3.515 \(\int \frac{1}{x^3 \left (a+b x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=92 \[ \frac{5 b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{7/2}}-\frac{5 \sqrt{a+b x^2}}{2 a^3 x^2}+\frac{5}{3 a^2 x^2 \sqrt{a+b x^2}}+\frac{1}{3 a x^2 \left (a+b x^2\right )^{3/2}} \]

[Out]

1/(3*a*x^2*(a + b*x^2)^(3/2)) + 5/(3*a^2*x^2*Sqrt[a + b*x^2]) - (5*Sqrt[a + b*x^
2])/(2*a^3*x^2) + (5*b*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(2*a^(7/2))

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Rubi [A]  time = 0.14283, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{5 b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{7/2}}-\frac{5 \sqrt{a+b x^2}}{2 a^3 x^2}+\frac{5}{3 a^2 x^2 \sqrt{a+b x^2}}+\frac{1}{3 a x^2 \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + b*x^2)^(5/2)),x]

[Out]

1/(3*a*x^2*(a + b*x^2)^(3/2)) + 5/(3*a^2*x^2*Sqrt[a + b*x^2]) - (5*Sqrt[a + b*x^
2])/(2*a^3*x^2) + (5*b*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(2*a^(7/2))

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Rubi in Sympy [A]  time = 14.0995, size = 85, normalized size = 0.92 \[ \frac{1}{3 a x^{2} \left (a + b x^{2}\right )^{\frac{3}{2}}} + \frac{5}{3 a^{2} x^{2} \sqrt{a + b x^{2}}} - \frac{5 \sqrt{a + b x^{2}}}{2 a^{3} x^{2}} + \frac{5 b \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{2 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(b*x**2+a)**(5/2),x)

[Out]

1/(3*a*x**2*(a + b*x**2)**(3/2)) + 5/(3*a**2*x**2*sqrt(a + b*x**2)) - 5*sqrt(a +
b*x**2)/(2*a**3*x**2) + 5*b*atanh(sqrt(a + b*x**2)/sqrt(a))/(2*a**(7/2))

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Mathematica [A]  time = 0.183753, size = 80, normalized size = 0.87 \[ \frac{-\frac{\sqrt{a} \left (3 a^2+20 a b x^2+15 b^2 x^4\right )}{x^2 \left (a+b x^2\right )^{3/2}}+15 b \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )-15 b \log (x)}{6 a^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(a + b*x^2)^(5/2)),x]

[Out]

(-((Sqrt[a]*(3*a^2 + 20*a*b*x^2 + 15*b^2*x^4))/(x^2*(a + b*x^2)^(3/2))) - 15*b*L
og[x] + 15*b*Log[a + Sqrt[a]*Sqrt[a + b*x^2]])/(6*a^(7/2))

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Maple [A]  time = 0.008, size = 78, normalized size = 0.9 \[ -{\frac{1}{2\,a{x}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{5\,b}{6\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{5\,b}{2\,{a}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{5\,b}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x^2+a)^(5/2),x)

[Out]

-1/2/a/x^2/(b*x^2+a)^(3/2)-5/6*b/a^2/(b*x^2+a)^(3/2)-5/2*b/a^3/(b*x^2+a)^(1/2)+5
/2*b/a^(7/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)

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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)^(5/2)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.255272, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (15 \, b^{2} x^{4} + 20 \, a b x^{2} + 3 \, a^{2}\right )} \sqrt{b x^{2} + a} \sqrt{a} - 15 \,{\left (b^{3} x^{6} + 2 \, a b^{2} x^{4} + a^{2} b x^{2}\right )} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right )}{12 \,{\left (a^{3} b^{2} x^{6} + 2 \, a^{4} b x^{4} + a^{5} x^{2}\right )} \sqrt{a}}, -\frac{{\left (15 \, b^{2} x^{4} + 20 \, a b x^{2} + 3 \, a^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-a} - 15 \,{\left (b^{3} x^{6} + 2 \, a b^{2} x^{4} + a^{2} b x^{2}\right )} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right )}{6 \,{\left (a^{3} b^{2} x^{6} + 2 \, a^{4} b x^{4} + a^{5} x^{2}\right )} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(5/2)*x^3),x, algorithm="fricas")

[Out]

