3.316 \(\int F^{a+\frac{b}{(c+d x)^2}} (c+d x)^7 \, dx\)

Optimal. Leaf size=31 \[ \frac{b^4 F^a \log ^4(F) \text{Gamma}\left (-4,-\frac{b \log (F)}{(c+d x)^2}\right )}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.0505083, 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} \[ \frac{b^4 F^a \log ^4(F) \text{Gamma}\left (-4,-\frac{b \log (F)}{(c+d x)^2}\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b/(c + d*x)^2)*(c + d*x)^7,x]

[Out]

(b^4*F^a*Gamma[-4, -((b*Log[F])/(c + d*x)^2)]*Log[F]^4)/(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)^7 \, dx &=\frac{b^4 F^a \Gamma \left (-4,-\frac{b \log (F)}{(c+d x)^2}\right ) \log ^4(F)}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.0075925, size = 31, normalized size = 1. \[ \frac{b^4 F^a \log ^4(F) \text{Gamma}\left (-4,-\frac{b \log (F)}{(c+d x)^2}\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b/(c + d*x)^2)*(c + d*x)^7,x]

[Out]

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

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Maple [B]  time = 0.064, size = 646, normalized size = 20.8 \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)^7,x)

[Out]

1/48*d^3*F^a*b^2*ln(F)^2*F^(b/(d*x+c)^2)*x^4+1/48*d*F^a*b^3*ln(F)^3*F^(b/(d*x+c)^2)*x^2+1/24*d^5*F^a*b*ln(F)*F
^(b/(d*x+c)^2)*x^6+1/24/d*F^a*b*ln(F)*F^(b/(d*x+c)^2)*c^6+1/48/d*F^a*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c^4+1/48/d*F^
a*b^3*ln(F)^3*F^(b/(d*x+c)^2)*c^2+1/4*F^a*b*ln(F)*F^(b/(d*x+c)^2)*c^5*x+1/12*F^a*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c
^3*x+1/24*F^a*b^3*ln(F)^3*F^(b/(d*x+c)^2)*c*x+1/12*d^2*F^a*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c*x^3+1/8*d*F^a*b^2*ln(
F)^2*F^(b/(d*x+c)^2)*c^2*x^2+1/4*d^4*F^a*b*ln(F)*F^(b/(d*x+c)^2)*c*x^5+5/8*d^3*F^a*b*ln(F)*F^(b/(d*x+c)^2)*c^2
*x^4+5/6*d^2*F^a*b*ln(F)*F^(b/(d*x+c)^2)*c^3*x^3+5/8*d*F^a*b*ln(F)*F^(b/(d*x+c)^2)*c^4*x^2+1/48/d*F^a*b^4*ln(F
)^4*Ei(1,-b*ln(F)/(d*x+c)^2)+d^6*F^a*F^(b/(d*x+c)^2)*c*x^7+7/2*d^5*F^a*F^(b/(d*x+c)^2)*c^2*x^6+7*d^4*F^a*F^(b/
(d*x+c)^2)*c^3*x^5+35/4*d^3*F^a*F^(b/(d*x+c)^2)*c^4*x^4+7*d^2*F^a*F^(b/(d*x+c)^2)*c^5*x^3+7/2*d*F^a*F^(b/(d*x+
c)^2)*c^6*x^2+F^a*F^(b/(d*x+c)^2)*c^7*x+1/8/d*F^a*F^(b/(d*x+c)^2)*c^8+1/8*d^7*F^a*F^(b/(d*x+c)^2)*x^8

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

[Out]

1/48*(6*F^a*d^7*x^8 + 48*F^a*c*d^6*x^7 + 2*(84*F^a*c^2*d^5 + F^a*b*d^5*log(F))*x^6 + 12*(28*F^a*c^3*d^4 + F^a*
b*c*d^4*log(F))*x^5 + (420*F^a*c^4*d^3 + 30*F^a*b*c^2*d^3*log(F) + F^a*b^2*d^3*log(F)^2)*x^4 + 4*(84*F^a*c^5*d
^2 + 10*F^a*b*c^3*d^2*log(F) + F^a*b^2*c*d^2*log(F)^2)*x^3 + (168*F^a*c^6*d + 30*F^a*b*c^4*d*log(F) + 6*F^a*b^
2*c^2*d*log(F)^2 + F^a*b^3*d*log(F)^3)*x^2 + 2*(24*F^a*c^7 + 6*F^a*b*c^5*log(F) + 2*F^a*b^2*c^3*log(F)^2 + F^a
*b^3*c*log(F)^3)*x)*F^(b/(d^2*x^2 + 2*c*d*x + c^2)) + integrate(1/24*(F^a*b^4*d^2*x^2*log(F)^4 + 2*F^a*b^4*c*d
*x*log(F)^4 - 6*F^a*b*c^8*log(F) - 2*F^a*b^2*c^6*log(F)^2 - F^a*b^3*c^4*log(F)^3)*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)^7,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)**7,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 )}^{7} 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)^7,x, algorithm="giac")

[Out]

integrate((d*x + c)^7*F^(a + b/(d*x + c)^2), x)