3.350 \(\int \frac{F^{a+\frac{b}{(c+d x)^3}}}{(c+d x)^{13}} \, dx\)

Optimal. Leaf size=123 \[ \frac{F^{a+\frac{b}{(c+d x)^3}}}{b^2 d \log ^2(F) (c+d x)^6}-\frac{2 F^{a+\frac{b}{(c+d x)^3}}}{b^3 d \log ^3(F) (c+d x)^3}+\frac{2 F^{a+\frac{b}{(c+d x)^3}}}{b^4 d \log ^4(F)}-\frac{F^{a+\frac{b}{(c+d x)^3}}}{3 b d \log (F) (c+d x)^9} \]

[Out]

(2*F^(a + b/(c + d*x)^3))/(b^4*d*Log[F]^4) - (2*F^(a + b/(c + d*x)^3))/(b^3*d*(c + d*x)^3*Log[F]^3) + F^(a + b
/(c + d*x)^3)/(b^2*d*(c + d*x)^6*Log[F]^2) - F^(a + b/(c + d*x)^3)/(3*b*d*(c + d*x)^9*Log[F])

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Rubi [A]  time = 0.185693, antiderivative size = 123, 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, Rules used = {2212, 2209} \[ \frac{F^{a+\frac{b}{(c+d x)^3}}}{b^2 d \log ^2(F) (c+d x)^6}-\frac{2 F^{a+\frac{b}{(c+d x)^3}}}{b^3 d \log ^3(F) (c+d x)^3}+\frac{2 F^{a+\frac{b}{(c+d x)^3}}}{b^4 d \log ^4(F)}-\frac{F^{a+\frac{b}{(c+d x)^3}}}{3 b d \log (F) (c+d x)^9} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b/(c + d*x)^3)/(c + d*x)^13,x]

[Out]

(2*F^(a + b/(c + d*x)^3))/(b^4*d*Log[F]^4) - (2*F^(a + b/(c + d*x)^3))/(b^3*d*(c + d*x)^3*Log[F]^3) + F^(a + b
/(c + d*x)^3)/(b^2*d*(c + d*x)^6*Log[F]^2) - F^(a + b/(c + d*x)^3)/(3*b*d*(c + d*x)^9*Log[F])

Rule 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int \frac{F^{a+\frac{b}{(c+d x)^3}}}{(c+d x)^{13}} \, dx &=-\frac{F^{a+\frac{b}{(c+d x)^3}}}{3 b d (c+d x)^9 \log (F)}-\frac{3 \int \frac{F^{a+\frac{b}{(c+d x)^3}}}{(c+d x)^{10}} \, dx}{b \log (F)}\\ &=\frac{F^{a+\frac{b}{(c+d x)^3}}}{b^2 d (c+d x)^6 \log ^2(F)}-\frac{F^{a+\frac{b}{(c+d x)^3}}}{3 b d (c+d x)^9 \log (F)}+\frac{6 \int \frac{F^{a+\frac{b}{(c+d x)^3}}}{(c+d x)^7} \, dx}{b^2 \log ^2(F)}\\ &=-\frac{2 F^{a+\frac{b}{(c+d x)^3}}}{b^3 d (c+d x)^3 \log ^3(F)}+\frac{F^{a+\frac{b}{(c+d x)^3}}}{b^2 d (c+d x)^6 \log ^2(F)}-\frac{F^{a+\frac{b}{(c+d x)^3}}}{3 b d (c+d x)^9 \log (F)}-\frac{6 \int \frac{F^{a+\frac{b}{(c+d x)^3}}}{(c+d x)^4} \, dx}{b^3 \log ^3(F)}\\ &=\frac{2 F^{a+\frac{b}{(c+d x)^3}}}{b^4 d \log ^4(F)}-\frac{2 F^{a+\frac{b}{(c+d x)^3}}}{b^3 d (c+d x)^3 \log ^3(F)}+\frac{F^{a+\frac{b}{(c+d x)^3}}}{b^2 d (c+d x)^6 \log ^2(F)}-\frac{F^{a+\frac{b}{(c+d x)^3}}}{3 b d (c+d x)^9 \log (F)}\\ \end{align*}

Mathematica [A]  time = 0.0333669, size = 73, normalized size = 0.59 \[ \frac{F^{a+\frac{b}{(c+d x)^3}} \left (-\frac{b^3 \log ^3(F)}{(c+d x)^9}+\frac{3 b^2 \log ^2(F)}{(c+d x)^6}-\frac{6 b \log (F)}{(c+d x)^3}+6\right )}{3 b^4 d \log ^4(F)} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b/(c + d*x)^3)/(c + d*x)^13,x]

[Out]

