3.344 \(\int F^{a+\frac{b}{(c+d x)^3}} (c+d x)^5 \, dx\)

Optimal. Leaf size=87 \[ -\frac{b^2 F^a \log ^2(F) \text{Ei}\left (\frac{b \log (F)}{(c+d x)^3}\right )}{6 d}+\frac{(c+d x)^6 F^{a+\frac{b}{(c+d x)^3}}}{6 d}+\frac{b \log (F) (c+d x)^3 F^{a+\frac{b}{(c+d x)^3}}}{6 d} \]

[Out]

(F^(a + b/(c + d*x)^3)*(c + d*x)^6)/(6*d) + (b*F^(a + b/(c + d*x)^3)*(c + d*x)^3*Log[F])/(6*d) - (b^2*F^a*ExpI
ntegralEi[(b*Log[F])/(c + d*x)^3]*Log[F]^2)/(6*d)

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Rubi [A]  time = 0.139198, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2214, 2210} \[ -\frac{b^2 F^a \log ^2(F) \text{Ei}\left (\frac{b \log (F)}{(c+d x)^3}\right )}{6 d}+\frac{(c+d x)^6 F^{a+\frac{b}{(c+d x)^3}}}{6 d}+\frac{b \log (F) (c+d x)^3 F^{a+\frac{b}{(c+d x)^3}}}{6 d} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b/(c + d*x)^3)*(c + d*x)^5,x]

[Out]

(F^(a + b/(c + d*x)^3)*(c + d*x)^6)/(6*d) + (b*F^(a + b/(c + d*x)^3)*(c + d*x)^3*Log[F])/(6*d) - (b^2*F^a*ExpI
ntegralEi[(b*Log[F])/(c + d*x)^3]*Log[F]^2)/(6*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)^5 \, dx &=\frac{F^{a+\frac{b}{(c+d x)^3}} (c+d x)^6}{6 d}+\frac{1}{2} (b \log (F)) \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)^6}{6 d}+\frac{b F^{a+\frac{b}{(c+d x)^3}} (c+d x)^3 \log (F)}{6 d}+\frac{1}{2} \left (b^2 \log ^2(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)^6}{6 d}+\frac{b F^{a+\frac{b}{(c+d x)^3}} (c+d x)^3 \log (F)}{6 d}-\frac{b^2 F^a \text{Ei}\left (\frac{b \log (F)}{(c+d x)^3}\right ) \log ^2(F)}{6 d}\\ \end{align*}

Mathematica [A]  time = 0.0599291, size = 71, normalized size = 0.82 \[ \frac{F^a \left (b \log (F) \left ((c+d x)^3 F^{\frac{b}{(c+d x)^3}}-b \log (F) \text{Ei}\left (\frac{b \log (F)}{(c+d x)^3}\right )\right )+(c+d x)^6 F^{\frac{b}{(c+d x)^3}}\right )}{6 d} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b/(c + d*x)^3)*(c + d*x)^5,x]

[Out]

(F^a*(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])))/(6*d)

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Maple [F]  time = 0.083, size = 0, normalized size = 0. \begin{align*} \int{F}^{a+{\frac{b}{ \left ( dx+c \right ) ^{3}}}} \left ( dx+c \right ) ^{5}\, 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)^5,x)

[Out]

int(F^(a+b/(d*x+c)^3)*(d*x+c)^5,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{6} \,{\left (F^{a} d^{5} x^{6} + 6 \, F^{a} c d^{4} x^{5} + 15 \, F^{a} c^{2} d^{3} x^{4} +{\left (20 \, F^{a} c^{3} d^{2} + F^{a} b d^{2} \log \left (F\right )\right )} x^{3} + 3 \,{\left (5 \, F^{a} c^{4} d + F^{a} b c d \log \left (F\right )\right )} x^{2} + 3 \,{\left (2 \, F^{a} c^{5} + F^{a} b c^{2} \log \left (F\right )\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^{2} d^{3} x^{3} \log \left (F\right )^{2} + 3 \, F^{a} b^{2} c d^{2} x^{2} \log \left (F\right )^{2} - F^{a} b c^{6} \log \left (F\right ) + 3 \, F^{a} b^{2} c^{2} d x \log \left (F\right )^{2}\right )} F^{\frac{b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{2 \,{\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)^5,x, algorithm="maxima")

[Out]

1/6*(F^a*d^5*x^6 + 6*F^a*c*d^4*x^5 + 15*F^a*c^2*d^3*x^4 + (20*F^a*c^3*d^2 + F^a*b*d^2*log(F))*x^3 + 3*(5*F^a*c
^4*d + F^a*b*c*d*log(F))*x^2 + 3*(2*F^a*c^5 + F^a*b*c^2*log(F))*x)*F^(b/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c
^3)) + integrate(1/2*(F^a*b^2*d^3*x^3*log(F)^2 + 3*F^a*b^2*c*d^2*x^2*log(F)^2 - F^a*b*c^6*log(F) + 3*F^a*b^2*c
^2*d*x*log(F)^2)*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 = 1.69202, size = 454, normalized size = 5.22 \begin{align*} -\frac{F^{a} b^{2}{\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 )^{2} -{\left (d^{6} x^{6} + 6 \, c d^{5} x^{5} + 15 \, c^{2} d^{4} x^{4} + 20 \, c^{3} d^{3} x^{3} + 15 \, c^{4} d^{2} x^{2} + 6 \, c^{5} d x + c^{6} +{\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 )} 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}}}}{6 \, 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)^5,x, algorithm="fricas")

[Out]

-1/6*(F^a*b^2*Ei(b*log(F)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))*log(F)^2 - (d^6*x^6 + 6*c*d^5*x^5 + 15*c^
2*d^4*x^4 + 20*c^3*d^3*x^3 + 15*c^4*d^2*x^2 + 6*c^5*d*x + c^6 + (b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c
^3)*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)**5,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 )}^{5} 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)^5,x, algorithm="giac")

[Out]

integrate((d*x + c)^5*F^(a + b/(d*x + c)^3), x)