3.288 \(\int \frac{F^{a+b (c+d x)^3}}{(c+d x)^4} \, dx\)

Optimal. Leaf size=53 \[ \frac{b F^a \log (F) \text{ExpIntegralEi}\left (b \log (F) (c+d x)^3\right )}{3 d}-\frac{F^{a+b (c+d x)^3}}{3 d (c+d x)^3} \]

[Out]

-F^(a + b*(c + d*x)^3)/(3*d*(c + d*x)^3) + (b*F^a*ExpIntegralEi[b*(c + d*x)^3*Lo
g[F]]*Log[F])/(3*d)

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Rubi [A]  time = 0.202287, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{b F^a \log (F) \text{ExpIntegralEi}\left (b \log (F) (c+d x)^3\right )}{3 d}-\frac{F^{a+b (c+d x)^3}}{3 d (c+d x)^3} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b*(c + d*x)^3)/(c + d*x)^4,x]

[Out]

-F^(a + b*(c + d*x)^3)/(3*d*(c + d*x)^3) + (b*F^a*ExpIntegralEi[b*(c + d*x)^3*Lo
g[F]]*Log[F])/(3*d)

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Rubi in Sympy [A]  time = 9.22558, size = 46, normalized size = 0.87 \[ \frac{F^{a} b \log{\left (F \right )} \operatorname{Ei}{\left (b \left (c + d x\right )^{3} \log{\left (F \right )} \right )}}{3 d} - \frac{F^{a + b \left (c + d x\right )^{3}}}{3 d \left (c + d x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(a+b*(d*x+c)**3)/(d*x+c)**4,x)

[Out]

F**a*b*log(F)*Ei(b*(c + d*x)**3*log(F))/(3*d) - F**(a + b*(c + d*x)**3)/(3*d*(c
+ d*x)**3)

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Mathematica [A]  time = 0.0494521, size = 47, normalized size = 0.89 \[ \frac{F^a \left (b \log (F) \text{ExpIntegralEi}\left (b \log (F) (c+d x)^3\right )-\frac{F^{b (c+d x)^3}}{(c+d x)^3}\right )}{3 d} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(a + b*(c + d*x)^3)/(c + d*x)^4,x]

[Out]

(F^a*(-(F^(b*(c + d*x)^3)/(c + d*x)^3) + b*ExpIntegralEi[b*(c + d*x)^3*Log[F]]*L
og[F]))/(3*d)

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Maple [F]  time = 0.06, size = 0, normalized size = 0. \[ \int{\frac{{F}^{a+b \left ( dx+c \right ) ^{3}}}{ \left ( dx+c \right ) ^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b*(d*x+c)^3)/(d*x+c)^4,x)

[Out]

int(F^(a+b*(d*x+c)^3)/(d*x+c)^4,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{{\left (d x + c\right )}^{3} b + a}}{{\left (d x + c\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((d*x + c)^3*b + a)/(d*x + c)^4,x, algorithm="maxima")

[Out]

integrate(F^((d*x + c)^3*b + a)/(d*x + c)^4, x)

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Fricas [A]  time = 0.259746, size = 198, normalized size = 3.74 \[ \frac{{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} F^{a}{\rm Ei}\left ({\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right )\right ) \log \left (F\right ) - F^{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}}{3 \,{\left (d^{4} x^{3} + 3 \, c d^{3} x^{2} + 3 \, c^{2} d^{2} x + c^{3} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((d*x + c)^3*b + a)/(d*x + c)^4,x, algorithm="fricas")

[Out]

1/3*((b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3)*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) - F^(b*d^3*x^3 + 3*b*c*d^2*x^2 +
3*b*c^2*d*x + b*c^3 + a))/(d^4*x^3 + 3*c*d^3*x^2 + 3*c^2*d^2*x + c^3*d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F**(a+b*(d*x+c)**3)/(d*x+c)**4,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{{\left (d x + c\right )}^{3} b + a}}{{\left (d x + c\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((d*x + c)^3*b + a)/(d*x + c)^4,x, algorithm="giac")

[Out]

integrate(F^((d*x + c)^3*b + a)/(d*x + c)^4, x)