3.626 \(\int \frac{1}{(d+e x)^{5/2} \left (a+c x^2\right )^2} \, dx\)

Optimal. Leaf size=930 \[ \frac{c d e \left (c d^2-19 a e^2\right )}{2 a \left (c d^2+a e^2\right )^3 \sqrt{d+e x}}+\frac{c^{3/4} e \left (c^2 d^4+34 a c e^2 d^2+\sqrt{c} \left (c d^2-19 a e^2\right ) \sqrt{c d^2+a e^2} d-7 a^2 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{4 \sqrt{2} a \left (c d^2+a e^2\right )^{7/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{c^{3/4} e \left (c^2 d^4+34 a c e^2 d^2+\sqrt{c} \left (c d^2-19 a e^2\right ) \sqrt{c d^2+a e^2} d-7 a^2 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{4 \sqrt{2} a \left (c d^2+a e^2\right )^{7/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{c^{3/4} e \left (c^2 d^4+34 a c e^2 d^2-\sqrt{c} \left (c d^2-19 a e^2\right ) \sqrt{c d^2+a e^2} d-7 a^2 e^4\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{8 \sqrt{2} a \left (c d^2+a e^2\right )^{7/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{c^{3/4} e \left (c^2 d^4+34 a c e^2 d^2-\sqrt{c} \left (c d^2-19 a e^2\right ) \sqrt{c d^2+a e^2} d-7 a^2 e^4\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{8 \sqrt{2} a \left (c d^2+a e^2\right )^{7/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \left (c x^2+a\right )}+\frac{e \left (3 c d^2-7 a e^2\right )}{6 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}} \]

[Out]

(e*(3*c*d^2 - 7*a*e^2))/(6*a*(c*d^2 + a*e^2)^2*(d + e*x)^(3/2)) + (c*d*e*(c*d^2
- 19*a*e^2))/(2*a*(c*d^2 + a*e^2)^3*Sqrt[d + e*x]) + (a*e + c*d*x)/(2*a*(c*d^2 +
a*e^2)*(d + e*x)^(3/2)*(a + c*x^2)) + (c^(3/4)*e*(c^2*d^4 + 34*a*c*d^2*e^2 - 7*
a^2*e^4 + Sqrt[c]*d*(c*d^2 - 19*a*e^2)*Sqrt[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 - Sqr
t[c*d^2 + a*e^2]]])/(4*Sqrt[2]*a*(c*d^2 + a*e^2)^(7/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d
^2 + a*e^2]]) - (c^(3/4)*e*(c^2*d^4 + 34*a*c*d^2*e^2 - 7*a^2*e^4 + Sqrt[c]*d*(c*
d^2 - 19*a*e^2)*Sqrt[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]]])/(4*
Sqrt[2]*a*(c*d^2 + a*e^2)^(7/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (c^(3/4
)*e*(c^2*d^4 + 34*a*c*d^2*e^2 - 7*a^2*e^4 - Sqrt[c]*d*(c*d^2 - 19*a*e^2)*Sqrt[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)])/(8*Sqrt[2]*a*(c*d^2 + a*e^2)^(
7/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (c^(3/4)*e*(c^2*d^4 + 34*a*c*d^2*e
^2 - 7*a^2*e^4 - Sqrt[c]*d*(c*d^2 - 19*a*e^2)*Sqrt[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)])/(8*Sqrt[2]*a*(c*d^2 + a*e^2)^(7/2)*Sqrt[Sqrt[c]*d + Sqrt[
c*d^2 + a*e^2]])

