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

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

[Out]

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

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Rubi [A]  time = 0.14883, 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 \[ \frac{(c+d x)^3 F^{a+\frac{b}{(c+d x)^3}}}{3 d}-\frac{b F^a \log (F) \text{ExpIntegralEi}\left (\frac{b \log (F)}{(c+d x)^3}\right )}{3 d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.5253, size = 46, normalized size = 0.87 \[ - \frac{F^{a} b \log{\left (F \right )} \operatorname{Ei}{\left (\frac{b \log{\left (F \right )}}{\left (c + d x\right )^{3}} \right )}}{3 d} + \frac{F^{a + \frac{b}{\left (c + d x\right )^{3}}} \left (c + d x\right )^{3}}{3 d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0594042, size = 47, normalized size = 0.89 \[ \frac{F^a \left ((c+d x)^3 F^{\frac{b}{(c+d x)^3}}-b \log (F) \text{ExpIntegralEi}\left (\frac{b \log (F)}{(c+d x)^3}\right )\right )}{3 d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.044, size = 0, normalized size = 0. \[ \int{F}^{a+{\frac{b}{ \left ( dx+c \right ) ^{3}}}} \left ( dx+c \right ) ^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{1}{3} \,{\left (F^{a} d^{2} x^{3} + 3 \, F^{a} c d x^{2} + 3 \, F^{a} c^{2} x\right )} F^{\frac{b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}} + \int \frac{{\left (F^{a} b d^{3} x^{3} \log \left (F\right ) + 3 \, F^{a} b c d^{2} x^{2} \log \left (F\right ) + 3 \, F^{a} b c^{2} d x \log \left (F\right )\right )} F^{\frac{b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(F^a*d^2*x^3 + 3*F^a*c*d*x^2 + 3*F^a*c^2*x)*F^(b/(d^3*x^3 + 3*c*d^2*x^2 + 3*
c^2*d*x + c^3)) + integrate((F^a*b*d^3*x^3*log(F) + 3*F^a*b*c*d^2*x^2*log(F) + 3
*F^a*b*c^2*d*x*log(F))*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 [A]  time = 0.28104, size = 190, normalized size = 3.58 \[ -\frac{F^{a} b{\rm Ei}\left (\frac{b \log \left (F\right )}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) \log \left (F\right ) -{\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )} F^{\frac{a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(F^a*b*Ei(b*log(F)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))*log(F) - (d^3
*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)*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. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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