3.82 \(\int f^{a+b x^2} x^{12} \, dx\)

Optimal. Leaf size=34 \[ -\frac{x^{13} f^a \text{Gamma}\left (\frac{13}{2},-b x^2 \log (f)\right )}{2 \left (-b x^2 \log (f)\right )^{13/2}} \]

[Out]

-(f^a*x^13*Gamma[13/2, -(b*x^2*Log[f])])/(2*(-(b*x^2*Log[f]))^(13/2))

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Rubi [A]  time = 0.0396286, 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 \[ -\frac{x^{13} f^a \text{Gamma}\left (\frac{13}{2},-b x^2 \log (f)\right )}{2 \left (-b x^2 \log (f)\right )^{13/2}} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b*x^2)*x^12,x]

[Out]

-(f^a*x^13*Gamma[13/2, -(b*x^2*Log[f])])/(2*(-(b*x^2*Log[f]))^(13/2))

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Rubi in Sympy [A]  time = 3.14685, size = 36, normalized size = 1.06 \[ - \frac{f^{a} x^{13} \Gamma{\left (\frac{13}{2},- b x^{2} \log{\left (f \right )} \right )}}{2 \left (- b x^{2} \log{\left (f \right )}\right )^{\frac{13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(b*x**2+a)*x**12,x)

[Out]

-f**a*x**13*Gamma(13/2, -b*x**2*log(f))/(2*(-b*x**2*log(f))**(13/2))

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Mathematica [B]  time = 0.0819979, size = 119, normalized size = 3.5 \[ \frac{f^a \left (2 \sqrt{b} x \sqrt{\log (f)} f^{b x^2} \left (32 b^5 x^{10} \log ^5(f)-176 b^4 x^8 \log ^4(f)+792 b^3 x^6 \log ^3(f)-2772 b^2 x^4 \log ^2(f)+6930 b x^2 \log (f)-10395\right )+10395 \sqrt{\pi } \text{Erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )\right )}{128 b^{13/2} \log ^{\frac{13}{2}}(f)} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b*x^2)*x^12,x]

[Out]

(f^a*(10395*Sqrt[Pi]*Erfi[Sqrt[b]*x*Sqrt[Log[f]]] + 2*Sqrt[b]*f^(b*x^2)*x*Sqrt[L
og[f]]*(-10395 + 6930*b*x^2*Log[f] - 2772*b^2*x^4*Log[f]^2 + 792*b^3*x^6*Log[f]^
3 - 176*b^4*x^8*Log[f]^4 + 32*b^5*x^10*Log[f]^5)))/(128*b^(13/2)*Log[f]^(13/2))

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Maple [A]  time = 0.118, size = 164, normalized size = 4.8 \[{\frac{{f}^{a}{f}^{b{x}^{2}}{x}^{11}}{2\,b\ln \left ( f \right ) }}-{\frac{11\,{f}^{a}{x}^{9}{f}^{b{x}^{2}}}{4\, \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}}+{\frac{99\,{f}^{a}{x}^{7}{f}^{b{x}^{2}}}{8\, \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}}-{\frac{693\,{f}^{a}{x}^{5}{f}^{b{x}^{2}}}{16\, \left ( \ln \left ( f \right ) \right ) ^{4}{b}^{4}}}+{\frac{3465\,{f}^{a}{x}^{3}{f}^{b{x}^{2}}}{32\, \left ( \ln \left ( f \right ) \right ) ^{5}{b}^{5}}}-{\frac{10395\,{f}^{a}x{f}^{b{x}^{2}}}{64\, \left ( \ln \left ( f \right ) \right ) ^{6}{b}^{6}}}+{\frac{10395\,{f}^{a}\sqrt{\pi }}{128\, \left ( \ln \left ( f \right ) \right ) ^{6}{b}^{6}}{\it Erf} \left ( \sqrt{-b\ln \left ( f \right ) }x \right ){\frac{1}{\sqrt{-b\ln \left ( f \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(b*x^2+a)*x^12,x)

[Out]

1/2*f^a*f^(b*x^2)*x^11/ln(f)/b-11/4*f^a/ln(f)^2/b^2*x^9*f^(b*x^2)+99/8*f^a/ln(f)
^3/b^3*x^7*f^(b*x^2)-693/16*f^a/ln(f)^4/b^4*x^5*f^(b*x^2)+3465/32*f^a/ln(f)^5/b^
5*x^3*f^(b*x^2)-10395/64*f^a/ln(f)^6/b^6*x*f^(b*x^2)+10395/128*f^a/ln(f)^6/b^6*P
i^(1/2)/(-b*ln(f))^(1/2)*erf((-b*ln(f))^(1/2)*x)

