3.420 \(\int \frac{\left (a+b x^2\right )^{9/2}}{x^7} \, dx\)

Optimal. Leaf size=126 \[ -\frac{105}{16} a^{3/2} b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )+\frac{35}{16} b^3 \left (a+b x^2\right )^{3/2}+\frac{105}{16} a b^3 \sqrt{a+b x^2}-\frac{21 b^2 \left (a+b x^2\right )^{5/2}}{16 x^2}-\frac{\left (a+b x^2\right )^{9/2}}{6 x^6}-\frac{3 b \left (a+b x^2\right )^{7/2}}{8 x^4} \]

[Out]

(105*a*b^3*Sqrt[a + b*x^2])/16 + (35*b^3*(a + b*x^2)^(3/2))/16 - (21*b^2*(a + b*
x^2)^(5/2))/(16*x^2) - (3*b*(a + b*x^2)^(7/2))/(8*x^4) - (a + b*x^2)^(9/2)/(6*x^
6) - (105*a^(3/2)*b^3*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/16

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Rubi [A]  time = 0.203708, antiderivative size = 126, 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 \[ -\frac{105}{16} a^{3/2} b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )+\frac{35}{16} b^3 \left (a+b x^2\right )^{3/2}+\frac{105}{16} a b^3 \sqrt{a+b x^2}-\frac{21 b^2 \left (a+b x^2\right )^{5/2}}{16 x^2}-\frac{\left (a+b x^2\right )^{9/2}}{6 x^6}-\frac{3 b \left (a+b x^2\right )^{7/2}}{8 x^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(9/2)/x^7,x]

[Out]

(105*a*b^3*Sqrt[a + b*x^2])/16 + (35*b^3*(a + b*x^2)^(3/2))/16 - (21*b^2*(a + b*
x^2)^(5/2))/(16*x^2) - (3*b*(a + b*x^2)^(7/2))/(8*x^4) - (a + b*x^2)^(9/2)/(6*x^
6) - (105*a^(3/2)*b^3*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/16

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Rubi in Sympy [A]  time = 19.117, size = 117, normalized size = 0.93 \[ - \frac{105 a^{\frac{3}{2}} b^{3} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{16} + \frac{105 a b^{3} \sqrt{a + b x^{2}}}{16} + \frac{35 b^{3} \left (a + b x^{2}\right )^{\frac{3}{2}}}{16} - \frac{21 b^{2} \left (a + b x^{2}\right )^{\frac{5}{2}}}{16 x^{2}} - \frac{3 b \left (a + b x^{2}\right )^{\frac{7}{2}}}{8 x^{4}} - \frac{\left (a + b x^{2}\right )^{\frac{9}{2}}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(9/2)/x**7,x)

[Out]

-105*a**(3/2)*b**3*atanh(sqrt(a + b*x**2)/sqrt(a))/16 + 105*a*b**3*sqrt(a + b*x*
*2)/16 + 35*b**3*(a + b*x**2)**(3/2)/16 - 21*b**2*(a + b*x**2)**(5/2)/(16*x**2)
- 3*b*(a + b*x**2)**(7/2)/(8*x**4) - (a + b*x**2)**(9/2)/(6*x**6)

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Mathematica [A]  time = 0.154709, size = 105, normalized size = 0.83 \[ \frac{1}{48} \left (-315 a^{3/2} b^3 \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+315 a^{3/2} b^3 \log (x)+\frac{\sqrt{a+b x^2} \left (-8 a^4-50 a^3 b x^2-165 a^2 b^2 x^4+208 a b^3 x^6+16 b^4 x^8\right )}{x^6}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(9/2)/x^7,x]

[Out]

((Sqrt[a + b*x^2]*(-8*a^4 - 50*a^3*b*x^2 - 165*a^2*b^2*x^4 + 208*a*b^3*x^6 + 16*
b^4*x^8))/x^6 + 315*a^(3/2)*b^3*Log[x] - 315*a^(3/2)*b^3*Log[a + Sqrt[a]*Sqrt[a
+ b*x^2]])/48

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Maple [A]  time = 0.018, size = 168, normalized size = 1.3 \[ -{\frac{1}{6\,a{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{5\,b}{24\,{a}^{2}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{35\,{b}^{2}}{48\,{a}^{3}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{35\,{b}^{3}}{48\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{9}{2}}}}+{\frac{15\,{b}^{3}}{16\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{21\,{b}^{3}}{16\,a} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{35\,{b}^{3}}{16} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{105\,{b}^{3}}{16}{a}^{{\frac{3}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) }+{\frac{105\,a{b}^{3}}{16}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(9/2)/x^7,x)

