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

Optimal. Leaf size=49 \[ -\frac {F^a \left (-b \log (F) (c+d x)^3\right )^{4/3} \Gamma \left (-\frac {4}{3},-b (c+d x)^3 \log (F)\right )}{3 d (c+d x)^4} \]

[Out]

-1/3*F^a*GAMMA(-4/3,-b*(d*x+c)^3*ln(F))*(-b*(d*x+c)^3*ln(F))^(4/3)/d/(d*x+c)^4

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Rubi [A]  time = 0.06, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2218} \[ -\frac {F^a \left (-b \log (F) (c+d x)^3\right )^{4/3} \text {Gamma}\left (-\frac {4}{3},-b \log (F) (c+d x)^3\right )}{3 d (c+d x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2218

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Simp[(F^a*(e + f*
x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ
[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin {align*} \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^5} \, dx &=-\frac {F^a \Gamma \left (-\frac {4}{3},-b (c+d x)^3 \log (F)\right ) \left (-b (c+d x)^3 \log (F)\right )^{4/3}}{3 d (c+d x)^4}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 49, normalized size = 1.00 \[ -\frac {F^a \left (-b \log (F) (c+d x)^3\right )^{4/3} \Gamma \left (-\frac {4}{3},-b (c+d x)^3 \log (F)\right )}{3 d (c+d x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.45, size = 226, normalized size = 4.61 \[ \frac {3 \, {\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4}\right )} \left (-b d^{3} \log \relax (F)\right )^{\frac {1}{3}} F^{a} \Gamma \left (\frac {2}{3}, -{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \relax (F)\right ) \log \relax (F) - {\left (3 \, {\left (b d^{4} x^{3} + 3 \, b c d^{3} x^{2} + 3 \, b c^{2} d^{2} x + b c^{3} d\right )} \log \relax (F) + d\right )} F^{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}}{4 \, {\left (d^{6} x^{4} + 4 \, c d^{5} x^{3} + 6 \, c^{2} d^{4} x^{2} + 4 \, c^{3} d^{3} x + c^{4} d^{2}\right )}} \]

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]

1/4*(3*(b*d^4*x^4 + 4*b*c*d^3*x^3 + 6*b*c^2*d^2*x^2 + 4*b*c^3*d*x + b*c^4)*(-b*d^3*log(F))^(1/3)*F^a*gamma(2/3
, -(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*d^4*x^3 + 3*b*c*d^3*x^2 + 3*b*c^2*
d^2*x + b*c^3*d)*log(F) + d)*F^(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a))/(d^6*x^4 + 4*c*d^5*x^3 +
6*c^2*d^4*x^2 + 4*c^3*d^3*x + c^4*d^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{{\left (d x + c\right )}^{3} b + a}}{{\left (d x + c\right )}^{5}}\,{d x} \]

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

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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {F^{a +\left (d x +c \right )^{3} b}}{\left (d x +c \right )^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{{\left (d x + c\right )}^{3} b + a}}{{\left (d x + c\right )}^{5}}\,{d x} \]

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]

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

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mupad [B]  time = 4.47, size = 130, normalized size = 2.65 \[ \frac {3\,F^a\,\Gamma \left (\frac {2}{3}\right )\,{\left (-b\,\ln \relax (F)\,{\left (c+d\,x\right )}^3\right )}^{4/3}}{4\,d\,{\left (c+d\,x\right )}^4}-\frac {F^a\,F^{b\,{\left (c+d\,x\right )}^3}}{4\,d\,{\left (c+d\,x\right )}^4}-\frac {3\,F^a\,\Gamma \left (\frac {2}{3},-b\,\ln \relax (F)\,{\left (c+d\,x\right )}^3\right )\,{\left (-b\,\ln \relax (F)\,{\left (c+d\,x\right )}^3\right )}^{4/3}}{4\,d\,{\left (c+d\,x\right )}^4}-\frac {3\,F^a\,F^{b\,{\left (c+d\,x\right )}^3}\,b\,\ln \relax (F)}{4\,d\,\left (c+d\,x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*F^a*gamma(2/3)*(-b*log(F)*(c + d*x)^3)^(4/3))/(4*d*(c + d*x)^4) - (F^a*F^(b*(c + d*x)^3))/(4*d*(c + d*x)^4)
- (3*F^a*igamma(2/3, -b*log(F)*(c + d*x)^3)*(-b*log(F)*(c + d*x)^3)^(4/3))/(4*d*(c + d*x)^4) - (3*F^a*F^(b*(c
+ d*x)^3)*b*log(F))/(4*d*(c + d*x))

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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)**5,x)

[Out]

Timed out

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