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Rubi [A]  time = 15.5284, antiderivative size = 930, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ \frac{c d e \left (c d^2-19 a e^2\right )}{2 a \left (c d^2+a e^2\right )^3 \sqrt{d+e x}}+\frac{c^{3/4} e \left (c^2 d^4+34 a c e^2 d^2+\sqrt{c} \left (c d^2-19 a e^2\right ) \sqrt{c d^2+a e^2} d-7 a^2 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{4 \sqrt{2} a \left (c d^2+a e^2\right )^{7/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{c^{3/4} e \left (c^2 d^4+34 a c e^2 d^2+\sqrt{c} \left (c d^2-19 a e^2\right ) \sqrt{c d^2+a e^2} d-7 a^2 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{4 \sqrt{2} a \left (c d^2+a e^2\right )^{7/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{c^{3/4} e \left (c^2 d^4+34 a c e^2 d^2-\sqrt{c} \left (c d^2-19 a e^2\right ) \sqrt{c d^2+a e^2} d-7 a^2 e^4\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{8 \sqrt{2} a \left (c d^2+a e^2\right )^{7/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{c^{3/4} e \left (c^2 d^4+34 a c e^2 d^2-\sqrt{c} \left (c d^2-19 a e^2\right ) \sqrt{c d^2+a e^2} d-7 a^2 e^4\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{8 \sqrt{2} a \left (c d^2+a e^2\right )^{7/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \left (c x^2+a\right )}+\frac{e \left (3 c d^2-7 a e^2\right )}{6 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(5/2)*(a + c*x^2)^2),x]

[Out]

(e*(3*c*d^2 - 7*a*e^2))/(6*a*(c*d^2 + a*e^2)^2*(d + e*x)^(3/2)) + (c*d*e*(c*d^2
- 19*a*e^2))/(2*a*(c*d^2 + a*e^2)^3*Sqrt[d + e*x]) + (a*e + c*d*x)/(2*a*(c*d^2 +
a*e^2)*(d + e*x)^(3/2)*(a + c*x^2)) + (c^(3/4)*e*(c^2*d^4 + 34*a*c*d^2*e^2 - 7*
a^2*e^4 + Sqrt[c]*d*(c*d^2 - 19*a*e^2)*Sqrt[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 - Sqr
t[c*d^2 + a*e^2]]])/(4*Sqrt[2]*a*(c*d^2 + a*e^2)^(7/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d
^2 + a*e^2]]) - (c^(3/4)*e*(c^2*d^4 + 34*a*c*d^2*e^2 - 7*a^2*e^4 + Sqrt[c]*d*(c*
d^2 - 19*a*e^2)*Sqrt[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]]])/(4*
Sqrt[2]*a*(c*d^2 + a*e^2)^(7/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (c^(3/4
)*e*(c^2*d^4 + 34*a*c*d^2*e^2 - 7*a^2*e^4 - Sqrt[c]*d*(c*d^2 - 19*a*e^2)*Sqrt[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)])/(8*Sqrt[2]*a*(c*d^2 + a*e^2)^(
7/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (c^(3/4)*e*(c^2*d^4 + 34*a*c*d^2*e
^2 - 7*a^2*e^4 - Sqrt[c]*d*(c*d^2 - 19*a*e^2)*Sqrt[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)])/(8*Sqrt[2]*a*(c*d^2 + a*e^2)^(7/2)*Sqrt[Sqrt[c]*d + Sqrt[
c*d^2 + a*e^2]])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(5/2)/(c*x**2+a)**2,x)

[Out]

Timed out

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Mathematica [C]  time = 1.70451, size = 349, normalized size = 0.38 \[ \frac{-\frac{2 \sqrt{a} \left (4 a^3 e^5+a^2 c e^3 \left (55 d^2+54 d e x+7 e^2 x^2\right )+a c^2 d e \left (-9 d^3-9 d^2 e x+61 d e^2 x^2+57 e^3 x^3\right )-3 c^3 d^3 x (d+e x)^2\right )}{\left (a+c x^2\right ) (d+e x)^{3/2} \left (a e^2+c d^2\right )^3}-\frac{3 i c \left (2 \sqrt{c} d-7 i \sqrt{a} e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-i \sqrt{a} \sqrt{c} e}}\right )}{\left (\sqrt{c} d-i \sqrt{a} e\right )^3 \sqrt{c d-i \sqrt{a} \sqrt{c} e}}+\frac{3 i c \left (2 \sqrt{c} d+7 i \sqrt{a} e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d+i \sqrt{a} \sqrt{c} e}}\right )}{\left (\sqrt{c} d+i \sqrt{a} e\right )^3 \sqrt{c d+i \sqrt{a} \sqrt{c} e}}}{12 a^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(5/2)*(a + c*x^2)^2),x]

[Out]

