3.72 \(\int f^{a+b x^2} x^7 \, dx\)

Optimal. Leaf size=86 \[ -\frac{3 x^4 f^{a+b x^2}}{2 b^2 \log ^2(f)}+\frac{3 x^2 f^{a+b x^2}}{b^3 \log ^3(f)}-\frac{3 f^{a+b x^2}}{b^4 \log ^4(f)}+\frac{x^6 f^{a+b x^2}}{2 b \log (f)} \]

[Out]

(-3*f^(a + b*x^2))/(b^4*Log[f]^4) + (3*f^(a + b*x^2)*x^2)/(b^3*Log[f]^3) - (3*f^(a + b*x^2)*x^4)/(2*b^2*Log[f]
^2) + (f^(a + b*x^2)*x^6)/(2*b*Log[f])

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Rubi [A]  time = 0.0929452, antiderivative size = 86, 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, 2209} \[ -\frac{3 x^4 f^{a+b x^2}}{2 b^2 \log ^2(f)}+\frac{3 x^2 f^{a+b x^2}}{b^3 \log ^3(f)}-\frac{3 f^{a+b x^2}}{b^4 \log ^4(f)}+\frac{x^6 f^{a+b x^2}}{2 b \log (f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^2)*x^7,x]

[Out]

(-3*f^(a + b*x^2))/(b^4*Log[f]^4) + (3*f^(a + b*x^2)*x^2)/(b^3*Log[f]^3) - (3*f^(a + b*x^2)*x^4)/(2*b^2*Log[f]
^2) + (f^(a + b*x^2)*x^6)/(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 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int f^{a+b x^2} x^7 \, dx &=\frac{f^{a+b x^2} x^6}{2 b \log (f)}-\frac{3 \int f^{a+b x^2} x^5 \, dx}{b \log (f)}\\ &=-\frac{3 f^{a+b x^2} x^4}{2 b^2 \log ^2(f)}+\frac{f^{a+b x^2} x^6}{2 b \log (f)}+\frac{6 \int f^{a+b x^2} x^3 \, dx}{b^2 \log ^2(f)}\\ &=\frac{3 f^{a+b x^2} x^2}{b^3 \log ^3(f)}-\frac{3 f^{a+b x^2} x^4}{2 b^2 \log ^2(f)}+\frac{f^{a+b x^2} x^6}{2 b \log (f)}-\frac{6 \int f^{a+b x^2} x \, dx}{b^3 \log ^3(f)}\\ &=-\frac{3 f^{a+b x^2}}{b^4 \log ^4(f)}+\frac{3 f^{a+b x^2} x^2}{b^3 \log ^3(f)}-\frac{3 f^{a+b x^2} x^4}{2 b^2 \log ^2(f)}+\frac{f^{a+b x^2} x^6}{2 b \log (f)}\\ \end{align*}

Mathematica [A]  time = 0.0105276, size = 53, normalized size = 0.62 \[ \frac{f^{a+b x^2} \left (b^3 x^6 \log ^3(f)-3 b^2 x^4 \log ^2(f)+6 b x^2 \log (f)-6\right )}{2 b^4 \log ^4(f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^2)*x^7,x]

[Out]

(f^(a + b*x^2)*(-6 + 6*b*x^2*Log[f] - 3*b^2*x^4*Log[f]^2 + b^3*x^6*Log[f]^3))/(2*b^4*Log[f]^4)

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Maple [A]  time = 0.006, size = 52, normalized size = 0.6 \begin{align*}{\frac{ \left ({b}^{3}{x}^{6} \left ( \ln \left ( f \right ) \right ) ^{3}-3\,{b}^{2}{x}^{4} \left ( \ln \left ( f \right ) \right ) ^{2}+6\,b{x}^{2}\ln \left ( f \right ) -6 \right ){f}^{b{x}^{2}+a}}{2\,{b}^{4} \left ( \ln \left ( f \right ) \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(b*x^2+a)*x^7,x)

[Out]

1/2*(b^3*x^6*ln(f)^3-3*b^2*x^4*ln(f)^2+6*b*x^2*ln(f)-6)*f^(b*x^2+a)/ln(f)^4/b^4

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Maxima [A]  time = 1.13667, size = 84, normalized size = 0.98 \begin{align*} \frac{{\left (b^{3} f^{a} x^{6} \log \left (f\right )^{3} - 3 \, b^{2} f^{a} x^{4} \log \left (f\right )^{2} + 6 \, b f^{a} x^{2} \log \left (f\right ) - 6 \, f^{a}\right )} f^{b x^{2}}}{2 \, 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^7,x, algorithm="maxima")

[Out]

1/2*(b^3*f^a*x^6*log(f)^3 - 3*b^2*f^a*x^4*log(f)^2 + 6*b*f^a*x^2*log(f) - 6*f^a)*f^(b*x^2)/(b^4*log(f)^4)

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Fricas [A]  time = 1.50238, size = 128, normalized size = 1.49 \begin{align*} \frac{{\left (b^{3} x^{6} \log \left (f\right )^{3} - 3 \, b^{2} x^{4} \log \left (f\right )^{2} + 6 \, b x^{2} \log \left (f\right ) - 6\right )} f^{b x^{2} + a}}{2 \, 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^7,x, algorithm="fricas")

[Out]

1/2*(b^3*x^6*log(f)^3 - 3*b^2*x^4*log(f)^2 + 6*b*x^2*log(f) - 6)*f^(b*x^2 + a)/(b^4*log(f)^4)

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Sympy [A]  time = 0.141825, size = 68, normalized size = 0.79 \begin{align*} \begin{cases} \frac{f^{a + b x^{2}} \left (b^{3} x^{6} \log{\left (f \right )}^{3} - 3 b^{2} x^{4} \log{\left (f \right )}^{2} + 6 b x^{2} \log{\left (f \right )} - 6\right )}{2 b^{4} \log{\left (f \right )}^{4}} & \text{for}\: 2 b^{4} \log{\left (f \right )}^{4} \neq 0 \\\frac{x^{8}}{8} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(b*x**2+a)*x**7,x)

[Out]

Piecewise((f**(a + b*x**2)*(b**3*x**6*log(f)**3 - 3*b**2*x**4*log(f)**2 + 6*b*x**2*log(f) - 6)/(2*b**4*log(f)*
*4), Ne(2*b**4*log(f)**4, 0)), (x**8/8, True))

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Giac [A]  time = 1.25016, size = 74, normalized size = 0.86 \begin{align*} \frac{{\left (b^{3} x^{6} \log \left (f\right )^{3} - 3 \, b^{2} x^{4} \log \left (f\right )^{2} + 6 \, b x^{2} \log \left (f\right ) - 6\right )} e^{\left (b x^{2} \log \left (f\right ) + a \log \left (f\right )\right )}}{2 \, 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^7,x, algorithm="giac")

[Out]

1/2*(b^3*x^6*log(f)^3 - 3*b^2*x^4*log(f)^2 + 6*b*x^2*log(f) - 6)*e^(b*x^2*log(f) + a*log(f))/(b^4*log(f)^4)