3.400 \(\int \frac{F^{a+\frac{b}{c+d x}}}{(e+f x)^4} \, dx\)

Optimal. Leaf size=460 \[ -\frac{b^3 d^3 f^2 \log ^3(F) F^{a-\frac{b f}{d e-c f}} \text{ExpIntegralEi}\left (\frac{b d \log (F) (e+f x)}{(c+d x) (d e-c f)}\right )}{6 (d e-c f)^6}+\frac{b^2 d^3 f \log ^2(F) F^{a-\frac{b f}{d e-c f}} \text{ExpIntegralEi}\left (\frac{b d \log (F) (e+f x)}{(c+d x) (d e-c f)}\right )}{(d e-c f)^5}+\frac{b^2 d^3 f \log ^2(F) F^{a+\frac{b}{c+d x}}}{6 (d e-c f)^5}-\frac{b^2 d^2 f \log ^2(F) F^{a+\frac{b}{c+d x}}}{6 (e+f x) (d e-c f)^4}-\frac{b d^3 \log (F) F^{a-\frac{b f}{d e-c f}} \text{ExpIntegralEi}\left (\frac{b d \log (F) (e+f x)}{(c+d x) (d e-c f)}\right )}{(d e-c f)^4}+\frac{d^3 F^{a+\frac{b}{c+d x}}}{3 f (d e-c f)^3}-\frac{5 b d^3 \log (F) F^{a+\frac{b}{c+d x}}}{6 (d e-c f)^4}+\frac{2 b d^2 \log (F) F^{a+\frac{b}{c+d x}}}{3 (e+f x) (d e-c f)^3}-\frac{F^{a+\frac{b}{c+d x}}}{3 f (e+f x)^3}+\frac{b d \log (F) F^{a+\frac{b}{c+d x}}}{6 (e+f x)^2 (d e-c f)^2} \]

[Out]

(d^3*F^(a + b/(c + d*x)))/(3*f*(d*e - c*f)^3) - F^(a + b/(c + d*x))/(3*f*(e + f*
x)^3) - (5*b*d^3*F^(a + b/(c + d*x))*Log[F])/(6*(d*e - c*f)^4) + (b*d*F^(a + b/(
c + d*x))*Log[F])/(6*(d*e - c*f)^2*(e + f*x)^2) + (2*b*d^2*F^(a + b/(c + d*x))*L
og[F])/(3*(d*e - c*f)^3*(e + f*x)) - (b*d^3*F^(a - (b*f)/(d*e - c*f))*ExpIntegra
lEi[(b*d*(e + f*x)*Log[F])/((d*e - c*f)*(c + d*x))]*Log[F])/(d*e - c*f)^4 + (b^2
*d^3*f*F^(a + b/(c + d*x))*Log[F]^2)/(6*(d*e - c*f)^5) - (b^2*d^2*f*F^(a + b/(c
+ d*x))*Log[F]^2)/(6*(d*e - c*f)^4*(e + f*x)) + (b^2*d^3*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)/(d*
e - c*f)^5 - (b^3*d^3*f^2*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]^3)/(6*(d*e - c*f)^6)

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Rubi [A]  time = 6.21191, antiderivative size = 460, normalized size of antiderivative = 1., number of steps used = 36, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{b^3 d^3 f^2 \log ^3(F) F^{a-\frac{b f}{d e-c f}} \text{ExpIntegralEi}\left (\frac{b d \log (F) (e+f x)}{(c+d x) (d e-c f)}\right )}{6 (d e-c f)^6}+\frac{b^2 d^3 f \log ^2(F) F^{a-\frac{b f}{d e-c f}} \text{ExpIntegralEi}\left (\frac{b d \log (F) (e+f x)}{(c+d x) (d e-c f)}\right )}{(d e-c f)^5}+\frac{b^2 d^3 f \log ^2(F) F^{a+\frac{b}{c+d x}}}{6 (d e-c f)^5}-\frac{b^2 d^2 f \log ^2(F) F^{a+\frac{b}{c+d x}}}{6 (e+f x) (d e-c f)^4}-\frac{b d^3 \log (F) F^{a-\frac{b f}{d e-c f}} \text{ExpIntegralEi}\left (\frac{b d \log (F) (e+f x)}{(c+d x) (d e-c f)}\right )}{(d e-c f)^4}+\frac{d^3 F^{a+\frac{b}{c+d x}}}{3 f (d e-c f)^3}-\frac{5 b d^3 \log (F) F^{a+\frac{b}{c+d x}}}{6 (d e-c f)^4}+\frac{2 b d^2 \log (F) F^{a+\frac{b}{c+d x}}}{3 (e+f x) (d e-c f)^3}-\frac{F^{a+\frac{b}{c+d x}}}{3 f (e+f x)^3}+\frac{b d \log (F) F^{a+\frac{b}{c+d x}}}{6 (e+f x)^2 (d e-c f)^2} \]