(F^(a + b/(c + d*x)^3)*(6 - (6*b*Log[F])/(c + d*x)^3 + (3*b^2*Log[F]^2)/(c + d*x)^6 - (b^3*Log[F]^3)/(c + d*x)
^9))/(3*b^4*d*Log[F]^4)

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Maple [B]  time = 0.185, size = 641, normalized size = 5.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b/(d*x+c)^3)/(d*x+c)^13,x)

[Out]

(-1/3*(-6*c^9+6*ln(F)*b*c^6-3*ln(F)^2*b^2*c^3+ln(F)^3*b^3)*c^3/b^4/ln(F)^4/d*exp((a+b/(d*x+c)^3)*ln(F))-c^2*(-
24*c^9+18*ln(F)*b*c^6-6*ln(F)^2*b^2*c^3+ln(F)^3*b^3)/b^4/ln(F)^4*x*exp((a+b/(d*x+c)^3)*ln(F))-1/3*d^2*(-1320*c
^9+504*ln(F)*b*c^6-60*ln(F)^2*b^2*c^3+ln(F)^3*b^3)/ln(F)^4/b^4*x^3*exp((a+b/(d*x+c)^3)*ln(F))+d^5*(1848*c^6-16
8*ln(F)*b*c^3+ln(F)^2*b^2)/ln(F)^4/b^4*x^6*exp((a+b/(d*x+c)^3)*ln(F))-2*d^8*(-220*c^3+b*ln(F))/ln(F)^4/b^4*x^9
*exp((a+b/(d*x+c)^3)*ln(F))+132*d^9*c^2/ln(F)^4/b^4*x^10*exp((a+b/(d*x+c)^3)*ln(F))+24*d^10*c/ln(F)^4/b^4*x^11
*exp((a+b/(d*x+c)^3)*ln(F))-c*d*(-132*c^9+72*ln(F)*b*c^6-15*ln(F)^2*b^2*c^3+ln(F)^3*b^3)/ln(F)^4/b^4*x^2*exp((
a+b/(d*x+c)^3)*ln(F))+3*c^2*d^3*(330*c^6-84*ln(F)*b*c^3+5*ln(F)^2*b^2)/ln(F)^4/b^4*x^4*exp((a+b/(d*x+c)^3)*ln(
F))+6*c*d^4*(264*c^6-42*ln(F)*b*c^3+ln(F)^2*b^2)/ln(F)^4/b^4*x^5*exp((a+b/(d*x+c)^3)*ln(F))-72*c^2*d^6*(-22*c^
3+b*ln(F))/ln(F)^4/b^4*x^7*exp((a+b/(d*x+c)^3)*ln(F))-18*c*d^7*(-55*c^3+b*ln(F))/ln(F)^4/b^4*x^8*exp((a+b/(d*x
+c)^3)*ln(F))+2*d^11/ln(F)^4/b^4*x^12*exp((a+b/(d*x+c)^3)*ln(F)))/(d*x+c)^12

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Maxima [B]  time = 1.08542, size = 684, normalized size = 5.56 \begin{align*} \frac{{\left (6 \, F^{a} d^{9} x^{9} + 54 \, F^{a} c d^{8} x^{8} + 216 \, F^{a} c^{2} d^{7} x^{7} + 6 \, F^{a} c^{9} - 6 \, F^{a} b c^{6} \log \left (F\right ) + 3 \, F^{a} b^{2} c^{3} \log \left (F\right )^{2} + 6 \,{\left (84 \, F^{a} c^{3} d^{6} - F^{a} b d^{6} \log \left (F\right )\right )} x^{6} - F^{a} b^{3} \log \left (F\right )^{3} + 36 \,{\left (21 \, F^{a} c^{4} d^{5} - F^{a} b c d^{5} \log \left (F\right )\right )} x^{5} + 18 \,{\left (42 \, F^{a} c^{5} d^{4} - 5 \, F^{a} b c^{2} d^{4} \log \left (F\right )\right )} x^{4} + 3 \,{\left (168 \, F^{a} c^{6} d^{3} - 40 \, F^{a} b c^{3} d^{3} \log \left (F\right ) + F^{a} b^{2} d^{3} \log \left (F\right )^{2}\right )} x^{3} + 9 \,{\left (24 \, F^{a} c^{7} d^{2} - 10 \, F^{a} b c^{4} d^{2} \log \left (F\right ) + F^{a} b^{2} c d^{2} \log \left (F\right )^{2}\right )} x^{2} + 9 \,{\left (6 \, F^{a} c^{8} d - 4 \, F^{a} b c^{5} d \log \left (F\right ) + F^{a} b^{2} c^{2} d \log \left (F\right )^{2}\right )} x\right )} F^{\frac{b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{3 \,{\left (b^{4} d^{10} x^{9} \log \left (F\right )^{4} + 9 \, b^{4} c d^{9} x^{8} \log \left (F\right )^{4} + 36 \, b^{4} c^{2} d^{8} x^{7} \log \left (F\right )^{4} + 84 \, b^{4} c^{3} d^{7} x^{6} \log \left (F\right )^{4} + 126 \, b^{4} c^{4} d^{6} x^{5} \log \left (F\right )^{4} + 126 \, b^{4} c^{5} d^{5} x^{4} \log \left (F\right )^{4} + 84 \, b^{4} c^{6} d^{4} x^{3} \log \left (F\right )^{4} + 36 \, b^{4} c^{7} d^{3} x^{2} \log \left (F\right )^{4} + 9 \, b^{4} c^{8} d^{2} x \log \left (F\right )^{4} + b^{4} c^{9} d \log \left (F\right )^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^3)/(d*x+c)^13,x, algorithm="maxima")

