3.84 \(\int f^{a+b x^2} x^8 \, dx\)

Optimal. Leaf size=128 \[ \frac{105 \sqrt{\pi } f^a \text{Erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )}{32 b^{9/2} \log ^{\frac{9}{2}}(f)}-\frac{7 x^5 f^{a+b x^2}}{4 b^2 \log ^2(f)}+\frac{35 x^3 f^{a+b x^2}}{8 b^3 \log ^3(f)}-\frac{105 x f^{a+b x^2}}{16 b^4 \log ^4(f)}+\frac{x^7 f^{a+b x^2}}{2 b \log (f)} \]

[Out]

(105*f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x*Sqrt[Log[f]]])/(32*b^(9/2)*Log[f]^(9/2)) - (105*f^(a + b*x^2)*x)/(16*b^4*Log[
f]^4) + (35*f^(a + b*x^2)*x^3)/(8*b^3*Log[f]^3) - (7*f^(a + b*x^2)*x^5)/(4*b^2*Log[f]^2) + (f^(a + b*x^2)*x^7)
/(2*b*Log[f])

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Rubi [A]  time = 0.142399, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2212, 2204} \[ \frac{105 \sqrt{\pi } f^a \text{Erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )}{32 b^{9/2} \log ^{\frac{9}{2}}(f)}-\frac{7 x^5 f^{a+b x^2}}{4 b^2 \log ^2(f)}+\frac{35 x^3 f^{a+b x^2}}{8 b^3 \log ^3(f)}-\frac{105 x f^{a+b x^2}}{16 b^4 \log ^4(f)}+\frac{x^7 f^{a+b x^2}}{2 b \log (f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^2)*x^8,x]

[Out]

(105*f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x*Sqrt[Log[f]]])/(32*b^(9/2)*Log[f]^(9/2)) - (105*f^(a + b*x^2)*x)/(16*b^4*Log[
f]^4) + (35*f^(a + b*x^2)*x^3)/(8*b^3*Log[f]^3) - (7*f^(a + b*x^2)*x^5)/(4*b^2*Log[f]^2) + (f^(a + b*x^2)*x^7)
/(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^8 \, dx &=\frac{f^{a+b x^2} x^7}{2 b \log (f)}-\frac{7 \int f^{a+b x^2} x^6 \, dx}{2 b \log (f)}\\ &=-\frac{7 f^{a+b x^2} x^5}{4 b^2 \log ^2(f)}+\frac{f^{a+b x^2} x^7}{2 b \log (f)}+\frac{35 \int f^{a+b x^2} x^4 \, dx}{4 b^2 \log ^2(f)}\\ &=\frac{35 f^{a+b x^2} x^3}{8 b^3 \log ^3(f)}-\frac{7 f^{a+b x^2} x^5}{4 b^2 \log ^2(f)}+\frac{f^{a+b x^2} x^7}{2 b \log (f)}-\frac{105 \int f^{a+b x^2} x^2 \, dx}{8 b^3 \log ^3(f)}\\ &=-\frac{105 f^{a+b x^2} x}{16 b^4 \log ^4(f)}+\frac{35 f^{a+b x^2} x^3}{8 b^3 \log ^3(f)}-\frac{7 f^{a+b x^2} x^5}{4 b^2 \log ^2(f)}+\frac{f^{a+b x^2} x^7}{2 b \log (f)}+\frac{105 \int f^{a+b x^2} \, dx}{16 b^4 \log ^4(f)}\\ &=\frac{105 f^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )}{32 b^{9/2} \log ^{\frac{9}{2}}(f)}-\frac{105 f^{a+b x^2} x}{16 b^4 \log ^4(f)}+\frac{35 f^{a+b x^2} x^3}{8 b^3 \log ^3(f)}-\frac{7 f^{a+b x^2} x^5}{4 b^2 \log ^2(f)}+\frac{f^{a+b x^2} x^7}{2 b \log (f)}\\ \end{align*}

Mathematica [A]  time = 0.0421417, size = 95, normalized size = 0.74 \[ \frac{f^a \left (2 \sqrt{b} x \sqrt{\log (f)} f^{b x^2} \left (8 b^3 x^6 \log ^3(f)-28 b^2 x^4 \log ^2(f)+70 b x^2 \log (f)-105\right )+105 \sqrt{\pi } \text{Erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )\right )}{32 b^{9/2} \log ^{\frac{9}{2}}(f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^2)*x^8,x]

[Out]

(f^a*(105*Sqrt[Pi]*Erfi[Sqrt[b]*x*Sqrt[Log[f]]] + 2*Sqrt[b]*f^(b*x^2)*x*Sqrt[Log[f]]*(-105 + 70*b*x^2*Log[f] -
28*b^2*x^4*Log[f]^2 + 8*b^3*x^6*Log[f]^3)))/(32*b^(9/2)*Log[f]^(9/2))

