3.290 \(\int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{10}} \, dx\)

Optimal. Leaf size=121 \[ \frac {b^3 F^a \log ^3(F) \text {Ei}\left (b (c+d x)^3 \log (F)\right )}{18 d}-\frac {b^2 \log ^2(F) F^{a+b (c+d x)^3}}{18 d (c+d x)^3}-\frac {F^{a+b (c+d x)^3}}{9 d (c+d x)^9}-\frac {b \log (F) F^{a+b (c+d x)^3}}{18 d (c+d x)^6} \]

[Out]

-1/9*F^(a+b*(d*x+c)^3)/d/(d*x+c)^9-1/18*b*F^(a+b*(d*x+c)^3)*ln(F)/d/(d*x+c)^6-1/18*b^2*F^(a+b*(d*x+c)^3)*ln(F)
^2/d/(d*x+c)^3+1/18*b^3*F^a*Ei(b*(d*x+c)^3*ln(F))*ln(F)^3/d

________________________________________________________________________________________

Rubi [A]  time = 0.26, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2214, 2210} \[ \frac {b^3 F^a \log ^3(F) \text {Ei}\left (b (c+d x)^3 \log (F)\right )}{18 d}-\frac {b^2 \log ^2(F) F^{a+b (c+d x)^3}}{18 d (c+d x)^3}-\frac {F^{a+b (c+d x)^3}}{9 d (c+d x)^9}-\frac {b \log (F) F^{a+b (c+d x)^3}}{18 d (c+d x)^6} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*(c + d*x)^3)/(c + d*x)^10,x]

[Out]

-F^(a + b*(c + d*x)^3)/(9*d*(c + d*x)^9) - (b*F^(a + b*(c + d*x)^3)*Log[F])/(18*d*(c + d*x)^6) - (b^2*F^(a + b
*(c + d*x)^3)*Log[F]^2)/(18*d*(c + d*x)^3) + (b^3*F^a*ExpIntegralEi[b*(c + d*x)^3*Log[F]]*Log[F]^3)/(18*d)

Rule 2210

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[(F^a*ExpIntegralEi[
b*(c + d*x)^n*Log[F]])/(f*n), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rule 2214

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*F^(a + b*(c + d*x)^n))/(d*(m + 1)), x] - Dist[(b*n*Log[F])/(m + 1), Int[(c + d*x)^(m + n)*F^(a + b*(c +
d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[-4, (m + 1)/n, 5] && IntegerQ[n
] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))

Rubi steps

\begin {align*} \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{10}} \, dx &=-\frac {F^{a+b (c+d x)^3}}{9 d (c+d x)^9}+\frac {1}{3} (b \log (F)) \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^7} \, dx\\ &=-\frac {F^{a+b (c+d x)^3}}{9 d (c+d x)^9}-\frac {b F^{a+b (c+d x)^3} \log (F)}{18 d (c+d x)^6}+\frac {1}{6} \left (b^2 \log ^2(F)\right ) \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^4} \, dx\\ &=-\frac {F^{a+b (c+d x)^3}}{9 d (c+d x)^9}-\frac {b F^{a+b (c+d x)^3} \log (F)}{18 d (c+d x)^6}-\frac {b^2 F^{a+b (c+d x)^3} \log ^2(F)}{18 d (c+d x)^3}+\frac {1}{6} \left (b^3 \log ^3(F)\right ) \int \frac {F^{a+b (c+d x)^3}}{c+d x} \, dx\\ &=-\frac {F^{a+b (c+d x)^3}}{9 d (c+d x)^9}-\frac {b F^{a+b (c+d x)^3} \log (F)}{18 d (c+d x)^6}-\frac {b^2 F^{a+b (c+d x)^3} \log ^2(F)}{18 d (c+d x)^3}+\frac {b^3 F^a \text {Ei}\left (b (c+d x)^3 \log (F)\right ) \log ^3(F)}{18 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.10, size = 80, normalized size = 0.66 \[ \frac {F^a \left (b^3 \log ^3(F) \text {Ei}\left (b (c+d x)^3 \log (F)\right )+\frac {F^{b (c+d x)^3} \left (-b^2 \log ^2(F) (c+d x)^6-b \log (F) (c+d x)^3-2\right )}{(c+d x)^9}\right )}{18 d} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*(c + d*x)^3)/(c + d*x)^10,x]

[Out]

(F^a*(b^3*ExpIntegralEi[b*(c + d*x)^3*Log[F]]*Log[F]^3 + (F^(b*(c + d*x)^3)*(-2 - b*(c + d*x)^3*Log[F] - b^2*(
c + d*x)^6*Log[F]^2))/(c + d*x)^9))/(18*d)

