqrt(x*e + d) - 27*(
2*a^3*c^2*d*e^6 + (4*a^5*c^6*d^2 + a^6*c^5*e^2)*sqrt(-e^10/(a^5*c^7)))*sqrt((a^5*c^3*sqrt(-e^10/(a^5*c^7)) - 1
6*c^2*d^5 - 20*a*c*d^3*e^2 - 5*a^2*d*e^4)/(a^5*c^3))) + 4*(6*c^2*d^2*x^3 + 10*a*c*d^2*x + (3*a*c*x^3 - a^2*x)*
e^2 + (a*c*d*x^2 - 7*a^2*d)*e)*sqrt(x*e + d))/(a^2*c^3*x^4 + 2*a^3*c^2*x^2 + a^4*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)**3,x)

[Out]

Timed out

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Giac [A]
time = 2.49, size = 586, normalized size = 0.69 \begin {gather*} -\frac {3 \, {\left (4 \, c^{4} d^{4} + 3 \, a c^{3} d^{2} e^{2} + {\left (2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} c^{2} - 2 \, {\left (\sqrt {-a c} c^{2} d^{3} e + \sqrt {-a c} a c d e^{3}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a^{2} c^{2} d + \sqrt {a^{4} c^{4} d^{2} - {\left (a^{2} c^{2} d^{2} + a^{3} c e^{2}\right )} a^{2} c^{2}}}{a^{2} c^{2}}}}\right )}{32 \, {\left (a^{3} c^{3} e - \sqrt {-a c} a^{2} c^{3} d\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e}} - \frac {3 \, {\left (4 \, c^{4} d^{4} + 3 \, a c^{3} d^{2} e^{2} + {\left (2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} c^{2} + 2 \, {\left (\sqrt {-a c} c^{2} d^{3} e + \sqrt {-a c} a c d e^{3}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a^{2} c^{2} d - \sqrt {a^{4} c^{4} d^{2} - {\left (a^{2} c^{2} d^{2} + a^{3} c e^{2}\right )} a^{2} c^{2}}}{a^{2} c^{2}}}}\right )}{32 \, {\left (a^{3} c^{3} e + \sqrt {-a c} a^{2} c^{3} d\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e}} + \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 + 3 \, {\left (x e + d\right )}^{\frac {7}{2}} a c e^{3} - 8 \, {\left (x e + d\right )}^{\frac {5}{2}} a c d e^{3} + 17 \, {\left (x e + d\right )}^{\frac {3}{2}} a c d^{2} e^{3} - 12 \, \sqrt {x e + d} a c d^{3} e^{3} - {\left (x e + d\right )}^{\frac {3}{2}} a^{2} e^{5} - 6 \, \sqrt {x e + d} a^{2} d e^{5}}{16 \, {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + a e^{2}\right )}^{2} a^{2} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)/(c*x^2+a)^3,x, algorithm="giac")

[Out]

-3/32*(4*c^4*d^4 + 3*a*c^3*d^2*e^2 + (2*a*c*d^2*e^2 + a^2*e^4)*c^2 - 2*(sqrt(-a*c)*c^2*d^3*e + sqrt(-a*c)*a*c*
d*e^3)*abs(c))*arctan(sqrt(x*e + d)/sqrt(-(a^2*c^2*d + sqrt(a^4*c^4*d^2 - (a^2*c^2*d^2 + a^3*c*e^2)*a^2*c^2))/
(a^2*c^2)))/((a^3*c^3*e - sqrt(-a*c)*a^2*c^3*d)*sqrt(-c^2*d + sqrt(-a*c)*c*e)) - 3/32*(4*c^4*d^4 + 3*a*c^3*d^2
*e^2 + (2*a*c*d^2*e^2 + a^2*e^4)*c^2 + 2*(sqrt(-a*c)*c^2*d^3*e + sqrt(-a*c)*a*c*d*e^3)*abs(c))*arctan(sqrt(x*e
+ d)/sqrt(-(a^2*c^2*d - sqrt(a^4*c^4*d^2 - (a^2*c^2*d^2 + a^3*c*e^2)*a^2*c^2))/(a^2*c^2)))/((a^3*c^3*e + sqrt
(-a*c)*a^2*c^3*d)*sqrt(-c^2*d - sqrt(-a*c)*c*e)) + 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 + 3*(x*e + d)^(7/2)*a*c*e^3 - 8*(x*e + d)^(5/
2)*a*c*d*e^3 + 17*(x*e + d)^(3/2)*a*c*d^2*e^3 - 12*sqrt(x*e + d)*a*c*d^3*e^3 - (x*e + d)^(3/2)*a^2*e^5 - 6*sqr
t(x*e + d)*a^2*d*e^5)/(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + a*e^2)^2*a^2*c)

