3.143 \(\int f^{a+\frac{b}{x^2}} x^6 \, dx\)

Optimal. Leaf size=119 \[ -\frac{8}{105} \sqrt{\pi } b^{7/2} f^a \log ^{\frac{7}{2}}(f) \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )+\frac{8}{105} b^3 x \log ^3(f) f^{a+\frac{b}{x^2}}+\frac{4}{105} b^2 x^3 \log ^2(f) f^{a+\frac{b}{x^2}}+\frac{1}{7} x^7 f^{a+\frac{b}{x^2}}+\frac{2}{35} b x^5 \log (f) f^{a+\frac{b}{x^2}} \]

[Out]

(f^(a + b/x^2)*x^7)/7 + (2*b*f^(a + b/x^2)*x^5*Log[f])/35 + (4*b^2*f^(a + b/x^2)
*x^3*Log[f]^2)/105 + (8*b^3*f^(a + b/x^2)*x*Log[f]^3)/105 - (8*b^(7/2)*f^a*Sqrt[
Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x]*Log[f]^(7/2))/105

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Rubi [A]  time = 0.193734, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{8}{105} \sqrt{\pi } b^{7/2} f^a \log ^{\frac{7}{2}}(f) \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )+\frac{8}{105} b^3 x \log ^3(f) f^{a+\frac{b}{x^2}}+\frac{4}{105} b^2 x^3 \log ^2(f) f^{a+\frac{b}{x^2}}+\frac{1}{7} x^7 f^{a+\frac{b}{x^2}}+\frac{2}{35} b x^5 \log (f) f^{a+\frac{b}{x^2}} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b/x^2)*x^6,x]

[Out]

(f^(a + b/x^2)*x^7)/7 + (2*b*f^(a + b/x^2)*x^5*Log[f])/35 + (4*b^2*f^(a + b/x^2)
*x^3*Log[f]^2)/105 + (8*b^3*f^(a + b/x^2)*x*Log[f]^3)/105 - (8*b^(7/2)*f^a*Sqrt[
Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x]*Log[f]^(7/2))/105

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Rubi in Sympy [A]  time = 17.9099, size = 117, normalized size = 0.98 \[ - \frac{8 \sqrt{\pi } b^{\frac{7}{2}} f^{a} \log{\left (f \right )}^{\frac{7}{2}} \operatorname{erfi}{\left (\frac{\sqrt{b} \sqrt{\log{\left (f \right )}}}{x} \right )}}{105} + \frac{8 b^{3} f^{a + \frac{b}{x^{2}}} x \log{\left (f \right )}^{3}}{105} + \frac{4 b^{2} f^{a + \frac{b}{x^{2}}} x^{3} \log{\left (f \right )}^{2}}{105} + \frac{2 b f^{a + \frac{b}{x^{2}}} x^{5} \log{\left (f \right )}}{35} + \frac{f^{a + \frac{b}{x^{2}}} x^{7}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(a+b/x**2)*x**6,x)

[Out]

-8*sqrt(pi)*b**(7/2)*f**a*log(f)**(7/2)*erfi(sqrt(b)*sqrt(log(f))/x)/105 + 8*b**
3*f**(a + b/x**2)*x*log(f)**3/105 + 4*b**2*f**(a + b/x**2)*x**3*log(f)**2/105 +
2*b*f**(a + b/x**2)*x**5*log(f)/35 + f**(a + b/x**2)*x**7/7

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Mathematica [A]  time = 0.0510242, size = 86, normalized size = 0.72 \[ \frac{1}{105} f^a \left (x f^{\frac{b}{x^2}} \left (8 b^3 \log ^3(f)+4 b^2 x^2 \log ^2(f)+6 b x^4 \log (f)+15 x^6\right )-8 \sqrt{\pi } b^{7/2} \log ^{\frac{7}{2}}(f) \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b/x^2)*x^6,x]

[Out]

