3.4.78 \(\int F^{a+b (c+d x)^n} (c+d x)^{-1-4 n} \, dx\) [378]

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

[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 = 0.03, 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 = {2250} \begin {gather*} -\frac {b^4 F^a \log ^4(F) \text {Gamma}\left (-4,-b \log (F) (c+d x)^n\right )}{d n} \end {gather*}

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 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]

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 \begin {gather*} -\frac {b^4 F^a \Gamma \left (-4,-b (c+d x)^n \log (F)\right ) \log ^4(F)}{d n} \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(174\) vs. \(2(34)=68\).
time = 0.10, size = 175, normalized size = 5.47

method	result	size
risch	\(-\frac {F^{b \left (d x +c \right )^{n}} F^{a} \left (d x +c \right )^{-4 n}}{4 n d}-\frac {\ln \left (F \right ) b \,F^{b \left (d x +c \right )^{n}} F^{a} \left (d x +c \right )^{-3 n}}{12 n d}-\frac {\ln \left (F \right )^{2} b^{2} F^{b \left (d x +c \right )^{n}} F^{a} \left (d x +c \right )^{-2 n}}{24 n d}-\frac {\ln \left (F \right )^{3} b^{3} F^{b \left (d x +c \right )^{n}} F^{a} \left (d x +c \right )^{-n}}{24 n d}-\frac {\ln \left (F \right )^{4} b^{4} F^{a} \expIntegral \left (1, -b \left (d x +c \right )^{n} \ln \left (F \right )\right )}{24 n d}\)	\(175\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (36) = 72\).
time = 0.11, size = 119, normalized size = 3.72 \begin {gather*} \frac {{\left (d x + c\right )}^{4 \, n} F^{a} b^{4} {\rm Ei}\left ({\left (d x + c\right )}^{n} b \log \left (F\right )\right ) \log \left (F\right )^{4} - {\left ({\left (d x + c\right )}^{3 \, n} b^{3} \log \left (F\right )^{3} + {\left (d x + c\right )}^{2 \, n} b^{2} \log \left (F\right )^{2} + 2 \, {\left (d x + c\right )}^{n} b \log \left (F\right ) + 6\right )} e^{\left ({\left (d x + c\right )}^{n} b \log \left (F\right ) + a \log \left (F\right )\right )}}{24 \, {\left (d x + c\right )}^{4 \, n} d n} \end {gather*}

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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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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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