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

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

[Out]

1/2*F^a*(d*x+c)^8*Ei(5,-b*ln(F)/(d*x+c)^2)/d

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Rubi [A]  time = 0.05, antiderivative size = 31, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 31, normalized size = 1.00 \[ \frac {b^4 F^a \log ^4(F) \Gamma \left (-4,-\frac {b \log (F)}{(c+d x)^2}\right )}{2 d} \]

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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fricas [B]  time = 0.44, size = 331, normalized size = 10.68 \[ -\frac {F^{a} b^{4} {\rm Ei}\left (\frac {b \log \relax (F)}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) \log \relax (F)^{4} - {\left (6 \, d^{8} x^{8} + 48 \, c d^{7} x^{7} + 168 \, c^{2} d^{6} x^{6} + 336 \, c^{3} d^{5} x^{5} + 420 \, c^{4} d^{4} x^{4} + 336 \, c^{5} d^{3} x^{3} + 168 \, c^{6} d^{2} x^{2} + 48 \, c^{7} d x + 6 \, c^{8} + {\left (b^{3} d^{2} x^{2} + 2 \, b^{3} c d x + b^{3} c^{2}\right )} \log \relax (F)^{3} + {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \relax (F)^{2} + 2 \, {\left (b d^{6} x^{6} + 6 \, b c d^{5} x^{5} + 15 \, b c^{2} d^{4} x^{4} + 20 \, b c^{3} d^{3} x^{3} + 15 \, b c^{4} d^{2} x^{2} + 6 \, b c^{5} d x + b c^{6}\right )} \log \relax (F)\right )} F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{48 \, d} \]

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]

-1/48*(F^a*b^4*Ei(b*log(F)/(d^2*x^2 + 2*c*d*x + c^2))*log(F)^4 - (6*d^8*x^8 + 48*c*d^7*x^7 + 168*c^2*d^6*x^6 +
336*c^3*d^5*x^5 + 420*c^4*d^4*x^4 + 336*c^5*d^3*x^3 + 168*c^6*d^2*x^2 + 48*c^7*d*x + 6*c^8 + (b^3*d^2*x^2 + 2
*b^3*c*d*x + b^3*c^2)*log(F)^3 + (b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b^2*c^4)
*log(F)^2 + 2*(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^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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{7} F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}\,{d x} \]

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)

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maple [B]  time = 0.10, size = 646, normalized size = 20.84 \[ \frac {d^{7} x^{8} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{8}+c \,d^{6} x^{7} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}+\frac {b \,d^{5} x^{6} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{24}+\frac {7 c^{2} d^{5} x^{6} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{2}+\frac {b c \,d^{4} x^{5} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{4}+7 c^{3} d^{4} x^{5} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}+\frac {b^{2} d^{3} x^{4} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{2}}{48}+\frac {5 b \,c^{2} d^{3} x^{4} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{8}+\frac {35 c^{4} d^{3} x^{4} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{4}+\frac {b^{2} c \,d^{2} x^{3} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{2}}{12}+\frac {5 b \,c^{3} d^{2} x^{3} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{6}+7 c^{5} d^{2} x^{3} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}+\frac {b^{3} d \,x^{2} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{3}}{48}+\frac {b^{2} c^{2} d \,x^{2} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{2}}{8}+\frac {5 b \,c^{4} d \,x^{2} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{8}+\frac {7 c^{6} d \,x^{2} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{2}+\frac {b^{3} c x \,F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{3}}{24}+\frac {b^{2} c^{3} x \,F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{2}}{12}+\frac {b \,c^{5} x \,F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{4}+c^{7} x \,F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}+\frac {b^{4} F^{a} \Ei \left (1, -\frac {b \ln \relax (F )}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )^{4}}{48 d}+\frac {b^{3} c^{2} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{3}}{48 d}+\frac {b^{2} c^{4} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{2}}{48 d}+\frac {b \,c^{6} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{24 d}+\frac {c^{8} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{8 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+1/(d*x+c)^2*b)*(d*x+c)^7,x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{48} \, {\left (6 \, F^{a} d^{7} x^{8} + 48 \, F^{a} c d^{6} x^{7} + 2 \, {\left (84 \, F^{a} c^{2} d^{5} + F^{a} b d^{5} \log \relax (F)\right )} x^{6} + 12 \, {\left (28 \, F^{a} c^{3} d^{4} + F^{a} b c d^{4} \log \relax (F)\right )} x^{5} + {\left (420 \, F^{a} c^{4} d^{3} + 30 \, F^{a} b c^{2} d^{3} \log \relax (F) + F^{a} b^{2} d^{3} \log \relax (F)^{2}\right )} x^{4} + 4 \, {\left (84 \, F^{a} c^{5} d^{2} + 10 \, F^{a} b c^{3} d^{2} \log \relax (F) + F^{a} b^{2} c d^{2} \log \relax (F)^{2}\right )} x^{3} + {\left (168 \, F^{a} c^{6} d + 30 \, F^{a} b c^{4} d \log \relax (F) + 6 \, F^{a} b^{2} c^{2} d \log \relax (F)^{2} + F^{a} b^{3} d \log \relax (F)^{3}\right )} x^{2} + 2 \, {\left (24 \, F^{a} c^{7} + 6 \, F^{a} b c^{5} \log \relax (F) + 2 \, F^{a} b^{2} c^{3} \log \relax (F)^{2} + F^{a} b^{3} c \log \relax (F)^{3}\right )} x\right )} F^{\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}} + \int \frac {{\left (F^{a} b^{4} d^{2} x^{2} \log \relax (F)^{4} + 2 \, F^{a} b^{4} c d x \log \relax (F)^{4} - 6 \, F^{a} b c^{8} \log \relax (F) - 2 \, F^{a} b^{2} c^{6} \log \relax (F)^{2} - F^{a} b^{3} c^{4} \log \relax (F)^{3}\right )} F^{\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{24 \, {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )}}\,{d x} \]

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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mupad [B]  time = 3.84, size = 120, normalized size = 3.87 \[ \frac {F^a\,b^4\,{\ln \relax (F)}^4\,\mathrm {expint}\left (-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^2}\right )}{48\,d}+\frac {F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^4\,{\ln \relax (F)}^4\,\left (\frac {{\left (c+d\,x\right )}^2}{24\,b\,\ln \relax (F)}+\frac {{\left (c+d\,x\right )}^4}{24\,b^2\,{\ln \relax (F)}^2}+\frac {{\left (c+d\,x\right )}^6}{12\,b^3\,{\ln \relax (F)}^3}+\frac {{\left (c+d\,x\right )}^8}{4\,b^4\,{\ln \relax (F)}^4}\right )}{2\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(F^a*b^4*log(F)^4*expint(-(b*log(F))/(c + d*x)^2))/(48*d) + (F^a*F^(b/(c + d*x)^2)*b^4*log(F)^4*((c + d*x)^2/(
24*b*log(F)) + (c + d*x)^4/(24*b^2*log(F)^2) + (c + d*x)^6/(12*b^3*log(F)^3) + (c + d*x)^8/(4*b^4*log(F)^4)))/
(2*d)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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