3.328 \(\int F^{a+\frac{b}{(c+d x)^2}} (c+d x)^8 \, dx\)

Optimal. Leaf size=49 \[ \frac{F^a (c+d x)^9 \left (-\frac{b \log (F)}{(c+d x)^2}\right )^{9/2} \text{Gamma}\left (-\frac{9}{2},-\frac{b \log (F)}{(c+d x)^2}\right )}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.0493251, antiderivative size = 49, 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} \[ \frac{F^a (c+d x)^9 \left (-\frac{b \log (F)}{(c+d x)^2}\right )^{9/2} \text{Gamma}\left (-\frac{9}{2},-\frac{b \log (F)}{(c+d x)^2}\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b/(c + d*x)^2)*(c + d*x)^8,x]

[Out]

(F^a*(c + d*x)^9*Gamma[-9/2, -((b*Log[F])/(c + d*x)^2)]*(-((b*Log[F])/(c + d*x)^2))^(9/2))/(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)^8 \, dx &=\frac{F^a (c+d x)^9 \Gamma \left (-\frac{9}{2},-\frac{b \log (F)}{(c+d x)^2}\right ) \left (-\frac{b \log (F)}{(c+d x)^2}\right )^{9/2}}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.025662, size = 49, normalized size = 1. \[ \frac{F^a (c+d x)^9 \left (-\frac{b \log (F)}{(c+d x)^2}\right )^{9/2} \text{Gamma}\left (-\frac{9}{2},-\frac{b \log (F)}{(c+d x)^2}\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b/(c + d*x)^2)*(c + d*x)^8,x]

[Out]

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

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Maple [B]  time = 0.079, size = 826, normalized size = 16.9 \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)^8,x)

[Out]

2/63/d*F^a*b*ln(F)*F^(b/(d*x+c)^2)*c^7+4/315/d*F^a*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c^5+8/945/d*F^a*b^3*ln(F)^3*F^(
b/(d*x+c)^2)*c^3+16/945/d*F^a*b^4*ln(F)^4*F^(b/(d*x+c)^2)*c+2/63*d^6*F^a*b*ln(F)*F^(b/(d*x+c)^2)*x^7+4/315*d^4
*F^a*b^2*ln(F)^2*F^(b/(d*x+c)^2)*x^5+8/945*d^2*F^a*b^3*ln(F)^3*F^(b/(d*x+c)^2)*x^3+8/315*F^a*b^3*ln(F)^3*F^(b/
(d*x+c)^2)*c^2*x+2/9*F^a*b*ln(F)*F^(b/(d*x+c)^2)*c^6*x+4/63*F^a*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c^4*x+16/945*F^a*b
^4*ln(F)^4*F^(b/(d*x+c)^2)*x+d^7*F^a*F^(b/(d*x+c)^2)*c*x^8+4*d^6*F^a*F^(b/(d*x+c)^2)*c^2*x^7+28/3*d^5*F^a*F^(b
/(d*x+c)^2)*c^3*x^6+14*d^4*F^a*F^(b/(d*x+c)^2)*c^4*x^5+14*d^3*F^a*F^(b/(d*x+c)^2)*c^5*x^4+28/3*d^2*F^a*F^(b/(d
*x+c)^2)*c^6*x^3+4*d*F^a*F^(b/(d*x+c)^2)*c^7*x^2-16/945/d*F^a*b^5*ln(F)^5*Pi^(1/2)/(-b*ln(F))^(1/2)*erf((-b*ln
(F))^(1/2)/(d*x+c))+2/9*d^5*F^a*b*ln(F)*F^(b/(d*x+c)^2)*c*x^6+2/3*d^4*F^a*b*ln(F)*F^(b/(d*x+c)^2)*c^2*x^5+10/9
*d^3*F^a*b*ln(F)*F^(b/(d*x+c)^2)*c^3*x^4+10/9*d^2*F^a*b*ln(F)*F^(b/(d*x+c)^2)*c^4*x^3+2/3*d*F^a*b*ln(F)*F^(b/(
d*x+c)^2)*c^5*x^2+4/63*d^3*F^a*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c*x^4+8/63*d^2*F^a*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c^2*
x^3+8/63*d*F^a*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c^3*x^2+8/315*d*F^a*b^3*ln(F)^3*F^(b/(d*x+c)^2)*c*x^2+F^a*F^(b/(d*x
+c)^2)*c^8*x+1/9/d*F^a*F^(b/(d*x+c)^2)*c^9+1/9*d^8*F^a*F^(b/(d*x+c)^2)*x^9

