3.343 \(\int F^{a+\frac{b}{(c+d x)^3}} (c+d x)^8 \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.191009, antiderivative size = 121, normalized size of antiderivative = 1., 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} \[ -\frac{b^3 F^a \log ^3(F) \text{Ei}\left (\frac{b \log (F)}{(c+d x)^3}\right )}{18 d}+\frac{b^2 \log ^2(F) (c+d x)^3 F^{a+\frac{b}{(c+d x)^3}}}{18 d}+\frac{(c+d x)^9 F^{a+\frac{b}{(c+d x)^3}}}{9 d}+\frac{b \log (F) (c+d x)^6 F^{a+\frac{b}{(c+d x)^3}}}{18 d} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b/(c + d*x)^3)*(c + d*x)^8,x]

[Out]

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

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]))

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]

Rubi steps

\begin{align*} \int F^{a+\frac{b}{(c+d x)^3}} (c+d x)^8 \, dx &=\frac{F^{a+\frac{b}{(c+d x)^3}} (c+d x)^9}{9 d}+\frac{1}{3} (b \log (F)) \int F^{a+\frac{b}{(c+d x)^3}} (c+d x)^5 \, dx\\ &=\frac{F^{a+\frac{b}{(c+d x)^3}} (c+d x)^9}{9 d}+\frac{b F^{a+\frac{b}{(c+d x)^3}} (c+d x)^6 \log (F)}{18 d}+\frac{1}{6} \left (b^2 \log ^2(F)\right ) \int F^{a+\frac{b}{(c+d x)^3}} (c+d x)^2 \, dx\\ &=\frac{F^{a+\frac{b}{(c+d x)^3}} (c+d x)^9}{9 d}+\frac{b F^{a+\frac{b}{(c+d x)^3}} (c+d x)^6 \log (F)}{18 d}+\frac{b^2 F^{a+\frac{b}{(c+d x)^3}} (c+d x)^3 \log ^2(F)}{18 d}+\frac{1}{6} \left (b^3 \log ^3(F)\right ) \int \frac{F^{a+\frac{b}{(c+d x)^3}}}{c+d x} \, dx\\ &=\frac{F^{a+\frac{b}{(c+d x)^3}} (c+d x)^9}{9 d}+\frac{b F^{a+\frac{b}{(c+d x)^3}} (c+d x)^6 \log (F)}{18 d}+\frac{b^2 F^{a+\frac{b}{(c+d x)^3}} (c+d x)^3 \log ^2(F)}{18 d}-\frac{b^3 F^a \text{Ei}\left (\frac{b \log (F)}{(c+d x)^3}\right ) \log ^3(F)}{18 d}\\ \end{align*}

Mathematica [A]  time = 0.206696, size = 96, normalized size = 0.79 \[ \frac{F^a \left (b \log (F) \left (b \log (F) \left ((c+d x)^3 F^{\frac{b}{(c+d x)^3}}-b \log (F) \text{Ei}\left (\frac{b \log (F)}{(c+d x)^3}\right )\right )+(c+d x)^6 F^{\frac{b}{(c+d x)^3}}\right )+2 (c+d x)^9 F^{\frac{b}{(c+d x)^3}}\right )}{18 d} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b/(c + d*x)^3)*(c + d*x)^8,x]

[Out]

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

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Maple [F]  time = 0.095, size = 0, normalized size = 0. \begin{align*} \int{F}^{a+{\frac{b}{ \left ( dx+c \right ) ^{3}}}} \left ( dx+c \right ) ^{8}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b/(d*x+c)^3)*(d*x+c)^8,x)

[Out]

