3.493 \(\int \frac{1}{x^{3/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx\)

Optimal. Leaf size=676 \[ -\frac{b^{9/4} (5 b c-13 a d) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{9/4} (b c-a d)^3}+\frac{b^{9/4} (5 b c-13 a d) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{9/4} (b c-a d)^3}+\frac{b^{9/4} (5 b c-13 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{9/4} (b c-a d)^3}-\frac{b^{9/4} (5 b c-13 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{9/4} (b c-a d)^3}-\frac{5 a^2 d^2-8 a b c d+5 b^2 c^2}{2 a^2 c^2 \sqrt{x} (b c-a d)^2}-\frac{d^{9/4} (13 b c-5 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{9/4} (b c-a d)^3}+\frac{d^{9/4} (13 b c-5 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{9/4} (b c-a d)^3}+\frac{d^{9/4} (13 b c-5 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{9/4} (b c-a d)^3}-\frac{d^{9/4} (13 b c-5 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{9/4} (b c-a d)^3}+\frac{b}{2 a \sqrt{x} \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}+\frac{d (a d+b c)}{2 a c \sqrt{x} \left (c+d x^2\right ) (b c-a d)^2} \]

[Out]

-(5*b^2*c^2 - 8*a*b*c*d + 5*a^2*d^2)/(2*a^2*c^2*(b*c - a*d)^2*Sqrt[x]) + (d*(b*c
+ a*d))/(2*a*c*(b*c - a*d)^2*Sqrt[x]*(c + d*x^2)) + b/(2*a*(b*c - a*d)*Sqrt[x]*
(a + b*x^2)*(c + d*x^2)) + (b^(9/4)*(5*b*c - 13*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)
*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(9/4)*(b*c - a*d)^3) - (b^(9/4)*(5*b*c - 13*a*d
)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(9/4)*(b*c - a*d)^
3) + (d^(9/4)*(13*b*c - 5*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4
*Sqrt[2]*c^(9/4)*(b*c - a*d)^3) - (d^(9/4)*(13*b*c - 5*a*d)*ArcTan[1 + (Sqrt[2]*
d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(9/4)*(b*c - a*d)^3) - (b^(9/4)*(5*b*c -
13*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*
a^(9/4)*(b*c - a*d)^3) + (b^(9/4)*(5*b*c - 13*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)
*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4)*(b*c - a*d)^3) - (d^(9/4)*(13*
b*c - 5*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt
[2]*c^(9/4)*(b*c - a*d)^3) + (d^(9/4)*(13*b*c - 5*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(
1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(9/4)*(b*c - a*d)^3)

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Rubi [A]  time = 2.31032, antiderivative size = 676, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458 \[ -\frac{b^{9/4} (5 b c-13 a d) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{9/4} (b c-a d)^3}+\frac{b^{9/4} (5 b c-13 a d) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{9/4} (b c-a d)^3}+\frac{b^{9/4} (5 b c-13 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{9/4} (b c-a d)^3}-\frac{b^{9/4} (5 b c-13 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{9/4} (b c-a d)^3}-\frac{5 a^2 d^2-8 a b c d+5 b^2 c^2}{2 a^2 c^2 \sqrt{x} (b c-a d)^2}-\frac{d^{9/4} (13 b c-5 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{9/4} (b c-a d)^3}+\frac{d^{9/4} (13 b c-5 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{9/4} (b c-a d)^3}+\frac{d^{9/4} (13 b c-5 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{9/4} (b c-a d)^3}-\frac{d^{9/4} (13 b c-5 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{9/4} (b c-a d)^3}+\frac{b}{2 a \sqrt{x} \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}+\frac{d (a d+b c)}{2 a c \sqrt{x} \left (c+d x^2\right ) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(3/2)*(a + b*x^2)^2*(c + d*x^2)^2),x]

[Out]

-(5*b^2*c^2 - 8*a*b*c*d + 5*a^2*d^2)/(2*a^2*c^2*(b*c - a*d)^2*Sqrt[x]) + (d*(b*c
+ a*d))/(2*a*c*(b*c - a*d)^2*Sqrt[x]*(c + d*x^2)) + b/(2*a*(b*c - a*d)*Sqrt[x]*
(a + b*x^2)*(c + d*x^2)) + (b^(9/4)*(5*b*c - 13*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)
*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(9/4)*(b*c - a*d)^3) - (b^(9/4)*(5*b*c - 13*a*d
)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(9/4)*(b*c - a*d)^
3) + (d^(9/4)*(13*b*c - 5*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4
*Sqrt[2]*c^(9/4)*(b*c - a*d)^3) - (d^(9/4)*(13*b*c - 5*a*d)*ArcTan[1 + (Sqrt[2]*
d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(9/4)*(b*c - a*d)^3) - (b^(9/4)*(5*b*c -
13*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*
a^(9/4)*(b*c - a*d)^3) + (b^(9/4)*(5*b*c - 13*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)
*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4)*(b*c - a*d)^3) - (d^(9/4)*(13*
b*c - 5*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt
[2]*c^(9/4)*(b*c - a*d)^3) + (d^(9/4)*(13*b*c - 5*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(
1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(9/4)*(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(1/x**(3/2)/(b*x**2+a)**2/(d*x**2+c)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 3.80456, size = 606, normalized size = 0.9 \[ \frac{1}{16} \left (\frac{\sqrt{2} b^{9/4} (13 a d-5 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{9/4} (b c-a d)^3}+\frac{\sqrt{2} b^{9/4} (13 a d-5 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{9/4} (a d-b c)^3}+\frac{2 \sqrt{2} b^{9/4} (13 a d-5 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{9/4} (a d-b c)^3}+\frac{2 \sqrt{2} b^{9/4} (13 a d-5 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{9/4} (b c-a d)^3}-\frac{8 b^3 x^{3/2}}{a^2 \left (a+b x^2\right ) (b c-a d)^2}-\frac{32}{a^2 c^2 \sqrt{x}}+\frac{\sqrt{2} d^{9/4} (13 b c-5 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{9/4} (a d-b c)^3}+\frac{\sqrt{2} d^{9/4} (13 b c-5 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{9/4} (b c-a d)^3}+\frac{2 \sqrt{2} d^{9/4} (13 b c-5 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{9/4} (b c-a d)^3}+\frac{2 \sqrt{2} d^{9/4} (5 a d-13 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{9/4} (b c-a d)^3}-\frac{8 d^3 x^{3/2}}{c^2 \left (c+d x^2\right ) (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(3/2)*(a + b*x^2)^2*(c + d*x^2)^2),x]