((-2*Sqrt[a]*(4*a^3*e^5 - 3*c^3*d^3*x*(d + e*x)^2 + a^2*c*e^3*(55*d^2 + 54*d*e*x
+ 7*e^2*x^2) + a*c^2*d*e*(-9*d^3 - 9*d^2*e*x + 61*d*e^2*x^2 + 57*e^3*x^3)))/((c
*d^2 + a*e^2)^3*(d + e*x)^(3/2)*(a + c*x^2)) - ((3*I)*c*(2*Sqrt[c]*d - (7*I)*Sqr
t[a]*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - I*Sqrt[a]*Sqrt[c]*e]])/((Sqrt
[c]*d - I*Sqrt[a]*e)^3*Sqrt[c*d - I*Sqrt[a]*Sqrt[c]*e]) + ((3*I)*c*(2*Sqrt[c]*d
+ (7*I)*Sqrt[a]*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d + I*Sqrt[a]*Sqrt[c]*
e]])/((Sqrt[c]*d + I*Sqrt[a]*e)^3*Sqrt[c*d + I*Sqrt[a]*Sqrt[c]*e]))/(12*a^(3/2))

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Maple [B]  time = 0.113, size = 12749, normalized size = 13.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(5/2)/(c*x^2+a)^2,x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + a\right )}^{2}{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)^2*(e*x + d)^(5/2)),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + a)^2*(e*x + d)^(5/2)), x)

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Fricas [A]  time = 2.40555, size = 10755, normalized size = 11.56 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)^2*(e*x + d)^(5/2)),x, algorithm="fricas")

[Out]

