3.316 \(\int F^{a+\frac{b}{(c+d x)^2}} (c+d x)^7 \, dx\)

Optimal. Leaf size=31 \[ \frac{b^4 F^a \log ^4(F) \text{Gamma}\left (-4,-\frac{b \log (F)}{(c+d x)^2}\right )}{2 d} \]

[Out]

(b^4*F^a*Gamma[-4, -((b*Log[F])/(c + d*x)^2)]*Log[F]^4)/(2*d)

_______________________________________________________________________________________

Rubi [A]  time = 0.0744386, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{b^4 F^a \log ^4(F) \text{Gamma}\left (-4,-\frac{b \log (F)}{(c+d x)^2}\right )}{2 d} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b/(c + d*x)^2)*(c + d*x)^7,x]

[Out]

(b^4*F^a*Gamma[-4, -((b*Log[F])/(c + d*x)^2)]*Log[F]^4)/(2*d)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.24837, size = 31, normalized size = 1. \[ \frac{F^{a} b^{4} \Gamma{\left (-4,- \frac{b \log{\left (F \right )}}{\left (c + d x\right )^{2}} \right )} \log{\left (F \right )}^{4}}{2 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(a+b/(d*x+c)**2)*(d*x+c)**7,x)

[Out]

F**a*b**4*Gamma(-4, -b*log(F)/(c + d*x)**2)*log(F)**4/(2*d)

_______________________________________________________________________________________

Mathematica [B]  time = 0.147699, size = 96, normalized size = 3.1 \[ \frac{F^a \left ((c+d x)^2 F^{\frac{b}{(c+d x)^2}} \left (b^3 \log ^3(F)+b^2 \log ^2(F) (c+d x)^2+2 b \log (F) (c+d x)^4+6 (c+d x)^6\right )-b^4 \log ^4(F) \text{ExpIntegralEi}\left (\frac{b \log (F)}{(c+d x)^2}\right )\right )}{48 d} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(a + b/(c + d*x)^2)*(c + d*x)^7,x]

[Out]

(F^a*(-(b^4*ExpIntegralEi[(b*Log[F])/(c + d*x)^2]*Log[F]^4) + F^(b/(c + d*x)^2)*
(c + d*x)^2*(6*(c + d*x)^6 + 2*b*(c + d*x)^4*Log[F] + b^2*(c + d*x)^2*Log[F]^2 +
b^3*Log[F]^3)))/(48*d)

_______________________________________________________________________________________

Maple [B]  time = 0.066, size = 646, normalized size = 20.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b/(d*x+c)^2)*(d*x+c)^7,x)

[Out]

1/8*F^a*d^7*F^(b/(d*x+c)^2)*x^8+F^a*F^(b/(d*x+c)^2)*c^7*x+1/8*F^a/d*F^(b/(d*x+c)
^2)*c^8+1/12*F^a*d^2*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c*x^3+1/8*F^a*d*b^2*ln(F)^2*F^(
b/(d*x+c)^2)*c^2*x^2+1/4*F^a*d^4*b*ln(F)*F^(b/(d*x+c)^2)*c*x^5+5/8*F^a*d^3*b*ln(
F)*F^(b/(d*x+c)^2)*c^2*x^4+5/6*F^a*d^2*b*ln(F)*F^(b/(d*x+c)^2)*c^3*x^3+5/8*F^a*d
*b*ln(F)*F^(b/(d*x+c)^2)*c^4*x^2+1/4*F^a*b*ln(F)*F^(b/(d*x+c)^2)*c^5*x+1/12*F^a*
b^2*ln(F)^2*F^(b/(d*x+c)^2)*c^3*x+1/24*F^a*b^3*ln(F)^3*F^(b/(d*x+c)^2)*c*x+1/24*
F^a/d*b*ln(F)*F^(b/(d*x+c)^2)*c^6+1/48*F^a/d*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c^4+1/4
8*F^a/d*b^3*ln(F)^3*F^(b/(d*x+c)^2)*c^2+1/24*F^a*d^5*b*ln(F)*F^(b/(d*x+c)^2)*x^6
+1/48*F^a*d^3*b^2*ln(F)^2*F^(b/(d*x+c)^2)*x^4+1/48*F^a*d*b^3*ln(F)^3*F^(b/(d*x+c
)^2)*x^2+F^a*d^6*F^(b/(d*x+c)^2)*c*x^7+7/2*F^a*d^5*F^(b/(d*x+c)^2)*c^2*x^6+7*F^a
*d^4*F^(b/(d*x+c)^2)*c^3*x^5+35/4*F^a*d^3*F^(b/(d*x+c)^2)*c^4*x^4+7*F^a*d^2*F^(b
/(d*x+c)^2)*c^5*x^3+7/2*F^a*d*F^(b/(d*x+c)^2)*c^6*x^2+1/48*F^a/d*b^4*ln(F)^4*Ei(
1,-b*ln(F)/(d*x+c)^2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{1}{48} \,{\left (6 \, F^{a} d^{7} x^{8} + 48 \, F^{a} c d^{6} x^{7} + 2 \,{\left (84 \, F^{a} c^{2} d^{5} + F^{a} b d^{5} \log \left (F\right )\right )} x^{6} + 12 \,{\left (28 \, F^{a} c^{3} d^{4} + F^{a} b c d^{4} \log \left (F\right )\right )} x^{5} +{\left (420 \, F^{a} c^{4} d^{3} + 30 \, F^{a} b c^{2} d^{3} \log \left (F\right ) + F^{a} b^{2} d^{3} \log \left (F\right )^{2}\right )} x^{4} + 4 \,{\left (84 \, F^{a} c^{5} d^{2} + 10 \, F^{a} b c^{3} d^{2} \log \left (F\right ) + F^{a} b^{2} c d^{2} \log \left (F\right )^{2}\right )} x^{3} +{\left (168 \, F^{a} c^{6} d + 30 \, F^{a} b c^{4} d \log \left (F\right ) + 6 \, F^{a} b^{2} c^{2} d \log \left (F\right )^{2} + F^{a} b^{3} d \log \left (F\right )^{3}\right )} x^{2} + 2 \,{\left (24 \, F^{a} c^{7} + 6 \, F^{a} b c^{5} \log \left (F\right ) + 2 \, F^{a} b^{2} c^{3} \log \left (F\right )^{2} + F^{a} b^{3} c \log \left (F\right )^{3}\right )} x\right )} F^{\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}} + \int \frac{{\left (F^{a} b^{4} d^{2} x^{2} \log \left (F\right )^{4} + 2 \, F^{a} b^{4} c d x \log \left (F\right )^{4} - 6 \, F^{a} b c^{8} \log \left (F\right ) - 2 \, F^{a} b^{2} c^{6} \log \left (F\right )^{2} - F^{a} b^{3} c^{4} \log \left (F\right )^{3}\right )} F^{\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{24 \,{\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^7*F^(a + b/(d*x + c)^2),x, algorithm="maxima")

