3.3.90 \(\int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{10}} \, dx\) [290]

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

[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

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Rubi [A]
time = 0.18, 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 = {2245, 2241} \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/9*F^(a + b*(c + d*x)^3)/(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 2241

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 2245

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*}

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Mathematica [A]
time = 0.25, size = 80, normalized size = 0.66 \begin {gather*} \frac {F^a \left (b^3 \text {Ei}\left (b (c+d x)^3 \log (F)\right ) \log ^3(F)+\frac {F^{b (c+d x)^3} \left (-2-b (c+d x)^3 \log (F)-b^2 (c+d x)^6 \log ^2(F)\right )}{(c+d x)^9}\right )}{18 d} \end {gather*}

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)

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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {F^{a +b \left (d x +c \right )^{3}}}{\left (d x +c \right )^{10}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 431 vs. \(2 (113) = 226\).
time = 0.40, size = 431, normalized size = 3.56 \begin {gather*} \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 \left (F\right )\right ) \log \left (F\right )^{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 \left (F\right )^{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 ) + 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 )}} \end {gather*}

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)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 4369 deep

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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)^10,x, algorithm="giac")

[Out]

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

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

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