3.93 \(\int \frac{f^{a+b x^2}}{x^{10}} \, dx\)

Optimal. Leaf size=34 \[ -\frac{f^a \left (-b x^2 \log (f)\right )^{9/2} \text{Gamma}\left (-\frac{9}{2},-b x^2 \log (f)\right )}{2 x^9} \]

[Out]

-(f^a*Gamma[-9/2, -(b*x^2*Log[f])]*(-(b*x^2*Log[f]))^(9/2))/(2*x^9)

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Rubi [A]  time = 0.0367308, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{f^a \left (-b x^2 \log (f)\right )^{9/2} \text{Gamma}\left (-\frac{9}{2},-b x^2 \log (f)\right )}{2 x^9} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b*x^2)/x^10,x]

[Out]

-(f^a*Gamma[-9/2, -(b*x^2*Log[f])]*(-(b*x^2*Log[f]))^(9/2))/(2*x^9)

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Rubi in Sympy [A]  time = 3.30615, size = 37, normalized size = 1.09 \[ - \frac{f^{a} \left (- b x^{2} \log{\left (f \right )}\right )^{\frac{9}{2}} \Gamma{\left (- \frac{9}{2},- b x^{2} \log{\left (f \right )} \right )}}{2 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-f**a*(-b*x**2*log(f))**(9/2)*Gamma(-9/2, -b*x**2*log(f))/(2*x**9)

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Mathematica [B]  time = 0.0759422, size = 101, normalized size = 2.97 \[ \frac{f^a \left (16 \sqrt{\pi } b^{9/2} x^9 \log ^{\frac{9}{2}}(f) \text{Erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )-f^{b x^2} \left (16 b^4 x^8 \log ^4(f)+8 b^3 x^6 \log ^3(f)+12 b^2 x^4 \log ^2(f)+30 b x^2 \log (f)+105\right )\right )}{945 x^9} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b*x^2)/x^10,x]

[Out]

(f^a*(16*b^(9/2)*Sqrt[Pi]*x^9*Erfi[Sqrt[b]*x*Sqrt[Log[f]]]*Log[f]^(9/2) - f^(b*x
^2)*(105 + 30*b*x^2*Log[f] + 12*b^2*x^4*Log[f]^2 + 8*b^3*x^6*Log[f]^3 + 16*b^4*x
^8*Log[f]^4)))/(945*x^9)

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Maple [A]  time = 0.054, size = 133, normalized size = 3.9 \[ -{\frac{{f}^{a}{f}^{b{x}^{2}}}{9\,{x}^{9}}}-{\frac{2\,{f}^{a}\ln \left ( f \right ) b{f}^{b{x}^{2}}}{63\,{x}^{7}}}-{\frac{4\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}{f}^{b{x}^{2}}}{315\,{x}^{5}}}-{\frac{8\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}{f}^{b{x}^{2}}}{945\,{x}^{3}}}-{\frac{16\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{4}{b}^{4}{f}^{b{x}^{2}}}{945\,x}}+{\frac{16\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{5}{b}^{5}\sqrt{\pi }}{945}{\it Erf} \left ( \sqrt{-b\ln \left ( f \right ) }x \right ){\frac{1}{\sqrt{-b\ln \left ( f \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(b*x^2+a)/x^10,x)

[Out]

-1/9*f^a*f^(b*x^2)/x^9-2/63*f^a*ln(f)*b*f^(b*x^2)/x^7-4/315*f^a*ln(f)^2*b^2*f^(b
*x^2)/x^5-8/945*f^a*ln(f)^3*b^3*f^(b*x^2)/x^3-16/945*f^a*ln(f)^4*b^4*f^(b*x^2)/x
+16/945*f^a*ln(f)^5*b^5*Pi^(1/2)/(-b*ln(f))^(1/2)*erf((-b*ln(f))^(1/2)*x)

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Maxima [A]  time = 0.828958, size = 38, normalized size = 1.12 \[ -\frac{\left (-b x^{2} \log \left (f\right )\right )^{\frac{9}{2}} f^{a} \Gamma \left (-\frac{9}{2}, -b x^{2} \log \left (f\right )\right )}{2 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(-b*x^2*log(f))^(9/2)*f^a*gamma(-9/2, -b*x^2*log(f))/x^9

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Fricas [A]  time = 0.252089, size = 142, normalized size = 4.18 \[ \frac{16 \, \sqrt{\pi } b^{5} f^{a} x^{9} \operatorname{erf}\left (\sqrt{-b \log \left (f\right )} x\right ) \log \left (f\right )^{5} -{\left (16 \, b^{4} x^{8} \log \left (f\right )^{4} + 8 \, b^{3} x^{6} \log \left (f\right )^{3} + 12 \, b^{2} x^{4} \log \left (f\right )^{2} + 30 \, b x^{2} \log \left (f\right ) + 105\right )} \sqrt{-b \log \left (f\right )} f^{b x^{2} + a}}{945 \, \sqrt{-b \log \left (f\right )} x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/945*(16*sqrt(pi)*b^5*f^a*x^9*erf(sqrt(-b*log(f))*x)*log(f)^5 - (16*b^4*x^8*log
(f)^4 + 8*b^3*x^6*log(f)^3 + 12*b^2*x^4*log(f)^2 + 30*b*x^2*log(f) + 105)*sqrt(-
b*log(f))*f^(b*x^2 + a))/(sqrt(-b*log(f))*x^9)

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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**(b*x**2+a)/x**10,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{b x^{2} + a}}{x^{10}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(f^(b*x^2 + a)/x^10, x)