int(F^(a+b/(d*x+c)^3)*(d*x+c)^8,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{18} \,{\left (2 \, F^{a} d^{8} x^{9} + 18 \, F^{a} c d^{7} x^{8} + 72 \, F^{a} c^{2} d^{6} x^{7} +{\left (168 \, F^{a} c^{3} d^{5} + F^{a} b d^{5} \log \left (F\right )\right )} x^{6} + 6 \,{\left (42 \, F^{a} c^{4} d^{4} + F^{a} b c d^{4} \log \left (F\right )\right )} x^{5} + 3 \,{\left (84 \, F^{a} c^{5} d^{3} + 5 \, F^{a} b c^{2} d^{3} \log \left (F\right )\right )} x^{4} +{\left (168 \, F^{a} c^{6} d^{2} + 20 \, F^{a} b c^{3} d^{2} \log \left (F\right ) + F^{a} b^{2} d^{2} \log \left (F\right )^{2}\right )} x^{3} + 3 \,{\left (24 \, F^{a} c^{7} d + 5 \, F^{a} b c^{4} d \log \left (F\right ) + F^{a} b^{2} c d \log \left (F\right )^{2}\right )} x^{2} + 3 \,{\left (6 \, F^{a} c^{8} + 2 \, F^{a} b c^{5} \log \left (F\right ) + F^{a} b^{2} c^{2} \log \left (F\right )^{2}\right )} x\right )} F^{\frac{b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}} + \int \frac{{\left (F^{a} b^{3} d^{3} x^{3} \log \left (F\right )^{3} - 2 \, F^{a} b c^{9} \log \left (F\right ) + 3 \, F^{a} b^{3} c d^{2} x^{2} \log \left (F\right )^{3} - F^{a} b^{2} c^{6} \log \left (F\right )^{2} + 3 \, F^{a} b^{3} c^{2} d x \log \left (F\right )^{3}\right )} F^{\frac{b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{6 \,{\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^3)*(d*x+c)^8,x, algorithm="maxima")

[Out]

1/18*(2*F^a*d^8*x^9 + 18*F^a*c*d^7*x^8 + 72*F^a*c^2*d^6*x^7 + (168*F^a*c^3*d^5 + F^a*b*d^5*log(F))*x^6 + 6*(42
*F^a*c^4*d^4 + F^a*b*c*d^4*log(F))*x^5 + 3*(84*F^a*c^5*d^3 + 5*F^a*b*c^2*d^3*log(F))*x^4 + (168*F^a*c^6*d^2 +
20*F^a*b*c^3*d^2*log(F) + F^a*b^2*d^2*log(F)^2)*x^3 + 3*(24*F^a*c^7*d + 5*F^a*b*c^4*d*log(F) + F^a*b^2*c*d*log
(F)^2)*x^2 + 3*(6*F^a*c^8 + 2*F^a*b*c^5*log(F) + F^a*b^2*c^2*log(F)^2)*x)*F^(b/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*
d*x + c^3)) + integrate(1/6*(F^a*b^3*d^3*x^3*log(F)^3 - 2*F^a*b*c^9*log(F) + 3*F^a*b^3*c*d^2*x^2*log(F)^3 - F^
a*b^2*c^6*log(F)^2 + 3*F^a*b^3*c^2*d*x*log(F)^3)*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]  time = 1.68754, size = 707, normalized size = 5.84 \begin{align*} -\frac{F^{a} b^{3}{\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 )^{3} -{\left (2 \, d^{9} x^{9} + 18 \, c d^{8} x^{8} + 72 \, c^{2} d^{7} x^{7} + 168 \, c^{3} d^{6} x^{6} + 252 \, c^{4} d^{5} x^{5} + 252 \, c^{5} d^{4} x^{4} + 168 \, c^{6} d^{3} x^{3} + 72 \, c^{7} d^{2} x^{2} + 18 \, c^{8} d x + 2 \, c^{9} +{\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\right )} \log \left (F\right )^{2} +{\left (b d^{6} x^{6} + 6 \, b c d^{5} x^{5} + 15 \, b c^{2} d^{4} x^{4} + 20 \, b c^{3} d^{3} x^{3} + 15 \, b c^{4} d^{2} x^{2} + 6 \, b c^{5} d x + b c^{6}\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}}}}{18 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^3)*(d*x+c)^8,x, algorithm="fricas")

[Out]

-1/18*(F^a*b^3*Ei(b*log(F)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))*log(F)^3 - (2*d^9*x^9 + 18*c*d^8*x^8 + 7
2*c^2*d^7*x^7 + 168*c^3*d^6*x^6 + 252*c^4*d^5*x^5 + 252*c^5*d^4*x^4 + 168*c^6*d^3*x^3 + 72*c^7*d^2*x^2 + 18*c^
8*d*x + 2*c^9 + (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*log(F)^2 + (b*d^6*x^6 + 6*b*c*d^5*x^
5 + 15*b*c^2*d^4*x^4 + 20*b*c^3*d^3*x^3 + 15*b*c^4*d^2*x^2 + 6*b*c^5*d*x + b*c^6)*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)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b/(d*x+c)**3)*(d*x+c)**8,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{8} F^{a + \frac{b}{{\left (d x + c\right )}^{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^3)*(d*x+c)^8,x, algorithm="giac")

[Out]

integrate((d*x + c)^8*F^(a + b/(d*x + c)^3), x)