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

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

[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 = 0.0875476, 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, Rules used = {2212, 2204} \[ -\frac{15 \sqrt{\pi } f^a \text{Erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )}{16 b^{7/2} \log ^{\frac{7}{2}}(f)}-\frac{5 x^3 f^{a+b x^2}}{4 b^2 \log ^2(f)}+\frac{15 x f^{a+b x^2}}{8 b^3 \log ^3(f)}+\frac{x^5 f^{a+b x^2}}{2 b \log (f)} \]

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])

Rule 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rubi steps

\begin{align*} \int f^{a+b x^2} x^6 \, dx &=\frac{f^{a+b x^2} x^5}{2 b \log (f)}-\frac{5 \int f^{a+b x^2} x^4 \, dx}{2 b \log (f)}\\ &=-\frac{5 f^{a+b x^2} x^3}{4 b^2 \log ^2(f)}+\frac{f^{a+b x^2} x^5}{2 b \log (f)}+\frac{15 \int f^{a+b x^2} x^2 \, dx}{4 b^2 \log ^2(f)}\\ &=\frac{15 f^{a+b x^2} x}{8 b^3 \log ^3(f)}-\frac{5 f^{a+b x^2} x^3}{4 b^2 \log ^2(f)}+\frac{f^{a+b x^2} x^5}{2 b \log (f)}-\frac{15 \int f^{a+b x^2} \, dx}{8 b^3 \log ^3(f)}\\ &=-\frac{15 f^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )}{16 b^{7/2} \log ^{\frac{7}{2}}(f)}+\frac{15 f^{a+b x^2} x}{8 b^3 \log ^3(f)}-\frac{5 f^{a+b x^2} x^3}{4 b^2 \log ^2(f)}+\frac{f^{a+b x^2} x^5}{2 b \log (f)}\\ \end{align*}

Mathematica [A]  time = 0.0347835, 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 = 0.029, size = 98, normalized size = 0.9 \begin{align*}{\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 ) }}}} \end{align*}

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 = 1.03444, size = 111, normalized size = 1.06 \begin{align*} \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}} \end{align*}

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*er
f(sqrt(-b*log(f))*x)/(sqrt(-b*log(f))*b^3*log(f)^3)

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Fricas [A]  time = 1.75294, size = 209, normalized size = 1.99 \begin{align*} \frac{15 \, \sqrt{\pi } \sqrt{-b \log \left (f\right )} f^{a} \operatorname{erf}\left (\sqrt{-b \log \left (f\right )} x\right ) + 2 \,{\left (4 \, b^{3} x^{5} \log \left (f\right )^{3} - 10 \, b^{2} x^{3} \log \left (f\right )^{2} + 15 \, b x \log \left (f\right )\right )} f^{b x^{2} + a}}{16 \, b^{4} \log \left (f\right )^{4}} \end{align*}

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

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

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 [A]  time = 1.24671, size = 108, normalized size = 1.03 \begin{align*} \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}} + \frac{{\left (4 \, b^{2} x^{5} \log \left (f\right )^{2} - 10 \, b x^{3} \log \left (f\right ) + 15 \, x\right )} e^{\left (b x^{2} \log \left (f\right ) + a \log \left (f\right )\right )}}{8 \, b^{3} \log \left (f\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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