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

Optimal. Leaf size=267 \[ \frac{b^2 d^2 f \log ^2(F) F^{a-\frac{b f}{d e-c f}} \text{ExpIntegralEi}\left (\frac{b d \log (F) (e+f x)}{(c+d x) (d e-c f)}\right )}{2 (d e-c f)^4}-\frac{b d^2 \log (F) F^{a-\frac{b f}{d e-c f}} \text{ExpIntegralEi}\left (\frac{b d \log (F) (e+f x)}{(c+d x) (d e-c f)}\right )}{(d e-c f)^3}+\frac{d^2 F^{a+\frac{b}{c+d x}}}{2 f (d e-c f)^2}-\frac{b d^2 \log (F) F^{a+\frac{b}{c+d x}}}{2 (d e-c f)^3}-\frac{F^{a+\frac{b}{c+d x}}}{2 f (e+f x)^2}+\frac{b d \log (F) F^{a+\frac{b}{c+d x}}}{2 (e+f x) (d e-c f)^2} \]

[Out]

(d^2*F^(a + b/(c + d*x)))/(2*f*(d*e - c*f)^2) - F^(a + b/(c + d*x))/(2*f*(e + f*
x)^2) - (b*d^2*F^(a + b/(c + d*x))*Log[F])/(2*(d*e - c*f)^3) + (b*d*F^(a + b/(c
+ d*x))*Log[F])/(2*(d*e - c*f)^2*(e + f*x)) - (b*d^2*F^(a - (b*f)/(d*e - c*f))*E
xpIntegralEi[(b*d*(e + f*x)*Log[F])/((d*e - c*f)*(c + d*x))]*Log[F])/(d*e - c*f)
^3 + (b^2*d^2*f*F^(a - (b*f)/(d*e - c*f))*ExpIntegralEi[(b*d*(e + f*x)*Log[F])/(
(d*e - c*f)*(c + d*x))]*Log[F]^2)/(2*(d*e - c*f)^4)

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Rubi [A]  time = 3.07482, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{b^2 d^2 f \log ^2(F) F^{a-\frac{b f}{d e-c f}} \text{ExpIntegralEi}\left (\frac{b d \log (F) (e+f x)}{(c+d x) (d e-c f)}\right )}{2 (d e-c f)^4}-\frac{b d^2 \log (F) F^{a-\frac{b f}{d e-c f}} \text{ExpIntegralEi}\left (\frac{b d \log (F) (e+f x)}{(c+d x) (d e-c f)}\right )}{(d e-c f)^3}+\frac{d^2 F^{a+\frac{b}{c+d x}}}{2 f (d e-c f)^2}-\frac{b d^2 \log (F) F^{a+\frac{b}{c+d x}}}{2 (d e-c f)^3}-\frac{F^{a+\frac{b}{c+d x}}}{2 f (e+f x)^2}+\frac{b d \log (F) F^{a+\frac{b}{c+d x}}}{2 (e+f x) (d e-c f)^2} \]

Antiderivative was successfully verified.

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

[Out]

