3.367 \(\int \frac {F^{a+b (c+d x)^n}}{(c+d x)^4} \, dx\)

Optimal. Leaf size=54 \[ -\frac {F^a \left (-b \log (F) (c+d x)^n\right )^{3/n} \Gamma \left (-\frac {3}{n},-b (c+d x)^n \log (F)\right )}{d n (c+d x)^3} \]

[Out]

-F^a*GAMMA(-3/n,-b*(d*x+c)^n*ln(F))*(-b*(d*x+c)^n*ln(F))^(3/n)/d/n/(d*x+c)^3

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Rubi [A]  time = 0.04, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2218} \[ -\frac {F^a \left (-b \log (F) (c+d x)^n\right )^{3/n} \text {Gamma}\left (-\frac {3}{n},-b \log (F) (c+d x)^n\right )}{d n (c+d x)^3} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*(c + d*x)^n)/(c + d*x)^4,x]

[Out]

-((F^a*Gamma[-3/n, -(b*(c + d*x)^n*Log[F])]*(-(b*(c + d*x)^n*Log[F]))^(3/n))/(d*n*(c + d*x)^3))

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 \frac {F^{a+b (c+d x)^n}}{(c+d x)^4} \, dx &=-\frac {F^a \Gamma \left (-\frac {3}{n},-b (c+d x)^n \log (F)\right ) \left (-b (c+d x)^n \log (F)\right )^{3/n}}{d n (c+d x)^3}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 54, normalized size = 1.00 \[ -\frac {F^a \left (-b \log (F) (c+d x)^n\right )^{3/n} \Gamma \left (-\frac {3}{n},-b (c+d x)^n \log (F)\right )}{d n (c+d x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*(c + d*x)^n)/(c + d*x)^4,x]

[Out]

-((F^a*Gamma[-3/n, -(b*(c + d*x)^n*Log[F])]*(-(b*(c + d*x)^n*Log[F]))^(3/n))/(d*n*(c + d*x)^3))

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fricas [F]  time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {F^{{\left (d x + c\right )}^{n} b + a}}{d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n)/(d*x+c)^4,x, algorithm="fricas")

[Out]

integral(F^((d*x + c)^n*b + a)/(d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2*x^2 + 4*c^3*d*x + c^4), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{{\left (d x + c\right )}^{n} b + a}}{{\left (d x + c\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n)/(d*x+c)^4,x, algorithm="giac")

[Out]

integrate(F^((d*x + c)^n*b + a)/(d*x + c)^4, x)

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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {F^{b \left (d x +c \right )^{n}+a}}{\left (d x +c \right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(b*(d*x+c)^n+a)/(d*x+c)^4,x)

[Out]

int(F^(b*(d*x+c)^n+a)/(d*x+c)^4,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{{\left (d x + c\right )}^{n} b + a}}{{\left (d x + c\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n)/(d*x+c)^4,x, algorithm="maxima")

[Out]

integrate(F^((d*x + c)^n*b + a)/(d*x + c)^4, x)

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mupad [B]  time = 3.62, size = 71, normalized size = 1.31 \[ -\frac {F^a\,{\mathrm {e}}^{\frac {b\,\ln \relax (F)\,{\left (c+d\,x\right )}^n}{2}}\,{\left (b\,\ln \relax (F)\,{\left (c+d\,x\right )}^n\right )}^{\frac {3}{2\,n}-\frac {1}{2}}\,{\mathrm {M}}_{\frac {3}{2\,n}+\frac {1}{2},-\frac {3}{2\,n}}\left (b\,\ln \relax (F)\,{\left (c+d\,x\right )}^n\right )}{3\,d\,{\left (c+d\,x\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a + b*(c + d*x)^n)/(c + d*x)^4,x)

[Out]

-(F^a*exp((b*log(F)*(c + d*x)^n)/2)*(b*log(F)*(c + d*x)^n)^(3/(2*n) - 1/2)*whittakerM(3/(2*n) + 1/2, -3/(2*n),
b*log(F)*(c + d*x)^n))/(3*d*(c + d*x)^3)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b*(d*x+c)**n)/(d*x+c)**4,x)

[Out]

Timed out

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