________________________________________________________________________________________

fricas [B]  time = 0.42, size = 431, normalized size = 3.56 \[ \frac {{\left (b^{3} d^{9} x^{9} + 9 \, b^{3} c d^{8} x^{8} + 36 \, b^{3} c^{2} d^{7} x^{7} + 84 \, b^{3} c^{3} d^{6} x^{6} + 126 \, b^{3} c^{4} d^{5} x^{5} + 126 \, b^{3} c^{5} d^{4} x^{4} + 84 \, b^{3} c^{6} d^{3} x^{3} + 36 \, b^{3} c^{7} d^{2} x^{2} + 9 \, b^{3} c^{8} d x + b^{3} c^{9}\right )} F^{a} {\rm Ei}\left ({\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \relax (F)\right ) \log \relax (F)^{3} - {\left ({\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 15 \, b^{2} c^{4} d^{2} x^{2} + 6 \, b^{2} c^{5} d x + b^{2} c^{6}\right )} \log \relax (F)^{2} + {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \relax (F) + 2\right )} F^{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}}{18 \, {\left (d^{10} x^{9} + 9 \, c d^{9} x^{8} + 36 \, c^{2} d^{8} x^{7} + 84 \, c^{3} d^{7} x^{6} + 126 \, c^{4} d^{6} x^{5} + 126 \, c^{5} d^{5} x^{4} + 84 \, c^{6} d^{4} x^{3} + 36 \, c^{7} d^{3} x^{2} + 9 \, c^{8} d^{2} x + c^{9} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^3)/(d*x+c)^10,x, algorithm="fricas")

[Out]

1/18*((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)*F^a*Ei((b*d^3*x^3 + 3*b*c*d^
2*x^2 + 3*b*c^2*d*x + b*c^3)*log(F))*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 + (b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*
x + b*c^3)*log(F) + 2)*F^(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a))/(d^10*x^9 + 9*c*d^9*x^8 + 36*c
^2*d^8*x^7 + 84*c^3*d^7*x^6 + 126*c^4*d^6*x^5 + 126*c^5*d^5*x^4 + 84*c^6*d^4*x^3 + 36*c^7*d^3*x^2 + 9*c^8*d^2*
x + c^9*d)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{{\left (d x + c\right )}^{3} b + a}}{{\left (d x + c\right )}^{10}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^3)/(d*x+c)^10,x, algorithm="giac")

[Out]

integrate(F^((d*x + c)^3*b + a)/(d*x + c)^10, x)

________________________________________________________________________________________

maple [F]  time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {F^{a +\left (d x +c \right )^{3} b}}{\left (d x +c \right )^{10}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+(d*x+c)^3*b)/(d*x+c)^10,x)

[Out]

int(F^(a+(d*x+c)^3*b)/(d*x+c)^10,x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{{\left (d x + c\right )}^{3} b + a}}{{\left (d x + c\right )}^{10}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^3)/(d*x+c)^10,x, algorithm="maxima")

[Out]

integrate(F^((d*x + c)^3*b + a)/(d*x + c)^10, x)

________________________________________________________________________________________

mupad [B]  time = 3.87, size = 104, normalized size = 0.86 \[ -\frac {F^a\,b^3\,{\ln \relax (F)}^3\,\mathrm {expint}\left (-b\,\ln \relax (F)\,{\left (c+d\,x\right )}^3\right )}{18\,d}-\frac {F^a\,F^{b\,{\left (c+d\,x\right )}^3}\,b^3\,{\ln \relax (F)}^3\,\left (\frac {1}{6\,b\,\ln \relax (F)\,{\left (c+d\,x\right )}^3}+\frac {1}{6\,b^2\,{\ln \relax (F)}^2\,{\left (c+d\,x\right )}^6}+\frac {1}{3\,b^3\,{\ln \relax (F)}^3\,{\left (c+d\,x\right )}^9}\right )}{3\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a + b*(c + d*x)^3)/(c + d*x)^10,x)

[Out]

- (F^a*b^3*log(F)^3*expint(-b*log(F)*(c + d*x)^3))/(18*d) - (F^a*F^(b*(c + d*x)^3)*b^3*log(F)^3*(1/(6*b*log(F)
*(c + d*x)^3) + 1/(6*b^2*log(F)^2*(c + d*x)^6) + 1/(3*b^3*log(F)^3*(c + d*x)^9)))/(3*d)

________________________________________________________________________________________

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)**3)/(d*x+c)**10,x)

[Out]

Timed out

________________________________________________________________________________________