3.324 \(\int \frac{F^{a+\frac{b}{(c+d x)^2}}}{(c+d x)^9} \, dx\)

Optimal. Leaf size=126 \[ \frac{3 F^{a+\frac{b}{(c+d x)^2}}}{b^4 d \log ^4(F)}-\frac{3 F^{a+\frac{b}{(c+d x)^2}}}{b^3 d \log ^3(F) (c+d x)^2}+\frac{3 F^{a+\frac{b}{(c+d x)^2}}}{2 b^2 d \log ^2(F) (c+d x)^4}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)^6} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.301123, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{3 F^{a+\frac{b}{(c+d x)^2}}}{b^4 d \log ^4(F)}-\frac{3 F^{a+\frac{b}{(c+d x)^2}}}{b^3 d \log ^3(F) (c+d x)^2}+\frac{3 F^{a+\frac{b}{(c+d x)^2}}}{2 b^2 d \log ^2(F) (c+d x)^4}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)^6} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b/(c + d*x)^2)/(c + d*x)^9,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 28.8236, size = 112, normalized size = 0.89 \[ - \frac{F^{a + \frac{b}{\left (c + d x\right )^{2}}}}{2 b d \left (c + d x\right )^{6} \log{\left (F \right )}} + \frac{3 F^{a + \frac{b}{\left (c + d x\right )^{2}}}}{2 b^{2} d \left (c + d x\right )^{4} \log{\left (F \right )}^{2}} - \frac{3 F^{a + \frac{b}{\left (c + d x\right )^{2}}}}{b^{3} d \left (c + d x\right )^{2} \log{\left (F \right )}^{3}} + \frac{3 F^{a + \frac{b}{\left (c + d x\right )^{2}}}}{b^{4} d \log{\left (F \right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(a+b/(d*x+c)**2)/(d*x+c)**9,x)

[Out]

-F**(a + b/(c + d*x)**2)/(2*b*d*(c + d*x)**6*log(F)) + 3*F**(a + b/(c + d*x)**2)
/(2*b**2*d*(c + d*x)**4*log(F)**2) - 3*F**(a + b/(c + d*x)**2)/(b**3*d*(c + d*x)
**2*log(F)**3) + 3*F**(a + b/(c + d*x)**2)/(b**4*d*log(F)**4)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0628498, size = 73, normalized size = 0.58 \[ \frac{F^{a+\frac{b}{(c+d x)^2}} \left (-\frac{b^3 \log ^3(F)}{(c+d x)^6}+\frac{3 b^2 \log ^2(F)}{(c+d x)^4}-\frac{6 b \log (F)}{(c+d x)^2}+6\right )}{2 b^4 d \log ^4(F)} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(a + b/(c + d*x)^2)/(c + d*x)^9,x]

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.128, size = 444, normalized size = 3.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b/(d*x+c)^2)/(d*x+c)^9,x)

[Out]

(3*d^7/ln(F)^4/b^4*x^8*exp((a+b/(d*x+c)^2)*ln(F))-c*(ln(F)^3*b^3-6*b^2*c^2*ln(F)
^2+18*ln(F)*b*c^4-24*c^6)/b^4/ln(F)^4*x*exp((a+b/(d*x+c)^2)*ln(F))-1/2*d*(ln(F)^
3*b^3-18*b^2*c^2*ln(F)^2+90*ln(F)*b*c^4-168*c^6)/ln(F)^4/b^4*x^2*exp((a+b/(d*x+c
)^2)*ln(F))+3/2*d^3*(ln(F)^2*b^2-30*ln(F)*b*c^2+140*c^4)/ln(F)^4/b^4*x^4*exp((a+
b/(d*x+c)^2)*ln(F))-3*d^5*(b*ln(F)-28*c^2)/ln(F)^4/b^4*x^6*exp((a+b/(d*x+c)^2)*l
n(F))+24*d^6*c/ln(F)^4/b^4*x^7*exp((a+b/(d*x+c)^2)*ln(F))-1/2*(ln(F)^3*b^3-3*b^2
*c^2*ln(F)^2+6*ln(F)*b*c^4-6*c^6)*c^2/b^4/ln(F)^4/d*exp((a+b/(d*x+c)^2)*ln(F))+6
*c*d^2*(ln(F)^2*b^2-10*ln(F)*b*c^2+28*c^4)/ln(F)^4/b^4*x^3*exp((a+b/(d*x+c)^2)*l
n(F))-6*c*d^4*(3*b*ln(F)-28*c^2)/ln(F)^4/b^4*x^5*exp((a+b/(d*x+c)^2)*ln(F)))/(d*
x+c)^8