1/24*(36*a*c^2*d^4*e - 220*a^2*c*d^2*e^3 - 16*a^3*e^5 + 12*(c^3*d^3*e^2 - 19*a*c
^2*d*e^4)*x^3 + 4*(6*c^3*d^4*e - 61*a*c^2*d^2*e^3 - 7*a^2*c*e^5)*x^2 + 3*(a^2*c^
3*d^7 + 3*a^3*c^2*d^5*e^2 + 3*a^4*c*d^3*e^4 + a^5*d*e^6 + (a*c^4*d^6*e + 3*a^2*c
^3*d^4*e^3 + 3*a^3*c^2*d^2*e^5 + a^4*c*e^7)*x^3 + (a*c^4*d^7 + 3*a^2*c^3*d^5*e^2
+ 3*a^3*c^2*d^3*e^4 + a^4*c*d*e^6)*x^2 + (a^2*c^3*d^6*e + 3*a^3*c^2*d^4*e^3 + 3
*a^4*c*d^2*e^5 + a^5*e^7)*x)*sqrt(e*x + d)*sqrt(-(4*c^6*d^9 + 63*a*c^5*d^7*e^2 +
189*a^2*c^4*d^5*e^4 - 1155*a^3*c^3*d^3*e^6 + 315*a^4*c^2*d*e^8 + (a^3*c^7*d^14
+ 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 + 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6
*e^8 + 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 + a^10*e^14)*sqrt(-(11025*c^9*d^12
*e^6 + 171990*a*c^8*d^10*e^8 + 494991*a^2*c^7*d^8*e^10 - 1360716*a^3*c^6*d^6*e^1
2 + 780831*a^4*c^5*d^4*e^14 - 82026*a^5*c^4*d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c
^14*d^28 + 14*a^4*c^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4 + 364*a^6*c^11*d^22*e^6 +
1001*a^7*c^10*d^20*e^8 + 2002*a^8*c^9*d^18*e^10 + 3003*a^9*c^8*d^16*e^12 + 3432
*a^10*c^7*d^14*e^14 + 3003*a^11*c^6*d^12*e^16 + 2002*a^12*c^5*d^10*e^18 + 1001*a
^13*c^4*d^8*e^20 + 364*a^14*c^3*d^6*e^22 + 91*a^15*c^2*d^4*e^24 + 14*a^16*c*d^2*
e^26 + a^17*e^28)))/(a^3*c^7*d^14 + 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 + 3
5*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 + 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12
+ a^10*e^14))*log((420*c^6*d^8*e^3 + 8421*a*c^5*d^6*e^5 + 36783*a^2*c^4*d^4*e^7
- 40817*a^3*c^3*d^2*e^9 + 2401*a^4*c^2*e^11)*sqrt(e*x + d) + (105*a^2*c^6*d^10*e
^4 + 4389*a^3*c^5*d^8*e^6 + 26274*a^4*c^4*d^6*e^8 - 34142*a^5*c^3*d^4*e^10 + 752
5*a^6*c^2*d^2*e^12 - 343*a^7*c*e^14 + 2*(a^3*c^9*d^19 + 15*a^4*c^8*d^17*e^2 + 64
*a^5*c^7*d^15*e^4 + 112*a^6*c^6*d^13*e^6 + 42*a^7*c^5*d^11*e^8 - 154*a^8*c^4*d^9
*e^10 - 280*a^9*c^3*d^7*e^12 - 216*a^10*c^2*d^5*e^14 - 83*a^11*c*d^3*e^16 - 13*a
^12*d*e^18)*sqrt(-(11025*c^9*d^12*e^6 + 171990*a*c^8*d^10*e^8 + 494991*a^2*c^7*d
^8*e^10 - 1360716*a^3*c^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 - 82026*a^5*c^4*d^2
*e^16 + 2401*a^6*c^3*e^18)/(a^3*c^14*d^28 + 14*a^4*c^13*d^26*e^2 + 91*a^5*c^12*d
^24*e^4 + 364*a^6*c^11*d^22*e^6 + 1001*a^7*c^10*d^20*e^8 + 2002*a^8*c^9*d^18*e^1
0 + 3003*a^9*c^8*d^16*e^12 + 3432*a^10*c^7*d^14*e^14 + 3003*a^11*c^6*d^12*e^16 +
2002*a^12*c^5*d^10*e^18 + 1001*a^13*c^4*d^8*e^20 + 364*a^14*c^3*d^6*e^22 + 91*a
^15*c^2*d^4*e^24 + 14*a^16*c*d^2*e^26 + a^17*e^28)))*sqrt(-(4*c^6*d^9 + 63*a*c^5
*d^7*e^2 + 189*a^2*c^4*d^5*e^4 - 1155*a^3*c^3*d^3*e^6 + 315*a^4*c^2*d*e^8 + (a^3
*c^7*d^14 + 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 + 35*a^6*c^4*d^8*e^6 + 35*a
^7*c^3*d^6*e^8 + 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 + a^10*e^14)*sqrt(-(1102
5*c^9*d^12*e^6 + 171990*a*c^8*d^10*e^8 + 494991*a^2*c^7*d^8*e^10 - 1360716*a^3*c
