3.772 \(\int f^{(a+b x)^n} (a+b x)^m \, dx\)

Optimal. Leaf size=56 \[ -\frac{(a+b x)^{m+1} \left (\log (f) \left (-(a+b x)^n\right )\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},\log (f) \left (-(a+b x)^n\right )\right )}{b n} \]

[Out]

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

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Rubi [A]  time = 0.0253398, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2218} \[ -\frac{(a+b x)^{m+1} \left (\log (f) \left (-(a+b x)^n\right )\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},\log (f) \left (-(a+b x)^n\right )\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x)^n*(a + b*x)^m,x]

[Out]

-(((a + b*x)^(1 + m)*Gamma[(1 + m)/n, -((a + b*x)^n*Log[f])])/(b*n*(-((a + b*x)^n*Log[f]))^((1 + m)/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^{(a+b x)^n} (a+b x)^m \, dx &=-\frac{(a+b x)^{1+m} \Gamma \left (\frac{1+m}{n},-(a+b x)^n \log (f)\right ) \left (-(a+b x)^n \log (f)\right )^{-\frac{1+m}{n}}}{b n}\\ \end{align*}

Mathematica [A]  time = 0.0137881, size = 56, normalized size = 1. \[ -\frac{(a+b x)^{m+1} \left (\log (f) \left (-(a+b x)^n\right )\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},\log (f) \left (-(a+b x)^n\right )\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x)^n*(a + b*x)^m,x]

[Out]

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

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Maple [F]  time = 0.34, size = 0, normalized size = 0. \begin{align*} \int{f}^{ \left ( bx+a \right ) ^{n}} \left ( bx+a \right ) ^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x + a)^m*f^((b*x + a)^n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x + a)^m*f^((b*x + a)^n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**((a + b*x)**n)*(a + b*x)**m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x + a)^m*f^((b*x + a)^n), x)