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

Optimal. Leaf size=108 \[ a^{9/2} \left (-\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )\right )+a^4 \sqrt{a+b x^2}+\frac{1}{3} a^3 \left (a+b x^2\right )^{3/2}+\frac{1}{5} a^2 \left (a+b x^2\right )^{5/2}+\frac{1}{7} a \left (a+b x^2\right )^{7/2}+\frac{1}{9} \left (a+b x^2\right )^{9/2} \]

[Out]

a^4*Sqrt[a + b*x^2] + (a^3*(a + b*x^2)^(3/2))/3 + (a^2*(a + b*x^2)^(5/2))/5 + (a
*(a + b*x^2)^(7/2))/7 + (a + b*x^2)^(9/2)/9 - a^(9/2)*ArcTanh[Sqrt[a + b*x^2]/Sq
rt[a]]

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Rubi [A]  time = 0.191432, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ a^{9/2} \left (-\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )\right )+a^4 \sqrt{a+b x^2}+\frac{1}{3} a^3 \left (a+b x^2\right )^{3/2}+\frac{1}{5} a^2 \left (a+b x^2\right )^{5/2}+\frac{1}{7} a \left (a+b x^2\right )^{7/2}+\frac{1}{9} \left (a+b x^2\right )^{9/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(9/2)/x,x]

[Out]

a^4*Sqrt[a + b*x^2] + (a^3*(a + b*x^2)^(3/2))/3 + (a^2*(a + b*x^2)^(5/2))/5 + (a
*(a + b*x^2)^(7/2))/7 + (a + b*x^2)^(9/2)/9 - a^(9/2)*ArcTanh[Sqrt[a + b*x^2]/Sq
rt[a]]

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Rubi in Sympy [A]  time = 17.796, size = 90, normalized size = 0.83 \[ - a^{\frac{9}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )} + a^{4} \sqrt{a + b x^{2}} + \frac{a^{3} \left (a + b x^{2}\right )^{\frac{3}{2}}}{3} + \frac{a^{2} \left (a + b x^{2}\right )^{\frac{5}{2}}}{5} + \frac{a \left (a + b x^{2}\right )^{\frac{7}{2}}}{7} + \frac{\left (a + b x^{2}\right )^{\frac{9}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(9/2)*atanh(sqrt(a + b*x**2)/sqrt(a)) + a**4*sqrt(a + b*x**2) + a**3*(a + b*
x**2)**(3/2)/3 + a**2*(a + b*x**2)**(5/2)/5 + a*(a + b*x**2)**(7/2)/7 + (a + b*x
**2)**(9/2)/9

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Mathematica [A]  time = 0.111024, size = 94, normalized size = 0.87 \[ -a^{9/2} \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+a^{9/2} \log (x)+\frac{1}{315} \sqrt{a+b x^2} \left (563 a^4+506 a^3 b x^2+408 a^2 b^2 x^4+185 a b^3 x^6+35 b^4 x^8\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x^2]*(563*a^4 + 506*a^3*b*x^2 + 408*a^2*b^2*x^4 + 185*a*b^3*x^6 + 35
*b^4*x^8))/315 + a^(9/2)*Log[x] - a^(9/2)*Log[a + Sqrt[a]*Sqrt[a + b*x^2]]

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Maple [A]  time = 0.007, size = 94, normalized size = 0.9 \[{\frac{1}{9} \left ( b{x}^{2}+a \right ) ^{{\frac{9}{2}}}}+{\frac{a}{7} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{{a}^{2}}{5} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{3}}{3} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{a}^{{\frac{9}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) +{a}^{4}\sqrt{b{x}^{2}+a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(9/2)/x,x)

[Out]

1/9*(b*x^2+a)^(9/2)+1/7*a*(b*x^2+a)^(7/2)+1/5*a^2*(b*x^2+a)^(5/2)+1/3*a^3*(b*x^2
+a)^(3/2)-a^(9/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)+a^4*(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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.263718, size = 1, normalized size = 0.01 \[ \left [\frac{1}{2} \, a^{\frac{9}{2}} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + \frac{1}{315} \,{\left (35 \, b^{4} x^{8} + 185 \, a b^{3} x^{6} + 408 \, a^{2} b^{2} x^{4} + 506 \, a^{3} b x^{2} + 563 \, a^{4}\right )} \sqrt{b x^{2} + a}, -\sqrt{-a} a^{4} \arctan \left (\frac{a}{\sqrt{b x^{2} + a} \sqrt{-a}}\right ) + \frac{1}{315} \,{\left (35 \, b^{4} x^{8} + 185 \, a b^{3} x^{6} + 408 \, a^{2} b^{2} x^{4} + 506 \, a^{3} b x^{2} + 563 \, a^{4}\right )} \sqrt{b x^{2} + a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*a^(9/2)*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 1/315*(35*b^4
*x^8 + 185*a*b^3*x^6 + 408*a^2*b^2*x^4 + 506*a^3*b*x^2 + 563*a^4)*sqrt(b*x^2 + a
), -sqrt(-a)*a^4*arctan(a/(sqrt(b*x^2 + a)*sqrt(-a))) + 1/315*(35*b^4*x^8 + 185*
a*b^3*x^6 + 408*a^2*b^2*x^4 + 506*a^3*b*x^2 + 563*a^4)*sqrt(b*x^2 + a)]

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Sympy [A]  time = 36.8011, size = 160, normalized size = 1.48 \[ \frac{563 a^{\frac{9}{2}} \sqrt{1 + \frac{b x^{2}}{a}}}{315} + \frac{a^{\frac{9}{2}} \log{\left (\frac{b x^{2}}{a} \right )}}{2} - a^{\frac{9}{2}} \log{\left (\sqrt{1 + \frac{b x^{2}}{a}} + 1 \right )} + \frac{506 a^{\frac{7}{2}} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}}}{315} + \frac{136 a^{\frac{5}{2}} b^{2} x^{4} \sqrt{1 + \frac{b x^{2}}{a}}}{105} + \frac{37 a^{\frac{3}{2}} b^{3} x^{6} \sqrt{1 + \frac{b x^{2}}{a}}}{63} + \frac{\sqrt{a} b^{4} x^{8} \sqrt{1 + \frac{b x^{2}}{a}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

563*a**(9/2)*sqrt(1 + b*x**2/a)/315 + a**(9/2)*log(b*x**2/a)/2 - a**(9/2)*log(sq
rt(1 + b*x**2/a) + 1) + 506*a**(7/2)*b*x**2*sqrt(1 + b*x**2/a)/315 + 136*a**(5/2
)*b**2*x**4*sqrt(1 + b*x**2/a)/105 + 37*a**(3/2)*b**3*x**6*sqrt(1 + b*x**2/a)/63
+ sqrt(a)*b**4*x**8*sqrt(1 + b*x**2/a)/9

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GIAC/XCAS [A]  time = 0.210249, size = 122, normalized size = 1.13 \[ \frac{a^{5} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \frac{1}{9} \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} + \frac{1}{7} \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a + \frac{1}{5} \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{2} + \frac{1}{3} \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3} + \sqrt{b x^{2} + a} a^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^5*arctan(sqrt(b*x^2 + a)/sqrt(-a))/sqrt(-a) + 1/9*(b*x^2 + a)^(9/2) + 1/7*(b*x
^2 + a)^(7/2)*a + 1/5*(b*x^2 + a)^(5/2)*a^2 + 1/3*(b*x^2 + a)^(3/2)*a^3 + sqrt(b
*x^2 + a)*a^4