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Mupad [B]
time = 0.85, size = 1028, normalized size = 1.22 \begin {gather*} \frac {\frac {3\,e\,\left (2\,c\,d^2+a\,e^2\right )\,{\left (d+e\,x\right )}^{7/2}}{16\,a^2}+\frac {{\left (d+e\,x\right )}^{3/2}\,\left (-a^2\,e^5+17\,a\,c\,d^2\,e^3+18\,c^2\,d^4\,e\right )}{16\,a^2\,c}-\frac {d\,\left (9\,c\,d^2\,e+4\,a\,e^3\right )\,{\left (d+e\,x\right )}^{5/2}}{8\,a^2}-\frac {3\,\sqrt {d+e\,x}\,\left (a^2\,d\,e^5+2\,a\,c\,d^3\,e^3+c^2\,d^5\,e\right )}{8\,a^2\,c}}{c^2\,{\left (d+e\,x\right )}^4+a^2\,e^4+c^2\,d^4+\left (6\,c^2\,d^2+2\,a\,c\,e^2\right )\,{\left (d+e\,x\right )}^2-\left (4\,c^2\,d^3+4\,a\,c\,d\,e^2\right )\,\left (d+e\,x\right )-4\,c^2\,d\,{\left (d+e\,x\right )}^3+2\,a\,c\,d^2\,e^2}-2\,\mathrm {atanh}\left (\frac {9\,e^8\,\sqrt {d+e\,x}\,\sqrt {-\frac {9\,d^5}{256\,a^5\,c}-\frac {45\,d\,e^4}{4096\,a^3\,c^3}-\frac {45\,d^3\,e^2}{1024\,a^4\,c^2}-\frac {9\,e^5\,\sqrt {-a^{15}\,c^7}}{4096\,a^{10}\,c^7}}}{32\,\left (\frac {27\,e^{11}}{2048\,a\,c^2}+\frac {27\,d^4\,e^7}{512\,a^3}+\frac {135\,d^2\,e^9}{2048\,a^2\,c}-\frac {27\,d\,e^{10}\,\sqrt {-a^{15}\,c^7}}{1024\,a^9\,c^5}-\frac {27\,d^3\,e^8\,\sqrt {-a^{15}\,c^7}}{1024\,a^{10}\,c^4}\right )}+\frac {9\,d\,e^7\,\sqrt {-a^{15}\,c^7}\,\sqrt {d+e\,x}\,\sqrt {-\frac {9\,d^5}{256\,a^5\,c}-\frac {45\,d\,e^4}{4096\,a^3\,c^3}-\frac {45\,d^3\,e^2}{1024\,a^4\,c^2}-\frac {9\,e^5\,\sqrt {-a^{15}\,c^7}}{4096\,a^{10}\,c^7}}}{32\,\left (\frac {27\,a^7\,c\,e^{11}}{2048}+\frac {27\,a^5\,c^3\,d^4\,e^7}{512}+\frac {135\,a^6\,c^2\,d^2\,e^9}{2048}-\frac {27\,d\,e^{10}\,\sqrt {-a^{15}\,c^7}}{1024\,a\,c^2}-\frac {27\,d^3\,e^8\,\sqrt {-a^{15}\,c^7}}{1024\,a^2\,c}\right )}\right )\,\sqrt {-\frac {9\,\left (e^5\,\sqrt {-a^{15}\,c^7}+16\,a^5\,c^6\,d^5+5\,a^7\,c^4\,d\,e^4+20\,a^6\,c^5\,d^3\,e^2\right )}{4096\,a^{10}\,c^7}}-2\,\mathrm {atanh}\left (\frac {9\,e^8\,\sqrt {d+e\,x}\,\sqrt {\frac {9\,e^5\,\sqrt {-a^{15}\,c^7}}{4096\,a^{10}\,c^7}-\frac {45\,d\,e^4}{4096\,a^3\,c^3}-\frac {45\,d^3\,e^2}{1024\,a^4\,c^2}-\frac {9\,d^5}{256\,a^5\,c}}}{32\,\left (\frac {27\,e^{11}}{2048\,a\,c^2}+\frac {27\,d^4\,e^7}{512\,a^3}+\frac {135\,d^2\,e^9}{2048\,a^2\,c}+\frac {27\,d\,e^{10}\,\sqrt {-a^{15}\,c^7}}{1024\,a^9\,c^5}+\frac {27\,d^3\,e^8\,\sqrt {-a^{15}\,c^7}}{1024\,a^{10}\,c^4}\right )}-\frac {9\,d\,e^7\,\sqrt {-a^{15}\,c^7}\,\sqrt {d+e\,x}\,\sqrt {\frac {9\,e^5\,\sqrt {-a^{15}\,c^7}}{4096\,a^{10}\,c^7}-\frac {45\,d\,e^4}{4096\,a^3\,c^3}-\frac {45\,d^3\,e^2}{1024\,a^4\,c^2}-\frac {9\,d^5}{256\,a^5\,c}}}{32\,\left (\frac {27\,a^7\,c\,e^{11}}{2048}+\frac {27\,a^5\,c^3\,d^4\,e^7}{512}+\frac {135\,a^6\,c^2\,d^2\,e^9}{2048}+\frac {27\,d\,e^{10}\,\sqrt {-a^{15}\,c^7}}{1024\,a\,c^2}+\frac {27\,d^3\,e^8\,\sqrt {-a^{15}\,c^7}}{1024\,a^2\,c}\right )}\right )\,\sqrt {-\frac {9\,\left (16\,a^5\,c^6\,d^5-e^5\,\sqrt {-a^{15}\,c^7}+5\,a^7\,c^4\,d\,e^4+20\,a^6\,c^5\,d^3\,e^2\right )}{4096\,a^{10}\,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)^3,x)

