3.4.42 \(\int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^{11} \, dx\) [342]

Optimal. Leaf size=31 \[ \frac {b^4 F^a \Gamma \left (-4,-\frac {b \log (F)}{(c+d x)^3}\right ) \log ^4(F)}{3 d} \]

[Out]

1/3*F^a*(d*x+c)^12*Ei(5,-b*ln(F)/(d*x+c)^3)/d

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Rubi [A]
time = 0.03, antiderivative size = 31, 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 = {2250} \begin {gather*} \frac {b^4 F^a \log ^4(F) \text {Gamma}\left (-4,-\frac {b \log (F)}{(c+d x)^3}\right )}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[F^(a + b/(c + d*x)^3)*(c + d*x)^11,x]

[Out]

(b^4*F^a*Gamma[-4, -((b*Log[F])/(c + d*x)^3)]*Log[F]^4)/(3*d)

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+\frac {b}{(c+d x)^3}} (c+d x)^{11} \, dx &=\frac {b^4 F^a \Gamma \left (-4,-\frac {b \log (F)}{(c+d x)^3}\right ) \log ^4(F)}{3 d}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 31, normalized size = 1.00 \begin {gather*} \frac {b^4 F^a \Gamma \left (-4,-\frac {b \log (F)}{(c+d x)^3}\right ) \log ^4(F)}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b/(c + d*x)^3)*(c + d*x)^11,x]

[Out]

(b^4*F^a*Gamma[-4, -((b*Log[F])/(c + d*x)^3)]*Log[F]^4)/(3*d)

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int F^{a +\frac {b}{\left (d x +c \right )^{3}}} \left (d x +c \right )^{11}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b/(d*x+c)^3)*(d*x+c)^11,x)

[Out]

int(F^(a+b/(d*x+c)^3)*(d*x+c)^11,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^(a+b/(d*x+c)^3)*(d*x+c)^11,x, algorithm="maxima")

[Out]

1/72*(6*F^a*d^11*x^12 + 72*F^a*c*d^10*x^11 + 396*F^a*c^2*d^9*x^10 + 2*(660*F^a*c^3*d^8 + F^a*b*d^8*log(F))*x^9
+ 18*(165*F^a*c^4*d^7 + F^a*b*c*d^7*log(F))*x^8 + 72*(66*F^a*c^5*d^6 + F^a*b*c^2*d^6*log(F))*x^7 + (5544*F^a*
c^6*d^5 + 168*F^a*b*c^3*d^5*log(F) + F^a*b^2*d^5*log(F)^2)*x^6 + 6*(792*F^a*c^7*d^4 + 42*F^a*b*c^4*d^4*log(F)
+ F^a*b^2*c*d^4*log(F)^2)*x^5 + 3*(990*F^a*c^8*d^3 + 84*F^a*b*c^5*d^3*log(F) + 5*F^a*b^2*c^2*d^3*log(F)^2)*x^4
+ (1320*F^a*c^9*d^2 + 168*F^a*b*c^6*d^2*log(F) + 20*F^a*b^2*c^3*d^2*log(F)^2 + F^a*b^3*d^2*log(F)^3)*x^3 + 3*
(132*F^a*c^10*d + 24*F^a*b*c^7*d*log(F) + 5*F^a*b^2*c^4*d*log(F)^2 + F^a*b^3*c*d*log(F)^3)*x^2 + 3*(24*F^a*c^1
1 + 6*F^a*b*c^8*log(F) + 2*F^a*b^2*c^5*log(F)^2 + F^a*b^3*c^2*log(F)^3)*x)*F^(b/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2
*d*x + c^3)) + integrate(-1/24*(6*F^a*b*c^12*log(F) - F^a*b^4*d^3*x^3*log(F)^4 + 2*F^a*b^2*c^9*log(F)^2 - 3*F^
a*b^4*c*d^2*x^2*log(F)^4 + F^a*b^3*c^6*log(F)^3 - 3*F^a*b^4*c^2*d*x*log(F)^4)*F^(b/(d^3*x^3 + 3*c*d^2*x^2 + 3*
c^2*d*x + c^3))/(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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 487 vs. \(2 (29) = 58\).
time = 0.13, size = 487, normalized size = 15.71 \begin {gather*} -\frac {F^{a} b^{4} {\rm Ei}\left (\frac {b \log \left (F\right )}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) \log \left (F\right )^{4} - {\left (6 \, d^{12} x^{12} + 72 \, c d^{11} x^{11} + 396 \, c^{2} d^{10} x^{10} + 1320 \, c^{3} d^{9} x^{9} + 2970 \, c^{4} d^{8} x^{8} + 4752 \, c^{5} d^{7} x^{7} + 5544 \, c^{6} d^{6} x^{6} + 4752 \, c^{7} d^{5} x^{5} + 2970 \, c^{8} d^{4} x^{4} + 1320 \, c^{9} d^{3} x^{3} + 396 \, c^{10} d^{2} x^{2} + 72 \, c^{11} d x + 6 \, c^{12} + {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \log \left (F\right )^{3} + {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 15 \, b^{2} c^{4} d^{2} x^{2} + 6 \, b^{2} c^{5} d x + b^{2} c^{6}\right )} \log \left (F\right )^{2} + 2 \, {\left (b d^{9} x^{9} + 9 \, b c d^{8} x^{8} + 36 \, b c^{2} d^{7} x^{7} + 84 \, b c^{3} d^{6} x^{6} + 126 \, b c^{4} d^{5} x^{5} + 126 \, b c^{5} d^{4} x^{4} + 84 \, b c^{6} d^{3} x^{3} + 36 \, b c^{7} d^{2} x^{2} + 9 \, b c^{8} d x + b c^{9}\right )} \log \left (F\right )\right )} F^{\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{72 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^3)*(d*x+c)^11,x, algorithm="fricas")

