3.423 \(\int \frac{\left (a+b x^2\right )^{9/2}}{x^{13}} \, dx\)

Optimal. Leaf size=155 \[ \frac{21 b^6 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{1024 a^{3/2}}-\frac{21 b^5 \sqrt{a+b x^2}}{1024 a x^2}-\frac{21 b^4 \sqrt{a+b x^2}}{512 x^4}-\frac{7 b^3 \left (a+b x^2\right )^{3/2}}{128 x^6}-\frac{21 b^2 \left (a+b x^2\right )^{5/2}}{320 x^8}-\frac{\left (a+b x^2\right )^{9/2}}{12 x^{12}}-\frac{3 b \left (a+b x^2\right )^{7/2}}{40 x^{10}} \]

[Out]

(-21*b^4*Sqrt[a + b*x^2])/(512*x^4) - (21*b^5*Sqrt[a + b*x^2])/(1024*a*x^2) - (7
*b^3*(a + b*x^2)^(3/2))/(128*x^6) - (21*b^2*(a + b*x^2)^(5/2))/(320*x^8) - (3*b*
(a + b*x^2)^(7/2))/(40*x^10) - (a + b*x^2)^(9/2)/(12*x^12) + (21*b^6*ArcTanh[Sqr
t[a + b*x^2]/Sqrt[a]])/(1024*a^(3/2))

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Rubi [A]  time = 0.264059, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{21 b^6 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{1024 a^{3/2}}-\frac{21 b^5 \sqrt{a+b x^2}}{1024 a x^2}-\frac{21 b^4 \sqrt{a+b x^2}}{512 x^4}-\frac{7 b^3 \left (a+b x^2\right )^{3/2}}{128 x^6}-\frac{21 b^2 \left (a+b x^2\right )^{5/2}}{320 x^8}-\frac{\left (a+b x^2\right )^{9/2}}{12 x^{12}}-\frac{3 b \left (a+b x^2\right )^{7/2}}{40 x^{10}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(9/2)/x^13,x]

[Out]

(-21*b^4*Sqrt[a + b*x^2])/(512*x^4) - (21*b^5*Sqrt[a + b*x^2])/(1024*a*x^2) - (7
*b^3*(a + b*x^2)^(3/2))/(128*x^6) - (21*b^2*(a + b*x^2)^(5/2))/(320*x^8) - (3*b*
(a + b*x^2)^(7/2))/(40*x^10) - (a + b*x^2)^(9/2)/(12*x^12) + (21*b^6*ArcTanh[Sqr
t[a + b*x^2]/Sqrt[a]])/(1024*a^(3/2))

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Rubi in Sympy [A]  time = 24.957, size = 144, normalized size = 0.93 \[ - \frac{21 b^{4} \sqrt{a + b x^{2}}}{512 x^{4}} - \frac{7 b^{3} \left (a + b x^{2}\right )^{\frac{3}{2}}}{128 x^{6}} - \frac{21 b^{2} \left (a + b x^{2}\right )^{\frac{5}{2}}}{320 x^{8}} - \frac{3 b \left (a + b x^{2}\right )^{\frac{7}{2}}}{40 x^{10}} - \frac{\left (a + b x^{2}\right )^{\frac{9}{2}}}{12 x^{12}} - \frac{21 b^{5} \sqrt{a + b x^{2}}}{1024 a x^{2}} + \frac{21 b^{6} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{1024 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(9/2)/x**13,x)

[Out]

-21*b**4*sqrt(a + b*x**2)/(512*x**4) - 7*b**3*(a + b*x**2)**(3/2)/(128*x**6) - 2
1*b**2*(a + b*x**2)**(5/2)/(320*x**8) - 3*b*(a + b*x**2)**(7/2)/(40*x**10) - (a
+ b*x**2)**(9/2)/(12*x**12) - 21*b**5*sqrt(a + b*x**2)/(1024*a*x**2) + 21*b**6*a
tanh(sqrt(a + b*x**2)/sqrt(a))/(1024*a**(3/2))

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Mathematica [A]  time = 0.141899, size = 123, normalized size = 0.79 \[ \frac{-\sqrt{a} \sqrt{a+b x^2} \left (1280 a^5+6272 a^4 b x^2+12144 a^3 b^2 x^4+11432 a^2 b^3 x^6+4910 a b^4 x^8+315 b^5 x^{10}\right )+315 b^6 x^{12} \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )-315 b^6 x^{12} \log (x)}{15360 a^{3/2} x^{12}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(9/2)/x^13,x]

[Out]

(-(Sqrt[a]*Sqrt[a + b*x^2]*(1280*a^5 + 6272*a^4*b*x^2 + 12144*a^3*b^2*x^4 + 1143
2*a^2*b^3*x^6 + 4910*a*b^4*x^8 + 315*b^5*x^10)) - 315*b^6*x^12*Log[x] + 315*b^6*
x^12*Log[a + Sqrt[a]*Sqrt[a + b*x^2]])/(15360*a^(3/2)*x^12)

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Maple [A]  time = 0.17, size = 233, normalized size = 1.5 \[ -{\frac{1}{12\,a{x}^{12}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{b}{120\,{a}^{2}{x}^{10}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{{b}^{2}}{960\,{a}^{3}{x}^{8}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{{b}^{3}}{1920\,{a}^{4}{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{{b}^{4}}{1536\,{a}^{5}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{7\,{b}^{5}}{3072\,{a}^{6}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{7\,{b}^{6}}{3072\,{a}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{9}{2}}}}-{\frac{3\,{b}^{6}}{1024\,{a}^{5}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{21\,{b}^{6}}{5120\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{7\,{b}^{6}}{1024\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{21\,{b}^{6}}{1024}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{21\,{b}^{6}}{1024\,{a}^{2}}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(9/2)/x^13,x)