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Maxima [A]  time = 0.794818, size = 171, normalized size = 5.03 \[ \frac{{\left (32 \, b^{5} f^{a} x^{11} \log \left (f\right )^{5} - 176 \, b^{4} f^{a} x^{9} \log \left (f\right )^{4} + 792 \, b^{3} f^{a} x^{7} \log \left (f\right )^{3} - 2772 \, b^{2} f^{a} x^{5} \log \left (f\right )^{2} + 6930 \, b f^{a} x^{3} \log \left (f\right ) - 10395 \, f^{a} x\right )} f^{b x^{2}}}{64 \, b^{6} \log \left (f\right )^{6}} + \frac{10395 \, \sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-b \log \left (f\right )} x\right )}{128 \, \sqrt{-b \log \left (f\right )} b^{6} \log \left (f\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(b*x^2 + a)*x^12,x, algorithm="maxima")

[Out]

1/64*(32*b^5*f^a*x^11*log(f)^5 - 176*b^4*f^a*x^9*log(f)^4 + 792*b^3*f^a*x^7*log(
f)^3 - 2772*b^2*f^a*x^5*log(f)^2 + 6930*b*f^a*x^3*log(f) - 10395*f^a*x)*f^(b*x^2
)/(b^6*log(f)^6) + 10395/128*sqrt(pi)*f^a*erf(sqrt(-b*log(f))*x)/(sqrt(-b*log(f)
)*b^6*log(f)^6)

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Fricas [A]  time = 0.255572, size = 153, normalized size = 4.5 \[ \frac{2 \,{\left (32 \, b^{5} x^{11} \log \left (f\right )^{5} - 176 \, b^{4} x^{9} \log \left (f\right )^{4} + 792 \, b^{3} x^{7} \log \left (f\right )^{3} - 2772 \, b^{2} x^{5} \log \left (f\right )^{2} + 6930 \, b x^{3} \log \left (f\right ) - 10395 \, x\right )} \sqrt{-b \log \left (f\right )} f^{b x^{2} + a} + 10395 \, \sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-b \log \left (f\right )} x\right )}{128 \, \sqrt{-b \log \left (f\right )} b^{6} \log \left (f\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(b*x^2 + a)*x^12,x, algorithm="fricas")

[Out]

1/128*(2*(32*b^5*x^11*log(f)^5 - 176*b^4*x^9*log(f)^4 + 792*b^3*x^7*log(f)^3 - 2
772*b^2*x^5*log(f)^2 + 6930*b*x^3*log(f) - 10395*x)*sqrt(-b*log(f))*f^(b*x^2 + a
) + 10395*sqrt(pi)*f^a*erf(sqrt(-b*log(f))*x))/(sqrt(-b*log(f))*b^6*log(f)^6)

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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**12,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.253143, size = 159, normalized size = 4.68 \[ \frac{{\left (32 \, b^{5} x^{11}{\rm ln}\left (f\right )^{5} - 176 \, b^{4} x^{9}{\rm ln}\left (f\right )^{4} + 792 \, b^{3} x^{7}{\rm ln}\left (f\right )^{3} - 2772 \, b^{2} x^{5}{\rm ln}\left (f\right )^{2} + 6930 \, b x^{3}{\rm ln}\left (f\right ) - 10395 \, x\right )} e^{\left (b x^{2}{\rm ln}\left (f\right ) + a{\rm ln}\left (f\right )\right )}}{64 \, b^{6}{\rm ln}\left (f\right )^{6}} - \frac{10395 \, \sqrt{\pi } \operatorname{erf}\left (-\sqrt{-b{\rm ln}\left (f\right )} x\right ) e^{\left (a{\rm ln}\left (f\right )\right )}}{128 \, \sqrt{-b{\rm ln}\left (f\right )} b^{6}{\rm ln}\left (f\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(b*x^2 + a)*x^12,x, algorithm="giac")

[Out]

1/64*(32*b^5*x^11*ln(f)^5 - 176*b^4*x^9*ln(f)^4 + 792*b^3*x^7*ln(f)^3 - 2772*b^2
*x^5*ln(f)^2 + 6930*b*x^3*ln(f) - 10395*x)*e^(b*x^2*ln(f) + a*ln(f))/(b^6*ln(f)^
6) - 10395/128*sqrt(pi)*erf(-sqrt(-b*ln(f))*x)*e^(a*ln(f))/(sqrt(-b*ln(f))*b^6*l
n(f)^6)