Antiderivative was successfully verified.

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

[Out]

(d^3*F^(a + b/(c + d*x)))/(3*f*(d*e - c*f)^3) - F^(a + b/(c + d*x))/(3*f*(e + f*
x)^3) - (5*b*d^3*F^(a + b/(c + d*x))*Log[F])/(6*(d*e - c*f)^4) + (b*d*F^(a + b/(
c + d*x))*Log[F])/(6*(d*e - c*f)^2*(e + f*x)^2) + (2*b*d^2*F^(a + b/(c + d*x))*L
og[F])/(3*(d*e - c*f)^3*(e + f*x)) - (b*d^3*F^(a - (b*f)/(d*e - c*f))*ExpIntegra
lEi[(b*d*(e + f*x)*Log[F])/((d*e - c*f)*(c + d*x))]*Log[F])/(d*e - c*f)^4 + (b^2
*d^3*f*F^(a + b/(c + d*x))*Log[F]^2)/(6*(d*e - c*f)^5) - (b^2*d^2*f*F^(a + b/(c
+ d*x))*Log[F]^2)/(6*(d*e - c*f)^4*(e + f*x)) + (b^2*d^3*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)/(d*
e - c*f)^5 - (b^3*d^3*f^2*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]^3)/(6*(d*e - c*f)^6)

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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)**4,x)

[Out]

