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

Optimal. Leaf size=31 \[ -\frac{b^5 F^a \log ^5(F) \text{Gamma}\left (-5,-\frac{b \log (F)}{(c+d x)^2}\right )}{2 d} \]

[Out]

-(b^5*F^a*Gamma[-5, -((b*Log[F])/(c + d*x)^2)]*Log[F]^5)/(2*d)

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Rubi [A]  time = 0.0519668, 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, Rules used = {2218} \[ -\frac{b^5 F^a \log ^5(F) \text{Gamma}\left (-5,-\frac{b \log (F)}{(c+d x)^2}\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^5*F^a*Gamma[-5, -((b*Log[F])/(c + d*x)^2)]*Log[F]^5)/(2*d)

Rule 2218

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

Rubi steps

\begin{align*} \int F^{a+\frac{b}{(c+d x)^2}} (c+d x)^9 \, dx &=-\frac{b^5 F^a \Gamma \left (-5,-\frac{b \log (F)}{(c+d x)^2}\right ) \log ^5(F)}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.008138, size = 31, normalized size = 1. \[ -\frac{b^5 F^a \log ^5(F) \text{Gamma}\left (-5,-\frac{b \log (F)}{(c+d x)^2}\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^5*F^a*Gamma[-5, -((b*Log[F])/(c + d*x)^2)]*Log[F]^5)/(2*d)

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Maple [B]  time = 0.096, size = 961, normalized size = 31. \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]

1/240*d*F^a*b^4*ln(F)^4*F^(b/(d*x+c)^2)*x^2+1/40*d^7*F^a*b*ln(F)*F^(b/(d*x+c)^2)*x^8+1/40/d*F^a*b*ln(F)*F^(b/(
d*x+c)^2)*c^8+1/120/d*F^a*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c^6+1/240/d*F^a*b^3*ln(F)^3*F^(b/(d*x+c)^2)*c^4+1/240/d*
F^a*b^4*ln(F)^4*F^(b/(d*x+c)^2)*c^2+1/120*d^5*F^a*b^2*ln(F)^2*F^(b/(d*x+c)^2)*x^6+1/240*d^3*F^a*b^3*ln(F)^3*F^
(b/(d*x+c)^2)*x^4+1/5*F^a*b*ln(F)*F^(b/(d*x+c)^2)*c^7*x+1/20*F^a*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c^5*x+1/60*F^a*b^
3*ln(F)^3*F^(b/(d*x+c)^2)*c^3*x+1/120*F^a*b^4*ln(F)^4*F^(b/(d*x+c)^2)*c*x+7/10*d^5*F^a*b*ln(F)*F^(b/(d*x+c)^2)
*c^2*x^6+1/5*d^6*F^a*b*ln(F)*F^(b/(d*x+c)^2)*c*x^7+1/10*d^9*F^a*F^(b/(d*x+c)^2)*x^10+1/10/d*F^a*F^(b/(d*x+c)^2
)*c^10+F^a*F^(b/(d*x+c)^2)*c^9*x+7/5*d^4*F^a*b*ln(F)*F^(b/(d*x+c)^2)*c^3*x^5+7/4*d^3*F^a*b*ln(F)*F^(b/(d*x+c)^
2)*c^4*x^4+7/5*d^2*F^a*b*ln(F)*F^(b/(d*x+c)^2)*c^5*x^3+7/10*d*F^a*b*ln(F)*F^(b/(d*x+c)^2)*c^6*x^2+1/20*d^4*F^a
*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c*x^5+1/8*d^3*F^a*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c^2*x^4+1/6*d^2*F^a*b^2*ln(F)^2*F^(
b/(d*x+c)^2)*c^3*x^3+1/8*d*F^a*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c^4*x^2+1/60*d^2*F^a*b^3*ln(F)^3*F^(b/(d*x+c)^2)*c*
x^3+1/40*d*F^a*b^3*ln(F)^3*F^(b/(d*x+c)^2)*c^2*x^2+21*d^3*F^a*F^(b/(d*x+c)^2)*c^6*x^4+12*d^2*F^a*F^(b/(d*x+c)^
2)*c^7*x^3+9/2*d*F^a*F^(b/(d*x+c)^2)*c^8*x^2+9/2*d^7*F^a*F^(b/(d*x+c)^2)*c^2*x^8+12*d^6*F^a*F^(b/(d*x+c)^2)*c^
3*x^7+21*d^5*F^a*F^(b/(d*x+c)^2)*c^4*x^6+126/5*d^4*F^a*F^(b/(d*x+c)^2)*c^5*x^5+1/240/d*F^a*b^5*ln(F)^5*Ei(1,-b
*ln(F)/(d*x+c)^2)+d^8*F^a*F^(b/(d*x+c)^2)*c*x^9