[Out]

-1/72*(F^a*b^4*Ei(b*log(F)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))*log(F)^4 - (6*d^12*x^12 + 72*c*d^11*x^11
+ 396*c^2*d^10*x^10 + 1320*c^3*d^9*x^9 + 2970*c^4*d^8*x^8 + 4752*c^5*d^7*x^7 + 5544*c^6*d^6*x^6 + 4752*c^7*d^
5*x^5 + 2970*c^8*d^4*x^4 + 1320*c^9*d^3*x^3 + 396*c^10*d^2*x^2 + 72*c^11*d*x + 6*c^12 + (b^3*d^3*x^3 + 3*b^3*c
*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*log(F)^3 + (b^2*d^6*x^6 + 6*b^2*c*d^5*x^5 + 15*b^2*c^2*d^4*x^4 + 20*b^2*c^
3*d^3*x^3 + 15*b^2*c^4*d^2*x^2 + 6*b^2*c^5*d*x + b^2*c^6)*log(F)^2 + 2*(b*d^9*x^9 + 9*b*c*d^8*x^8 + 36*b*c^2*d
^7*x^7 + 84*b*c^3*d^6*x^6 + 126*b*c^4*d^5*x^5 + 126*b*c^5*d^4*x^4 + 84*b*c^6*d^3*x^3 + 36*b*c^7*d^2*x^2 + 9*b*
c^8*d*x + b*c^9)*log(F))*F^((a*d^3*x^3 + 3*a*c*d^2*x^2 + 3*a*c^2*d*x + a*c^3 + b)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c
^2*d*x + c^3)))/d

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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)**3)*(d*x+c)**11,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)^3)*(d*x+c)^11,x, algorithm="giac")

[Out]

integrate((d*x + c)^11*F^(a + b/(d*x + c)^3), x)

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Mupad [B]
time = 3.83, size = 120, normalized size = 3.87 \begin {gather*} \frac {F^a\,b^4\,{\ln \left (F\right )}^4\,\mathrm {expint}\left (-\frac {b\,\ln \left (F\right )}{{\left (c+d\,x\right )}^3}\right )}{72\,d}+\frac {F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^3}}\,b^4\,{\ln \left (F\right )}^4\,\left (\frac {{\left (c+d\,x\right )}^3}{24\,b\,\ln \left (F\right )}+\frac {{\left (c+d\,x\right )}^6}{24\,b^2\,{\ln \left (F\right )}^2}+\frac {{\left (c+d\,x\right )}^9}{12\,b^3\,{\ln \left (F\right )}^3}+\frac {{\left (c+d\,x\right )}^{12}}{4\,b^4\,{\ln \left (F\right )}^4}\right )}{3\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a + b/(c + d*x)^3)*(c + d*x)^11,x)

[Out]

(F^a*b^4*log(F)^4*expint(-(b*log(F))/(c + d*x)^3))/(72*d) + (F^a*F^(b/(c + d*x)^3)*b^4*log(F)^4*((c + d*x)^3/(
24*b*log(F)) + (c + d*x)^6/(24*b^2*log(F)^2) + (c + d*x)^9/(12*b^3*log(F)^3) + (c + d*x)^12/(4*b^4*log(F)^4)))
/(3*d)

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