3.1942 \(\int \frac{x^3}{\left (a+\frac{b}{x^2}\right )^{5/2}} \, dx\)

Optimal. Leaf size=116 \[ \frac{35 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{8 a^{9/2}}-\frac{35 b x^2 \sqrt{a+\frac{b}{x^2}}}{8 a^4}+\frac{35 x^4 \sqrt{a+\frac{b}{x^2}}}{12 a^3}-\frac{7 x^4}{3 a^2 \sqrt{a+\frac{b}{x^2}}}-\frac{x^4}{3 a \left (a+\frac{b}{x^2}\right )^{3/2}} \]

[Out]

(-35*b*Sqrt[a + b/x^2]*x^2)/(8*a^4) - x^4/(3*a*(a + b/x^2)^(3/2)) - (7*x^4)/(3*a
^2*Sqrt[a + b/x^2]) + (35*Sqrt[a + b/x^2]*x^4)/(12*a^3) + (35*b^2*ArcTanh[Sqrt[a
+ b/x^2]/Sqrt[a]])/(8*a^(9/2))

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Rubi [A]  time = 0.183621, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{35 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{8 a^{9/2}}-\frac{35 b x^2 \sqrt{a+\frac{b}{x^2}}}{8 a^4}+\frac{35 x^4 \sqrt{a+\frac{b}{x^2}}}{12 a^3}-\frac{7 x^4}{3 a^2 \sqrt{a+\frac{b}{x^2}}}-\frac{x^4}{3 a \left (a+\frac{b}{x^2}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b/x^2)^(5/2),x]

[Out]

(-35*b*Sqrt[a + b/x^2]*x^2)/(8*a^4) - x^4/(3*a*(a + b/x^2)^(3/2)) - (7*x^4)/(3*a
^2*Sqrt[a + b/x^2]) + (35*Sqrt[a + b/x^2]*x^4)/(12*a^3) + (35*b^2*ArcTanh[Sqrt[a
+ b/x^2]/Sqrt[a]])/(8*a^(9/2))

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Rubi in Sympy [A]  time = 17.2595, size = 107, normalized size = 0.92 \[ - \frac{x^{4}}{3 a \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}} - \frac{7 x^{4}}{3 a^{2} \sqrt{a + \frac{b}{x^{2}}}} + \frac{35 x^{4} \sqrt{a + \frac{b}{x^{2}}}}{12 a^{3}} - \frac{35 b x^{2} \sqrt{a + \frac{b}{x^{2}}}}{8 a^{4}} + \frac{35 b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{2}}}}{\sqrt{a}} \right )}}{8 a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(a+b/x**2)**(5/2),x)

[Out]

-x**4/(3*a*(a + b/x**2)**(3/2)) - 7*x**4/(3*a**2*sqrt(a + b/x**2)) + 35*x**4*sqr
t(a + b/x**2)/(12*a**3) - 35*b*x**2*sqrt(a + b/x**2)/(8*a**4) + 35*b**2*atanh(sq
rt(a + b/x**2)/sqrt(a))/(8*a**(9/2))

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Mathematica [A]  time = 0.106279, size = 110, normalized size = 0.95 \[ \frac{\sqrt{a} x \left (6 a^3 x^6-21 a^2 b x^4-140 a b^2 x^2-105 b^3\right )+105 b^2 \left (a x^2+b\right )^{3/2} \log \left (\sqrt{a} \sqrt{a x^2+b}+a x\right )}{24 a^{9/2} x \sqrt{a+\frac{b}{x^2}} \left (a x^2+b\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(a + b/x^2)^(5/2),x]

[Out]

(Sqrt[a]*x*(-105*b^3 - 140*a*b^2*x^2 - 21*a^2*b*x^4 + 6*a^3*x^6) + 105*b^2*(b +
a*x^2)^(3/2)*Log[a*x + Sqrt[a]*Sqrt[b + a*x^2]])/(24*a^(9/2)*Sqrt[a + b/x^2]*x*(
b + a*x^2))

