3.378 \(\int F^{a+b (c+d x)^n} (c+d x)^{-1-4 n} \, dx\)

Optimal. Leaf size=32 \[ -\frac {b^4 F^a \log ^4(F) \Gamma \left (-4,-b (c+d x)^n \log (F)\right )}{d n} \]

[Out]

-F^a/((d*x+c)^n)^4*Ei(5,-b*(d*x+c)^n*ln(F))/d/n

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Rubi [A]  time = 0.04, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2218} \[ -\frac {b^4 F^a \log ^4(F) \text {Gamma}\left (-4,-b \log (F) (c+d x)^n\right )}{d n} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*(c + d*x)^n)*(c + d*x)^(-1 - 4*n),x]

[Out]

-((b^4*F^a*Gamma[-4, -(b*(c + d*x)^n*Log[F])]*Log[F]^4)/(d*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 (c+d x)^n} (c+d x)^{-1-4 n} \, dx &=-\frac {b^4 F^a \Gamma \left (-4,-b (c+d x)^n \log (F)\right ) \log ^4(F)}{d n}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 32, normalized size = 1.00 \[ -\frac {b^4 F^a \log ^4(F) \Gamma \left (-4,-b (c+d x)^n \log (F)\right )}{d n} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*(c + d*x)^n)*(c + d*x)^(-1 - 4*n),x]

[Out]

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

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fricas [B]  time = 0.42, size = 119, normalized size = 3.72 \[ \frac {{\left (d x + c\right )}^{4 \, n} F^{a} b^{4} {\rm Ei}\left ({\left (d x + c\right )}^{n} b \log \relax (F)\right ) \log \relax (F)^{4} - {\left ({\left (d x + c\right )}^{3 \, n} b^{3} \log \relax (F)^{3} + {\left (d x + c\right )}^{2 \, n} b^{2} \log \relax (F)^{2} + 2 \, {\left (d x + c\right )}^{n} b \log \relax (F) + 6\right )} e^{\left ({\left (d x + c\right )}^{n} b \log \relax (F) + a \log \relax (F)\right )}}{24 \, {\left (d x + c\right )}^{4 \, n} d n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1-4*n),x, algorithm="fricas")

[Out]

1/24*((d*x + c)^(4*n)*F^a*b^4*Ei((d*x + c)^n*b*log(F))*log(F)^4 - ((d*x + c)^(3*n)*b^3*log(F)^3 + (d*x + c)^(2
*n)*b^2*log(F)^2 + 2*(d*x + c)^n*b*log(F) + 6)*e^((d*x + c)^n*b*log(F) + a*log(F)))/((d*x + c)^(4*n)*d*n)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{-4 \, n - 1} F^{{\left (d x + c\right )}^{n} b + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1-4*n),x, algorithm="giac")

[Out]

integrate((d*x + c)^(-4*n - 1)*F^((d*x + c)^n*b + a), x)

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maple [B]  time = 0.12, size = 175, normalized size = 5.47 \[ -\frac {b^{4} F^{a} \Ei \left (1, -b \left (d x +c \right )^{n} \ln \relax (F )\right ) \ln \relax (F )^{4}}{24 d n}-\frac {b^{3} F^{a} F^{b \left (d x +c \right )^{n}} \left (d x +c \right )^{-n} \ln \relax (F )^{3}}{24 d n}-\frac {b^{2} F^{a} F^{b \left (d x +c \right )^{n}} \left (d x +c \right )^{-2 n} \ln \relax (F )^{2}}{24 d n}-\frac {b \,F^{a} F^{b \left (d x +c \right )^{n}} \left (d x +c \right )^{-3 n} \ln \relax (F )}{12 d n}-\frac {F^{a} F^{b \left (d x +c \right )^{n}} \left (d x +c \right )^{-4 n}}{4 d n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(b*(d*x+c)^n+a)*(d*x+c)^(-1-4*n),x)

[Out]

-1/4/n/d*F^(b*(d*x+c)^n)*F^a/((d*x+c)^n)^4-1/12/n/d*ln(F)*b*F^(b*(d*x+c)^n)*F^a/((d*x+c)^n)^3-1/24/n/d*ln(F)^2
*b^2*F^(b*(d*x+c)^n)*F^a/((d*x+c)^n)^2-1/24/n/d*ln(F)^3*b^3*F^(b*(d*x+c)^n)*F^a/((d*x+c)^n)-1/24/n/d*ln(F)^4*b
^4*F^a*Ei(1,-b*(d*x+c)^n*ln(F))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{-4 \, n - 1} F^{{\left (d x + c\right )}^{n} b + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1-4*n),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(-4*n - 1)*F^((d*x + c)^n*b + a), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {F^{a+b\,{\left (c+d\,x\right )}^n}}{{\left (c+d\,x\right )}^{4\,n+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a + b*(c + d*x)^n)/(c + d*x)^(4*n + 1),x)

[Out]

int(F^(a + b*(c + d*x)^n)/(c + d*x)^(4*n + 1), x)

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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)**(-1-4*n),x)

[Out]

Timed out

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