_______________________________________________________________________________________

Maxima [A]  time = 0.783292, size = 471, normalized size = 3.74 \[ \frac{{\left (6 \, F^{a} d^{6} x^{6} + 36 \, F^{a} c d^{5} x^{5} + 6 \, F^{a} c^{6} - 6 \, F^{a} b c^{4} \log \left (F\right ) + 3 \, F^{a} b^{2} c^{2} \log \left (F\right )^{2} - F^{a} b^{3} \log \left (F\right )^{3} + 6 \,{\left (15 \, F^{a} c^{2} d^{4} - F^{a} b d^{4} \log \left (F\right )\right )} x^{4} + 24 \,{\left (5 \, F^{a} c^{3} d^{3} - F^{a} b c d^{3} \log \left (F\right )\right )} x^{3} + 3 \,{\left (30 \, F^{a} c^{4} d^{2} - 12 \, F^{a} b c^{2} d^{2} \log \left (F\right ) + F^{a} b^{2} d^{2} \log \left (F\right )^{2}\right )} x^{2} + 6 \,{\left (6 \, F^{a} c^{5} d - 4 \, F^{a} b c^{3} d \log \left (F\right ) + F^{a} b^{2} c d \log \left (F\right )^{2}\right )} x\right )} F^{\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \,{\left (b^{4} d^{7} x^{6} \log \left (F\right )^{4} + 6 \, b^{4} c d^{6} x^{5} \log \left (F\right )^{4} + 15 \, b^{4} c^{2} d^{5} x^{4} \log \left (F\right )^{4} + 20 \, b^{4} c^{3} d^{4} x^{3} \log \left (F\right )^{4} + 15 \, b^{4} c^{4} d^{3} x^{2} \log \left (F\right )^{4} + 6 \, b^{4} c^{5} d^{2} x \log \left (F\right )^{4} + b^{4} c^{6} d \log \left (F\right )^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(a + b/(d*x + c)^2)/(d*x + c)^9,x, algorithm="maxima")

[Out]

1/2*(6*F^a*d^6*x^6 + 36*F^a*c*d^5*x^5 + 6*F^a*c^6 - 6*F^a*b*c^4*log(F) + 3*F^a*b
^2*c^2*log(F)^2 - F^a*b^3*log(F)^3 + 6*(15*F^a*c^2*d^4 - F^a*b*d^4*log(F))*x^4 +
24*(5*F^a*c^3*d^3 - F^a*b*c*d^3*log(F))*x^3 + 3*(30*F^a*c^4*d^2 - 12*F^a*b*c^2*
d^2*log(F) + F^a*b^2*d^2*log(F)^2)*x^2 + 6*(6*F^a*c^5*d - 4*F^a*b*c^3*d*log(F) +
F^a*b^2*c*d*log(F)^2)*x)*F^(b/(d^2*x^2 + 2*c*d*x + c^2))/(b^4*d^7*x^6*log(F)^4
+ 6*b^4*c*d^6*x^5*log(F)^4 + 15*b^4*c^2*d^5*x^4*log(F)^4 + 20*b^4*c^3*d^4*x^3*lo
g(F)^4 + 15*b^4*c^4*d^3*x^2*log(F)^4 + 6*b^4*c^5*d^2*x*log(F)^4 + b^4*c^6*d*log(
F)^4)

_______________________________________________________________________________________

Fricas [A]  time = 0.27602, size = 387, normalized size = 3.07 \[ \frac{{\left (6 \, d^{6} x^{6} + 36 \, c d^{5} x^{5} + 90 \, c^{2} d^{4} x^{4} + 120 \, c^{3} d^{3} x^{3} + 90 \, c^{4} d^{2} x^{2} + 36 \, c^{5} d x + 6 \, c^{6} - b^{3} \log \left (F\right )^{3} + 3 \,{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (F\right )^{2} - 6 \,{\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4}\right )} \log \left (F\right )\right )} F^{\frac{a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \,{\left (b^{4} d^{7} x^{6} + 6 \, b^{4} c d^{6} x^{5} + 15 \, b^{4} c^{2} d^{5} x^{4} + 20 \, b^{4} c^{3} d^{4} x^{3} + 15 \, b^{4} c^{4} d^{3} x^{2} + 6 \, b^{4} c^{5} d^{2} x + b^{4} c^{6} d\right )} \log \left (F\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(a + b/(d*x + c)^2)/(d*x + c)^9,x, algorithm="fricas")

