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

Optimal. Leaf size=105 \[ -\frac{15 \sqrt{\pi } f^a \text{Erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )}{16 b^{7/2} \log ^{\frac{7}{2}}(f)}+\frac{15 x f^{a+b x^2}}{8 b^3 \log ^3(f)}-\frac{5 x^3 f^{a+b x^2}}{4 b^2 \log ^2(f)}+\frac{x^5 f^{a+b x^2}}{2 b \log (f)} \]

[Out]

(-15*f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x*Sqrt[Log[f]]])/(16*b^(7/2)*Log[f]^(7/2)) + (15*
f^(a + b*x^2)*x)/(8*b^3*Log[f]^3) - (5*f^(a + b*x^2)*x^3)/(4*b^2*Log[f]^2) + (f^
(a + b*x^2)*x^5)/(2*b*Log[f])

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Rubi [A]  time = 0.1527, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{15 \sqrt{\pi } f^a \text{Erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )}{16 b^{7/2} \log ^{\frac{7}{2}}(f)}+\frac{15 x f^{a+b x^2}}{8 b^3 \log ^3(f)}-\frac{5 x^3 f^{a+b x^2}}{4 b^2 \log ^2(f)}+\frac{x^5 f^{a+b x^2}}{2 b \log (f)} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b*x^2)*x^6,x]

[Out]

(-15*f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x*Sqrt[Log[f]]])/(16*b^(7/2)*Log[f]^(7/2)) + (15*
f^(a + b*x^2)*x)/(8*b^3*Log[f]^3) - (5*f^(a + b*x^2)*x^3)/(4*b^2*Log[f]^2) + (f^
(a + b*x^2)*x^5)/(2*b*Log[f])

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Rubi in Sympy [A]  time = 16.3757, size = 102, normalized size = 0.97 \[ \frac{f^{a + b x^{2}} x^{5}}{2 b \log{\left (f \right )}} - \frac{5 f^{a + b x^{2}} x^{3}}{4 b^{2} \log{\left (f \right )}^{2}} + \frac{15 f^{a + b x^{2}} x}{8 b^{3} \log{\left (f \right )}^{3}} - \frac{15 \sqrt{\pi } f^{a} \operatorname{erfi}{\left (\sqrt{b} x \sqrt{\log{\left (f \right )}} \right )}}{16 b^{\frac{7}{2}} \log{\left (f \right )}^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

f**(a + b*x**2)*x**5/(2*b*log(f)) - 5*f**(a + b*x**2)*x**3/(4*b**2*log(f)**2) +
15*f**(a + b*x**2)*x/(8*b**3*log(f)**3) - 15*sqrt(pi)*f**a*erfi(sqrt(b)*x*sqrt(l
og(f)))/(16*b**(7/2)*log(f)**(7/2))

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Mathematica [A]  time = 0.0503708, size = 83, normalized size = 0.79 \[ \frac{f^a \left (2 \sqrt{b} x \sqrt{\log (f)} f^{b x^2} \left (4 b^2 x^4 \log ^2(f)-10 b x^2 \log (f)+15\right )-15 \sqrt{\pi } \text{Erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )\right )}{16 b^{7/2} \log ^{\frac{7}{2}}(f)} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b*x^2)*x^6,x]

[Out]

(f^a*(-15*Sqrt[Pi]*Erfi[Sqrt[b]*x*Sqrt[Log[f]]] + 2*Sqrt[b]*f^(b*x^2)*x*Sqrt[Log
[f]]*(15 - 10*b*x^2*Log[f] + 4*b^2*x^4*Log[f]^2)))/(16*b^(7/2)*Log[f]^(7/2))