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Maple [A]  time = 0.02, size = 98, normalized size = 0.8 \[{\frac{a{x}^{2}+b}{24\,{x}^{5}} \left ( 6\,{x}^{7}{a}^{9/2}-21\,{a}^{7/2}{x}^{5}b-140\,{a}^{5/2}{x}^{3}{b}^{2}-105\,{a}^{3/2}x{b}^{3}+105\,\ln \left ( \sqrt{a}x+\sqrt{a{x}^{2}+b} \right ) \left ( a{x}^{2}+b \right ) ^{3/2}a{b}^{2} \right ) \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{5}{2}}}{a}^{-{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(a+b/x^2)^(5/2),x)

[Out]

1/24*(a*x^2+b)*(6*x^7*a^(9/2)-21*a^(7/2)*x^5*b-140*a^(5/2)*x^3*b^2-105*a^(3/2)*x
*b^3+105*ln(a^(1/2)*x+(a*x^2+b)^(1/2))*(a*x^2+b)^(3/2)*a*b^2)/((a*x^2+b)/x^2)^(5
/2)/x^5/a^(11/2)

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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/(a + b/x^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.277646, size = 1, normalized size = 0.01 \[ \left [\frac{105 \,{\left (a^{2} b^{2} x^{4} + 2 \, a b^{3} x^{2} + b^{4}\right )} \sqrt{a} \log \left (-2 \, a x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} -{\left (2 \, a x^{2} + b\right )} \sqrt{a}\right ) + 2 \,{\left (6 \, a^{4} x^{8} - 21 \, a^{3} b x^{6} - 140 \, a^{2} b^{2} x^{4} - 105 \, a b^{3} x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{48 \,{\left (a^{7} x^{4} + 2 \, a^{6} b x^{2} + a^{5} b^{2}\right )}}, -\frac{105 \,{\left (a^{2} b^{2} x^{4} + 2 \, a b^{3} x^{2} + b^{4}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a}}{\sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) -{\left (6 \, a^{4} x^{8} - 21 \, a^{3} b x^{6} - 140 \, a^{2} b^{2} x^{4} - 105 \, a b^{3} x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{24 \,{\left (a^{7} x^{4} + 2 \, a^{6} b x^{2} + a^{5} b^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(a + b/x^2)^(5/2),x, algorithm="fricas")

[Out]

[1/48*(105*(a^2*b^2*x^4 + 2*a*b^3*x^2 + b^4)*sqrt(a)*log(-2*a*x^2*sqrt((a*x^2 +
b)/x^2) - (2*a*x^2 + b)*sqrt(a)) + 2*(6*a^4*x^8 - 21*a^3*b*x^6 - 140*a^2*b^2*x^4
- 105*a*b^3*x^2)*sqrt((a*x^2 + b)/x^2))/(a^7*x^4 + 2*a^6*b*x^2 + a^5*b^2), -1/2
4*(105*(a^2*b^2*x^4 + 2*a*b^3*x^2 + b^4)*sqrt(-a)*arctan(sqrt(-a)/sqrt((a*x^2 +
b)/x^2)) - (6*a^4*x^8 - 21*a^3*b*x^6 - 140*a^2*b^2*x^4 - 105*a*b^3*x^2)*sqrt((a*
x^2 + b)/x^2))/(a^7*x^4 + 2*a^6*b*x^2 + a^5*b^2)]