[Out]

-1/6/a/x^6*(b*x^2+a)^(11/2)-5/24*b/a^2/x^4*(b*x^2+a)^(11/2)-35/48*b^2/a^3/x^2*(b
*x^2+a)^(11/2)+35/48*b^3/a^3*(b*x^2+a)^(9/2)+15/16*b^3/a^2*(b*x^2+a)^(7/2)+21/16
*b^3/a*(b*x^2+a)^(5/2)+35/16*b^3*(b*x^2+a)^(3/2)-105/16*b^3*a^(3/2)*ln((2*a+2*a^
(1/2)*(b*x^2+a)^(1/2))/x)+105/16*a*b^3*(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^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251017, size = 1, normalized size = 0.01 \[ \left [\frac{315 \, a^{\frac{3}{2}} b^{3} x^{6} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (16 \, b^{4} x^{8} + 208 \, a b^{3} x^{6} - 165 \, a^{2} b^{2} x^{4} - 50 \, a^{3} b x^{2} - 8 \, a^{4}\right )} \sqrt{b x^{2} + a}}{96 \, x^{6}}, -\frac{315 \, \sqrt{-a} a b^{3} x^{6} \arctan \left (\frac{a}{\sqrt{b x^{2} + a} \sqrt{-a}}\right ) -{\left (16 \, b^{4} x^{8} + 208 \, a b^{3} x^{6} - 165 \, a^{2} b^{2} x^{4} - 50 \, a^{3} b x^{2} - 8 \, a^{4}\right )} \sqrt{b x^{2} + a}}{48 \, x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(9/2)/x^7,x, algorithm="fricas")

[Out]

[1/96*(315*a^(3/2)*b^3*x^6*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) +
2*(16*b^4*x^8 + 208*a*b^3*x^6 - 165*a^2*b^2*x^4 - 50*a^3*b*x^2 - 8*a^4)*sqrt(b*
x^2 + a))/x^6, -1/48*(315*sqrt(-a)*a*b^3*x^6*arctan(a/(sqrt(b*x^2 + a)*sqrt(-a))
) - (16*b^4*x^8 + 208*a*b^3*x^6 - 165*a^2*b^2*x^4 - 50*a^3*b*x^2 - 8*a^4)*sqrt(b
*x^2 + a))/x^6]

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Sympy [A]  time = 33.4413, size = 175, normalized size = 1.39 \[ - \frac{105 a^{\frac{3}{2}} b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{16} - \frac{a^{5}}{6 \sqrt{b} x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{29 a^{4} \sqrt{b}}{24 x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{215 a^{3} b^{\frac{3}{2}}}{48 x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{43 a^{2} b^{\frac{5}{2}}}{48 x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{14 a b^{\frac{7}{2}} x}{3 \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{b^{\frac{9}{2}} x^{3}}{3 \sqrt{\frac{a}{b x^{2}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(9/2)/x**7,x)

[Out]

-105*a**(3/2)*b**3*asinh(sqrt(a)/(sqrt(b)*x))/16 - a**5/(6*sqrt(b)*x**7*sqrt(a/(
b*x**2) + 1)) - 29*a**4*sqrt(b)/(24*x**5*sqrt(a/(b*x**2) + 1)) - 215*a**3*b**(3/
2)/(48*x**3*sqrt(a/(b*x**2) + 1)) + 43*a**2*b**(5/2)/(48*x*sqrt(a/(b*x**2) + 1))
+ 14*a*b**(7/2)*x/(3*sqrt(a/(b*x**2) + 1)) + b**(9/2)*x**3/(3*sqrt(a/(b*x**2) +
1))

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GIAC/XCAS [A]  time = 0.212559, size = 143, normalized size = 1.13 \[ \frac{1}{48} \,{\left (\frac{315 \, a^{2} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 16 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} + 192 \, \sqrt{b x^{2} + a} a - \frac{165 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{2} - 280 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3} + 123 \, \sqrt{b x^{2} + a} a^{4}}{b^{3} x^{6}}\right )} b^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(9/2)/x^7,x, algorithm="giac")

[Out]

1/48*(315*a^2*arctan(sqrt(b*x^2 + a)/sqrt(-a))/sqrt(-a) + 16*(b*x^2 + a)^(3/2) +
192*sqrt(b*x^2 + a)*a - (165*(b*x^2 + a)^(5/2)*a^2 - 280*(b*x^2 + a)^(3/2)*a^3
+ 123*sqrt(b*x^2 + a)*a^4)/(b^3*x^6))*b^3