3.490 \(\int \frac{x^{3/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx\)

Optimal. Leaf size=601 \[ -\frac{b^{3/4} (7 a d+b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{3/4} (b c-a d)^3}+\frac{b^{3/4} (7 a d+b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{3/4} (b c-a d)^3}-\frac{b^{3/4} (7 a d+b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{3/4} (b c-a d)^3}+\frac{b^{3/4} (7 a d+b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{3/4} (b c-a d)^3}+\frac{d^{3/4} (a d+7 b c) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{3/4} (b c-a d)^3}-\frac{d^{3/4} (a d+7 b c) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{3/4} (b c-a d)^3}+\frac{d^{3/4} (a d+7 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{3/4} (b c-a d)^3}-\frac{d^{3/4} (a d+7 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{3/4} (b c-a d)^3}-\frac{d \sqrt{x}}{\left (c+d x^2\right ) (b c-a d)^2}-\frac{\sqrt{x}}{2 \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)} \]

[Out]

-((d*Sqrt[x])/((b*c - a*d)^2*(c + d*x^2))) - Sqrt[x]/(2*(b*c - a*d)*(a + b*x^2)*
(c + d*x^2)) - (b^(3/4)*(b*c + 7*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/
4)])/(4*Sqrt[2]*a^(3/4)*(b*c - a*d)^3) + (b^(3/4)*(b*c + 7*a*d)*ArcTan[1 + (Sqrt
[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(3/4)*(b*c - a*d)^3) + (d^(3/4)*(7*b
*c + a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(3/4)*(b*c
- a*d)^3) - (d^(3/4)*(7*b*c + a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)
])/(4*Sqrt[2]*c^(3/4)*(b*c - a*d)^3) - (b^(3/4)*(b*c + 7*a*d)*Log[Sqrt[a] - Sqrt
[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(3/4)*(b*c - a*d)^3) + (b
^(3/4)*(b*c + 7*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])
/(8*Sqrt[2]*a^(3/4)*(b*c - a*d)^3) + (d^(3/4)*(7*b*c + a*d)*Log[Sqrt[c] - Sqrt[2
]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(3/4)*(b*c - a*d)^3) - (d^(
3/4)*(7*b*c + a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(
8*Sqrt[2]*c^(3/4)*(b*c - a*d)^3)

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Rubi [A]  time = 1.41324, antiderivative size = 601, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ -\frac{b^{3/4} (7 a d+b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{3/4} (b c-a d)^3}+\frac{b^{3/4} (7 a d+b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{3/4} (b c-a d)^3}-\frac{b^{3/4} (7 a d+b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{3/4} (b c-a d)^3}+\frac{b^{3/4} (7 a d+b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{3/4} (b c-a d)^3}+\frac{d^{3/4} (a d+7 b c) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{3/4} (b c-a d)^3}-\frac{d^{3/4} (a d+7 b c) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{3/4} (b c-a d)^3}+\frac{d^{3/4} (a d+7 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{3/4} (b c-a d)^3}-\frac{d^{3/4} (a d+7 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{3/4} (b c-a d)^3}-\frac{d \sqrt{x}}{\left (c+d x^2\right ) (b c-a d)^2}-\frac{\sqrt{x}}{2 \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)/((a + b*x^2)^2*(c + d*x^2)^2),x]

[Out]

-((d*Sqrt[x])/((b*c - a*d)^2*(c + d*x^2))) - Sqrt[x]/(2*(b*c - a*d)*(a + b*x^2)*
(c + d*x^2)) - (b^(3/4)*(b*c + 7*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/
4)])/(4*Sqrt[2]*a^(3/4)*(b*c - a*d)^3) + (b^(3/4)*(b*c + 7*a*d)*ArcTan[1 + (Sqrt
[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(3/4)*(b*c - a*d)^3) + (d^(3/4)*(7*b
*c + a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(3/4)*(b*c
- a*d)^3) - (d^(3/4)*(7*b*c + a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)
])/(4*Sqrt[2]*c^(3/4)*(b*c - a*d)^3) - (b^(3/4)*(b*c + 7*a*d)*Log[Sqrt[a] - Sqrt
[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(3/4)*(b*c - a*d)^3) + (b
^(3/4)*(b*c + 7*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])
/(8*Sqrt[2]*a^(3/4)*(b*c - a*d)^3) + (d^(3/4)*(7*b*c + a*d)*Log[Sqrt[c] - Sqrt[2
]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(3/4)*(b*c - a*d)^3) - (d^(
3/4)*(7*b*c + a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(
8*Sqrt[2]*c^(3/4)*(b*c - a*d)^3)

