\(\int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{13}} \, dx\) [291]

Optimal result

Integrand size = 21, antiderivative size = 31 \[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{13}} \, dx=-\frac {b^4 F^a \Gamma \left (-4,-b (c+d x)^3 \log (F)\right ) \log ^4(F)}{3 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2250} \[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{13}} \, dx=-\frac {b^4 F^a \log ^4(F) \Gamma \left (-4,-b (c+d x)^3 \log (F)\right )}{3 d} \]

[In]

Int[F^(a + b*(c + d*x)^3)/(c + d*x)^13,x]

[Out]

-1/3*(b^4*F^a*Gamma[-4, -(b*(c + d*x)^3*Log[F])]*Log[F]^4)/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*} \text {integral}& = -\frac {b^4 F^a \Gamma \left (-4,-b (c+d x)^3 \log (F)\right ) \log ^4(F)}{3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{13}} \, dx=-\frac {b^4 F^a \Gamma \left (-4,-b (c+d x)^3 \log (F)\right ) \log ^4(F)}{3 d} \]

[In]

Integrate[F^(a + b*(c + d*x)^3)/(c + d*x)^13,x]

[Out]

-1/3*(b^4*F^a*Gamma[-4, -(b*(c + d*x)^3*Log[F])]*Log[F]^4)/d

Maple [F]

\[\int \frac {F^{a +b \left (d x +c \right )^{3}}}{\left (d x +c \right )^{13}}d x\]

[In]

int(F^(a+b*(d*x+c)^3)/(d*x+c)^13,x)

[Out]

int(F^(a+b*(d*x+c)^3)/(d*x+c)^13,x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 636 vs. \(2 (29) = 58\).

Time = 0.29 (sec) , antiderivative size = 636, normalized size of antiderivative = 20.52 \[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{13}} \, dx=\frac {{\left (b^{4} d^{12} x^{12} + 12 \, b^{4} c d^{11} x^{11} + 66 \, b^{4} c^{2} d^{10} x^{10} + 220 \, b^{4} c^{3} d^{9} x^{9} + 495 \, b^{4} c^{4} d^{8} x^{8} + 792 \, b^{4} c^{5} d^{7} x^{7} + 924 \, b^{4} c^{6} d^{6} x^{6} + 792 \, b^{4} c^{7} d^{5} x^{5} + 495 \, b^{4} c^{8} d^{4} x^{4} + 220 \, b^{4} c^{9} d^{3} x^{3} + 66 \, b^{4} c^{10} d^{2} x^{2} + 12 \, b^{4} c^{11} d x + b^{4} c^{12}\right )} F^{a} {\rm Ei}\left ({\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right )\right ) \log \left (F\right )^{4} - {\left ({\left (b^{3} d^{9} x^{9} + 9 \, b^{3} c d^{8} x^{8} + 36 \, b^{3} c^{2} d^{7} x^{7} + 84 \, b^{3} c^{3} d^{6} x^{6} + 126 \, b^{3} c^{4} d^{5} x^{5} + 126 \, b^{3} c^{5} d^{4} x^{4} + 84 \, b^{3} c^{6} d^{3} x^{3} + 36 \, b^{3} c^{7} d^{2} x^{2} + 9 \, b^{3} c^{8} d x + b^{3} c^{9}\right )} \log \left (F\right )^{3} + {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 15 \, b^{2} c^{4} d^{2} x^{2} + 6 \, b^{2} c^{5} d x + b^{2} c^{6}\right )} \log \left (F\right )^{2} + 2 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right ) + 6\right )} F^{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}}{72 \, {\left (d^{13} x^{12} + 12 \, c d^{12} x^{11} + 66 \, c^{2} d^{11} x^{10} + 220 \, c^{3} d^{10} x^{9} + 495 \, c^{4} d^{9} x^{8} + 792 \, c^{5} d^{8} x^{7} + 924 \, c^{6} d^{7} x^{6} + 792 \, c^{7} d^{6} x^{5} + 495 \, c^{8} d^{5} x^{4} + 220 \, c^{9} d^{4} x^{3} + 66 \, c^{10} d^{3} x^{2} + 12 \, c^{11} d^{2} x + c^{12} d\right )}} \]

[In]

integrate(F^(a+b*(d*x+c)^3)/(d*x+c)^13,x, algorithm="fricas")

