3.433 \(\int \frac{\left (a+b x^2\right )^{9/2}}{x^{10}} \, dx\)

Optimal. Leaf size=124 \[ b^{9/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{b^4 \sqrt{a+b x^2}}{x}-\frac{b^3 \left (a+b x^2\right )^{3/2}}{3 x^3}-\frac{b^2 \left (a+b x^2\right )^{5/2}}{5 x^5}-\frac{\left (a+b x^2\right )^{9/2}}{9 x^9}-\frac{b \left (a+b x^2\right )^{7/2}}{7 x^7} \]

[Out]

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

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Rubi [A]  time = 0.150345, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ b^{9/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{b^4 \sqrt{a+b x^2}}{x}-\frac{b^3 \left (a+b x^2\right )^{3/2}}{3 x^3}-\frac{b^2 \left (a+b x^2\right )^{5/2}}{5 x^5}-\frac{\left (a+b x^2\right )^{9/2}}{9 x^9}-\frac{b \left (a+b x^2\right )^{7/2}}{7 x^7} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(9/2)/x^10,x]

[Out]

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

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Rubi in Sympy [A]  time = 18.2268, size = 107, normalized size = 0.86 \[ b^{\frac{9}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )} - \frac{b^{4} \sqrt{a + b x^{2}}}{x} - \frac{b^{3} \left (a + b x^{2}\right )^{\frac{3}{2}}}{3 x^{3}} - \frac{b^{2} \left (a + b x^{2}\right )^{\frac{5}{2}}}{5 x^{5}} - \frac{b \left (a + b x^{2}\right )^{\frac{7}{2}}}{7 x^{7}} - \frac{\left (a + b x^{2}\right )^{\frac{9}{2}}}{9 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(9/2)/x**10,x)

[Out]

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

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Mathematica [A]  time = 0.0839914, size = 90, normalized size = 0.73 \[ b^{9/2} \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )-\frac{\sqrt{a+b x^2} \left (35 a^4+185 a^3 b x^2+408 a^2 b^2 x^4+506 a b^3 x^6+563 b^4 x^8\right )}{315 x^9} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(9/2)/x^10,x]

[Out]

-(Sqrt[a + b*x^2]*(35*a^4 + 185*a^3*b*x^2 + 408*a^2*b^2*x^4 + 506*a*b^3*x^6 + 56
3*b^4*x^8))/(315*x^9) + b^(9/2)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]]

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Maple [B]  time = 0.047, size = 206, normalized size = 1.7 \[ -{\frac{1}{9\,a{x}^{9}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{2\,b}{63\,{a}^{2}{x}^{7}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{8\,{b}^{2}}{315\,{a}^{3}{x}^{5}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{16\,{b}^{3}}{315\,{a}^{4}{x}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{128\,{b}^{4}}{315\,{a}^{5}x} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{128\,{b}^{5}x}{315\,{a}^{5}} \left ( b{x}^{2}+a \right ) ^{{\frac{9}{2}}}}+{\frac{16\,{b}^{5}x}{35\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{8\,{b}^{5}x}{15\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{2\,{b}^{5}x}{3\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{5}x}{a}\sqrt{b{x}^{2}+a}}+{b}^{{\frac{9}{2}}}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(9/2)/x^10,x)

[Out]