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Maple [A]  time = 0.039, size = 120, normalized size = 0.9 \begin{align*}{\frac{{f}^{a}{x}^{7}{f}^{b{x}^{2}}}{2\,b\ln \left ( f \right ) }}-{\frac{7\,{f}^{a}{x}^{5}{f}^{b{x}^{2}}}{4\, \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}}+{\frac{35\,{f}^{a}{x}^{3}{f}^{b{x}^{2}}}{8\, \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}}-{\frac{105\,{f}^{a}x{f}^{b{x}^{2}}}{16\,{b}^{4} \left ( \ln \left ( f \right ) \right ) ^{4}}}+{\frac{105\,{f}^{a}\sqrt{\pi }}{32\,{b}^{4} \left ( \ln \left ( f \right ) \right ) ^{4}}{\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^8,x)

[Out]

1/2*f^a/ln(f)/b*x^7*f^(b*x^2)-7/4*f^a/ln(f)^2/b^2*x^5*f^(b*x^2)+35/8*f^a/ln(f)^3/b^3*x^3*f^(b*x^2)-105/16*f^a/
ln(f)^4/b^4*x*f^(b*x^2)+105/32*f^a/ln(f)^4/b^4*Pi^(1/2)/(-b*ln(f))^(1/2)*erf((-b*ln(f))^(1/2)*x)

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Maxima [A]  time = 1.07931, size = 131, normalized size = 1.02 \begin{align*} \frac{{\left (8 \, b^{3} f^{a} x^{7} \log \left (f\right )^{3} - 28 \, b^{2} f^{a} x^{5} \log \left (f\right )^{2} + 70 \, b f^{a} x^{3} \log \left (f\right ) - 105 \, f^{a} x\right )} f^{b x^{2}}}{16 \, b^{4} \log \left (f\right )^{4}} + \frac{105 \, \sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-b \log \left (f\right )} x\right )}{32 \, \sqrt{-b \log \left (f\right )} 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^8,x, algorithm="maxima")

[Out]

1/16*(8*b^3*f^a*x^7*log(f)^3 - 28*b^2*f^a*x^5*log(f)^2 + 70*b*f^a*x^3*log(f) - 105*f^a*x)*f^(b*x^2)/(b^4*log(f
)^4) + 105/32*sqrt(pi)*f^a*erf(sqrt(-b*log(f))*x)/(sqrt(-b*log(f))*b^4*log(f)^4)

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Fricas [A]  time = 1.75507, size = 243, normalized size = 1.9 \begin{align*} -\frac{105 \, \sqrt{\pi } \sqrt{-b \log \left (f\right )} f^{a} \operatorname{erf}\left (\sqrt{-b \log \left (f\right )} x\right ) - 2 \,{\left (8 \, b^{4} x^{7} \log \left (f\right )^{4} - 28 \, b^{3} x^{5} \log \left (f\right )^{3} + 70 \, b^{2} x^{3} \log \left (f\right )^{2} - 105 \, b x \log \left (f\right )\right )} f^{b x^{2} + a}}{32 \, b^{5} \log \left (f\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^2+a)*x^8,x, algorithm="fricas")

[Out]

-1/32*(105*sqrt(pi)*sqrt(-b*log(f))*f^a*erf(sqrt(-b*log(f))*x) - 2*(8*b^4*x^7*log(f)^4 - 28*b^3*x^5*log(f)^3 +
70*b^2*x^3*log(f)^2 - 105*b*x*log(f))*f^(b*x^2 + a))/(b^5*log(f)^5)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + b x^{2}} x^{8}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(b*x**2+a)*x**8,x)

[Out]

Integral(f**(a + b*x**2)*x**8, x)

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Giac [A]  time = 1.2209, size = 124, normalized size = 0.97 \begin{align*} -\frac{105 \, \sqrt{\pi } f^{a} \operatorname{erf}\left (-\sqrt{-b \log \left (f\right )} x\right )}{32 \, \sqrt{-b \log \left (f\right )} b^{4} \log \left (f\right )^{4}} + \frac{{\left (8 \, b^{3} x^{7} \log \left (f\right )^{3} - 28 \, b^{2} x^{5} \log \left (f\right )^{2} + 70 \, b x^{3} \log \left (f\right ) - 105 \, x\right )} e^{\left (b x^{2} \log \left (f\right ) + a \log \left (f\right )\right )}}{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^8,x, algorithm="giac")

[Out]

-105/32*sqrt(pi)*f^a*erf(-sqrt(-b*log(f))*x)/(sqrt(-b*log(f))*b^4*log(f)^4) + 1/16*(8*b^3*x^7*log(f)^3 - 28*b^
2*x^5*log(f)^2 + 70*b*x^3*log(f) - 105*x)*e^(b*x^2*log(f) + a*log(f))/(b^4*log(f)^4)