[Out]

1/72*((b^4*d^12*x^12 + 12*b^4*c*d^11*x^11 + 66*b^4*c^2*d^10*x^10 + 220*b^4*c^3*d^9*x^9 + 495*b^4*c^4*d^8*x^8 +
792*b^4*c^5*d^7*x^7 + 924*b^4*c^6*d^6*x^6 + 792*b^4*c^7*d^5*x^5 + 495*b^4*c^8*d^4*x^4 + 220*b^4*c^9*d^3*x^3 +
66*b^4*c^10*d^2*x^2 + 12*b^4*c^11*d*x + b^4*c^12)*F^a*Ei((b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3)*lo
g(F))*log(F)^4 - ((b^3*d^9*x^9 + 9*b^3*c*d^8*x^8 + 36*b^3*c^2*d^7*x^7 + 84*b^3*c^3*d^6*x^6 + 126*b^3*c^4*d^5*x
^5 + 126*b^3*c^5*d^4*x^4 + 84*b^3*c^6*d^3*x^3 + 36*b^3*c^7*d^2*x^2 + 9*b^3*c^8*d*x + b^3*c^9)*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^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3)*log(F) + 6)*F^(b*d^3*x^3 + 3*b*c*d^2*x^2
+ 3*b*c^2*d*x + b*c^3 + a))/(d^13*x^12 + 12*c*d^12*x^11 + 66*c^2*d^11*x^10 + 220*c^3*d^10*x^9 + 495*c^4*d^9*x^
8 + 792*c^5*d^8*x^7 + 924*c^6*d^7*x^6 + 792*c^7*d^6*x^5 + 495*c^8*d^5*x^4 + 220*c^9*d^4*x^3 + 66*c^10*d^3*x^2
+ 12*c^11*d^2*x + c^12*d)

Sympy [F]

\[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{13}} \, dx=\int \frac {F^{a + b \left (c + d x\right )^{3}}}{\left (c + d x\right )^{13}}\, dx \]

[In]

integrate(F**(a+b*(d*x+c)**3)/(d*x+c)**13,x)

[Out]

Integral(F**(a + b*(c + d*x)**3)/(c + d*x)**13, x)

Maxima [F]

\[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{13}} \, dx=\int { \frac {F^{{\left (d x + c\right )}^{3} b + a}}{{\left (d x + c\right )}^{13}} \,d x } \]

[In]

integrate(F^(a+b*(d*x+c)^3)/(d*x+c)^13,x, algorithm="maxima")

[Out]

integrate(F^((d*x + c)^3*b + a)/(d*x + c)^13, x)

Giac [F]

\[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{13}} \, dx=\int { \frac {F^{{\left (d x + c\right )}^{3} b + a}}{{\left (d x + c\right )}^{13}} \,d x } \]

[In]

integrate(F^(a+b*(d*x+c)^3)/(d*x+c)^13,x, algorithm="giac")

[Out]

integrate(F^((d*x + c)^3*b + a)/(d*x + c)^13, x)

Mupad [B] (verification not implemented)

Time = 0.72 (sec) , antiderivative size = 120, normalized size of antiderivative = 3.87 \[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{13}} \, dx=-\frac {F^a\,b^4\,{\ln \left (F\right )}^4\,\mathrm {expint}\left (-b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3\right )}{72\,d}-\frac {F^a\,F^{b\,{\left (c+d\,x\right )}^3}\,b^4\,{\ln \left (F\right )}^4\,\left (\frac {1}{24\,b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3}+\frac {1}{24\,b^2\,{\ln \left (F\right )}^2\,{\left (c+d\,x\right )}^6}+\frac {1}{12\,b^3\,{\ln \left (F\right )}^3\,{\left (c+d\,x\right )}^9}+\frac {1}{4\,b^4\,{\ln \left (F\right )}^4\,{\left (c+d\,x\right )}^{12}}\right )}{3\,d} \]

[In]

int(F^(a + b*(c + d*x)^3)/(c + d*x)^13,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*(1/(24*b*log(F
)*(c + d*x)^3) + 1/(24*b^2*log(F)^2*(c + d*x)^6) + 1/(12*b^3*log(F)^3*(c + d*x)^9) + 1/(4*b^4*log(F)^4*(c + d*
x)^12)))/(3*d)