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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(x**(3/2)/(b*x**2+a)**2/(d*x**2+c)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 2.75995, size = 575, normalized size = 0.96 \[ \frac{1}{16} \left (\frac{\sqrt{2} b^{3/4} (7 a d+b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{3/4} (a d-b c)^3}+\frac{\sqrt{2} b^{3/4} (7 a d+b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{3/4} (b c-a d)^3}+\frac{2 \sqrt{2} b^{3/4} (7 a d+b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{3/4} (a d-b c)^3}-\frac{2 \sqrt{2} b^{3/4} (7 a d+b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{3/4} (a d-b c)^3}+\frac{\sqrt{2} d^{3/4} (a d+7 b c) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{3/4} (b c-a d)^3}+\frac{\sqrt{2} d^{3/4} (a d+7 b c) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{3/4} (a d-b c)^3}+\frac{2 \sqrt{2} d^{3/4} (a d+7 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{3/4} (b c-a d)^3}-\frac{2 \sqrt{2} d^{3/4} (a d+7 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{3/4} (b c-a d)^3}-\frac{8 b \sqrt{x}}{\left (a+b x^2\right ) (b c-a d)^2}-\frac{8 d \sqrt{x}}{\left (c+d x^2\right ) (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-8*b*Sqrt[x])/((b*c - a*d)^2*(a + b*x^2)) - (8*d*Sqrt[x])/((b*c - a*d)^2*(c +
d*x^2)) + (2*Sqrt[2]*b^(3/4)*(b*c + 7*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/
a^(1/4)])/(a^(3/4)*(-(b*c) + a*d)^3) - (2*Sqrt[2]*b^(3/4)*(b*c + 7*a*d)*ArcTan[1
+ (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(3/4)*(-(b*c) + a*d)^3) + (2*Sqrt[2]*d
^(3/4)*(7*b*c + a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(3/4)*(b*
c - a*d)^3) - (2*Sqrt[2]*d^(3/4)*(7*b*c + a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[
x])/c^(1/4)])/(c^(3/4)*(b*c - a*d)^3) + (Sqrt[2]*b^(3/4)*(b*c + 7*a*d)*Log[Sqrt[
a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(3/4)*(-(b*c) + a*d)^3) +
(Sqrt[2]*b^(3/4)*(b*c + 7*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + S
qrt[b]*x])/(a^(3/4)*(b*c - a*d)^3) + (Sqrt[2]*d^(3/4)*(7*b*c + a*d)*Log[Sqrt[c]
- Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(3/4)*(b*c - a*d)^3) + (Sqrt[
2]*d^(3/4)*(7*b*c + a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]
*x])/(c^(3/4)*(-(b*c) + a*d)^3))/16

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Maple [A]  time = 0.031, size = 770, normalized size = 1.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)/(b*x^2+a)^2/(d*x^2+c)^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3/2)/((b*x^2 + a)^2*(d*x^2 + c)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 75.9416, size = 6036, normalized size = 10.04 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3/2)/((b*x^2 + a)^2*(d*x^2 + c)^2),x, algorithm="fricas")

[Out]