(f^a*(-8*b^(7/2)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x]*Log[f]^(7/2) + f^(b/x^2
)*x*(15*x^6 + 6*b*x^4*Log[f] + 4*b^2*x^2*Log[f]^2 + 8*b^3*Log[f]^3)))/105

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Maple [A]  time = 0.036, size = 111, normalized size = 0.9 \[{\frac{{f}^{a}{x}^{7}}{7}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{2\,{f}^{a}\ln \left ( f \right ) b{x}^{5}}{35}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{4\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}{x}^{3}}{105}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{8\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}x}{105}{f}^{{\frac{b}{{x}^{2}}}}}-{\frac{8\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{4}{b}^{4}\sqrt{\pi }}{105}{\it Erf} \left ({\frac{1}{x}\sqrt{-b\ln \left ( f \right ) }} \right ){\frac{1}{\sqrt{-b\ln \left ( f \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(a+b/x^2)*x^6,x)

[Out]

1/7*f^a*f^(b/x^2)*x^7+2/35*f^a*ln(f)*b*f^(b/x^2)*x^5+4/105*f^a*ln(f)^2*b^2*f^(b/
x^2)*x^3+8/105*f^a*ln(f)^3*b^3*f^(b/x^2)*x-8/105*f^a*ln(f)^4*b^4*Pi^(1/2)/(-b*ln
(f))^(1/2)*erf((-b*ln(f))^(1/2)/x)

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Maxima [A]  time = 0.831959, size = 134, normalized size = 1.13 \[ -\frac{8 \, \sqrt{\pi } b^{4} f^{a}{\left (\operatorname{erf}\left (\sqrt{-\frac{b \log \left (f\right )}{x^{2}}}\right ) - 1\right )} \log \left (f\right )^{4}}{105 \, x \sqrt{-\frac{b \log \left (f\right )}{x^{2}}}} + \frac{1}{105} \,{\left (15 \, f^{a} x^{7} + 6 \, b f^{a} x^{5} \log \left (f\right ) + 4 \, b^{2} f^{a} x^{3} \log \left (f\right )^{2} + 8 \, b^{3} f^{a} x \log \left (f\right )^{3}\right )} f^{\frac{b}{x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x^2)*x^6,x, algorithm="maxima")

[Out]

-8/105*sqrt(pi)*b^4*f^a*(erf(sqrt(-b*log(f)/x^2)) - 1)*log(f)^4/(x*sqrt(-b*log(f
)/x^2)) + 1/105*(15*f^a*x^7 + 6*b*f^a*x^5*log(f) + 4*b^2*f^a*x^3*log(f)^2 + 8*b^
3*f^a*x*log(f)^3)*f^(b/x^2)

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Fricas [A]  time = 0.265439, size = 128, normalized size = 1.08 \[ -\frac{8 \, \sqrt{\pi } b^{4} f^{a} \operatorname{erf}\left (\frac{\sqrt{-b \log \left (f\right )}}{x}\right ) \log \left (f\right )^{4} -{\left (15 \, x^{7} + 6 \, b x^{5} \log \left (f\right ) + 4 \, b^{2} x^{3} \log \left (f\right )^{2} + 8 \, b^{3} x \log \left (f\right )^{3}\right )} \sqrt{-b \log \left (f\right )} f^{\frac{a x^{2} + b}{x^{2}}}}{105 \, \sqrt{-b \log \left (f\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x^2)*x^6,x, algorithm="fricas")

[Out]

-1/105*(8*sqrt(pi)*b^4*f^a*erf(sqrt(-b*log(f))/x)*log(f)^4 - (15*x^7 + 6*b*x^5*l
og(f) + 4*b^2*x^3*log(f)^2 + 8*b^3*x*log(f)^3)*sqrt(-b*log(f))*f^((a*x^2 + b)/x^
2))/sqrt(-b*log(f))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(a+b/x**2)*x**6,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int f^{a + \frac{b}{x^{2}}} x^{6}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x^2)*x^6,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^2)*x^6, x)