(d^2*F^(a + b/(c + d*x)))/(2*f*(d*e - c*f)^2) - F^(a + b/(c + d*x))/(2*f*(e + f*
x)^2) - (b*d^2*F^(a + b/(c + d*x))*Log[F])/(2*(d*e - c*f)^3) + (b*d*F^(a + b/(c
+ d*x))*Log[F])/(2*(d*e - c*f)^2*(e + f*x)) - (b*d^2*F^(a - (b*f)/(d*e - c*f))*E
xpIntegralEi[(b*d*(e + f*x)*Log[F])/((d*e - c*f)*(c + d*x))]*Log[F])/(d*e - c*f)
^3 + (b^2*d^2*f*F^(a - (b*f)/(d*e - c*f))*ExpIntegralEi[(b*d*(e + f*x)*Log[F])/(
(d*e - c*f)*(c + d*x))]*Log[F]^2)/(2*(d*e - c*f)^4)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.552767, size = 0, normalized size = 0. \[ \int \frac{F^{a+\frac{b}{c+d x}}}{(e+f x)^3} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [B]  time = 0.069, size = 521, normalized size = 2. \[ -{\frac{\ln \left ( F \right ) b{d}^{2}}{ \left ( cf-ed \right ) ^{3}}{F}^{{\frac{xda+ac+b}{dx+c}}} \left ({\frac{b\ln \left ( F \right ) }{dx+c}}+\ln \left ( F \right ) a-{\frac{\ln \left ( F \right ) acf}{cf-ed}}+{\frac{\ln \left ( F \right ) ade}{cf-ed}}-{\frac{\ln \left ( F \right ) bf}{cf-ed}} \right ) ^{-1}}-{\frac{\ln \left ( F \right ) b{d}^{2}}{ \left ( cf-ed \right ) ^{3}}{F}^{{\frac{acf-ade+bf}{cf-ed}}}{\it Ei} \left ( 1,-{\frac{b\ln \left ( F \right ) }{dx+c}}-\ln \left ( F \right ) a-{\frac{-\ln \left ( F \right ) acf+\ln \left ( F \right ) ade-\ln \left ( F \right ) bf}{cf-ed}} \right ) }-{\frac{ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{d}^{2}f}{2\, \left ( cf-ed \right ) ^{4}}{F}^{{\frac{xda+ac+b}{dx+c}}} \left ({\frac{b\ln \left ( F \right ) }{dx+c}}+\ln \left ( F \right ) a-{\frac{\ln \left ( F \right ) acf}{cf-ed}}+{\frac{\ln \left ( F \right ) ade}{cf-ed}}-{\frac{\ln \left ( F \right ) bf}{cf-ed}} \right ) ^{-2}}-{\frac{ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{d}^{2}f}{2\, \left ( cf-ed \right ) ^{4}}{F}^{{\frac{xda+ac+b}{dx+c}}} \left ({\frac{b\ln \left ( F \right ) }{dx+c}}+\ln \left ( F \right ) a-{\frac{\ln \left ( F \right ) acf}{cf-ed}}+{\frac{\ln \left ( F \right ) ade}{cf-ed}}-{\frac{\ln \left ( F \right ) bf}{cf-ed}} \right ) ^{-1}}-{\frac{ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{d}^{2}f}{2\, \left ( cf-ed \right ) ^{4}}{F}^{{\frac{acf-ade+bf}{cf-ed}}}{\it Ei} \left ( 1,-{\frac{b\ln \left ( F \right ) }{dx+c}}-\ln \left ( F \right ) a-{\frac{-\ln \left ( F \right ) acf+\ln \left ( F \right ) ade-\ln \left ( F \right ) bf}{cf-ed}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*d^2*ln(F)/(c*f-d*e)^3*F^((a*d*x+a*c+b)/(d*x+c))/(b*ln(F)/(d*x+c)+ln(F)*a-1/(c
*f-d*e)*ln(F)*a*c*f+1/(c*f-d*e)*ln(F)*a*d*e-1/(c*f-d*e)*ln(F)*b*f)-b*d^2*ln(F)/(
c*f-d*e)^3*F^((a*c*f-a*d*e+b*f)/(c*f-d*e))*Ei(1,-b*ln(F)/(d*x+c)-ln(F)*a-(-ln(F)
*a*c*f+ln(F)*a*d*e-ln(F)*b*f)/(c*f-d*e))-1/2*b^2*d^2*ln(F)^2*f/(c*f-d*e)^4*F^((a
*d*x+a*c+b)/(d*x+c))/(b*ln(F)/(d*x+c)+ln(F)*a-1/(c*f-d*e)*ln(F)*a*c*f+1/(c*f-d*e
)*ln(F)*a*d*e-1/(c*f-d*e)*ln(F)*b*f)^2-1/2*b^2*d^2*ln(F)^2*f/(c*f-d*e)^4*F^((a*d
