3.432 \(\int \frac{\left (a+b x^2\right )^{9/2}}{x^8} \, dx\)

Optimal. Leaf size=126 \[ \frac{9}{2} a b^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )+\frac{9}{2} b^4 x \sqrt{a+b x^2}-\frac{3 b^3 \left (a+b x^2\right )^{3/2}}{x}-\frac{3 b^2 \left (a+b x^2\right )^{5/2}}{5 x^3}-\frac{\left (a+b x^2\right )^{9/2}}{7 x^7}-\frac{9 b \left (a+b x^2\right )^{7/2}}{35 x^5} \]

[Out]

(9*b^4*x*Sqrt[a + b*x^2])/2 - (3*b^3*(a + b*x^2)^(3/2))/x - (3*b^2*(a + b*x^2)^(
5/2))/(5*x^3) - (9*b*(a + b*x^2)^(7/2))/(35*x^5) - (a + b*x^2)^(9/2)/(7*x^7) + (
9*a*b^(7/2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/2

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Rubi [A]  time = 0.143521, antiderivative size = 126, 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 \[ \frac{9}{2} a b^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )+\frac{9}{2} b^4 x \sqrt{a+b x^2}-\frac{3 b^3 \left (a+b x^2\right )^{3/2}}{x}-\frac{3 b^2 \left (a+b x^2\right )^{5/2}}{5 x^3}-\frac{\left (a+b x^2\right )^{9/2}}{7 x^7}-\frac{9 b \left (a+b x^2\right )^{7/2}}{35 x^5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(9/2)/x^8,x]

[Out]

(9*b^4*x*Sqrt[a + b*x^2])/2 - (3*b^3*(a + b*x^2)^(3/2))/x - (3*b^2*(a + b*x^2)^(
5/2))/(5*x^3) - (9*b*(a + b*x^2)^(7/2))/(35*x^5) - (a + b*x^2)^(9/2)/(7*x^7) + (
9*a*b^(7/2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/2

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Rubi in Sympy [A]  time = 16.4963, size = 117, normalized size = 0.93 \[ \frac{9 a b^{\frac{7}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{2} + \frac{9 b^{4} x \sqrt{a + b x^{2}}}{2} - \frac{3 b^{3} \left (a + b x^{2}\right )^{\frac{3}{2}}}{x} - \frac{3 b^{2} \left (a + b x^{2}\right )^{\frac{5}{2}}}{5 x^{3}} - \frac{9 b \left (a + b x^{2}\right )^{\frac{7}{2}}}{35 x^{5}} - \frac{\left (a + b x^{2}\right )^{\frac{9}{2}}}{7 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(9/2)/x**8,x)

[Out]

9*a*b**(7/2)*atanh(sqrt(b)*x/sqrt(a + b*x**2))/2 + 9*b**4*x*sqrt(a + b*x**2)/2 -
3*b**3*(a + b*x**2)**(3/2)/x - 3*b**2*(a + b*x**2)**(5/2)/(5*x**3) - 9*b*(a + b
*x**2)**(7/2)/(35*x**5) - (a + b*x**2)**(9/2)/(7*x**7)

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Mathematica [A]  time = 0.0850259, size = 94, normalized size = 0.75 \[ \frac{9}{2} a b^{7/2} \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )-\frac{\sqrt{a+b x^2} \left (10 a^4+58 a^3 b x^2+156 a^2 b^2 x^4+388 a b^3 x^6-35 b^4 x^8\right )}{70 x^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(9/2)/x^8,x]

[Out]

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

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Maple [A]  time = 0.022, size = 186, normalized size = 1.5 \[ -{\frac{1}{7\,a{x}^{7}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{4\,b}{35\,{a}^{2}{x}^{5}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{8\,{b}^{2}}{35\,{a}^{3}{x}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{64\,{b}^{3}}{35\,{a}^{4}x} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{64\,{b}^{4}x}{35\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{9}{2}}}}+{\frac{72\,{b}^{4}x}{35\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{12\,{b}^{4}x}{5\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+3\,{\frac{{b}^{4}x \left ( b{x}^{2}+a \right ) ^{3/2}}{a}}+{\frac{9\,{b}^{4}x}{2}\sqrt{b{x}^{2}+a}}+{\frac{9\,a}{2}{b}^{{\frac{7}{2}}}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(9/2)/x^8,x)