[Out]

1/3*(6*F^a*d^9*x^9 + 54*F^a*c*d^8*x^8 + 216*F^a*c^2*d^7*x^7 + 6*F^a*c^9 - 6*F^a*b*c^6*log(F) + 3*F^a*b^2*c^3*l
og(F)^2 + 6*(84*F^a*c^3*d^6 - F^a*b*d^6*log(F))*x^6 - F^a*b^3*log(F)^3 + 36*(21*F^a*c^4*d^5 - F^a*b*c*d^5*log(
F))*x^5 + 18*(42*F^a*c^5*d^4 - 5*F^a*b*c^2*d^4*log(F))*x^4 + 3*(168*F^a*c^6*d^3 - 40*F^a*b*c^3*d^3*log(F) + F^
a*b^2*d^3*log(F)^2)*x^3 + 9*(24*F^a*c^7*d^2 - 10*F^a*b*c^4*d^2*log(F) + F^a*b^2*c*d^2*log(F)^2)*x^2 + 9*(6*F^a
*c^8*d - 4*F^a*b*c^5*d*log(F) + F^a*b^2*c^2*d*log(F)^2)*x)*F^(b/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))/(b^
4*d^10*x^9*log(F)^4 + 9*b^4*c*d^9*x^8*log(F)^4 + 36*b^4*c^2*d^8*x^7*log(F)^4 + 84*b^4*c^3*d^7*x^6*log(F)^4 + 1
26*b^4*c^4*d^6*x^5*log(F)^4 + 126*b^4*c^5*d^5*x^4*log(F)^4 + 84*b^4*c^6*d^4*x^3*log(F)^4 + 36*b^4*c^7*d^3*x^2*
log(F)^4 + 9*b^4*c^8*d^2*x*log(F)^4 + b^4*c^9*d*log(F)^4)

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Fricas [B]  time = 1.81621, size = 900, normalized size = 7.32 \begin{align*} \frac{{\left (6 \, d^{9} x^{9} + 54 \, c d^{8} x^{8} + 216 \, c^{2} d^{7} x^{7} + 504 \, c^{3} d^{6} x^{6} + 756 \, c^{4} d^{5} x^{5} + 756 \, c^{5} d^{4} x^{4} + 504 \, c^{6} d^{3} x^{3} + 216 \, c^{7} d^{2} x^{2} + 54 \, c^{8} d x + 6 \, c^{9} - b^{3} \log \left (F\right )^{3} + 3 \,{\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\right )} \log \left (F\right )^{2} - 6 \,{\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^{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 \,{\left (b^{4} d^{10} x^{9} + 9 \, b^{4} c d^{9} x^{8} + 36 \, b^{4} c^{2} d^{8} x^{7} + 84 \, b^{4} c^{3} d^{7} x^{6} + 126 \, b^{4} c^{4} d^{6} x^{5} + 126 \, b^{4} c^{5} d^{5} x^{4} + 84 \, b^{4} c^{6} d^{4} x^{3} + 36 \, b^{4} c^{7} d^{3} x^{2} + 9 \, b^{4} c^{8} d^{2} x + b^{4} c^{9} d\right )} \log \left (F\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^3)/(d*x+c)^13,x, algorithm="fricas")

[Out]

1/3*(6*d^9*x^9 + 54*c*d^8*x^8 + 216*c^2*d^7*x^7 + 504*c^3*d^6*x^6 + 756*c^4*d^5*x^5 + 756*c^5*d^4*x^4 + 504*c^
6*d^3*x^3 + 216*c^7*d^2*x^2 + 54*c^8*d*x + 6*c^9 - b^3*log(F)^3 + 3*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2
*d*x + b^2*c^3)*log(F)^2 - 6*(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^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))/((b^4*d^10*x^9 + 9*b^4*c*d^9*x^8 + 36*b^4*c^2*d^8*x^7 + 84*b^4*c^3*d^7*x^6 + 126*b^4*c
^4*d^6*x^5 + 126*b^4*c^5*d^5*x^4 + 84*b^4*c^6*d^4*x^3 + 36*b^4*c^7*d^3*x^2 + 9*b^4*c^8*d^2*x + b^4*c^9*d)*log(
F)^4)