*x+a*c+b)/(d*x+c))/(b*ln(F)/(d*x+c)+ln(F)*a-1/(c*f-d*e)*ln(F)*a*c*f+1/(c*f-d*e)*
ln(F)*a*d*e-1/(c*f-d*e)*ln(F)*b*f)-1/2*b^2*d^2*ln(F)^2*f/(c*f-d*e)^4*F^((a*c*f-a
*d*e+b*f)/(c*f-d*e))*Ei(1,-b*ln(F)/(d*x+c)-ln(F)*a-(-ln(F)*a*c*f+ln(F)*a*d*e-ln(
F)*b*f)/(c*f-d*e))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.292265, size = 749, normalized size = 2.81 \[ \frac{{\left ({\left (b^{2} d^{2} f^{3} x^{2} + 2 \, b^{2} d^{2} e f^{2} x + b^{2} d^{2} e^{2} f\right )} \log \left (F\right )^{2} - 2 \,{\left (b d^{3} e^{3} - b c d^{2} e^{2} f +{\left (b d^{3} e f^{2} - b c d^{2} f^{3}\right )} x^{2} + 2 \,{\left (b d^{3} e^{2} f - b c d^{2} e f^{2}\right )} x\right )} \log \left (F\right )\right )} F^{\frac{a d e -{\left (a c + b\right )} f}{d e - c f}}{\rm Ei}\left (\frac{{\left (b d f x + b d e\right )} \log \left (F\right )}{c d e - c^{2} f +{\left (d^{2} e - c d f\right )} x}\right ) +{\left (2 \, c d^{3} e^{3} - 5 \, c^{2} d^{2} e^{2} f + 4 \, c^{3} d e f^{2} - c^{4} f^{3} +{\left (d^{4} e^{2} f - 2 \, c d^{3} e f^{2} + c^{2} d^{2} f^{3}\right )} x^{2} + 2 \,{\left (d^{4} e^{3} - 2 \, c d^{3} e^{2} f + c^{2} d^{2} e f^{2}\right )} x -{\left (b c d^{2} e^{2} f - b c^{2} d e f^{2} +{\left (b d^{3} e f^{2} - b c d^{2} f^{3}\right )} x^{2} +{\left (b d^{3} e^{2} f - b c^{2} d f^{3}\right )} x\right )} \log \left (F\right )\right )} F^{\frac{a d x + a c + b}{d x + c}}}{2 \,{\left (d^{4} e^{6} - 4 \, c d^{3} e^{5} f + 6 \, c^{2} d^{2} e^{4} f^{2} - 4 \, c^{3} d e^{3} f^{3} + c^{4} e^{2} f^{4} +{\left (d^{4} e^{4} f^{2} - 4 \, c d^{3} e^{3} f^{3} + 6 \, c^{2} d^{2} e^{2} f^{4} - 4 \, c^{3} d e f^{5} + c^{4} f^{6}\right )} x^{2} + 2 \,{\left (d^{4} e^{5} f - 4 \, c d^{3} e^{4} f^{2} + 6 \, c^{2} d^{2} e^{3} f^{3} - 4 \, c^{3} d e^{2} f^{4} + c^{4} e f^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(((b^2*d^2*f^3*x^2 + 2*b^2*d^2*e*f^2*x + b^2*d^2*e^2*f)*log(F)^2 - 2*(b*d^3*
e^3 - b*c*d^2*e^2*f + (b*d^3*e*f^2 - b*c*d^2*f^3)*x^2 + 2*(b*d^3*e^2*f - b*c*d^2
*e*f^2)*x)*log(F))*F^((a*d*e - (a*c + b)*f)/(d*e - c*f))*Ei((b*d*f*x + b*d*e)*lo
g(F)/(c*d*e - c^2*f + (d^2*e - c*d*f)*x)) + (2*c*d^3*e^3 - 5*c^2*d^2*e^2*f + 4*c
^3*d*e*f^2 - c^4*f^3 + (d^4*e^2*f - 2*c*d^3*e*f^2 + c^2*d^2*f^3)*x^2 + 2*(d^4*e^
3 - 2*c*d^3*e^2*f + c^2*d^2*e*f^2)*x - (b*c*d^2*e^2*f - b*c^2*d*e*f^2 + (b*d^3*e
*f^2 - b*c*d^2*f^3)*x^2 + (b*d^3*e^2*f - b*c^2*d*f^3)*x)*log(F))*F^((a*d*x + a*c
+ b)/(d*x + c)))/(d^4*e^6 - 4*c*d^3*e^5*f + 6*c^2*d^2*e^4*f^2 - 4*c^3*d*e^3*f^3
+ c^4*e^2*f^4 + (d^4*e^4*f^2 - 4*c*d^3*e^3*f^3 + 6*c^2*d^2*e^2*f^4 - 4*c^3*d*e*
f^5 + c^4*f^6)*x^2 + 2*(d^4*e^5*f - 4*c*d^3*e^4*f^2 + 6*c^2*d^2*e^3*f^3 - 4*c^3*
d*e^2*f^4 + c^4*e*f^5)*x)

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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))/(f*x+e)**3,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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