^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 - 82026*a^5*c^4*d^2*e^16 + 2401*a^6*c^3*e^
18)/(a^3*c^14*d^28 + 14*a^4*c^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4 + 364*a^6*c^11*
d^22*e^6 + 1001*a^7*c^10*d^20*e^8 + 2002*a^8*c^9*d^18*e^10 + 3003*a^9*c^8*d^16*e
^12 + 3432*a^10*c^7*d^14*e^14 + 3003*a^11*c^6*d^12*e^16 + 2002*a^12*c^5*d^10*e^1
8 + 1001*a^13*c^4*d^8*e^20 + 364*a^14*c^3*d^6*e^22 + 91*a^15*c^2*d^4*e^24 + 14*a
^16*c*d^2*e^26 + a^17*e^28)))/(a^3*c^7*d^14 + 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^
10*e^4 + 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 + 21*a^8*c^2*d^4*e^10 + 7*a^9*c
*d^2*e^12 + a^10*e^14))) - 3*(a^2*c^3*d^7 + 3*a^3*c^2*d^5*e^2 + 3*a^4*c*d^3*e^4
+ a^5*d*e^6 + (a*c^4*d^6*e + 3*a^2*c^3*d^4*e^3 + 3*a^3*c^2*d^2*e^5 + a^4*c*e^7)*
x^3 + (a*c^4*d^7 + 3*a^2*c^3*d^5*e^2 + 3*a^3*c^2*d^3*e^4 + a^4*c*d*e^6)*x^2 + (a
^2*c^3*d^6*e + 3*a^3*c^2*d^4*e^3 + 3*a^4*c*d^2*e^5 + a^5*e^7)*x)*sqrt(e*x + d)*s
qrt(-(4*c^6*d^9 + 63*a*c^5*d^7*e^2 + 189*a^2*c^4*d^5*e^4 - 1155*a^3*c^3*d^3*e^6
+ 315*a^4*c^2*d*e^8 + (a^3*c^7*d^14 + 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 +
35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 + 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^1
2 + a^10*e^14)*sqrt(-(11025*c^9*d^12*e^6 + 171990*a*c^8*d^10*e^8 + 494991*a^2*c^
7*d^8*e^10 - 1360716*a^3*c^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 - 82026*a^5*c^4*
d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c^14*d^28 + 14*a^4*c^13*d^26*e^2 + 91*a^5*c^1
2*d^24*e^4 + 364*a^6*c^11*d^22*e^6 + 1001*a^7*c^10*d^20*e^8 + 2002*a^8*c^9*d^18*
e^10 + 3003*a^9*c^8*d^16*e^12 + 3432*a^10*c^7*d^14*e^14 + 3003*a^11*c^6*d^12*e^1
6 + 2002*a^12*c^5*d^10*e^18 + 1001*a^13*c^4*d^8*e^20 + 364*a^14*c^3*d^6*e^22 + 9
1*a^15*c^2*d^4*e^24 + 14*a^16*c*d^2*e^26 + a^17*e^28)))/(a^3*c^7*d^14 + 7*a^4*c^
6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 + 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 + 21*
a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 + a^10*e^14))*log((420*c^6*d^8*e^3 + 8421*a*
c^5*d^6*e^5 + 36783*a^2*c^4*d^4*e^7 - 40817*a^3*c^3*d^2*e^9 + 2401*a^4*c^2*e^11)
*sqrt(e*x + d) - (105*a^2*c^6*d^10*e^4 + 4389*a^3*c^5*d^8*e^6 + 26274*a^4*c^4*d^
6*e^8 - 34142*a^5*c^3*d^4*e^10 + 7525*a^6*c^2*d^2*e^12 - 343*a^7*c*e^14 + 2*(a^3
*c^9*d^19 + 15*a^4*c^8*d^17*e^2 + 64*a^5*c^7*d^15*e^4 + 112*a^6*c^6*d^13*e^6 + 4
2*a^7*c^5*d^11*e^8 - 154*a^8*c^4*d^9*e^10 - 280*a^9*c^3*d^7*e^12 - 216*a^10*c^2*
d^5*e^14 - 83*a^11*c*d^3*e^16 - 13*a^12*d*e^18)*sqrt(-(11025*c^9*d^12*e^6 + 1719
90*a*c^8*d^10*e^8 + 494991*a^2*c^7*d^8*e^10 - 1360716*a^3*c^6*d^6*e^12 + 780831*
a^4*c^5*d^4*e^14 - 82026*a^5*c^4*d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c^14*d^28 +
14*a^4*c^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4 + 364*a^6*c^11*d^22*e^6 + 1001*a^7*c
^10*d^20*e^8 + 2002*a^8*c^9*d^18*e^10 + 3003*a^9*c^8*d^16*e^12 + 3432*a^10*c^7*d