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Sympy [B]  time = 0.62872, size = 484, normalized size = 3.93 \begin{align*} \frac{F^{a + \frac{b}{\left (c + d x\right )^{3}}} \left (- b^{3} \log{\left (F \right )}^{3} + 3 b^{2} c^{3} \log{\left (F \right )}^{2} + 9 b^{2} c^{2} d x \log{\left (F \right )}^{2} + 9 b^{2} c d^{2} x^{2} \log{\left (F \right )}^{2} + 3 b^{2} d^{3} x^{3} \log{\left (F \right )}^{2} - 6 b c^{6} \log{\left (F \right )} - 36 b c^{5} d x \log{\left (F \right )} - 90 b c^{4} d^{2} x^{2} \log{\left (F \right )} - 120 b c^{3} d^{3} x^{3} \log{\left (F \right )} - 90 b c^{2} d^{4} x^{4} \log{\left (F \right )} - 36 b c d^{5} x^{5} \log{\left (F \right )} - 6 b d^{6} x^{6} \log{\left (F \right )} + 6 c^{9} + 54 c^{8} d x + 216 c^{7} d^{2} x^{2} + 504 c^{6} d^{3} x^{3} + 756 c^{5} d^{4} x^{4} + 756 c^{4} d^{5} x^{5} + 504 c^{3} d^{6} x^{6} + 216 c^{2} d^{7} x^{7} + 54 c d^{8} x^{8} + 6 d^{9} x^{9}\right )}{3 b^{4} c^{9} d \log{\left (F \right )}^{4} + 27 b^{4} c^{8} d^{2} x \log{\left (F \right )}^{4} + 108 b^{4} c^{7} d^{3} x^{2} \log{\left (F \right )}^{4} + 252 b^{4} c^{6} d^{4} x^{3} \log{\left (F \right )}^{4} + 378 b^{4} c^{5} d^{5} x^{4} \log{\left (F \right )}^{4} + 378 b^{4} c^{4} d^{6} x^{5} \log{\left (F \right )}^{4} + 252 b^{4} c^{3} d^{7} x^{6} \log{\left (F \right )}^{4} + 108 b^{4} c^{2} d^{8} x^{7} \log{\left (F \right )}^{4} + 27 b^{4} c d^{9} x^{8} \log{\left (F \right )}^{4} + 3 b^{4} d^{10} x^{9} \log{\left (F \right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b/(d*x+c)**3)/(d*x+c)**13,x)

[Out]

F**(a + b/(c + d*x)**3)*(-b**3*log(F)**3 + 3*b**2*c**3*log(F)**2 + 9*b**2*c**2*d*x*log(F)**2 + 9*b**2*c*d**2*x
**2*log(F)**2 + 3*b**2*d**3*x**3*log(F)**2 - 6*b*c**6*log(F) - 36*b*c**5*d*x*log(F) - 90*b*c**4*d**2*x**2*log(
F) - 120*b*c**3*d**3*x**3*log(F) - 90*b*c**2*d**4*x**4*log(F) - 36*b*c*d**5*x**5*log(F) - 6*b*d**6*x**6*log(F)
+ 6*c**9 + 54*c**8*d*x + 216*c**7*d**2*x**2 + 504*c**6*d**3*x**3 + 756*c**5*d**4*x**4 + 756*c**4*d**5*x**5 +
504*c**3*d**6*x**6 + 216*c**2*d**7*x**7 + 54*c*d**8*x**8 + 6*d**9*x**9)/(3*b**4*c**9*d*log(F)**4 + 27*b**4*c**
8*d**2*x*log(F)**4 + 108*b**4*c**7*d**3*x**2*log(F)**4 + 252*b**4*c**6*d**4*x**3*log(F)**4 + 378*b**4*c**5*d**
5*x**4*log(F)**4 + 378*b**4*c**4*d**6*x**5*log(F)**4 + 252*b**4*c**3*d**7*x**6*log(F)**4 + 108*b**4*c**2*d**8*
x**7*log(F)**4 + 27*b**4*c*d**9*x**8*log(F)**4 + 3*b**4*d**10*x**9*log(F)**4)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{a + \frac{b}{{\left (d x + c\right )}^{3}}}}{{\left (d x + c\right )}^{13}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^3)/(d*x+c)^13,x, algorithm="giac")

[Out]

integrate(F^(a + b/(d*x + c)^3)/(d*x + c)^13, x)