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{945} \,{\left (105 \, F^{a} d^{8} x^{9} + 945 \, F^{a} c d^{7} x^{8} + 30 \,{\left (126 \, F^{a} c^{2} d^{6} + F^{a} b d^{6} \log \left (F\right )\right )} x^{7} + 210 \,{\left (42 \, F^{a} c^{3} d^{5} + F^{a} b c d^{5} \log \left (F\right )\right )} x^{6} + 6 \,{\left (2205 \, F^{a} c^{4} d^{4} + 105 \, F^{a} b c^{2} d^{4} \log \left (F\right ) + 2 \, F^{a} b^{2} d^{4} \log \left (F\right )^{2}\right )} x^{5} + 30 \,{\left (441 \, F^{a} c^{5} d^{3} + 35 \, F^{a} b c^{3} d^{3} \log \left (F\right ) + 2 \, F^{a} b^{2} c d^{3} \log \left (F\right )^{2}\right )} x^{4} + 2 \,{\left (4410 \, F^{a} c^{6} d^{2} + 525 \, F^{a} b c^{4} d^{2} \log \left (F\right ) + 60 \, F^{a} b^{2} c^{2} d^{2} \log \left (F\right )^{2} + 4 \, F^{a} b^{3} d^{2} \log \left (F\right )^{3}\right )} x^{3} + 6 \,{\left (630 \, F^{a} c^{7} d + 105 \, F^{a} b c^{5} d \log \left (F\right ) + 20 \, F^{a} b^{2} c^{3} d \log \left (F\right )^{2} + 4 \, F^{a} b^{3} c d \log \left (F\right )^{3}\right )} x^{2} +{\left (945 \, F^{a} c^{8} + 210 \, F^{a} b c^{6} \log \left (F\right ) + 60 \, F^{a} b^{2} c^{4} \log \left (F\right )^{2} + 24 \, F^{a} b^{3} c^{2} \log \left (F\right )^{3} + 16 \, F^{a} b^{4} \log \left (F\right )^{4}\right )} x\right )} F^{\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}} + \int \frac{2 \,{\left (16 \, F^{a} b^{5} d x \log \left (F\right )^{5} - 105 \, F^{a} b c^{9} \log \left (F\right ) - 30 \, F^{a} b^{2} c^{7} \log \left (F\right )^{2} - 12 \, F^{a} b^{3} c^{5} \log \left (F\right )^{3} - 8 \, F^{a} b^{4} c^{3} \log \left (F\right )^{4}\right )} F^{\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{945 \,{\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)^8,x, algorithm="maxima")

[Out]