^14*e^14 + 3003*a^11*c^6*d^12*e^16 + 2002*a^12*c^5*d^10*e^18 + 1001*a^13*c^4*d^8
*e^20 + 364*a^14*c^3*d^6*e^22 + 91*a^15*c^2*d^4*e^24 + 14*a^16*c*d^2*e^26 + a^17
*e^28)))*sqrt(-(4*c^6*d^9 + 63*a*c^5*d^7*e^2 + 189*a^2*c^4*d^5*e^4 - 1155*a^3*c^
3*d^3*e^6 + 315*a^4*c^2*d*e^8 + (a^3*c^7*d^14 + 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*
d^10*e^4 + 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 + 21*a^8*c^2*d^4*e^10 + 7*a^9
*c*d^2*e^12 + a^10*e^14)*sqrt(-(11025*c^9*d^12*e^6 + 171990*a*c^8*d^10*e^8 + 494
991*a^2*c^7*d^8*e^10 - 1360716*a^3*c^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 - 8202
6*a^5*c^4*d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c^14*d^28 + 14*a^4*c^13*d^26*e^2 +
91*a^5*c^12*d^24*e^4 + 364*a^6*c^11*d^22*e^6 + 1001*a^7*c^10*d^20*e^8 + 2002*a^8
*c^9*d^18*e^10 + 3003*a^9*c^8*d^16*e^12 + 3432*a^10*c^7*d^14*e^14 + 3003*a^11*c^
6*d^12*e^16 + 2002*a^12*c^5*d^10*e^18 + 1001*a^13*c^4*d^8*e^20 + 364*a^14*c^3*d^
6*e^22 + 91*a^15*c^2*d^4*e^24 + 14*a^16*c*d^2*e^26 + a^17*e^28)))/(a^3*c^7*d^14
+ 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 + 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6
*e^8 + 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 + a^10*e^14))) + 3*(a^2*c^3*d^7 +
3*a^3*c^2*d^5*e^2 + 3*a^4*c*d^3*e^4 + a^5*d*e^6 + (a*c^4*d^6*e + 3*a^2*c^3*d^4*e
^3 + 3*a^3*c^2*d^2*e^5 + a^4*c*e^7)*x^3 + (a*c^4*d^7 + 3*a^2*c^3*d^5*e^2 + 3*a^3
*c^2*d^3*e^4 + a^4*c*d*e^6)*x^2 + (a^2*c^3*d^6*e + 3*a^3*c^2*d^4*e^3 + 3*a^4*c*d
^2*e^5 + a^5*e^7)*x)*sqrt(e*x + d)*sqrt(-(4*c^6*d^9 + 63*a*c^5*d^7*e^2 + 189*a^2
*c^4*d^5*e^4 - 1155*a^3*c^3*d^3*e^6 + 315*a^4*c^2*d*e^8 - (a^3*c^7*d^14 + 7*a^4*
c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 + 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 + 2
1*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 + a^10*e^14)*sqrt(-(11025*c^9*d^12*e^6 + 1
71990*a*c^8*d^10*e^8 + 494991*a^2*c^7*d^8*e^10 - 1360716*a^3*c^6*d^6*e^12 + 7808
31*a^4*c^5*d^4*e^14 - 82026*a^5*c^4*d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c^14*d^28
+ 14*a^4*c^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4 + 364*a^6*c^11*d^22*e^6 + 1001*a^
7*c^10*d^20*e^8 + 2002*a^8*c^9*d^18*e^10 + 3003*a^9*c^8*d^16*e^12 + 3432*a^10*c^
7*d^14*e^14 + 3003*a^11*c^6*d^12*e^16 + 2002*a^12*c^5*d^10*e^18 + 1001*a^13*c^4*
d^8*e^20 + 364*a^14*c^3*d^6*e^22 + 91*a^15*c^2*d^4*e^24 + 14*a^16*c*d^2*e^26 + a
^17*e^28)))/(a^3*c^7*d^14 + 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 + 35*a^6*c^
4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 + 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 + a^10*e
^14))*log((420*c^6*d^8*e^3 + 8421*a*c^5*d^6*e^5 + 36783*a^2*c^4*d^4*e^7 - 40817*
a^3*c^3*d^2*e^9 + 2401*a^4*c^2*e^11)*sqrt(e*x + d) + (105*a^2*c^6*d^10*e^4 + 438
9*a^3*c^5*d^8*e^6 + 26274*a^4*c^4*d^6*e^8 - 34142*a^5*c^3*d^4*e^10 + 7525*a^6*c^
2*d^2*e^12 - 343*a^7*c*e^14 - 2*(a^3*c^9*d^19 + 15*a^4*c^8*d^17*e^2 + 64*a^5*c^7