[Out]

(-32/(a^2*c^2*Sqrt[x]) - (8*b^3*x^(3/2))/(a^2*(b*c - a*d)^2*(a + b*x^2)) - (8*d^
3*x^(3/2))/(c^2*(b*c - a*d)^2*(c + d*x^2)) + (2*Sqrt[2]*b^(9/4)*(-5*b*c + 13*a*d
)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(9/4)*(-(b*c) + a*d)^3) + (2
*Sqrt[2]*b^(9/4)*(-5*b*c + 13*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]
)/(a^(9/4)*(b*c - a*d)^3) + (2*Sqrt[2]*d^(9/4)*(13*b*c - 5*a*d)*ArcTan[1 - (Sqrt
[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(9/4)*(b*c - a*d)^3) + (2*Sqrt[2]*d^(9/4)*(-13
*b*c + 5*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(9/4)*(b*c - a*d
)^3) + (Sqrt[2]*b^(9/4)*(-5*b*c + 13*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*
Sqrt[x] + Sqrt[b]*x])/(a^(9/4)*(b*c - a*d)^3) + (Sqrt[2]*b^(9/4)*(-5*b*c + 13*a*
d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(9/4)*(-(b*c)
+ a*d)^3) + (Sqrt[2]*d^(9/4)*(13*b*c - 5*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1
/4)*Sqrt[x] + Sqrt[d]*x])/(c^(9/4)*(-(b*c) + a*d)^3) + (Sqrt[2]*d^(9/4)*(13*b*c
- 5*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(9/4)*(b
*c - a*d)^3))/16

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Maple [A]  time = 0.036, size = 825, normalized size = 1.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(3/2)/(b*x^2+a)^2/(d*x^2+c)^2,x)

[Out]

-1/2*d^4/c^2/(a*d-b*c)^3*x^(3/2)/(d*x^2+c)*a+1/2*d^3/c/(a*d-b*c)^3*x^(3/2)/(d*x^
2+c)*b-5/16*d^3/c^2/(a*d-b*c)^3/(c/d)^(1/4)*2^(1/2)*a*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)))-5/8*d^3/c^2/(a
*d-b*c)^3/(c/d)^(1/4)*2^(1/2)*a*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)-5/8*d^3/c^
2/(a*d-b*c)^3/(c/d)^(1/4)*2^(1/2)*a*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)+13/16*
d^2/c/(a*d-b*c)^3/(c/d)^(1/4)*2^(1/2)*b*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)))+13/8*d^2/c/(a*d-b*c)^3/(c/d)
^(1/4)*2^(1/2)*b*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+13/8*d^2/c/(a*d-b*c)^3/(c
/d)^(1/4)*2^(1/2)*b*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)-2/a^2/c^2/x^(1/2)-1/2*
b^3/a/(a*d-b*c)^3*x^(3/2)/(b*x^2+a)*d+1/2*b^4/a^2/(a*d-b*c)^3*x^(3/2)/(b*x^2+a)*
c-13/16*b^2/a/(a*d-b*c)^3/(a/b)^(1/4)*2^(1/2)*d*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)))-13/8*b^2/a/(a*d-b*c)
^3/(a/b)^(1/4)*2^(1/2)*d*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)-13/8*b^2/a/(a*d-b
*c)^3/(a/b)^(1/4)*2^(1/2)*d*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)+5/16*b^3/a^2/(
a*d-b*c)^3/(a/b)^(1/4)*2^(1/2)*c*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)))+5/8*b^3/a^2/(a*d-b*c)^3/(a/b)^(1/4)
*2^(1/2)*c*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+5/8*b^3/a^2/(a*d-b*c)^3/(a/b)^(
1/4)*2^(1/2)*c*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)

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

[Out]

Exception raised: ValueError

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

[Out]

Timed out

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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/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. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{2} x^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^2*x^(3/2)),x, algorithm="giac")

[Out]

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