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

Optimal. Leaf size=34 \[ -\frac{x^{11} f^a \text{Gamma}\left (\frac{11}{2},-b x^2 \log (f)\right )}{2 \left (-b x^2 \log (f)\right )^{11/2}} \]

[Out]

-(f^a*x^11*Gamma[11/2, -(b*x^2*Log[f])])/(2*(-(b*x^2*Log[f]))^(11/2))

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Rubi [A]  time = 0.0391461, 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 \[ -\frac{x^{11} f^a \text{Gamma}\left (\frac{11}{2},-b x^2 \log (f)\right )}{2 \left (-b x^2 \log (f)\right )^{11/2}} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b*x^2)*x^10,x]

[Out]

-(f^a*x^11*Gamma[11/2, -(b*x^2*Log[f])])/(2*(-(b*x^2*Log[f]))^(11/2))

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Rubi in Sympy [A]  time = 3.15146, size = 36, normalized size = 1.06 \[ - \frac{f^{a} x^{11} \Gamma{\left (\frac{11}{2},- b x^{2} \log{\left (f \right )} \right )}}{2 \left (- b x^{2} \log{\left (f \right )}\right )^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-f**a*x**11*Gamma(11/2, -b*x**2*log(f))/(2*(-b*x**2*log(f))**(11/2))

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Mathematica [B]  time = 0.0707825, size = 107, normalized size = 3.15 \[ \frac{f^a \left (2 \sqrt{b} x \sqrt{\log (f)} f^{b x^2} \left (16 b^4 x^8 \log ^4(f)-72 b^3 x^6 \log ^3(f)+252 b^2 x^4 \log ^2(f)-630 b x^2 \log (f)+945\right )-945 \sqrt{\pi } \text{Erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )\right )}{64 b^{11/2} \log ^{\frac{11}{2}}(f)} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b*x^2)*x^10,x]

[Out]

(f^a*(-945*Sqrt[Pi]*Erfi[Sqrt[b]*x*Sqrt[Log[f]]] + 2*Sqrt[b]*f^(b*x^2)*x*Sqrt[Lo
g[f]]*(945 - 630*b*x^2*Log[f] + 252*b^2*x^4*Log[f]^2 - 72*b^3*x^6*Log[f]^3 + 16*
b^4*x^8*Log[f]^4)))/(64*b^(11/2)*Log[f]^(11/2))

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Maple [A]  time = 0.044, size = 142, normalized size = 4.2 \[{\frac{{f}^{a}{x}^{9}{f}^{b{x}^{2}}}{2\,b\ln \left ( f \right ) }}-{\frac{9\,{f}^{a}{x}^{7}{f}^{b{x}^{2}}}{4\, \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}}+{\frac{63\,{f}^{a}{x}^{5}{f}^{b{x}^{2}}}{8\, \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}}-{\frac{315\,{f}^{a}{x}^{3}{f}^{b{x}^{2}}}{16\, \left ( \ln \left ( f \right ) \right ) ^{4}{b}^{4}}}+{\frac{945\,{f}^{a}x{f}^{b{x}^{2}}}{32\, \left ( \ln \left ( f \right ) \right ) ^{5}{b}^{5}}}-{\frac{945\,{f}^{a}\sqrt{\pi }}{64\, \left ( \ln \left ( f \right ) \right ) ^{5}{b}^{5}}{\it Erf} \left ( \sqrt{-b\ln \left ( f \right ) }x \right ){\frac{1}{\sqrt{-b\ln \left ( f \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(b*x^2+a)*x^10,x)

[Out]

1/2*f^a/ln(f)/b*x^9*f^(b*x^2)-9/4*f^a/ln(f)^2/b^2*x^7*f^(b*x^2)+63/8*f^a/ln(f)^3
/b^3*x^5*f^(b*x^2)-315/16*f^a/ln(f)^4/b^4*x^3*f^(b*x^2)+945/32*f^a/ln(f)^5/b^5*x
*f^(b*x^2)-945/64*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 = 0.780448, size = 151, normalized size = 4.44 \[ \frac{{\left (16 \, b^{4} f^{a} x^{9} \log \left (f\right )^{4} - 72 \, b^{3} f^{a} x^{7} \log \left (f\right )^{3} + 252 \, b^{2} f^{a} x^{5} \log \left (f\right )^{2} - 630 \, b f^{a} x^{3} \log \left (f\right ) + 945 \, f^{a} x\right )} f^{b x^{2}}}{32 \, b^{5} \log \left (f\right )^{5}} - \frac{945 \, \sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-b \log \left (f\right )} x\right )}{64 \, \sqrt{-b \log \left (f\right )} b^{5} \log \left (f\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/32*(16*b^4*f^a*x^9*log(f)^4 - 72*b^3*f^a*x^7*log(f)^3 + 252*b^2*f^a*x^5*log(f)
^2 - 630*b*f^a*x^3*log(f) + 945*f^a*x)*f^(b*x^2)/(b^5*log(f)^5) - 945/64*sqrt(pi
)*f^a*erf(sqrt(-b*log(f))*x)/(sqrt(-b*log(f))*b^5*log(f)^5)

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Fricas [A]  time = 0.249502, size = 136, normalized size = 4. \[ \frac{2 \,{\left (16 \, b^{4} x^{9} \log \left (f\right )^{4} - 72 \, b^{3} x^{7} \log \left (f\right )^{3} + 252 \, b^{2} x^{5} \log \left (f\right )^{2} - 630 \, b x^{3} \log \left (f\right ) + 945 \, x\right )} \sqrt{-b \log \left (f\right )} f^{b x^{2} + a} - 945 \, \sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-b \log \left (f\right )} x\right )}{64 \, \sqrt{-b \log \left (f\right )} b^{5} \log \left (f\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/64*(2*(16*b^4*x^9*log(f)^4 - 72*b^3*x^7*log(f)^3 + 252*b^2*x^5*log(f)^2 - 630*
b*x^3*log(f) + 945*x)*sqrt(-b*log(f))*f^(b*x^2 + a) - 945*sqrt(pi)*f^a*erf(sqrt(
-b*log(f))*x))/(sqrt(-b*log(f))*b^5*log(f)^5)

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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 [A]  time = 0.246913, size = 143, normalized size = 4.21 \[ \frac{{\left (16 \, b^{4} x^{9}{\rm ln}\left (f\right )^{4} - 72 \, b^{3} x^{7}{\rm ln}\left (f\right )^{3} + 252 \, b^{2} x^{5}{\rm ln}\left (f\right )^{2} - 630 \, b x^{3}{\rm ln}\left (f\right ) + 945 \, x\right )} e^{\left (b x^{2}{\rm ln}\left (f\right ) + a{\rm ln}\left (f\right )\right )}}{32 \, b^{5}{\rm ln}\left (f\right )^{5}} + \frac{945 \, \sqrt{\pi } \operatorname{erf}\left (-\sqrt{-b{\rm ln}\left (f\right )} x\right ) e^{\left (a{\rm ln}\left (f\right )\right )}}{64 \, \sqrt{-b{\rm ln}\left (f\right )} b^{5}{\rm ln}\left (f\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/32*(16*b^4*x^9*ln(f)^4 - 72*b^3*x^7*ln(f)^3 + 252*b^2*x^5*ln(f)^2 - 630*b*x^3*
ln(f) + 945*x)*e^(b*x^2*ln(f) + a*ln(f))/(b^5*ln(f)^5) + 945/64*sqrt(pi)*erf(-sq
rt(-b*ln(f))*x)*e^(a*ln(f))/(sqrt(-b*ln(f))*b^5*ln(f)^5)