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

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

[Out]

-(b*x+a)^(1+m)*GAMMA((1+m)/n,-(b*x+a)^n*ln(f))/b/n/((-(b*x+a)^n*ln(f))^((1+m)/n))

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Rubi [A]  time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 56, normalized size = 1.00 \[ -\frac {(a+b x)^{m+1} \left (\log (f) \left (-(a+b x)^n\right )\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-(a+b x)^n \log (f)\right )}{b n} \]

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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fricas [F]  time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x + a\right )}^{m} f^{\left ({\left (b x + a\right )}^{n}\right )}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )}^{m} f^{\left ({\left (b x + a\right )}^{n}\right )}\,{d x} \]

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)

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maple [F]  time = 0.25, size = 0, normalized size = 0.00 \[ \int f^{\left (b x +a \right )^{n}} \left (b x +a \right )^{m}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )}^{m} f^{\left ({\left (b x + a\right )}^{n}\right )}\,{d x} \]

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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mupad [B]  time = 4.01, size = 94, normalized size = 1.68 \[ \frac {f^{{\left (a+b\,x\right )}^n}\,{\mathrm {e}}^{-\frac {\ln \relax (f)\,{\left (a+b\,x\right )}^n}{2}}\,{\left (a+b\,x\right )}^{m+1}\,{\mathrm {M}}_{1-\frac {m+n+1}{2\,n},\frac {m+n+1}{2\,n}-\frac {1}{2}}\left (\ln \relax (f)\,{\left (a+b\,x\right )}^n\right )}{b\,\left (m+1\right )\,{\left (\ln \relax (f)\,{\left (a+b\,x\right )}^n\right )}^{\frac {m+n+1}{2\,n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^((a + b*x)^n)*(a + b*x)^m,x)

[Out]

(f^((a + b*x)^n)*exp(-(log(f)*(a + b*x)^n)/2)*(a + b*x)^(m + 1)*whittakerM(1 - (m + n + 1)/(2*n), (m + n + 1)/
(2*n) - 1/2, log(f)*(a + b*x)^n))/(b*(m + 1)*(log(f)*(a + b*x)^n)^((m + n + 1)/(2*n)))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{\left (a + b x\right )^{n}} \left (a + b x\right )^{m}\, dx \]

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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