[Out]

1/48*(6*F^a*d^7*x^8 + 48*F^a*c*d^6*x^7 + 2*(84*F^a*c^2*d^5 + F^a*b*d^5*log(F))*x
^6 + 12*(28*F^a*c^3*d^4 + F^a*b*c*d^4*log(F))*x^5 + (420*F^a*c^4*d^3 + 30*F^a*b*
c^2*d^3*log(F) + F^a*b^2*d^3*log(F)^2)*x^4 + 4*(84*F^a*c^5*d^2 + 10*F^a*b*c^3*d^
2*log(F) + F^a*b^2*c*d^2*log(F)^2)*x^3 + (168*F^a*c^6*d + 30*F^a*b*c^4*d*log(F)
+ 6*F^a*b^2*c^2*d*log(F)^2 + F^a*b^3*d*log(F)^3)*x^2 + 2*(24*F^a*c^7 + 6*F^a*b*c
^5*log(F) + 2*F^a*b^2*c^3*log(F)^2 + F^a*b^3*c*log(F)^3)*x)*F^(b/(d^2*x^2 + 2*c*
d*x + c^2)) + integrate(1/24*(F^a*b^4*d^2*x^2*log(F)^4 + 2*F^a*b^4*c*d*x*log(F)^
4 - 6*F^a*b*c^8*log(F) - 2*F^a*b^2*c^6*log(F)^2 - F^a*b^3*c^4*log(F)^3)*F^(b/(d^
2*x^2 + 2*c*d*x + c^2))/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.264485, size = 447, normalized size = 14.42 \[ -\frac{F^{a} b^{4}{\rm Ei}\left (\frac{b \log \left (F\right )}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) \log \left (F\right )^{4} -{\left (6 \, d^{8} x^{8} + 48 \, c d^{7} x^{7} + 168 \, c^{2} d^{6} x^{6} + 336 \, c^{3} d^{5} x^{5} + 420 \, c^{4} d^{4} x^{4} + 336 \, c^{5} d^{3} x^{3} + 168 \, c^{6} d^{2} x^{2} + 48 \, c^{7} d x + 6 \, c^{8} +{\left (b^{3} d^{2} x^{2} + 2 \, b^{3} c d x + b^{3} c^{2}\right )} \log \left (F\right )^{3} +{\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \left (F\right )^{2} + 2 \,{\left (b d^{6} x^{6} + 6 \, b c d^{5} x^{5} + 15 \, b c^{2} d^{4} x^{4} + 20 \, b c^{3} d^{3} x^{3} + 15 \, b c^{4} d^{2} x^{2} + 6 \, b c^{5} d x + b c^{6}\right )} \log \left (F\right )\right )} F^{\frac{a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{48 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^7*F^(a + b/(d*x + c)^2),x, algorithm="fricas")

[Out]

-1/48*(F^a*b^4*Ei(b*log(F)/(d^2*x^2 + 2*c*d*x + c^2))*log(F)^4 - (6*d^8*x^8 + 48
*c*d^7*x^7 + 168*c^2*d^6*x^6 + 336*c^3*d^5*x^5 + 420*c^4*d^4*x^4 + 336*c^5*d^3*x
^3 + 168*c^6*d^2*x^2 + 48*c^7*d*x + 6*c^8 + (b^3*d^2*x^2 + 2*b^3*c*d*x + b^3*c^2
)*log(F)^3 + (b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x
+ b^2*c^4)*log(F)^2 + 2*(b*d^6*x^6 + 6*b*c*d^5*x^5 + 15*b*c^2*d^4*x^4 + 20*b*c^3
*d^3*x^3 + 15*b*c^4*d^2*x^2 + 6*b*c^5*d*x + b*c^6)*log(F))*F^((a*d^2*x^2 + 2*a*c
*d*x + a*c^2 + b)/(d^2*x^2 + 2*c*d*x + c^2)))/d

_______________________________________________________________________________________

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)**2)*(d*x+c)**7,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (d x + c\right )}^{7} F^{a + \frac{b}{{\left (d x + c\right )}^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^7*F^(a + b/(d*x + c)^2),x, algorithm="giac")

[Out]

integrate((d*x + c)^7*F^(a + b/(d*x + c)^2), x)