Timed out

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Mathematica [A]  time = 0.31183, size = 0, normalized size = 0. \[ \int \frac{F^{a+\frac{b}{c+d x}}}{(e+f x)^4} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [B]  time = 0.115, size = 952, normalized size = 2.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*d^3*ln(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)+b*d^3*ln(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))+1/3*b^3*d^3*ln(F)^3*f^2/(c*f-d*e)^6*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)^3+1/6*b^3*d^3*ln(F)^3*f^2/(c*f-d*e)^6*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+1/6*b^3*d^3*ln(F)^3*f^2/(c*f-d*e)^6*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)+1/6*b^3*d^3*ln(F)^3*f^2/(c*f-d*e)^6*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))+b^2*d^3*ln(F)^2*f/(c*f-d*e)^5*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+b^2*d^3*ln(F)^2*f/(c*f-d*e)^5*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)*l
n(F)*b*f)+b^2*d^3*ln(F)^2*f/(c*f-d*e)^5*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. \[ \int \frac{F^{a + \frac{b}{d x + c}}}{{\left (f x + e\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.284307, size = 1858, normalized size = 4.04 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(((b^3*d^3*f^5*x^3 + 3*b^3*d^3*e*f^4*x^2 + 3*b^3*d^3*e^2*f^3*x + b^3*d^3*e^
3*f^2)*log(F)^3 - 6*(b^2*d^4*e^4*f - b^2*c*d^3*e^3*f^2 + (b^2*d^4*e*f^4 - b^2*c*
d^3*f^5)*x^3 + 3*(b^2*d^4*e^2*f^3 - b^2*c*d^3*e*f^4)*x^2 + 3*(b^2*d^4*e^3*f^2 -
b^2*c*d^3*e^2*f^3)*x)*log(F)^2 + 6*(b*d^5*e^5 - 2*b*c*d^4*e^4*f + b*c^2*d^3*e^3*
f^2 + (b*d^5*e^2*f^3 - 2*b*c*d^4*e*f^4 + b*c^2*d^3*f^5)*x^3 + 3*(b*d^5*e^3*f^2 -
2*b*c*d^4*e^2*f^3 + b*c^2*d^3*e*f^4)*x^2 + 3*(b*d^5*e^4*f - 2*b*c*d^4*e^3*f^2 +
b*c^2*d^3*e^2*f^3)*x)*log(F))*F^((a*d*e - (a*c + b)*f)/(d*e - c*f))*Ei((b*d*f*x
+ b*d*e)*log(F)/(c*d*e - c^2*f + (d^2*e - c*d*f)*x)) - (6*c*d^5*e^5 - 24*c^2*d^
4*e^4*f + 38*c^3*d^3*e^3*f^2 - 30*c^4*d^2*e^2*f^3 + 12*c^5*d*e*f^4 - 2*c^6*f^5 +
2*(d^6*e^3*f^2 - 3*c*d^5*e^2*f^3 + 3*c^2*d^4*e*f^4 - c^3*d^3*f^5)*x^3 + 6*(d^6*
e^4*f - 3*c*d^5*e^3*f^2 + 3*c^2*d^4*e^2*f^3 - c^3*d^3*e*f^4)*x^2 + (b^2*c*d^3*e^
3*f^2 - b^2*c^2*d^2*e^2*f^3 + (b^2*d^4*e*f^4 - b^2*c*d^3*f^5)*x^3 + (2*b^2*d^4*e
^2*f^3 - b^2*c*d^3*e*f^4 - b^2*c^2*d^2*f^5)*x^2 + (b^2*d^4*e^3*f^2 + b^2*c*d^3*e
^2*f^3 - 2*b^2*c^2*d^2*e*f^4)*x)*log(F)^2 + 6*(d^6*e^5 - 3*c*d^5*e^4*f + 3*c^2*d
^4*e^3*f^2 - c^3*d^3*e^2*f^3)*x - (6*b*c*d^4*e^4*f - 13*b*c^2*d^3*e^3*f^2 + 8*b*
c^3*d^2*e^2*f^3 - b*c^4*d*e*f^4 + 5*(b*d^5*e^2*f^3 - 2*b*c*d^4*e*f^4 + b*c^2*d^3
*f^5)*x^3 + (11*b*d^5*e^3*f^2 - 18*b*c*d^4*e^2*f^3 + 3*b*c^2*d^3*e*f^4 + 4*b*c^3
*d^2*f^5)*x^2 + (6*b*d^5*e^4*f - 2*b*c*d^4*e^3*f^2 - 15*b*c^2*d^3*e^2*f^3 + 12*b
*c^3*d^2*e*f^4 - b*c^4*d*f^5)*x)*log(F))*F^((a*d*x + a*c + b)/(d*x + c)))/(d^6*e
^9 - 6*c*d^5*e^8*f + 15*c^2*d^4*e^7*f^2 - 20*c^3*d^3*e^6*f^3 + 15*c^4*d^2*e^5*f^
4 - 6*c^5*d*e^4*f^5 + c^6*e^3*f^6 + (d^6*e^6*f^3 - 6*c*d^5*e^5*f^4 + 15*c^2*d^4*
e^4*f^5 - 20*c^3*d^3*e^3*f^6 + 15*c^4*d^2*e^2*f^7 - 6*c^5*d*e*f^8 + c^6*f^9)*x^3
+ 3*(d^6*e^7*f^2 - 6*c*d^5*e^6*f^3 + 15*c^2*d^4*e^5*f^4 - 20*c^3*d^3*e^4*f^5 +
15*c^4*d^2*e^3*f^6 - 6*c^5*d*e^2*f^7 + c^6*e*f^8)*x^2 + 3*(d^6*e^8*f - 6*c*d^5*e
^7*f^2 + 15*c^2*d^4*e^6*f^3 - 20*c^3*d^3*e^5*f^4 + 15*c^4*d^2*e^4*f^5 - 6*c^5*d*
e^3*f^6 + c^6*e^2*f^7)*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)**4,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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