*d^15*e^4 + 112*a^6*c^6*d^13*e^6 + 42*a^7*c^5*d^11*e^8 - 154*a^8*c^4*d^9*e^10 -
280*a^9*c^3*d^7*e^12 - 216*a^10*c^2*d^5*e^14 - 83*a^11*c*d^3*e^16 - 13*a^12*d*e^
18)*sqrt(-(11025*c^9*d^12*e^6 + 171990*a*c^8*d^10*e^8 + 494991*a^2*c^7*d^8*e^10
- 1360716*a^3*c^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 - 82026*a^5*c^4*d^2*e^16 +
2401*a^6*c^3*e^18)/(a^3*c^14*d^28 + 14*a^4*c^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4
+ 364*a^6*c^11*d^22*e^6 + 1001*a^7*c^10*d^20*e^8 + 2002*a^8*c^9*d^18*e^10 + 3003
*a^9*c^8*d^16*e^12 + 3432*a^10*c^7*d^14*e^14 + 3003*a^11*c^6*d^12*e^16 + 2002*a^
12*c^5*d^10*e^18 + 1001*a^13*c^4*d^8*e^20 + 364*a^14*c^3*d^6*e^22 + 91*a^15*c^2*
d^4*e^24 + 14*a^16*c*d^2*e^26 + a^17*e^28)))*sqrt(-(4*c^6*d^9 + 63*a*c^5*d^7*e^2
+ 189*a^2*c^4*d^5*e^4 - 1155*a^3*c^3*d^3*e^6 + 315*a^4*c^2*d*e^8 - (a^3*c^7*d^1
4 + 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 + 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d
^6*e^8 + 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 + a^10*e^14)*sqrt(-(11025*c^9*d^
12*e^6 + 171990*a*c^8*d^10*e^8 + 494991*a^2*c^7*d^8*e^10 - 1360716*a^3*c^6*d^6*e
^12 + 780831*a^4*c^5*d^4*e^14 - 82026*a^5*c^4*d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3
*c^14*d^28 + 14*a^4*c^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4 + 364*a^6*c^11*d^22*e^6
+ 1001*a^7*c^10*d^20*e^8 + 2002*a^8*c^9*d^18*e^10 + 3003*a^9*c^8*d^16*e^12 + 34
32*a^10*c^7*d^14*e^14 + 3003*a^11*c^6*d^12*e^16 + 2002*a^12*c^5*d^10*e^18 + 1001
*a^13*c^4*d^8*e^20 + 364*a^14*c^3*d^6*e^22 + 91*a^15*c^2*d^4*e^24 + 14*a^16*c*d^
2*e^26 + a^17*e^28)))/(a^3*c^7*d^14 + 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 +
35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 + 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^1
2 + a^10*e^14))) - 3*(a^2*c^3*d^7 + 3*a^3*c^2*d^5*e^2 + 3*a^4*c*d^3*e^4 + a^5*d*
e^6 + (a*c^4*d^6*e + 3*a^2*c^3*d^4*e^3 + 3*a^3*c^2*d^2*e^5 + a^4*c*e^7)*x^3 + (a
*c^4*d^7 + 3*a^2*c^3*d^5*e^2 + 3*a^3*c^2*d^3*e^4 + a^4*c*d*e^6)*x^2 + (a^2*c^3*d
^6*e + 3*a^3*c^2*d^4*e^3 + 3*a^4*c*d^2*e^5 + a^5*e^7)*x)*sqrt(e*x + d)*sqrt(-(4*
c^6*d^9 + 63*a*c^5*d^7*e^2 + 189*a^2*c^4*d^5*e^4 - 1155*a^3*c^3*d^3*e^6 + 315*a^
4*c^2*d*e^8 - (a^3*c^7*d^14 + 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 + 35*a^6*
c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 + 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 + a^10
*e^14)*sqrt(-(11025*c^9*d^12*e^6 + 171990*a*c^8*d^10*e^8 + 494991*a^2*c^7*d^8*e^
10 - 1360716*a^3*c^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 - 82026*a^5*c^4*d^2*e^16
+ 2401*a^6*c^3*e^18)/(a^3*c^14*d^28 + 14*a^4*c^13*d^26*e^2 + 91*a^5*c^12*d^24*e
^4 + 364*a^6*c^11*d^22*e^6 + 1001*a^7*c^10*d^20*e^8 + 2002*a^8*c^9*d^18*e^10 + 3
003*a^9*c^8*d^16*e^12 + 3432*a^10*c^7*d^14*e^14 + 3003*a^11*c^6*d^12*e^16 + 2002
*a^12*c^5*d^10*e^18 + 1001*a^13*c^4*d^8*e^20 + 364*a^14*c^3*d^6*e^22 + 91*a^15*c