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Sympy [A]  time = 26.95, size = 432, normalized size = 3.72 \[ \frac{6 a^{\frac{89}{2}} b^{75} x^{7}}{24 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{2} \sqrt{\frac{a x^{2}}{b} + 1} + 24 a^{\frac{91}{2}} b^{\frac{153}{2}} \sqrt{\frac{a x^{2}}{b} + 1}} - \frac{21 a^{\frac{87}{2}} b^{76} x^{5}}{24 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{2} \sqrt{\frac{a x^{2}}{b} + 1} + 24 a^{\frac{91}{2}} b^{\frac{153}{2}} \sqrt{\frac{a x^{2}}{b} + 1}} - \frac{140 a^{\frac{85}{2}} b^{77} x^{3}}{24 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{2} \sqrt{\frac{a x^{2}}{b} + 1} + 24 a^{\frac{91}{2}} b^{\frac{153}{2}} \sqrt{\frac{a x^{2}}{b} + 1}} - \frac{105 a^{\frac{83}{2}} b^{78} x}{24 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{2} \sqrt{\frac{a x^{2}}{b} + 1} + 24 a^{\frac{91}{2}} b^{\frac{153}{2}} \sqrt{\frac{a x^{2}}{b} + 1}} + \frac{105 a^{42} b^{\frac{155}{2}} x^{2} \sqrt{\frac{a x^{2}}{b} + 1} \operatorname{asinh}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{24 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{2} \sqrt{\frac{a x^{2}}{b} + 1} + 24 a^{\frac{91}{2}} b^{\frac{153}{2}} \sqrt{\frac{a x^{2}}{b} + 1}} + \frac{105 a^{41} b^{\frac{157}{2}} \sqrt{\frac{a x^{2}}{b} + 1} \operatorname{asinh}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{24 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{2} \sqrt{\frac{a x^{2}}{b} + 1} + 24 a^{\frac{91}{2}} b^{\frac{153}{2}} \sqrt{\frac{a x^{2}}{b} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(a+b/x**2)**(5/2),x)

[Out]

6*a**(89/2)*b**75*x**7/(24*a**(93/2)*b**(151/2)*x**2*sqrt(a*x**2/b + 1) + 24*a**
(91/2)*b**(153/2)*sqrt(a*x**2/b + 1)) - 21*a**(87/2)*b**76*x**5/(24*a**(93/2)*b*
*(151/2)*x**2*sqrt(a*x**2/b + 1) + 24*a**(91/2)*b**(153/2)*sqrt(a*x**2/b + 1)) -
140*a**(85/2)*b**77*x**3/(24*a**(93/2)*b**(151/2)*x**2*sqrt(a*x**2/b + 1) + 24*
a**(91/2)*b**(153/2)*sqrt(a*x**2/b + 1)) - 105*a**(83/2)*b**78*x/(24*a**(93/2)*b
**(151/2)*x**2*sqrt(a*x**2/b + 1) + 24*a**(91/2)*b**(153/2)*sqrt(a*x**2/b + 1))
+ 105*a**42*b**(155/2)*x**2*sqrt(a*x**2/b + 1)*asinh(sqrt(a)*x/sqrt(b))/(24*a**(
93/2)*b**(151/2)*x**2*sqrt(a*x**2/b + 1) + 24*a**(91/2)*b**(153/2)*sqrt(a*x**2/b
+ 1)) + 105*a**41*b**(157/2)*sqrt(a*x**2/b + 1)*asinh(sqrt(a)*x/sqrt(b))/(24*a*
*(93/2)*b**(151/2)*x**2*sqrt(a*x**2/b + 1) + 24*a**(91/2)*b**(153/2)*sqrt(a*x**2
/b + 1))

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GIAC/XCAS [A]  time = 0.272943, size = 193, normalized size = 1.66 \[ -\frac{1}{24} \, b^{2}{\left (\frac{8 \,{\left (a + \frac{9 \,{\left (a x^{2} + b\right )}}{x^{2}}\right )} x^{2}}{{\left (a x^{2} + b\right )} a^{4} \sqrt{\frac{a x^{2} + b}{x^{2}}}} + \frac{105 \, \arctan \left (\frac{\sqrt{\frac{a x^{2} + b}{x^{2}}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{4}} - \frac{3 \,{\left (13 \, a \sqrt{\frac{a x^{2} + b}{x^{2}}} - \frac{11 \,{\left (a x^{2} + b\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{x^{2}}\right )}}{{\left (a - \frac{a x^{2} + b}{x^{2}}\right )}^{2} a^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(a + b/x^2)^(5/2),x, algorithm="giac")

[Out]

-1/24*b^2*(8*(a + 9*(a*x^2 + b)/x^2)*x^2/((a*x^2 + b)*a^4*sqrt((a*x^2 + b)/x^2))
+ 105*arctan(sqrt((a*x^2 + b)/x^2)/sqrt(-a))/(sqrt(-a)*a^4) - 3*(13*a*sqrt((a*x
^2 + b)/x^2) - 11*(a*x^2 + b)*sqrt((a*x^2 + b)/x^2)/x^2)/((a - (a*x^2 + b)/x^2)^
2*a^4))