[Out]

-1/12/a/x^12*(b*x^2+a)^(11/2)+1/120*b/a^2/x^10*(b*x^2+a)^(11/2)+1/960*b^2/a^3/x^
8*(b*x^2+a)^(11/2)+1/1920*b^3/a^4/x^6*(b*x^2+a)^(11/2)+1/1536*b^4/a^5/x^4*(b*x^2
+a)^(11/2)+7/3072*b^5/a^6/x^2*(b*x^2+a)^(11/2)-7/3072*b^6/a^6*(b*x^2+a)^(9/2)-3/
1024*b^6/a^5*(b*x^2+a)^(7/2)-21/5120*b^6/a^4*(b*x^2+a)^(5/2)-7/1024*b^6/a^3*(b*x
^2+a)^(3/2)+21/1024*b^6/a^(3/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)-21/1024*b^
6/a^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^13,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.361881, size = 1, normalized size = 0.01 \[ \left [\frac{315 \, b^{6} x^{12} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) - 2 \,{\left (315 \, b^{5} x^{10} + 4910 \, a b^{4} x^{8} + 11432 \, a^{2} b^{3} x^{6} + 12144 \, a^{3} b^{2} x^{4} + 6272 \, a^{4} b x^{2} + 1280 \, a^{5}\right )} \sqrt{b x^{2} + a} \sqrt{a}}{30720 \, a^{\frac{3}{2}} x^{12}}, \frac{315 \, b^{6} x^{12} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (315 \, b^{5} x^{10} + 4910 \, a b^{4} x^{8} + 11432 \, a^{2} b^{3} x^{6} + 12144 \, a^{3} b^{2} x^{4} + 6272 \, a^{4} b x^{2} + 1280 \, a^{5}\right )} \sqrt{b x^{2} + a} \sqrt{-a}}{15360 \, \sqrt{-a} a x^{12}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(9/2)/x^13,x, algorithm="fricas")

[Out]

[1/30720*(315*b^6*x^12*log(-((b*x^2 + 2*a)*sqrt(a) + 2*sqrt(b*x^2 + a)*a)/x^2) -
2*(315*b^5*x^10 + 4910*a*b^4*x^8 + 11432*a^2*b^3*x^6 + 12144*a^3*b^2*x^4 + 6272
*a^4*b*x^2 + 1280*a^5)*sqrt(b*x^2 + a)*sqrt(a))/(a^(3/2)*x^12), 1/15360*(315*b^6
*x^12*arctan(sqrt(-a)/sqrt(b*x^2 + a)) - (315*b^5*x^10 + 4910*a*b^4*x^8 + 11432*
a^2*b^3*x^6 + 12144*a^3*b^2*x^4 + 6272*a^4*b*x^2 + 1280*a^5)*sqrt(b*x^2 + a)*sqr
t(-a))/(sqrt(-a)*a*x^12)]

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Sympy [A]  time = 64.9683, size = 204, normalized size = 1.32 \[ - \frac{a^{5}}{12 \sqrt{b} x^{13} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{59 a^{4} \sqrt{b}}{120 x^{11} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{1151 a^{3} b^{\frac{3}{2}}}{960 x^{9} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{2947 a^{2} b^{\frac{5}{2}}}{1920 x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{8171 a b^{\frac{7}{2}}}{7680 x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{1045 b^{\frac{9}{2}}}{3072 x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{21 b^{\frac{11}{2}}}{1024 a x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{21 b^{6} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{1024 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(9/2)/x**13,x)

[Out]

-a**5/(12*sqrt(b)*x**13*sqrt(a/(b*x**2) + 1)) - 59*a**4*sqrt(b)/(120*x**11*sqrt(
a/(b*x**2) + 1)) - 1151*a**3*b**(3/2)/(960*x**9*sqrt(a/(b*x**2) + 1)) - 2947*a**
2*b**(5/2)/(1920*x**7*sqrt(a/(b*x**2) + 1)) - 8171*a*b**(7/2)/(7680*x**5*sqrt(a/
(b*x**2) + 1)) - 1045*b**(9/2)/(3072*x**3*sqrt(a/(b*x**2) + 1)) - 21*b**(11/2)/(
1024*a*x*sqrt(a/(b*x**2) + 1)) + 21*b**6*asinh(sqrt(a)/(sqrt(b)*x))/(1024*a**(3/
2))

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GIAC/XCAS [A]  time = 0.213854, size = 165, normalized size = 1.06 \[ -\frac{1}{15360} \, b^{6}{\left (\frac{315 \, \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{315 \,{\left (b x^{2} + a\right )}^{\frac{11}{2}} + 3335 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} a - 5058 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a^{2} + 4158 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{3} - 1785 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{4} + 315 \, \sqrt{b x^{2} + a} a^{5}}{a b^{6} x^{12}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(9/2)/x^13,x, algorithm="giac")

[Out]

-1/15360*b^6*(315*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a) + (315*(b*x^2 +
a)^(11/2) + 3335*(b*x^2 + a)^(9/2)*a - 5058*(b*x^2 + a)^(7/2)*a^2 + 4158*(b*x^2
+ a)^(5/2)*a^3 - 1785*(b*x^2 + a)^(3/2)*a^4 + 315*sqrt(b*x^2 + a)*a^5)/(a*b^6*x^
12))