-1/9/a/x^9*(b*x^2+a)^(11/2)-2/63*b/a^2/x^7*(b*x^2+a)^(11/2)-8/315*b^2/a^3/x^5*(b
*x^2+a)^(11/2)-16/315*b^3/a^4/x^3*(b*x^2+a)^(11/2)-128/315*b^4/a^5/x*(b*x^2+a)^(
11/2)+128/315*b^5/a^5*x*(b*x^2+a)^(9/2)+16/35*b^5/a^4*x*(b*x^2+a)^(7/2)+8/15*b^5
/a^3*x*(b*x^2+a)^(5/2)+2/3*b^5/a^2*x*(b*x^2+a)^(3/2)+b^5/a*x*(b*x^2+a)^(1/2)+b^(
9/2)*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^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.272093, size = 1, normalized size = 0.01 \[ \left [\frac{315 \, b^{\frac{9}{2}} x^{9} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) - 2 \,{\left (563 \, b^{4} x^{8} + 506 \, a b^{3} x^{6} + 408 \, a^{2} b^{2} x^{4} + 185 \, a^{3} b x^{2} + 35 \, a^{4}\right )} \sqrt{b x^{2} + a}}{630 \, x^{9}}, \frac{315 \, \sqrt{-b} b^{4} x^{9} \arctan \left (\frac{b x}{\sqrt{b x^{2} + a} \sqrt{-b}}\right ) -{\left (563 \, b^{4} x^{8} + 506 \, a b^{3} x^{6} + 408 \, a^{2} b^{2} x^{4} + 185 \, a^{3} b x^{2} + 35 \, a^{4}\right )} \sqrt{b x^{2} + a}}{315 \, x^{9}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(9/2)/x^10,x, algorithm="fricas")

[Out]

[1/630*(315*b^(9/2)*x^9*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(563
*b^4*x^8 + 506*a*b^3*x^6 + 408*a^2*b^2*x^4 + 185*a^3*b*x^2 + 35*a^4)*sqrt(b*x^2
+ a))/x^9, 1/315*(315*sqrt(-b)*b^4*x^9*arctan(b*x/(sqrt(b*x^2 + a)*sqrt(-b))) -
(563*b^4*x^8 + 506*a*b^3*x^6 + 408*a^2*b^2*x^4 + 185*a^3*b*x^2 + 35*a^4)*sqrt(b*
x^2 + a))/x^9]

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Sympy [A]  time = 42.9649, size = 160, normalized size = 1.29 \[ - \frac{a^{4} \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{9 x^{8}} - \frac{37 a^{3} b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{63 x^{6}} - \frac{136 a^{2} b^{\frac{5}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{105 x^{4}} - \frac{506 a b^{\frac{7}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{315 x^{2}} - \frac{563 b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{315} - \frac{b^{\frac{9}{2}} \log{\left (\frac{a}{b x^{2}} \right )}}{2} + b^{\frac{9}{2}} \log{\left (\sqrt{\frac{a}{b x^{2}} + 1} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(9/2)/x**10,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.21991, size = 373, normalized size = 3.01 \[ -\frac{1}{2} \, b^{\frac{9}{2}}{\rm ln}\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right ) + \frac{2 \,{\left (1575 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{16} a b^{\frac{9}{2}} - 6300 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{14} a^{2} b^{\frac{9}{2}} + 21000 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} a^{3} b^{\frac{9}{2}} - 31500 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} a^{4} b^{\frac{9}{2}} + 39438 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} a^{5} b^{\frac{9}{2}} - 26292 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a^{6} b^{\frac{9}{2}} + 13968 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{7} b^{\frac{9}{2}} - 3492 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{8} b^{\frac{9}{2}} + 563 \, a^{9} b^{\frac{9}{2}}\right )}}{315 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(9/2)/x^10,x, algorithm="giac")

[Out]

-1/2*b^(9/2)*ln((sqrt(b)*x - sqrt(b*x^2 + a))^2) + 2/315*(1575*(sqrt(b)*x - sqrt
(b*x^2 + a))^16*a*b^(9/2) - 6300*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a^2*b^(9/2) +
21000*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^3*b^(9/2) - 31500*(sqrt(b)*x - sqrt(b*x
^2 + a))^10*a^4*b^(9/2) + 39438*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^5*b^(9/2) - 26
292*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^6*b^(9/2) + 13968*(sqrt(b)*x - sqrt(b*x^2
+ a))^4*a^7*b^(9/2) - 3492*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^8*b^(9/2) + 563*a^9
*b^(9/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^9