[Out]

((3*e*(a*e^2 + 2*c*d^2)*(d + e*x)^(7/2))/(16*a^2) + ((d + e*x)^(3/2)*(18*c^2*d^4*e - a^2*e^5 + 17*a*c*d^2*e^3)
)/(16*a^2*c) - (d*(4*a*e^3 + 9*c*d^2*e)*(d + e*x)^(5/2))/(8*a^2) - (3*(d + e*x)^(1/2)*(a^2*d*e^5 + c^2*d^5*e +
2*a*c*d^3*e^3))/(8*a^2*c))/(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) - 2*atanh((9*e^8*(d + e*x)^(1/2)*(- (9*d
^5)/(256*a^5*c) - (45*d*e^4)/(4096*a^3*c^3) - (45*d^3*e^2)/(1024*a^4*c^2) - (9*e^5*(-a^15*c^7)^(1/2))/(4096*a^
10*c^7))^(1/2))/(32*((27*e^11)/(2048*a*c^2) + (27*d^4*e^7)/(512*a^3) + (135*d^2*e^9)/(2048*a^2*c) - (27*d*e^10
*(-a^15*c^7)^(1/2))/(1024*a^9*c^5) - (27*d^3*e^8*(-a^15*c^7)^(1/2))/(1024*a^10*c^4))) + (9*d*e^7*(-a^15*c^7)^(
1/2)*(d + e*x)^(1/2)*(- (9*d^5)/(256*a^5*c) - (45*d*e^4)/(4096*a^3*c^3) - (45*d^3*e^2)/(1024*a^4*c^2) - (9*e^5
*(-a^15*c^7)^(1/2))/(4096*a^10*c^7))^(1/2))/(32*((27*a^7*c*e^11)/2048 + (27*a^5*c^3*d^4*e^7)/512 + (135*a^6*c^
2*d^2*e^9)/2048 - (27*d*e^10*(-a^15*c^7)^(1/2))/(1024*a*c^2) - (27*d^3*e^8*(-a^15*c^7)^(1/2))/(1024*a^2*c))))*
(-(9*(e^5*(-a^15*c^7)^(1/2) + 16*a^5*c^6*d^5 + 5*a^7*c^4*d*e^4 + 20*a^6*c^5*d^3*e^2))/(4096*a^10*c^7))^(1/2) -
2*atanh((9*e^8*(d + e*x)^(1/2)*((9*e^5*(-a^15*c^7)^(1/2))/(4096*a^10*c^7) - (45*d*e^4)/(4096*a^3*c^3) - (45*d
^3*e^2)/(1024*a^4*c^2) - (9*d^5)/(256*a^5*c))^(1/2))/(32*((27*e^11)/(2048*a*c^2) + (27*d^4*e^7)/(512*a^3) + (1
35*d^2*e^9)/(2048*a^2*c) + (27*d*e^10*(-a^15*c^7)^(1/2))/(1024*a^9*c^5) + (27*d^3*e^8*(-a^15*c^7)^(1/2))/(1024
*a^10*c^4))) - (9*d*e^7*(-a^15*c^7)^(1/2)*(d + e*x)^(1/2)*((9*e^5*(-a^15*c^7)^(1/2))/(4096*a^10*c^7) - (45*d*e
^4)/(4096*a^3*c^3) - (45*d^3*e^2)/(1024*a^4*c^2) - (9*d^5)/(256*a^5*c))^(1/2))/(32*((27*a^7*c*e^11)/2048 + (27
*a^5*c^3*d^4*e^7)/512 + (135*a^6*c^2*d^2*e^9)/2048 + (27*d*e^10*(-a^15*c^7)^(1/2))/(1024*a*c^2) + (27*d^3*e^8*
(-a^15*c^7)^(1/2))/(1024*a^2*c))))*(-(9*(16*a^5*c^6*d^5 - e^5*(-a^15*c^7)^(1/2) + 5*a^7*c^4*d*e^4 + 20*a^6*c^5
*d^3*e^2))/(4096*a^10*c^7))^(1/2)

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