^2*d^4*e^24 + 14*a^16*c*d^2*e^26 + a^17*e^28)))/(a^3*c^7*d^14 + 7*a^4*c^6*d^12*e
^2 + 21*a^5*c^5*d^10*e^4 + 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 + 21*a^8*c^2*
d^4*e^10 + 7*a^9*c*d^2*e^12 + a^10*e^14))*log((420*c^6*d^8*e^3 + 8421*a*c^5*d^6*
e^5 + 36783*a^2*c^4*d^4*e^7 - 40817*a^3*c^3*d^2*e^9 + 2401*a^4*c^2*e^11)*sqrt(e*
x + d) - (105*a^2*c^6*d^10*e^4 + 4389*a^3*c^5*d^8*e^6 + 26274*a^4*c^4*d^6*e^8 -
34142*a^5*c^3*d^4*e^10 + 7525*a^6*c^2*d^2*e^12 - 343*a^7*c*e^14 - 2*(a^3*c^9*d^1
9 + 15*a^4*c^8*d^17*e^2 + 64*a^5*c^7*d^15*e^4 + 112*a^6*c^6*d^13*e^6 + 42*a^7*c^
5*d^11*e^8 - 154*a^8*c^4*d^9*e^10 - 280*a^9*c^3*d^7*e^12 - 216*a^10*c^2*d^5*e^14
- 83*a^11*c*d^3*e^16 - 13*a^12*d*e^18)*sqrt(-(11025*c^9*d^12*e^6 + 171990*a*c^8
*d^10*e^8 + 494991*a^2*c^7*d^8*e^10 - 1360716*a^3*c^6*d^6*e^12 + 780831*a^4*c^5*
d^4*e^14 - 82026*a^5*c^4*d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c^14*d^28 + 14*a^4*c
^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4 + 364*a^6*c^11*d^22*e^6 + 1001*a^7*c^10*d^20
*e^8 + 2002*a^8*c^9*d^18*e^10 + 3003*a^9*c^8*d^16*e^12 + 3432*a^10*c^7*d^14*e^14
+ 3003*a^11*c^6*d^12*e^16 + 2002*a^12*c^5*d^10*e^18 + 1001*a^13*c^4*d^8*e^20 +
364*a^14*c^3*d^6*e^22 + 91*a^15*c^2*d^4*e^24 + 14*a^16*c*d^2*e^26 + a^17*e^28)))
*sqrt(-(4*c^6*d^9 + 63*a*c^5*d^7*e^2 + 189*a^2*c^4*d^5*e^4 - 1155*a^3*c^3*d^3*e^
6 + 315*a^4*c^2*d*e^8 - (a^3*c^7*d^14 + 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4
+ 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 + 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e
^12 + a^10*e^14)*sqrt(-(11025*c^9*d^12*e^6 + 171990*a*c^8*d^10*e^8 + 494991*a^2*
c^7*d^8*e^10 - 1360716*a^3*c^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 - 82026*a^5*c^
4*d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c^14*d^28 + 14*a^4*c^13*d^26*e^2 + 91*a^5*c
^12*d^24*e^4 + 364*a^6*c^11*d^22*e^6 + 1001*a^7*c^10*d^20*e^8 + 2002*a^8*c^9*d^1
8*e^10 + 3003*a^9*c^8*d^16*e^12 + 3432*a^10*c^7*d^14*e^14 + 3003*a^11*c^6*d^12*e
^16 + 2002*a^12*c^5*d^10*e^18 + 1001*a^13*c^4*d^8*e^20 + 364*a^14*c^3*d^6*e^22 +
91*a^15*c^2*d^4*e^24 + 14*a^16*c*d^2*e^26 + a^17*e^28)))/(a^3*c^7*d^14 + 7*a^4*
c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 + 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 + 2
1*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 + a^10*e^14))) + 12*(c^3*d^5 + 3*a*c^2*d^3
*e^2 - 18*a^2*c*d*e^4)*x)/((a^2*c^3*d^7 + 3*a^3*c^2*d^5*e^2 + 3*a^4*c*d^3*e^4 +
a^5*d*e^6 + (a*c^4*d^6*e + 3*a^2*c^3*d^4*e^3 + 3*a^3*c^2*d^2*e^5 + a^4*c*e^7)*x^
3 + (a*c^4*d^7 + 3*a^2*c^3*d^5*e^2 + 3*a^3*c^2*d^3*e^4 + a^4*c*d*e^6)*x^2 + (a^2
*c^3*d^6*e + 3*a^3*c^2*d^4*e^3 + 3*a^4*c*d^2*e^5 + a^5*e^7)*x)*sqrt(e*x + d))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**(5/2)/(c*x**2+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)^2*(e*x + d)^(5/2)),x, algorithm="giac")

[Out]

Exception raised: TypeError