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

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

[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])

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Rubi [A]  time = 0.191042, 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, Rules used = {2212, 2209} \[ \frac{3 F^{a+\frac{b}{(c+d x)^2}}}{2 b^2 d \log ^2(F) (c+d x)^4}-\frac{3 F^{a+\frac{b}{(c+d x)^2}}}{b^3 d \log ^3(F) (c+d x)^2}+\frac{3 F^{a+\frac{b}{(c+d x)^2}}}{b^4 d \log ^4(F)}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)^6} \]

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])

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)^2}}}{(c+d x)^9} \, dx &=-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d (c+d x)^6 \log (F)}-\frac{3 \int \frac{F^{a+\frac{b}{(c+d x)^2}}}{(c+d x)^7} \, dx}{b \log (F)}\\ &=\frac{3 F^{a+\frac{b}{(c+d x)^2}}}{2 b^2 d (c+d x)^4 \log ^2(F)}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d (c+d x)^6 \log (F)}+\frac{6 \int \frac{F^{a+\frac{b}{(c+d x)^2}}}{(c+d x)^5} \, dx}{b^2 \log ^2(F)}\\ &=-\frac{3 F^{a+\frac{b}{(c+d x)^2}}}{b^3 d (c+d x)^2 \log ^3(F)}+\frac{3 F^{a+\frac{b}{(c+d x)^2}}}{2 b^2 d (c+d x)^4 \log ^2(F)}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d (c+d x)^6 \log (F)}-\frac{6 \int \frac{F^{a+\frac{b}{(c+d x)^2}}}{(c+d x)^3} \, dx}{b^3 \log ^3(F)}\\ &=\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 (c+d x)^2 \log ^3(F)}+\frac{3 F^{a+\frac{b}{(c+d x)^2}}}{2 b^2 d (c+d x)^4 \log ^2(F)}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d (c+d x)^6 \log (F)}\\ \end{align*}

Mathematica [A]  time = 0.0421996, size = 81, normalized size = 0.64 \[ \frac{F^{a+\frac{b}{(c+d x)^2}} \left (3 b^2 \log ^2(F) (c+d x)^2-b^3 \log ^3(F)-6 b \log (F) (c+d x)^4+6 (c+d x)^6\right )}{2 b^4 d \log ^4(F) (c+d x)^6} \]

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

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Maple [B]  time = 0.093, size = 444, normalized size = 3.5 \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)^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*ln(F)^2*b^2*c^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*ln(F)^2*b^2*c^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)*ln(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*ln(F)^2*b^2*c^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)*ln(F))-6*c*d^4*(3*b*l
n(F)-28*c^2)/ln(F)^4/b^4*x^5*exp((a+b/(d*x+c)^2)*ln(F)))/(d*x+c)^8

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Maxima [B]  time = 1.04967, size = 471, normalized size = 3.74 \begin{align*} \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 )}} \end{align*}

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*log(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)

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Fricas [B]  time = 2.00736, size = 610, normalized size = 4.84 \begin{align*} \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}} \end{align*}

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*l
og(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)

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Sympy [B]  time = 0.431121, size = 333, normalized size = 2.64 \begin{align*} \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}} \end{align*}

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*l
og(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)

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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 )}^{2}}}}{{\left (d x + c\right )}^{9}}\,{d x} \end{align*}

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)