3.249 \(\int f^{c (a+b x)^n} x \, dx\)

Optimal. Leaf size=99 \[ \frac{a (a+b x) \left (-c \log (f) (a+b x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-c \log (f) (a+b x)^n\right )}{b^2 n}-\frac{(a+b x)^2 \left (-c \log (f) (a+b x)^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},-c \log (f) (a+b x)^n\right )}{b^2 n} \]

[Out]

-(((a + b*x)^2*Gamma[2/n, -(c*(a + b*x)^n*Log[f])])/(b^2*n*(-(c*(a + b*x)^n*Log[f]))^(2/n))) + (a*(a + b*x)*Ga
mma[n^(-1), -(c*(a + b*x)^n*Log[f])])/(b^2*n*(-(c*(a + b*x)^n*Log[f]))^n^(-1))

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Rubi [A]  time = 0.0584529, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2226, 2208, 2218} \[ \frac{a (a+b x) \left (-c \log (f) (a+b x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-c \log (f) (a+b x)^n\right )}{b^2 n}-\frac{(a+b x)^2 \left (-c \log (f) (a+b x)^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},-c \log (f) (a+b x)^n\right )}{b^2 n} \]

Antiderivative was successfully verified.

[In]

Int[f^(c*(a + b*x)^n)*x,x]

[Out]

-(((a + b*x)^2*Gamma[2/n, -(c*(a + b*x)^n*Log[f])])/(b^2*n*(-(c*(a + b*x)^n*Log[f]))^(2/n))) + (a*(a + b*x)*Ga
mma[n^(-1), -(c*(a + b*x)^n*Log[f])])/(b^2*n*(-(c*(a + b*x)^n*Log[f]))^n^(-1))

Rule 2226

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

Rule 2208

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

Rule 2218

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

Rubi steps

\begin{align*} \int f^{c (a+b x)^n} x \, dx &=\int \left (-\frac{a f^{c (a+b x)^n}}{b}+\frac{f^{c (a+b x)^n} (a+b x)}{b}\right ) \, dx\\ &=\frac{\int f^{c (a+b x)^n} (a+b x) \, dx}{b}-\frac{a \int f^{c (a+b x)^n} \, dx}{b}\\ &=-\frac{(a+b x)^2 \Gamma \left (\frac{2}{n},-c (a+b x)^n \log (f)\right ) \left (-c (a+b x)^n \log (f)\right )^{-2/n}}{b^2 n}+\frac{a (a+b x) \Gamma \left (\frac{1}{n},-c (a+b x)^n \log (f)\right ) \left (-c (a+b x)^n \log (f)\right )^{-1/n}}{b^2 n}\\ \end{align*}

Mathematica [A]  time = 0.0330934, size = 91, normalized size = 0.92 \[ -\frac{(a+b x) \left (-c \log (f) (a+b x)^n\right )^{-2/n} \left ((a+b x) \text{Gamma}\left (\frac{2}{n},-c \log (f) (a+b x)^n\right )-a \left (-c \log (f) (a+b x)^n\right )^{\frac{1}{n}} \text{Gamma}\left (\frac{1}{n},-c \log (f) (a+b x)^n\right )\right )}{b^2 n} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(c*(a + b*x)^n)*x,x]

[Out]

-(((a + b*x)*((a + b*x)*Gamma[2/n, -(c*(a + b*x)^n*Log[f])] - a*Gamma[n^(-1), -(c*(a + b*x)^n*Log[f])]*(-(c*(a
+ b*x)^n*Log[f]))^n^(-1)))/(b^2*n*(-(c*(a + b*x)^n*Log[f]))^(2/n)))

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Maple [F]  time = 0.031, size = 0, normalized size = 0. \begin{align*} \int{f}^{c \left ( bx+a \right ) ^{n}}x\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(c*(b*x+a)^n)*x,x)

[Out]

int(f^(c*(b*x+a)^n)*x,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{{\left (b x + a\right )}^{n} c} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*(b*x+a)^n)*x,x, algorithm="maxima")

[Out]

integrate(f^((b*x + a)^n*c)*x, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (f^{{\left (b x + a\right )}^{n} c} x, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*(b*x+a)^n)*x,x, algorithm="fricas")

[Out]

integral(f^((b*x + a)^n*c)*x, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{c \left (a + b x\right )^{n}} x\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(c*(b*x+a)**n)*x,x)

[Out]

Integral(f**(c*(a + b*x)**n)*x, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{{\left (b x + a\right )}^{n} c} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*(b*x+a)^n)*x,x, algorithm="giac")

[Out]

integrate(f^((b*x + a)^n*c)*x, x)