1/8*(4*(a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2
*b*d^3)*x^4 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^2)*(-(b^7*c^4 +
28*a*b^6*c^3*d + 294*a^2*b^5*c^2*d^2 + 1372*a^3*b^4*c*d^3 + 2401*a^4*b^3*d^4)/(a
^3*b^12*c^12 - 12*a^4*b^11*c^11*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3 +
495*a^7*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b^5*
c^5*d^7 + 495*a^11*b^4*c^4*d^8 - 220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 1
2*a^14*b*c*d^11 + a^15*d^12))^(1/4)*arctan(-(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3
*b*c*d^2 - a^4*d^3)*(-(b^7*c^4 + 28*a*b^6*c^3*d + 294*a^2*b^5*c^2*d^2 + 1372*a^3
*b^4*c*d^3 + 2401*a^4*b^3*d^4)/(a^3*b^12*c^12 - 12*a^4*b^11*c^11*d + 66*a^5*b^10
*c^10*d^2 - 220*a^6*b^9*c^9*d^3 + 495*a^7*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 92
4*a^9*b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^11*b^4*c^4*d^8 - 220*a^12*b^3*c
^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b*c*d^11 + a^15*d^12))^(1/4)/((b^2*c + 7
*a*b*d)*sqrt(x) + sqrt((b^4*c^2 + 14*a*b^3*c*d + 49*a^2*b^2*d^2)*x + (a^2*b^6*c^
6 - 6*a^3*b^5*c^5*d + 15*a^4*b^4*c^4*d^2 - 20*a^5*b^3*c^3*d^3 + 15*a^6*b^2*c^2*d
^4 - 6*a^7*b*c*d^5 + a^8*d^6)*sqrt(-(b^7*c^4 + 28*a*b^6*c^3*d + 294*a^2*b^5*c^2*
d^2 + 1372*a^3*b^4*c*d^3 + 2401*a^4*b^3*d^4)/(a^3*b^12*c^12 - 12*a^4*b^11*c^11*d
+ 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3 + 495*a^7*b^8*c^8*d^4 - 792*a^8*b^
7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^11*b^4*c^4*d^8 -
220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b*c*d^11 + a^15*d^12)))))
- 4*(a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*
d^3)*x^4 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^2)*(-(2401*b^4*c^4*
d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6 + a^4*d^7)/(b^12
*c^15 - 12*a*b^11*c^14*d + 66*a^2*b^10*c^13*d^2 - 220*a^3*b^9*c^12*d^3 + 495*a^4
*b^8*c^11*d^4 - 792*a^5*b^7*c^10*d^5 + 924*a^6*b^6*c^9*d^6 - 792*a^7*b^5*c^8*d^7
+ 495*a^8*b^4*c^7*d^8 - 220*a^9*b^3*c^6*d^9 + 66*a^10*b^2*c^5*d^10 - 12*a^11*b*
c^4*d^11 + a^12*c^3*d^12))^(1/4)*arctan(-(b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*
d^2 - a^3*c*d^3)*(-(2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5
+ 28*a^3*b*c*d^6 + a^4*d^7)/(b^12*c^15 - 12*a*b^11*c^14*d + 66*a^2*b^10*c^13*d^2
- 220*a^3*b^9*c^12*d^3 + 495*a^4*b^8*c^11*d^4 - 792*a^5*b^7*c^10*d^5 + 924*a^6*
b^6*c^9*d^6 - 792*a^7*b^5*c^8*d^7 + 495*a^8*b^4*c^7*d^8 - 220*a^9*b^3*c^6*d^9 +
66*a^10*b^2*c^5*d^10 - 12*a^11*b*c^4*d^11 + a^12*c^3*d^12))^(1/4)/((7*b*c*d + a*
d^2)*sqrt(x) + sqrt((49*b^2*c^2*d^2 + 14*a*b*c*d^3 + a^2*d^4)*x + (b^6*c^8 - 6*a
*b^5*c^7*d + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a^
5*b*c^3*d^5 + a^6*c^2*d^6)*sqrt(-(2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^
2*b^2*c^2*d^5 + 28*a^3*b*c*d^6 + a^4*d^7)/(b^12*c^15 - 12*a*b^11*c^14*d + 66*a^2
*b^10*c^13*d^2 - 220*a^3*b^9*c^12*d^3 + 495*a^4*b^8*c^11*d^4 - 792*a^5*b^7*c^10*
d^5 + 924*a^6*b^6*c^9*d^6 - 792*a^7*b^5*c^8*d^7 + 495*a^8*b^4*c^7*d^8 - 220*a^9*
b^3*c^6*d^9 + 66*a^10*b^2*c^5*d^10 - 12*a^11*b*c^4*d^11 + a^12*c^3*d^12))))) + (
a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*
x^4 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^2)*(-(b^7*c^4 + 28*a*b^6
*c^3*d + 294*a^2*b^5*c^2*d^2 + 1372*a^3*b^4*c*d^3 + 2401*a^4*b^3*d^4)/(a^3*b^12*
c^12 - 12*a^4*b^11*c^11*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3 + 495*a^7
*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7
+ 495*a^11*b^4*c^4*d^8 - 220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b
*c*d^11 + a^15*d^12))^(1/4)*log((b^2*c + 7*a*b*d)*sqrt(x) + (a*b^3*c^3 - 3*a^2*b
^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*(-(b^7*c^4 + 28*a*b^6*c^3*d + 294*a^2*b^5*c^
