3.360 \(\int \frac{\sqrt{a+b x^2}}{x^7} \, dx\)

Optimal. Leaf size=95 \[ -\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{16 a^{5/2}}+\frac{b^2 \sqrt{a+b x^2}}{16 a^2 x^2}-\frac{\sqrt{a+b x^2}}{6 x^6}-\frac{b \sqrt{a+b x^2}}{24 a x^4} \]

[Out]

-Sqrt[a + b*x^2]/(6*x^6) - (b*Sqrt[a + b*x^2])/(24*a*x^4) + (b^2*Sqrt[a + b*x^2]
)/(16*a^2*x^2) - (b^3*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(16*a^(5/2))

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Rubi [A]  time = 0.152887, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{16 a^{5/2}}+\frac{b^2 \sqrt{a+b x^2}}{16 a^2 x^2}-\frac{\sqrt{a+b x^2}}{6 x^6}-\frac{b \sqrt{a+b x^2}}{24 a x^4} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x^2]/x^7,x]

[Out]

-Sqrt[a + b*x^2]/(6*x^6) - (b*Sqrt[a + b*x^2])/(24*a*x^4) + (b^2*Sqrt[a + b*x^2]
)/(16*a^2*x^2) - (b^3*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(16*a^(5/2))

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Rubi in Sympy [A]  time = 14.2184, size = 82, normalized size = 0.86 \[ - \frac{\sqrt{a + b x^{2}}}{6 x^{6}} - \frac{b \sqrt{a + b x^{2}}}{24 a x^{4}} + \frac{b^{2} \sqrt{a + b x^{2}}}{16 a^{2} x^{2}} - \frac{b^{3} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{16 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(1/2)/x**7,x)

[Out]

-sqrt(a + b*x**2)/(6*x**6) - b*sqrt(a + b*x**2)/(24*a*x**4) + b**2*sqrt(a + b*x*
*2)/(16*a**2*x**2) - b**3*atanh(sqrt(a + b*x**2)/sqrt(a))/(16*a**(5/2))

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Mathematica [A]  time = 0.0742489, size = 91, normalized size = 0.96 \[ -\frac{b^3 \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{16 a^{5/2}}+\frac{b^3 \log (x)}{16 a^{5/2}}+\left (\frac{b^2}{16 a^2 x^2}-\frac{b}{24 a x^4}-\frac{1}{6 x^6}\right ) \sqrt{a+b x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x^2]/x^7,x]

[Out]

(-1/(6*x^6) - b/(24*a*x^4) + b^2/(16*a^2*x^2))*Sqrt[a + b*x^2] + (b^3*Log[x])/(1
6*a^(5/2)) - (b^3*Log[a + Sqrt[a]*Sqrt[a + b*x^2]])/(16*a^(5/2))

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Maple [A]  time = 0.011, size = 105, normalized size = 1.1 \[ -{\frac{1}{6\,a{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{b}{8\,{a}^{2}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}}{16\,{a}^{3}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{{b}^{3}}{16\,{a}^{3}}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(1/2)/x^7,x)

[Out]

-1/6/a/x^6*(b*x^2+a)^(3/2)+1/8*b/a^2/x^4*(b*x^2+a)^(3/2)-1/16*b^2/a^3/x^2*(b*x^2
+a)^(3/2)-1/16*b^3/a^(5/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)+1/16*b^3/a^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(sqrt(b*x^2 + a)/x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.25732, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{3} x^{6} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left (3 \, b^{2} x^{4} - 2 \, a b x^{2} - 8 \, a^{2}\right )} \sqrt{b x^{2} + a} \sqrt{a}}{96 \, a^{\frac{5}{2}} x^{6}}, -\frac{3 \, b^{3} x^{6} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (3 \, b^{2} x^{4} - 2 \, a b x^{2} - 8 \, a^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-a}}{48 \, \sqrt{-a} a^{2} x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(3*b^3*x^6*log(-((b*x^2 + 2*a)*sqrt(a) - 2*sqrt(b*x^2 + a)*a)/x^2) + 2*(3*
b^2*x^4 - 2*a*b*x^2 - 8*a^2)*sqrt(b*x^2 + a)*sqrt(a))/(a^(5/2)*x^6), -1/48*(3*b^
3*x^6*arctan(sqrt(-a)/sqrt(b*x^2 + a)) - (3*b^2*x^4 - 2*a*b*x^2 - 8*a^2)*sqrt(b*
x^2 + a)*sqrt(-a))/(sqrt(-a)*a^2*x^6)]

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Sympy [A]  time = 19.3802, size = 117, normalized size = 1.23 \[ - \frac{a}{6 \sqrt{b} x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{5 \sqrt{b}}{24 x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{b^{\frac{3}{2}}}{48 a x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{b^{\frac{5}{2}}}{16 a^{2} x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{16 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(1/2)/x**7,x)

[Out]

-a/(6*sqrt(b)*x**7*sqrt(a/(b*x**2) + 1)) - 5*sqrt(b)/(24*x**5*sqrt(a/(b*x**2) +
1)) + b**(3/2)/(48*a*x**3*sqrt(a/(b*x**2) + 1)) + b**(5/2)/(16*a**2*x*sqrt(a/(b*
x**2) + 1)) - b**3*asinh(sqrt(a)/(sqrt(b)*x))/(16*a**(5/2))

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GIAC/XCAS [A]  time = 0.210707, size = 108, normalized size = 1.14 \[ \frac{1}{48} \, b^{3}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} - 8 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a - 3 \, \sqrt{b x^{2} + a} a^{2}}{a^{2} b^{3} x^{6}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*b^3*(3*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^2) + (3*(b*x^2 + a)^(5/
2) - 8*(b*x^2 + a)^(3/2)*a - 3*sqrt(b*x^2 + a)*a^2)/(a^2*b^3*x^6))