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

Optimal. Leaf size=178 \[ -\frac{b^3 \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{3/2} (b c-a d)^{5/2}}-\frac{\sqrt{c+d x^2} (b c-4 a d) (3 b c-2 a d)}{3 a c^3 x (b c-a d)^2}-\frac{d (7 b c-4 a d)}{3 c^2 x \sqrt{c+d x^2} (b c-a d)^2}-\frac{d}{3 c x \left (c+d x^2\right )^{3/2} (b c-a d)} \]

[Out]

-d/(3*c*(b*c - a*d)*x*(c + d*x^2)^(3/2)) - (d*(7*b*c - 4*a*d))/(3*c^2*(b*c - a*d
)^2*x*Sqrt[c + d*x^2]) - ((b*c - 4*a*d)*(3*b*c - 2*a*d)*Sqrt[c + d*x^2])/(3*a*c^
3*(b*c - a*d)^2*x) - (b^3*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])
/(a^(3/2)*(b*c - a*d)^(5/2))

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Rubi [A]  time = 0.686313, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{b^3 \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{3/2} (b c-a d)^{5/2}}-\frac{\sqrt{c+d x^2} (b c-4 a d) (3 b c-2 a d)}{3 a c^3 x (b c-a d)^2}-\frac{d (7 b c-4 a d)}{3 c^2 x \sqrt{c+d x^2} (b c-a d)^2}-\frac{d}{3 c x \left (c+d x^2\right )^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

-d/(3*c*(b*c - a*d)*x*(c + d*x^2)^(3/2)) - (d*(7*b*c - 4*a*d))/(3*c^2*(b*c - a*d
)^2*x*Sqrt[c + d*x^2]) - ((b*c - 4*a*d)*(3*b*c - 2*a*d)*Sqrt[c + d*x^2])/(3*a*c^
3*(b*c - a*d)^2*x) - (b^3*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])
/(a^(3/2)*(b*c - a*d)^(5/2))

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Rubi in Sympy [A]  time = 99.8846, size = 153, normalized size = 0.86 \[ \frac{d}{3 c x \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{d \left (4 a d - 7 b c\right )}{3 c^{2} x \sqrt{c + d x^{2}} \left (a d - b c\right )^{2}} - \frac{\sqrt{c + d x^{2}} \left (2 a d - 3 b c\right ) \left (4 a d - b c\right )}{3 a c^{3} x \left (a d - b c\right )^{2}} - \frac{b^{3} \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{a^{\frac{3}{2}} \left (a d - b c\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d/(3*c*x*(c + d*x**2)**(3/2)*(a*d - b*c)) + d*(4*a*d - 7*b*c)/(3*c**2*x*sqrt(c +
d*x**2)*(a*d - b*c)**2) - sqrt(c + d*x**2)*(2*a*d - 3*b*c)*(4*a*d - b*c)/(3*a*c
**3*x*(a*d - b*c)**2) - b**3*atanh(x*sqrt(a*d - b*c)/(sqrt(a)*sqrt(c + d*x**2)))
/(a**(3/2)*(a*d - b*c)**(5/2))

