3.16 \(\int \frac{F^{c (a+b x)}}{d^3+3 d^2 e x+3 d e^2 x^2+e^3 x^3} \, dx\)

Optimal. Leaf size=95 \[ \frac{b^2 c^2 \log ^2(F) F^{c \left (a-\frac{b d}{e}\right )} \text{ExpIntegralEi}\left (\frac{b c \log (F) (d+e x)}{e}\right )}{2 e^3}-\frac{b c \log (F) F^{c (a+b x)}}{2 e^2 (d+e x)}-\frac{F^{c (a+b x)}}{2 e (d+e x)^2} \]

[Out]

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

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Rubi [A]  time = 0.159986, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{b^2 c^2 \log ^2(F) F^{c \left (a-\frac{b d}{e}\right )} \text{ExpIntegralEi}\left (\frac{b c \log (F) (d+e x)}{e}\right )}{2 e^3}-\frac{b c \log (F) F^{c (a+b x)}}{2 e^2 (d+e x)}-\frac{F^{c (a+b x)}}{2 e (d+e x)^2} \]

Antiderivative was successfully verified.

[In]  Int[F^(c*(a + b*x))/(d^3 + 3*d^2*e*x + 3*d*e^2*x^2 + e^3*x^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 34.8704, size = 87, normalized size = 0.92 \[ - \frac{F^{c \left (a + b x\right )} b c \log{\left (F \right )}}{2 e^{2} \left (d + e x\right )} - \frac{F^{c \left (a + b x\right )}}{2 e \left (d + e x\right )^{2}} + \frac{F^{\frac{c \left (a e - b d\right )}{e}} b^{2} c^{2} \log{\left (F \right )}^{2} \operatorname{Ei}{\left (\frac{b c \left (d + e x\right ) \log{\left (F \right )}}{e} \right )}}{2 e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(c*(b*x+a))/(e**3*x**3+3*d*e**2*x**2+3*d**2*e*x+d**3),x)

[Out]

-F**(c*(a + b*x))*b*c*log(F)/(2*e**2*(d + e*x)) - F**(c*(a + b*x))/(2*e*(d + e*x
)**2) + F**(c*(a*e - b*d)/e)*b**2*c**2*log(F)**2*Ei(b*c*(d + e*x)*log(F)/e)/(2*e
**3)

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Mathematica [A]  time = 0.0240768, size = 88, normalized size = 0.93 \[ \frac{F^{c \left (a-\frac{b d}{e}\right )} \left (b^2 c^2 \log ^2(F) (d+e x)^2 \text{ExpIntegralEi}\left (\frac{b c \log (F) (d+e x)}{e}\right )-e F^{\frac{b c (d+e x)}{e}} (b c \log (F) (d+e x)+e)\right )}{2 e^3 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(c*(a + b*x))/(d^3 + 3*d^2*e*x + 3*d*e^2*x^2 + e^3*x^3),x]

[Out]

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

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Maple [A]  time = 0.068, size = 151, normalized size = 1.6 \[ -{\frac{{b}^{2}{c}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}{F}^{c \left ( bx+a \right ) }}{2\,{e}^{3}} \left ( bcx\ln \left ( F \right ) +{\frac{\ln \left ( F \right ) bcd}{e}} \right ) ^{-2}}-{\frac{{b}^{2}{c}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}{F}^{c \left ( bx+a \right ) }}{2\,{e}^{3}} \left ( bcx\ln \left ( F \right ) +{\frac{\ln \left ( F \right ) bcd}{e}} \right ) ^{-1}}-{\frac{{b}^{2}{c}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}}{2\,{e}^{3}}{F}^{{\frac{ \left ( ea-bd \right ) c}{e}}}{\it Ei} \left ( 1,-bcx\ln \left ( F \right ) -\ln \left ( F \right ) ac-{\frac{-eac\ln \left ( F \right ) +\ln \left ( F \right ) bcd}{e}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(c*(b*x+a))/(e^3*x^3+3*d*e^2*x^2+3*d^2*e*x+d^3),x)

[Out]

-1/2*b^2*c^2*ln(F)^2/e^3*F^(c*(b*x+a))/(b*c*x*ln(F)+1/e*ln(F)*b*c*d)^2-1/2*b^2*c
^2*ln(F)^2/e^3*F^(c*(b*x+a))/(b*c*x*ln(F)+1/e*ln(F)*b*c*d)-1/2*b^2*c^2*ln(F)^2/e
^3*F^(c*(a*e-b*d)/e)*Ei(1,-b*c*x*ln(F)-ln(F)*a*c-(-e*a*c*ln(F)+ln(F)*b*c*d)/e)

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Maxima [A]  time = 0.811642, size = 59, normalized size = 0.62 \[ -\frac{F^{a c} exp_integral_e\left (3, -\frac{{\left (e x + d\right )} b c \log \left (F\right )}{e}\right )}{{\left (e x + d\right )}^{2} F^{\frac{b c d}{e}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((b*x + a)*c)/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3),x, algorithm="maxima")

[Out]

-F^(a*c)*exp_integral_e(3, -(e*x + d)*b*c*log(F)/e)/((e*x + d)^2*F^(b*c*d/e)*e)

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Fricas [A]  time = 0.243352, size = 181, normalized size = 1.91 \[ \frac{\frac{{\left (b^{2} c^{2} e^{2} x^{2} + 2 \, b^{2} c^{2} d e x + b^{2} c^{2} d^{2}\right )}{\rm Ei}\left (\frac{{\left (b c e x + b c d\right )} \log \left (F\right )}{e}\right ) \log \left (F\right )^{2}}{F^{\frac{b c d - a c e}{e}}} -{\left (e^{2} +{\left (b c e^{2} x + b c d e\right )} \log \left (F\right )\right )} F^{b c x + a c}}{2 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((b*x + a)*c)/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3),x, algorithm="fricas")

[Out]

1/2*((b^2*c^2*e^2*x^2 + 2*b^2*c^2*d*e*x + b^2*c^2*d^2)*Ei((b*c*e*x + b*c*d)*log(
F)/e)*log(F)^2/F^((b*c*d - a*c*e)/e) - (e^2 + (b*c*e^2*x + b*c*d*e)*log(F))*F^(b
*c*x + a*c))/(e^5*x^2 + 2*d*e^4*x + d^2*e^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F**(c*(b*x+a))/(e**3*x**3+3*d*e**2*x**2+3*d**2*e*x+d**3),x)

[Out]

Integral(F**(c*(a + b*x))/(d + e*x)**3, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{{\left (b x + a\right )} c}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((b*x + a)*c)/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3),x, algorithm="giac")

[Out]

integrate(F^((b*x + a)*c)/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3), x)