[Out]

-1/7/a/x^7*(b*x^2+a)^(11/2)-4/35*b/a^2/x^5*(b*x^2+a)^(11/2)-8/35*b^2/a^3/x^3*(b*
x^2+a)^(11/2)-64/35*b^3/a^4/x*(b*x^2+a)^(11/2)+64/35*b^4/a^4*x*(b*x^2+a)^(9/2)+7
2/35*b^4/a^3*x*(b*x^2+a)^(7/2)+12/5*b^4/a^2*x*(b*x^2+a)^(5/2)+3*b^4/a*x*(b*x^2+a
)^(3/2)+9/2*b^4*x*(b*x^2+a)^(1/2)+9/2*b^(7/2)*a*ln(x*b^(1/2)+(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^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.270387, size = 1, normalized size = 0.01 \[ \left [\frac{315 \, a b^{\frac{7}{2}} x^{7} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 2 \,{\left (35 \, b^{4} x^{8} - 388 \, a b^{3} x^{6} - 156 \, a^{2} b^{2} x^{4} - 58 \, a^{3} b x^{2} - 10 \, a^{4}\right )} \sqrt{b x^{2} + a}}{140 \, x^{7}}, \frac{315 \, a \sqrt{-b} b^{3} x^{7} \arctan \left (\frac{b x}{\sqrt{b x^{2} + a} \sqrt{-b}}\right ) +{\left (35 \, b^{4} x^{8} - 388 \, a b^{3} x^{6} - 156 \, a^{2} b^{2} x^{4} - 58 \, a^{3} b x^{2} - 10 \, a^{4}\right )} \sqrt{b x^{2} + a}}{70 \, x^{7}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(9/2)/x^8,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 37.6491, size = 167, normalized size = 1.33 \[ - \frac{a^{4} \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{7 x^{6}} - \frac{29 a^{3} b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{35 x^{4}} - \frac{78 a^{2} b^{\frac{5}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{35 x^{2}} - \frac{194 a b^{\frac{7}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{35} - \frac{9 a b^{\frac{7}{2}} \log{\left (\frac{a}{b x^{2}} \right )}}{4} + \frac{9 a b^{\frac{7}{2}} \log{\left (\sqrt{\frac{a}{b x^{2}} + 1} + 1 \right )}}{2} + \frac{b^{\frac{9}{2}} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(9/2)/x**8,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.216359, size = 324, normalized size = 2.57 \[ \frac{1}{2} \, \sqrt{b x^{2} + a} b^{4} x - \frac{9}{4} \, a b^{\frac{7}{2}}{\rm ln}\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right ) + \frac{4 \,{\left (175 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} a^{2} b^{\frac{7}{2}} - 700 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} a^{3} b^{\frac{7}{2}} + 1575 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} a^{4} b^{\frac{7}{2}} - 1820 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a^{5} b^{\frac{7}{2}} + 1337 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{6} b^{\frac{7}{2}} - 504 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{7} b^{\frac{7}{2}} + 97 \, a^{8} b^{\frac{7}{2}}\right )}}{35 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(9/2)/x^8,x, algorithm="giac")

[Out]

1/2*sqrt(b*x^2 + a)*b^4*x - 9/4*a*b^(7/2)*ln((sqrt(b)*x - sqrt(b*x^2 + a))^2) +
4/35*(175*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^2*b^(7/2) - 700*(sqrt(b)*x - sqrt(b
*x^2 + a))^10*a^3*b^(7/2) + 1575*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^4*b^(7/2) - 1
820*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^5*b^(7/2) + 1337*(sqrt(b)*x - sqrt(b*x^2 +
a))^4*a^6*b^(7/2) - 504*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^7*b^(7/2) + 97*a^8*b^
(7/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^7