3.86 \(\int f^{a+b x^2} x^4 \, dx\)

Optimal. Leaf size=82 \[ \frac{3 \sqrt{\pi } f^a \text{Erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )}{8 b^{5/2} \log ^{\frac{5}{2}}(f)}-\frac{3 x f^{a+b x^2}}{4 b^2 \log ^2(f)}+\frac{x^3 f^{a+b x^2}}{2 b \log (f)} \]

[Out]

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

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Rubi [A]  time = 0.0593823, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2212, 2204} \[ \frac{3 \sqrt{\pi } f^a \text{Erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )}{8 b^{5/2} \log ^{\frac{5}{2}}(f)}-\frac{3 x f^{a+b x^2}}{4 b^2 \log ^2(f)}+\frac{x^3 f^{a+b x^2}}{2 b \log (f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^2)*x^4,x]

[Out]

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

Mathematica [A]  time = 0.0269446, size = 71, normalized size = 0.87 \[ \frac{f^a \left (3 \sqrt{\pi } \text{Erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )+2 \sqrt{b} x \sqrt{\log (f)} f^{b x^2} \left (2 b x^2 \log (f)-3\right )\right )}{8 b^{5/2} \log ^{\frac{5}{2}}(f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^2)*x^4,x]

[Out]

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

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Maple [A]  time = 0.023, size = 76, normalized size = 0.9 \begin{align*}{\frac{{f}^{a}{x}^{3}{f}^{b{x}^{2}}}{2\,b\ln \left ( f \right ) }}-{\frac{3\,{f}^{a}x{f}^{b{x}^{2}}}{4\, \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}}+{\frac{3\,{f}^{a}\sqrt{\pi }}{8\, \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}{\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^4,x)

[Out]

1/2*f^a/ln(f)/b*x^3*f^(b*x^2)-3/4*f^a/ln(f)^2/b^2*x*f^(b*x^2)+3/8*f^a/ln(f)^2/b^2*Pi^(1/2)/(-b*ln(f))^(1/2)*er
f((-b*ln(f))^(1/2)*x)

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Maxima [A]  time = 1.14074, size = 90, normalized size = 1.1 \begin{align*} \frac{{\left (2 \, b f^{a} x^{3} \log \left (f\right ) - 3 \, f^{a} x\right )} f^{b x^{2}}}{4 \, b^{2} \log \left (f\right )^{2}} + \frac{3 \, \sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-b \log \left (f\right )} x\right )}{8 \, \sqrt{-b \log \left (f\right )} b^{2} \log \left (f\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^2+a)*x^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.71272, size = 177, normalized size = 2.16 \begin{align*} -\frac{3 \, \sqrt{\pi } \sqrt{-b \log \left (f\right )} f^{a} \operatorname{erf}\left (\sqrt{-b \log \left (f\right )} x\right ) - 2 \,{\left (2 \, b^{2} x^{3} \log \left (f\right )^{2} - 3 \, b x \log \left (f\right )\right )} f^{b x^{2} + a}}{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^4,x, algorithm="fricas")

[Out]

-1/8*(3*sqrt(pi)*sqrt(-b*log(f))*f^a*erf(sqrt(-b*log(f))*x) - 2*(2*b^2*x^3*log(f)^2 - 3*b*x*log(f))*f^(b*x^2 +
a))/(b^3*log(f)^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + b x^{2}} x^{4}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(b*x**2+a)*x**4,x)

[Out]

Integral(f**(a + b*x**2)*x**4, x)

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Giac [A]  time = 1.33054, size = 92, normalized size = 1.12 \begin{align*} -\frac{3 \, \sqrt{\pi } f^{a} \operatorname{erf}\left (-\sqrt{-b \log \left (f\right )} x\right )}{8 \, \sqrt{-b \log \left (f\right )} b^{2} \log \left (f\right )^{2}} + \frac{{\left (2 \, b x^{3} \log \left (f\right ) - 3 \, x\right )} e^{\left (b x^{2} \log \left (f\right ) + a \log \left (f\right )\right )}}{4 \, b^{2} \log \left (f\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^2+a)*x^4,x, algorithm="giac")

[Out]

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