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{240} \,{\left (24 \, F^{a} d^{9} x^{10} + 240 \, F^{a} c d^{8} x^{9} + 6 \,{\left (180 \, F^{a} c^{2} d^{7} + F^{a} b d^{7} \log \left (F\right )\right )} x^{8} + 48 \,{\left (60 \, F^{a} c^{3} d^{6} + F^{a} b c d^{6} \log \left (F\right )\right )} x^{7} + 2 \,{\left (2520 \, F^{a} c^{4} d^{5} + 84 \, F^{a} b c^{2} d^{5} \log \left (F\right ) + F^{a} b^{2} d^{5} \log \left (F\right )^{2}\right )} x^{6} + 12 \,{\left (504 \, F^{a} c^{5} d^{4} + 28 \, F^{a} b c^{3} d^{4} \log \left (F\right ) + F^{a} b^{2} c d^{4} \log \left (F\right )^{2}\right )} x^{5} +{\left (5040 \, F^{a} c^{6} d^{3} + 420 \, F^{a} b c^{4} d^{3} \log \left (F\right ) + 30 \, F^{a} b^{2} c^{2} d^{3} \log \left (F\right )^{2} + F^{a} b^{3} d^{3} \log \left (F\right )^{3}\right )} x^{4} + 4 \,{\left (720 \, F^{a} c^{7} d^{2} + 84 \, F^{a} b c^{5} d^{2} \log \left (F\right ) + 10 \, F^{a} b^{2} c^{3} d^{2} \log \left (F\right )^{2} + F^{a} b^{3} c d^{2} \log \left (F\right )^{3}\right )} x^{3} +{\left (1080 \, F^{a} c^{8} d + 168 \, F^{a} b c^{6} d \log \left (F\right ) + 30 \, F^{a} b^{2} c^{4} d \log \left (F\right )^{2} + 6 \, F^{a} b^{3} c^{2} d \log \left (F\right )^{3} + F^{a} b^{4} d \log \left (F\right )^{4}\right )} x^{2} + 2 \,{\left (120 \, F^{a} c^{9} + 24 \, F^{a} b c^{7} \log \left (F\right ) + 6 \, F^{a} b^{2} c^{5} \log \left (F\right )^{2} + 2 \, F^{a} b^{3} c^{3} \log \left (F\right )^{3} + F^{a} b^{4} c \log \left (F\right )^{4}\right )} x\right )} F^{\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}} + \int \frac{{\left (F^{a} b^{5} d^{2} x^{2} \log \left (F\right )^{5} + 2 \, F^{a} b^{5} c d x \log \left (F\right )^{5} - 24 \, F^{a} b c^{10} \log \left (F\right ) - 6 \, F^{a} b^{2} c^{8} \log \left (F\right )^{2} - 2 \, F^{a} b^{3} c^{6} \log \left (F\right )^{3} - F^{a} b^{4} c^{4} \log \left (F\right )^{4}\right )} F^{\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{120 \,{\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )}}\,{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="maxima")

[Out]

1/240*(24*F^a*d^9*x^10 + 240*F^a*c*d^8*x^9 + 6*(180*F^a*c^2*d^7 + F^a*b*d^7*log(F))*x^8 + 48*(60*F^a*c^3*d^6 +
F^a*b*c*d^6*log(F))*x^7 + 2*(2520*F^a*c^4*d^5 + 84*F^a*b*c^2*d^5*log(F) + F^a*b^2*d^5*log(F)^2)*x^6 + 12*(504
*F^a*c^5*d^4 + 28*F^a*b*c^3*d^4*log(F) + F^a*b^2*c*d^4*log(F)^2)*x^5 + (5040*F^a*c^6*d^3 + 420*F^a*b*c^4*d^3*l
og(F) + 30*F^a*b^2*c^2*d^3*log(F)^2 + F^a*b^3*d^3*log(F)^3)*x^4 + 4*(720*F^a*c^7*d^2 + 84*F^a*b*c^5*d^2*log(F)
+ 10*F^a*b^2*c^3*d^2*log(F)^2 + F^a*b^3*c*d^2*log(F)^3)*x^3 + (1080*F^a*c^8*d + 168*F^a*b*c^6*d*log(F) + 30*F
^a*b^2*c^4*d*log(F)^2 + 6*F^a*b^3*c^2*d*log(F)^3 + F^a*b^4*d*log(F)^4)*x^2 + 2*(120*F^a*c^9 + 24*F^a*b*c^7*log
(F) + 6*F^a*b^2*c^5*log(F)^2 + 2*F^a*b^3*c^3*log(F)^3 + F^a*b^4*c*log(F)^4)*x)*F^(b/(d^2*x^2 + 2*c*d*x + c^2))
+ integrate(1/120*(F^a*b^5*d^2*x^2*log(F)^5 + 2*F^a*b^5*c*d*x*log(F)^5 - 24*F^a*b*c^10*log(F) - 6*F^a*b^2*c^8
*log(F)^2 - 2*F^a*b^3*c^6*log(F)^3 - F^a*b^4*c^4*log(F)^4)*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)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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]

Exception raised: TypeError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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]

Timed out

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