[Out]

1/2*(6*d^6*x^6 + 36*c*d^5*x^5 + 90*c^2*d^4*x^4 + 120*c^3*d^3*x^3 + 90*c^4*d^2*x^
2 + 36*c^5*d*x + 6*c^6 - b^3*log(F)^3 + 3*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*
log(F)^2 - 6*(b*d^4*x^4 + 4*b*c*d^3*x^3 + 6*b*c^2*d^2*x^2 + 4*b*c^3*d*x + b*c^4)
*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))/((b^4
*d^7*x^6 + 6*b^4*c*d^6*x^5 + 15*b^4*c^2*d^5*x^4 + 20*b^4*c^3*d^4*x^3 + 15*b^4*c^
4*d^3*x^2 + 6*b^4*c^5*d^2*x + b^4*c^6*d)*log(F)^4)

_______________________________________________________________________________________

Sympy [A]  time = 0.660359, size = 333, normalized size = 2.64 \[ \frac{F^{a + \frac{b}{\left (c + d x\right )^{2}}} \left (- b^{3} \log{\left (F \right )}^{3} + 3 b^{2} c^{2} \log{\left (F \right )}^{2} + 6 b^{2} c d x \log{\left (F \right )}^{2} + 3 b^{2} d^{2} x^{2} \log{\left (F \right )}^{2} - 6 b c^{4} \log{\left (F \right )} - 24 b c^{3} d x \log{\left (F \right )} - 36 b c^{2} d^{2} x^{2} \log{\left (F \right )} - 24 b c d^{3} x^{3} \log{\left (F \right )} - 6 b d^{4} x^{4} \log{\left (F \right )} + 6 c^{6} + 36 c^{5} d x + 90 c^{4} d^{2} x^{2} + 120 c^{3} d^{3} x^{3} + 90 c^{2} d^{4} x^{4} + 36 c d^{5} x^{5} + 6 d^{6} x^{6}\right )}{2 b^{4} c^{6} d \log{\left (F \right )}^{4} + 12 b^{4} c^{5} d^{2} x \log{\left (F \right )}^{4} + 30 b^{4} c^{4} d^{3} x^{2} \log{\left (F \right )}^{4} + 40 b^{4} c^{3} d^{4} x^{3} \log{\left (F \right )}^{4} + 30 b^{4} c^{2} d^{5} x^{4} \log{\left (F \right )}^{4} + 12 b^{4} c d^{6} x^{5} \log{\left (F \right )}^{4} + 2 b^{4} d^{7} x^{6} \log{\left (F \right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F**(a+b/(d*x+c)**2)/(d*x+c)**9,x)

[Out]

F**(a + b/(c + d*x)**2)*(-b**3*log(F)**3 + 3*b**2*c**2*log(F)**2 + 6*b**2*c*d*x*
log(F)**2 + 3*b**2*d**2*x**2*log(F)**2 - 6*b*c**4*log(F) - 24*b*c**3*d*x*log(F)
- 36*b*c**2*d**2*x**2*log(F) - 24*b*c*d**3*x**3*log(F) - 6*b*d**4*x**4*log(F) +
6*c**6 + 36*c**5*d*x + 90*c**4*d**2*x**2 + 120*c**3*d**3*x**3 + 90*c**2*d**4*x**
4 + 36*c*d**5*x**5 + 6*d**6*x**6)/(2*b**4*c**6*d*log(F)**4 + 12*b**4*c**5*d**2*x
*log(F)**4 + 30*b**4*c**4*d**3*x**2*log(F)**4 + 40*b**4*c**3*d**4*x**3*log(F)**4
+ 30*b**4*c**2*d**5*x**4*log(F)**4 + 12*b**4*c*d**6*x**5*log(F)**4 + 2*b**4*d**
7*x**6*log(F)**4)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{a + \frac{b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{9}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(a + b/(d*x + c)^2)/(d*x + c)^9,x, algorithm="giac")

[Out]

integrate(F^(a + b/(d*x + c)^2)/(d*x + c)^9, x)