[-1/12*(2*(15*b^2*x^4 + 20*a*b*x^2 + 3*a^2)*sqrt(b*x^2 + a)*sqrt(a) - 15*(b^3*x^
6 + 2*a*b^2*x^4 + a^2*b*x^2)*log(-((b*x^2 + 2*a)*sqrt(a) + 2*sqrt(b*x^2 + a)*a)/
x^2))/((a^3*b^2*x^6 + 2*a^4*b*x^4 + a^5*x^2)*sqrt(a)), -1/6*((15*b^2*x^4 + 20*a*
b*x^2 + 3*a^2)*sqrt(b*x^2 + a)*sqrt(-a) - 15*(b^3*x^6 + 2*a*b^2*x^4 + a^2*b*x^2)
*arctan(sqrt(-a)/sqrt(b*x^2 + a)))/((a^3*b^2*x^6 + 2*a^4*b*x^4 + a^5*x^2)*sqrt(-
a))]

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Sympy [A]  time = 18.0549, size = 864, normalized size = 9.39 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(b*x**2+a)**(5/2),x)

[Out]

-6*a**17*sqrt(1 + b*x**2/a)/(12*a**(39/2)*x**2 + 36*a**(37/2)*b*x**4 + 36*a**(35
/2)*b**2*x**6 + 12*a**(33/2)*b**3*x**8) - 46*a**16*b*x**2*sqrt(1 + b*x**2/a)/(12
*a**(39/2)*x**2 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*x**6 + 12*a**(33/2)*b*
*3*x**8) - 15*a**16*b*x**2*log(b*x**2/a)/(12*a**(39/2)*x**2 + 36*a**(37/2)*b*x**
4 + 36*a**(35/2)*b**2*x**6 + 12*a**(33/2)*b**3*x**8) + 30*a**16*b*x**2*log(sqrt(
1 + b*x**2/a) + 1)/(12*a**(39/2)*x**2 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*
x**6 + 12*a**(33/2)*b**3*x**8) - 70*a**15*b**2*x**4*sqrt(1 + b*x**2/a)/(12*a**(3
9/2)*x**2 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*x**6 + 12*a**(33/2)*b**3*x**
8) - 45*a**15*b**2*x**4*log(b*x**2/a)/(12*a**(39/2)*x**2 + 36*a**(37/2)*b*x**4 +
36*a**(35/2)*b**2*x**6 + 12*a**(33/2)*b**3*x**8) + 90*a**15*b**2*x**4*log(sqrt(
1 + b*x**2/a) + 1)/(12*a**(39/2)*x**2 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*
x**6 + 12*a**(33/2)*b**3*x**8) - 30*a**14*b**3*x**6*sqrt(1 + b*x**2/a)/(12*a**(3
9/2)*x**2 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*x**6 + 12*a**(33/2)*b**3*x**
8) - 45*a**14*b**3*x**6*log(b*x**2/a)/(12*a**(39/2)*x**2 + 36*a**(37/2)*b*x**4 +
36*a**(35/2)*b**2*x**6 + 12*a**(33/2)*b**3*x**8) + 90*a**14*b**3*x**6*log(sqrt(
1 + b*x**2/a) + 1)/(12*a**(39/2)*x**2 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*
x**6 + 12*a**(33/2)*b**3*x**8) - 15*a**13*b**4*x**8*log(b*x**2/a)/(12*a**(39/2)*
x**2 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*x**6 + 12*a**(33/2)*b**3*x**8) +
30*a**13*b**4*x**8*log(sqrt(1 + b*x**2/a) + 1)/(12*a**(39/2)*x**2 + 36*a**(37/2)
*b*x**4 + 36*a**(35/2)*b**2*x**6 + 12*a**(33/2)*b**3*x**8)

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GIAC/XCAS [A]  time = 0.230744, size = 100, normalized size = 1.09 \[ -\frac{1}{6} \, b{\left (\frac{15 \, \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} + \frac{2 \,{\left (6 \, b x^{2} + 7 \, a\right )}}{{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3}} + \frac{3 \, \sqrt{b x^{2} + a}}{a^{3} b x^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(5/2)*x^3),x, algorithm="giac")

[Out]

-1/6*b*(15*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^3) + 2*(6*b*x^2 + 7*a)/(
(b*x^2 + a)^(3/2)*a^3) + 3*sqrt(b*x^2 + a)/(a^3*b*x^2))