3.418 \(\int \frac{\left (a+b x^2\right )^{9/2}}{x^3} \, dx\)

Optimal. Leaf size=118 \[ -\frac{9}{2} a^{7/2} b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )+\frac{9}{2} a^3 b \sqrt{a+b x^2}+\frac{3}{2} a^2 b \left (a+b x^2\right )^{3/2}-\frac{\left (a+b x^2\right )^{9/2}}{2 x^2}+\frac{9}{14} b \left (a+b x^2\right )^{7/2}+\frac{9}{10} a b \left (a+b x^2\right )^{5/2} \]

[Out]

(9*a^3*b*Sqrt[a + b*x^2])/2 + (3*a^2*b*(a + b*x^2)^(3/2))/2 + (9*a*b*(a + b*x^2)
^(5/2))/10 + (9*b*(a + b*x^2)^(7/2))/14 - (a + b*x^2)^(9/2)/(2*x^2) - (9*a^(7/2)
*b*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/2

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Rubi [A]  time = 0.192969, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{9}{2} a^{7/2} b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )+\frac{9}{2} a^3 b \sqrt{a+b x^2}+\frac{3}{2} a^2 b \left (a+b x^2\right )^{3/2}-\frac{\left (a+b x^2\right )^{9/2}}{2 x^2}+\frac{9}{14} b \left (a+b x^2\right )^{7/2}+\frac{9}{10} a b \left (a+b x^2\right )^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(9/2)/x^3,x]

[Out]

(9*a^3*b*Sqrt[a + b*x^2])/2 + (3*a^2*b*(a + b*x^2)^(3/2))/2 + (9*a*b*(a + b*x^2)
^(5/2))/10 + (9*b*(a + b*x^2)^(7/2))/14 - (a + b*x^2)^(9/2)/(2*x^2) - (9*a^(7/2)
*b*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/2

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Rubi in Sympy [A]  time = 18.0144, size = 110, normalized size = 0.93 \[ - \frac{9 a^{\frac{7}{2}} b \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{2} + \frac{9 a^{3} b \sqrt{a + b x^{2}}}{2} + \frac{3 a^{2} b \left (a + b x^{2}\right )^{\frac{3}{2}}}{2} + \frac{9 a b \left (a + b x^{2}\right )^{\frac{5}{2}}}{10} + \frac{9 b \left (a + b x^{2}\right )^{\frac{7}{2}}}{14} - \frac{\left (a + b x^{2}\right )^{\frac{9}{2}}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(9/2)/x**3,x)

[Out]

-9*a**(7/2)*b*atanh(sqrt(a + b*x**2)/sqrt(a))/2 + 9*a**3*b*sqrt(a + b*x**2)/2 +
3*a**2*b*(a + b*x**2)**(3/2)/2 + 9*a*b*(a + b*x**2)**(5/2)/10 + 9*b*(a + b*x**2)
**(7/2)/14 - (a + b*x**2)**(9/2)/(2*x**2)

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Mathematica [A]  time = 0.14314, size = 104, normalized size = 0.88 \[ -\frac{9}{2} a^{7/2} b \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+\frac{9}{2} a^{7/2} b \log (x)+\frac{\sqrt{a+b x^2} \left (-35 a^4+388 a^3 b x^2+156 a^2 b^2 x^4+58 a b^3 x^6+10 b^4 x^8\right )}{70 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(9/2)/x^3,x]

[Out]

(Sqrt[a + b*x^2]*(-35*a^4 + 388*a^3*b*x^2 + 156*a^2*b^2*x^4 + 58*a*b^3*x^6 + 10*
b^4*x^8))/(70*x^2) + (9*a^(7/2)*b*Log[x])/2 - (9*a^(7/2)*b*Log[a + Sqrt[a]*Sqrt[
a + b*x^2]])/2

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Maple [A]  time = 0.008, size = 118, normalized size = 1. \[ -{\frac{1}{2\,a{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{b}{2\,a} \left ( b{x}^{2}+a \right ) ^{{\frac{9}{2}}}}+{\frac{9\,b}{14} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{9\,ab}{10} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{3\,{a}^{2}b}{2} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{9\,b}{2}{a}^{{\frac{7}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) }+{\frac{9\,{a}^{3}b}{2}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(9/2)/x^3,x)

[Out]

