3.3.49 \(\int f^{c (a+b x)^n} x \, dx\) [249]

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

[Out]

-(b*x+a)^2*GAMMA(2/n,-c*(b*x+a)^n*ln(f))/b^2/n/((-c*(b*x+a)^n*ln(f))^(2/n))+a*(b*x+a)*GAMMA(1/n,-c*(b*x+a)^n*l
n(f))/b^2/n/((-c*(b*x+a)^n*ln(f))^(1/n))

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Rubi [A]
time = 0.04, antiderivative size = 99, normalized size of antiderivative = 1.00, 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 = {2258, 2239, 2250} \begin {gather*} \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} \end {gather*}

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 2239

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 2250

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

Rule 2258

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]

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*}

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Mathematica [A]
time = 0.02, size = 91, normalized size = 0.92 \begin {gather*} -\frac {(a+b x) \left (-c (a+b x)^n \log (f)\right )^{-2/n} \left ((a+b x) \Gamma \left (\frac {2}{n},-c (a+b x)^n \log (f)\right )-a \Gamma \left (\frac {1}{n},-c (a+b x)^n \log (f)\right ) \left (-c (a+b x)^n \log (f)\right )^{\frac {1}{n}}\right )}{b^2 n} \end {gather*}

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 = 0.01, size = 0, normalized size = 0.00 \[\int f^{c \left (b x +a \right )^{n}} x\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int f^{c \left (a + b x\right )^{n}} x\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int f^{c\,{\left (a+b\,x\right )}^n}\,x \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(c*(a + b*x)^n)*x,x)

[Out]

int(f^(c*(a + b*x)^n)*x, x)

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