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Mathematica [A]  time = 0.434817, size = 143, normalized size = 0.8 \[ \frac{\sqrt{c+d x^2} \left (\frac{d^2 x^2 (8 b c-5 a d)}{\left (c+d x^2\right ) (b c-a d)^2}+\frac{c d^2 x^2}{\left (c+d x^2\right )^2 (b c-a d)}-\frac{3}{a}\right )}{3 c^3 x}-\frac{b^3 \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{3/2} (b c-a d)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.025, size = 1192, normalized size = 6.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/a/c/x/(d*x^2+c)^(3/2)-4/3/a*d/c^2*x/(d*x^2+c)^(3/2)-8/3/a*d/c^3*x/(d*x^2+c)^(
1/2)+1/6*b^2/a/(-a*b)^(1/2)/(a*d-b*c)/((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)
/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)-1/6*b/a*d/(a*d-b*c)/c/((x-1/b*(-a*b)^
(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)*x-1/3*b/a*
d/(a*d-b*c)/c^2/((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2)
)-(a*d-b*c)/b)^(1/2)*x-1/2*b^3/a/(-a*b)^(1/2)/(a*d-b*c)^2/((x-1/b*(-a*b)^(1/2))^
2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+1/2*b^2/a/(a*d-b*
c)^2/c/((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*
c)/b)^(1/2)*x*d+1/2*b^3/a/(-a*b)^(1/2)/(a*d-b*c)^2/(-(a*d-b*c)/b)^(1/2)*ln((-2*(
a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x-1/
b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/
(x-1/b*(-a*b)^(1/2)))-1/6*b^2/a/(-a*b)^(1/2)/(a*d-b*c)/((x+1/b*(-a*b)^(1/2))^2*d
-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)-1/6*b/a*d/(a*d-b*c)/
c/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)
^(3/2)*x-1/3*b/a*d/(a*d-b*c)/c^2/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x
+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x+1/2*b^3/a/(-a*b)^(1/2)/(a*d-b*c)^2/((x+1
/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+
1/2*b^2/a/(a*d-b*c)^2/c/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*
b)^(1/2))-(a*d-b*c)/b)^(1/2)*x*d-1/2*b^3/a/(-a*b)^(1/2)/(a*d-b*c)^2/(-(a*d-b*c)/
b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c
)/b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*
d-b*c)/b)^(1/2))/(x+1/b*(-a*b)^(1/2)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.67363, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left (3 \, b^{2} c^{4} - 6 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} +{\left (3 \, b^{2} c^{2} d^{2} - 14 \, a b c d^{3} + 8 \, a^{2} d^{4}\right )} x^{4} + 3 \,{\left (2 \, b^{2} c^{3} d - 7 \, a b c^{2} d^{2} + 4 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d} \sqrt{d x^{2} + c} - 3 \,{\left (b^{3} c^{3} d^{2} x^{5} + 2 \, b^{3} c^{4} d x^{3} + b^{3} c^{5} x\right )} \log \left (\frac{{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d} - 4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{3} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{12 \,{\left ({\left (a b^{2} c^{5} d^{2} - 2 \, a^{2} b c^{4} d^{3} + a^{3} c^{3} d^{4}\right )} x^{5} + 2 \,{\left (a b^{2} c^{6} d - 2 \, a^{2} b c^{5} d^{2} + a^{3} c^{4} d^{3}\right )} x^{3} +{\left (a b^{2} c^{7} - 2 \, a^{2} b c^{6} d + a^{3} c^{5} d^{2}\right )} x\right )} \sqrt{-a b c + a^{2} d}}, -\frac{2 \,{\left (3 \, b^{2} c^{4} - 6 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} +{\left (3 \, b^{2} c^{2} d^{2} - 14 \, a b c d^{3} + 8 \, a^{2} d^{4}\right )} x^{4} + 3 \,{\left (2 \, b^{2} c^{3} d - 7 \, a b c^{2} d^{2} + 4 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} + 3 \,{\left (b^{3} c^{3} d^{2} x^{5} + 2 \, b^{3} c^{4} d x^{3} + b^{3} c^{5} x\right )} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} x}\right )}{6 \,{\left ({\left (a b^{2} c^{5} d^{2} - 2 \, a^{2} b c^{4} d^{3} + a^{3} c^{3} d^{4}\right )} x^{5} + 2 \,{\left (a b^{2} c^{6} d - 2 \, a^{2} b c^{5} d^{2} + a^{3} c^{4} d^{3}\right )} x^{3} +{\left (a b^{2} c^{7} - 2 \, a^{2} b c^{6} d + a^{3} c^{5} d^{2}\right )} x\right )} \sqrt{a b c - a^{2} d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/12*(4*(3*b^2*c^4 - 6*a*b*c^3*d + 3*a^2*c^2*d^2 + (3*b^2*c^2*d^2 - 14*a*b*c*d
^3 + 8*a^2*d^4)*x^4 + 3*(2*b^2*c^3*d - 7*a*b*c^2*d^2 + 4*a^2*c*d^3)*x^2)*sqrt(-a
*b*c + a^2*d)*sqrt(d*x^2 + c) - 3*(b^3*c^3*d^2*x^5 + 2*b^3*c^4*d*x^3 + b^3*c^5*x
)*log((((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c
*d)*x^2)*sqrt(-a*b*c + a^2*d) - 4*((a*b^2*c^2 - 3*a^2*b*c*d + 2*a^3*d^2)*x^3 - (
a^2*b*c^2 - a^3*c*d)*x)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)))/(((a*b^2*
c^5*d^2 - 2*a^2*b*c^4*d^3 + a^3*c^3*d^4)*x^5 + 2*(a*b^2*c^6*d - 2*a^2*b*c^5*d^2
+ a^3*c^4*d^3)*x^3 + (a*b^2*c^7 - 2*a^2*b*c^6*d + a^3*c^5*d^2)*x)*sqrt(-a*b*c +
a^2*d)), -1/6*(2*(3*b^2*c^4 - 6*a*b*c^3*d + 3*a^2*c^2*d^2 + (3*b^2*c^2*d^2 - 14*
a*b*c*d^3 + 8*a^2*d^4)*x^4 + 3*(2*b^2*c^3*d - 7*a*b*c^2*d^2 + 4*a^2*c*d^3)*x^2)*
sqrt(a*b*c - a^2*d)*sqrt(d*x^2 + c) + 3*(b^3*c^3*d^2*x^5 + 2*b^3*c^4*d*x^3 + b^3
*c^5*x)*arctan(1/2*((b*c - 2*a*d)*x^2 - a*c)/(sqrt(a*b*c - a^2*d)*sqrt(d*x^2 + c
)*x)))/(((a*b^2*c^5*d^2 - 2*a^2*b*c^4*d^3 + a^3*c^3*d^4)*x^5 + 2*(a*b^2*c^6*d -
2*a^2*b*c^5*d^2 + a^3*c^4*d^3)*x^3 + (a*b^2*c^7 - 2*a^2*b*c^6*d + a^3*c^5*d^2)*x
)*sqrt(a*b*c - a^2*d))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**2*(a + b*x**2)*(c + d*x**2)**(5/2)), x)

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

[Out]

Exception raised: TypeError