1/945*(105*F^a*d^8*x^9 + 945*F^a*c*d^7*x^8 + 30*(126*F^a*c^2*d^6 + F^a*b*d^6*log(F))*x^7 + 210*(42*F^a*c^3*d^5
+ F^a*b*c*d^5*log(F))*x^6 + 6*(2205*F^a*c^4*d^4 + 105*F^a*b*c^2*d^4*log(F) + 2*F^a*b^2*d^4*log(F)^2)*x^5 + 30
*(441*F^a*c^5*d^3 + 35*F^a*b*c^3*d^3*log(F) + 2*F^a*b^2*c*d^3*log(F)^2)*x^4 + 2*(4410*F^a*c^6*d^2 + 525*F^a*b*
c^4*d^2*log(F) + 60*F^a*b^2*c^2*d^2*log(F)^2 + 4*F^a*b^3*d^2*log(F)^3)*x^3 + 6*(630*F^a*c^7*d + 105*F^a*b*c^5*
d*log(F) + 20*F^a*b^2*c^3*d*log(F)^2 + 4*F^a*b^3*c*d*log(F)^3)*x^2 + (945*F^a*c^8 + 210*F^a*b*c^6*log(F) + 60*
F^a*b^2*c^4*log(F)^2 + 24*F^a*b^3*c^2*log(F)^3 + 16*F^a*b^4*log(F)^4)*x)*F^(b/(d^2*x^2 + 2*c*d*x + c^2)) + int
egrate(2/945*(16*F^a*b^5*d*x*log(F)^5 - 105*F^a*b*c^9*log(F) - 30*F^a*b^2*c^7*log(F)^2 - 12*F^a*b^3*c^5*log(F)
^3 - 8*F^a*b^4*c^3*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 [B]  time = 1.6409, size = 934, normalized size = 19.06 \begin{align*} \frac{16 \, \sqrt{\pi } F^{a} b^{4} d \sqrt{-\frac{b \log \left (F\right )}{d^{2}}} \operatorname{erf}\left (\frac{d \sqrt{-\frac{b \log \left (F\right )}{d^{2}}}}{d x + c}\right ) \log \left (F\right )^{4} +{\left (105 \, d^{9} x^{9} + 945 \, c d^{8} x^{8} + 3780 \, c^{2} d^{7} x^{7} + 8820 \, c^{3} d^{6} x^{6} + 13230 \, c^{4} d^{5} x^{5} + 13230 \, c^{5} d^{4} x^{4} + 8820 \, c^{6} d^{3} x^{3} + 3780 \, c^{7} d^{2} x^{2} + 945 \, c^{8} d x + 105 \, c^{9} + 16 \,{\left (b^{4} d x + b^{4} c\right )} \log \left (F\right )^{4} + 8 \,{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \log \left (F\right )^{3} + 12 \,{\left (b^{2} d^{5} x^{5} + 5 \, b^{2} c d^{4} x^{4} + 10 \, b^{2} c^{2} d^{3} x^{3} + 10 \, b^{2} c^{3} d^{2} x^{2} + 5 \, b^{2} c^{4} d x + b^{2} c^{5}\right )} \log \left (F\right )^{2} + 30 \,{\left (b d^{7} x^{7} + 7 \, b c d^{6} x^{6} + 21 \, b c^{2} d^{5} x^{5} + 35 \, b c^{3} d^{4} x^{4} + 35 \, b c^{4} d^{3} x^{3} + 21 \, b c^{5} d^{2} x^{2} + 7 \, b c^{6} d x + b c^{7}\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}}}}{945 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^2)*(d*x+c)^8,x, algorithm="fricas")

[Out]

1/945*(16*sqrt(pi)*F^a*b^4*d*sqrt(-b*log(F)/d^2)*erf(d*sqrt(-b*log(F)/d^2)/(d*x + c))*log(F)^4 + (105*d^9*x^9
+ 945*c*d^8*x^8 + 3780*c^2*d^7*x^7 + 8820*c^3*d^6*x^6 + 13230*c^4*d^5*x^5 + 13230*c^5*d^4*x^4 + 8820*c^6*d^3*x
^3 + 3780*c^7*d^2*x^2 + 945*c^8*d*x + 105*c^9 + 16*(b^4*d*x + b^4*c)*log(F)^4 + 8*(b^3*d^3*x^3 + 3*b^3*c*d^2*x
^2 + 3*b^3*c^2*d*x + b^3*c^3)*log(F)^3 + 12*(b^2*d^5*x^5 + 5*b^2*c*d^4*x^4 + 10*b^2*c^2*d^3*x^3 + 10*b^2*c^3*d
^2*x^2 + 5*b^2*c^4*d*x + b^2*c^5)*log(F)^2 + 30*(b*d^7*x^7 + 7*b*c*d^6*x^6 + 21*b*c^2*d^5*x^5 + 35*b*c^3*d^4*x
^4 + 35*b*c^4*d^3*x^3 + 21*b*c^5*d^2*x^2 + 7*b*c^6*d*x + b*c^7)*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

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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)**8,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 )}^{8} 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)^8,x, algorithm="giac")

[Out]

integrate((d*x + c)^8*F^(a + b/(d*x + c)^2), x)