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Maple [A]  time = 0.033, size = 98, normalized size = 0.9 \[{\frac{{f}^{a}{x}^{5}{f}^{b{x}^{2}}}{2\,b\ln \left ( f \right ) }}-{\frac{5\,{f}^{a}{x}^{3}{f}^{b{x}^{2}}}{4\, \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}}+{\frac{15\,{f}^{a}x{f}^{b{x}^{2}}}{8\, \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}}-{\frac{15\,{f}^{a}\sqrt{\pi }}{16\, \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}{\it Erf} \left ( \sqrt{-b\ln \left ( f \right ) }x \right ){\frac{1}{\sqrt{-b\ln \left ( f \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(b*x^2+a)*x^6,x)

[Out]

1/2*f^a/ln(f)/b*x^5*f^(b*x^2)-5/4*f^a/ln(f)^2/b^2*x^3*f^(b*x^2)+15/8*f^a/ln(f)^3
/b^3*x*f^(b*x^2)-15/16*f^a/ln(f)^3/b^3*Pi^(1/2)/(-b*ln(f))^(1/2)*erf((-b*ln(f))^
(1/2)*x)

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Maxima [A]  time = 0.814119, size = 111, normalized size = 1.06 \[ \frac{{\left (4 \, b^{2} f^{a} x^{5} \log \left (f\right )^{2} - 10 \, b f^{a} x^{3} \log \left (f\right ) + 15 \, f^{a} x\right )} f^{b x^{2}}}{8 \, b^{3} \log \left (f\right )^{3}} - \frac{15 \, \sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-b \log \left (f\right )} x\right )}{16 \, \sqrt{-b \log \left (f\right )} b^{3} \log \left (f\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(4*b^2*f^a*x^5*log(f)^2 - 10*b*f^a*x^3*log(f) + 15*f^a*x)*f^(b*x^2)/(b^3*log
(f)^3) - 15/16*sqrt(pi)*f^a*erf(sqrt(-b*log(f))*x)/(sqrt(-b*log(f))*b^3*log(f)^3
)

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Fricas [A]  time = 0.251412, size = 104, normalized size = 0.99 \[ \frac{2 \,{\left (4 \, b^{2} x^{5} \log \left (f\right )^{2} - 10 \, b x^{3} \log \left (f\right ) + 15 \, x\right )} \sqrt{-b \log \left (f\right )} f^{b x^{2} + a} - 15 \, \sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-b \log \left (f\right )} x\right )}{16 \, \sqrt{-b \log \left (f\right )} b^{3} \log \left (f\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*(2*(4*b^2*x^5*log(f)^2 - 10*b*x^3*log(f) + 15*x)*sqrt(-b*log(f))*f^(b*x^2 +
a) - 15*sqrt(pi)*f^a*erf(sqrt(-b*log(f))*x))/(sqrt(-b*log(f))*b^3*log(f)^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int f^{a + b x^{2}} x^{6}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(b*x**2+a)*x**6,x)

[Out]

Integral(f**(a + b*x**2)*x**6, x)

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GIAC/XCAS [A]  time = 0.232396, size = 111, normalized size = 1.06 \[ \frac{{\left (4 \, b^{2} x^{5}{\rm ln}\left (f\right )^{2} - 10 \, b x^{3}{\rm ln}\left (f\right ) + 15 \, x\right )} e^{\left (b x^{2}{\rm ln}\left (f\right ) + a{\rm ln}\left (f\right )\right )}}{8 \, b^{3}{\rm ln}\left (f\right )^{3}} + \frac{15 \, \sqrt{\pi } \operatorname{erf}\left (-\sqrt{-b{\rm ln}\left (f\right )} x\right ) e^{\left (a{\rm ln}\left (f\right )\right )}}{16 \, \sqrt{-b{\rm ln}\left (f\right )} b^{3}{\rm ln}\left (f\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(4*b^2*x^5*ln(f)^2 - 10*b*x^3*ln(f) + 15*x)*e^(b*x^2*ln(f) + a*ln(f))/(b^3*l
n(f)^3) + 15/16*sqrt(pi)*erf(-sqrt(-b*ln(f))*x)*e^(a*ln(f))/(sqrt(-b*ln(f))*b^3*
ln(f)^3)