2*d^2 + 1372*a^3*b^4*c*d^3 + 2401*a^4*b^3*d^4)/(a^3*b^12*c^12 - 12*a^4*b^11*c^11
*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3 + 495*a^7*b^8*c^8*d^4 - 792*a^8*
b^7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^11*b^4*c^4*d^8
- 220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b*c*d^11 + a^15*d^12))^(
1/4)) - (a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^
2*b*d^3)*x^4 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^2)*(-(b^7*c^4 +
28*a*b^6*c^3*d + 294*a^2*b^5*c^2*d^2 + 1372*a^3*b^4*c*d^3 + 2401*a^4*b^3*d^4)/(
a^3*b^12*c^12 - 12*a^4*b^11*c^11*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3
+ 495*a^7*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b^5
*c^5*d^7 + 495*a^11*b^4*c^4*d^8 - 220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 -
12*a^14*b*c*d^11 + a^15*d^12))^(1/4)*log((b^2*c + 7*a*b*d)*sqrt(x) - (a*b^3*c^3
- 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*(-(b^7*c^4 + 28*a*b^6*c^3*d + 294*a
^2*b^5*c^2*d^2 + 1372*a^3*b^4*c*d^3 + 2401*a^4*b^3*d^4)/(a^3*b^12*c^12 - 12*a^4*
b^11*c^11*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3 + 495*a^7*b^8*c^8*d^4 -
792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^11*b^4
*c^4*d^8 - 220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b*c*d^11 + a^15
*d^12))^(1/4)) - (a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c
*d^2 + a^2*b*d^3)*x^4 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^2)*(-(
2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6 + a
^4*d^7)/(b^12*c^15 - 12*a*b^11*c^14*d + 66*a^2*b^10*c^13*d^2 - 220*a^3*b^9*c^12*
d^3 + 495*a^4*b^8*c^11*d^4 - 792*a^5*b^7*c^10*d^5 + 924*a^6*b^6*c^9*d^6 - 792*a^
7*b^5*c^8*d^7 + 495*a^8*b^4*c^7*d^8 - 220*a^9*b^3*c^6*d^9 + 66*a^10*b^2*c^5*d^10
- 12*a^11*b*c^4*d^11 + a^12*c^3*d^12))^(1/4)*log((7*b*c*d + a*d^2)*sqrt(x) + (b
^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)*(-(2401*b^4*c^4*d^3 + 1372
*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6 + a^4*d^7)/(b^12*c^15 - 12
*a*b^11*c^14*d + 66*a^2*b^10*c^13*d^2 - 220*a^3*b^9*c^12*d^3 + 495*a^4*b^8*c^11*
d^4 - 792*a^5*b^7*c^10*d^5 + 924*a^6*b^6*c^9*d^6 - 792*a^7*b^5*c^8*d^7 + 495*a^8
*b^4*c^7*d^8 - 220*a^9*b^3*c^6*d^9 + 66*a^10*b^2*c^5*d^10 - 12*a^11*b*c^4*d^11 +
a^12*c^3*d^12))^(1/4)) + (a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^2*d -
2*a*b^2*c*d^2 + a^2*b*d^3)*x^4 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)
*x^2)*(-(2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*
c*d^6 + a^4*d^7)/(b^12*c^15 - 12*a*b^11*c^14*d + 66*a^2*b^10*c^13*d^2 - 220*a^3*
b^9*c^12*d^3 + 495*a^4*b^8*c^11*d^4 - 792*a^5*b^7*c^10*d^5 + 924*a^6*b^6*c^9*d^6
- 792*a^7*b^5*c^8*d^7 + 495*a^8*b^4*c^7*d^8 - 220*a^9*b^3*c^6*d^9 + 66*a^10*b^2
*c^5*d^10 - 12*a^11*b*c^4*d^11 + a^12*c^3*d^12))^(1/4)*log((7*b*c*d + a*d^2)*sqr
t(x) - (b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)*(-(2401*b^4*c^4*d
^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6 + a^4*d^7)/(b^12*
c^15 - 12*a*b^11*c^14*d + 66*a^2*b^10*c^13*d^2 - 220*a^3*b^9*c^12*d^3 + 495*a^4*
b^8*c^11*d^4 - 792*a^5*b^7*c^10*d^5 + 924*a^6*b^6*c^9*d^6 - 792*a^7*b^5*c^8*d^7
+ 495*a^8*b^4*c^7*d^8 - 220*a^9*b^3*c^6*d^9 + 66*a^10*b^2*c^5*d^10 - 12*a^11*b*c
^4*d^11 + a^12*c^3*d^12))^(1/4)) - 4*(2*b*d*x^2 + b*c + a*d)*sqrt(x))/(a*b^2*c^3
- 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^4 + (b^
3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^2)

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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(x**(3/2)/(b*x**2+a)**2/(d*x**2+c)**2,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{3}{2}}}{{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3/2)/((b*x^2 + a)^2*(d*x^2 + c)^2),x, algorithm="giac")

[Out]

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