-1/2/a/x^2*(b*x^2+a)^(11/2)+1/2*b/a*(b*x^2+a)^(9/2)+9/14*b*(b*x^2+a)^(7/2)+9/10*
a*b*(b*x^2+a)^(5/2)+3/2*a^2*b*(b*x^2+a)^(3/2)-9/2*b*a^(7/2)*ln((2*a+2*a^(1/2)*(b
*x^2+a)^(1/2))/x)+9/2*a^3*b*(b*x^2+a)^(1/2)

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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((b*x^2 + a)^(9/2)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.259711, size = 1, normalized size = 0.01 \[ \left [\frac{315 \, a^{\frac{7}{2}} b x^{2} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (10 \, b^{4} x^{8} + 58 \, a b^{3} x^{6} + 156 \, a^{2} b^{2} x^{4} + 388 \, a^{3} b x^{2} - 35 \, a^{4}\right )} \sqrt{b x^{2} + a}}{140 \, x^{2}}, -\frac{315 \, \sqrt{-a} a^{3} b x^{2} \arctan \left (\frac{a}{\sqrt{b x^{2} + a} \sqrt{-a}}\right ) -{\left (10 \, b^{4} x^{8} + 58 \, a b^{3} x^{6} + 156 \, a^{2} b^{2} x^{4} + 388 \, a^{3} b x^{2} - 35 \, a^{4}\right )} \sqrt{b x^{2} + a}}{70 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(9/2)/x^3,x, algorithm="fricas")

[Out]

[1/140*(315*a^(7/2)*b*x^2*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) +
2*(10*b^4*x^8 + 58*a*b^3*x^6 + 156*a^2*b^2*x^4 + 388*a^3*b*x^2 - 35*a^4)*sqrt(b*
x^2 + a))/x^2, -1/70*(315*sqrt(-a)*a^3*b*x^2*arctan(a/(sqrt(b*x^2 + a)*sqrt(-a))
) - (10*b^4*x^8 + 58*a*b^3*x^6 + 156*a^2*b^2*x^4 + 388*a^3*b*x^2 - 35*a^4)*sqrt(
b*x^2 + a))/x^2]

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Sympy [A]  time = 37.8905, size = 167, normalized size = 1.42 \[ - \frac{a^{\frac{9}{2}} \sqrt{1 + \frac{b x^{2}}{a}}}{2 x^{2}} + \frac{194 a^{\frac{7}{2}} b \sqrt{1 + \frac{b x^{2}}{a}}}{35} + \frac{9 a^{\frac{7}{2}} b \log{\left (\frac{b x^{2}}{a} \right )}}{4} - \frac{9 a^{\frac{7}{2}} b \log{\left (\sqrt{1 + \frac{b x^{2}}{a}} + 1 \right )}}{2} + \frac{78 a^{\frac{5}{2}} b^{2} x^{2} \sqrt{1 + \frac{b x^{2}}{a}}}{35} + \frac{29 a^{\frac{3}{2}} b^{3} x^{4} \sqrt{1 + \frac{b x^{2}}{a}}}{35} + \frac{\sqrt{a} b^{4} x^{6} \sqrt{1 + \frac{b x^{2}}{a}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(9/2)/x**3,x)

[Out]

-a**(9/2)*sqrt(1 + b*x**2/a)/(2*x**2) + 194*a**(7/2)*b*sqrt(1 + b*x**2/a)/35 + 9
*a**(7/2)*b*log(b*x**2/a)/4 - 9*a**(7/2)*b*log(sqrt(1 + b*x**2/a) + 1)/2 + 78*a*
*(5/2)*b**2*x**2*sqrt(1 + b*x**2/a)/35 + 29*a**(3/2)*b**3*x**4*sqrt(1 + b*x**2/a
)/35 + sqrt(a)*b**4*x**6*sqrt(1 + b*x**2/a)/7

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GIAC/XCAS [A]  time = 0.211968, size = 136, normalized size = 1.15 \[ \frac{1}{70} \,{\left (\frac{315 \, a^{4} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 10 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} + 28 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a + 70 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2} + 280 \, \sqrt{b x^{2} + a} a^{3} - \frac{35 \, \sqrt{b x^{2} + a} a^{4}}{b x^{2}}\right )} b \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(9/2)/x^3,x, algorithm="giac")

[Out]

1/70*(315*a^4*arctan(sqrt(b*x^2 + a)/sqrt(-a))/sqrt(-a) + 10*(b*x^2 + a)^(7/2) +
28*(b*x^2 + a)^(5/2)*a + 70*(b*x^2 + a)^(3/2)*a^2 + 280*sqrt(b*x^2 + a)*a^3 - 3
5*sqrt(b*x^2 + a)*a^4/(b*x^2))*b