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

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

[Out]

1/2*F^a*(d*x+c)^9*(-32/945*Pi^(1/2)*erfc((-b*ln(F)/(d*x+c)^2)^(1/2))+32/945/(-b*ln(F)/(d*x+c)^2)^(1/2)*exp(b*l
n(F)/(d*x+c)^2)-16/945/(-b*ln(F)/(d*x+c)^2)^(3/2)*exp(b*ln(F)/(d*x+c)^2)+8/315/(-b*ln(F)/(d*x+c)^2)^(5/2)*exp(
b*ln(F)/(d*x+c)^2)-4/63/(-b*ln(F)/(d*x+c)^2)^(7/2)*exp(b*ln(F)/(d*x+c)^2)+2/9/(-b*ln(F)/(d*x+c)^2)^(9/2)*exp(b
*ln(F)/(d*x+c)^2))*(-b*ln(F)/(d*x+c)^2)^(9/2)/d

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Rubi [A]  time = 0.05, antiderivative size = 49, 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} \[ \frac {F^a (c+d x)^9 \left (-\frac {b \log (F)}{(c+d x)^2}\right )^{9/2} \text {Gamma}\left (-\frac {9}{2},-\frac {b \log (F)}{(c+d x)^2}\right )}{2 d} \]

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*}

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Mathematica [A]  time = 0.03, size = 49, normalized size = 1.00 \[ \frac {F^a (c+d x)^9 \left (-\frac {b \log (F)}{(c+d x)^2}\right )^{9/2} \Gamma \left (-\frac {9}{2},-\frac {b \log (F)}{(c+d x)^2}\right )}{2 d} \]

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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fricas [B]  time = 0.41, size = 413, normalized size = 8.43 \[ \frac {16 \, \sqrt {\pi } F^{a} b^{4} d \sqrt {-\frac {b \log \relax (F)}{d^{2}}} \operatorname {erf}\left (\frac {d \sqrt {-\frac {b \log \relax (F)}{d^{2}}}}{d x + c}\right ) \log \relax (F)^{4} + {\left (105 \, d^{9} x^{9} + 945 \, c d^{8} x^{8} + 3780 \, c^{2} d^{7} x^{7} + 8820 \, c^{3} d^{6} x^{6} + 13230 \, c^{4} d^{5} x^{5} + 13230 \, c^{5} d^{4} x^{4} + 8820 \, c^{6} d^{3} x^{3} + 3780 \, c^{7} d^{2} x^{2} + 945 \, c^{8} d x + 105 \, c^{9} + 16 \, {\left (b^{4} d x + b^{4} c\right )} \log \relax (F)^{4} + 8 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \log \relax (F)^{3} + 12 \, {\left (b^{2} d^{5} x^{5} + 5 \, b^{2} c d^{4} x^{4} + 10 \, b^{2} c^{2} d^{3} x^{3} + 10 \, b^{2} c^{3} d^{2} x^{2} + 5 \, b^{2} c^{4} d x + b^{2} c^{5}\right )} \log \relax (F)^{2} + 30 \, {\left (b d^{7} x^{7} + 7 \, b c d^{6} x^{6} + 21 \, b c^{2} d^{5} x^{5} + 35 \, b c^{3} d^{4} x^{4} + 35 \, b c^{4} d^{3} x^{3} + 21 \, b c^{5} d^{2} x^{2} + 7 \, b c^{6} d x + b c^{7}\right )} \log \relax (F)\right )} F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{945 \, d} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{8} F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.12, size = 826, normalized size = 16.86 \[ \frac {d^{8} x^{9} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{9}+c \,d^{7} x^{8} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}+\frac {2 b \,d^{6} x^{7} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{63}+4 c^{2} d^{6} x^{7} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}+\frac {2 b c \,d^{5} x^{6} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{9}+\frac {28 c^{3} d^{5} x^{6} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{3}+\frac {4 b^{2} d^{4} x^{5} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{2}}{315}+\frac {2 b \,c^{2} d^{4} x^{5} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{3}+14 c^{4} d^{4} x^{5} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}+\frac {4 b^{2} c \,d^{3} x^{4} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{2}}{63}+\frac {10 b \,c^{3} d^{3} x^{4} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{9}+14 c^{5} d^{3} x^{4} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}+\frac {8 b^{3} d^{2} x^{3} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{3}}{945}+\frac {8 b^{2} c^{2} d^{2} x^{3} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{2}}{63}+\frac {10 b \,c^{4} d^{2} x^{3} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{9}+\frac {28 c^{6} d^{2} x^{3} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{3}+\frac {8 b^{3} c d \,x^{2} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{3}}{315}+\frac {8 b^{2} c^{3} d \,x^{2} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{2}}{63}+\frac {2 b \,c^{5} d \,x^{2} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{3}+4 c^{7} d \,x^{2} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}-\frac {16 \sqrt {\pi }\, b^{5} F^{a} \erf \left (\frac {\sqrt {-b \ln \relax (F )}}{d x +c}\right ) \ln \relax (F )^{5}}{945 \sqrt {-b \ln \relax (F )}\, d}+\frac {16 b^{4} x \,F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{4}}{945}+\frac {8 b^{3} c^{2} x \,F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{3}}{315}+\frac {4 b^{2} c^{4} x \,F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{2}}{63}+\frac {2 b \,c^{6} x \,F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{9}+c^{8} x \,F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}+\frac {16 b^{4} c \,F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{4}}{945 d}+\frac {8 b^{3} c^{3} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{3}}{945 d}+\frac {4 b^{2} c^{5} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{2}}{315 d}+\frac {2 b \,c^{7} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{63 d}+\frac {c^{9} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{9 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+1/(d*x+c)^2*b)*(d*x+c)^8,x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{945} \, {\left (105 \, F^{a} d^{8} x^{9} + 945 \, F^{a} c d^{7} x^{8} + 30 \, {\left (126 \, F^{a} c^{2} d^{6} + F^{a} b d^{6} \log \relax (F)\right )} x^{7} + 210 \, {\left (42 \, F^{a} c^{3} d^{5} + F^{a} b c d^{5} \log \relax (F)\right )} x^{6} + 6 \, {\left (2205 \, F^{a} c^{4} d^{4} + 105 \, F^{a} b c^{2} d^{4} \log \relax (F) + 2 \, F^{a} b^{2} d^{4} \log \relax (F)^{2}\right )} x^{5} + 30 \, {\left (441 \, F^{a} c^{5} d^{3} + 35 \, F^{a} b c^{3} d^{3} \log \relax (F) + 2 \, F^{a} b^{2} c d^{3} \log \relax (F)^{2}\right )} x^{4} + 2 \, {\left (4410 \, F^{a} c^{6} d^{2} + 525 \, F^{a} b c^{4} d^{2} \log \relax (F) + 60 \, F^{a} b^{2} c^{2} d^{2} \log \relax (F)^{2} + 4 \, F^{a} b^{3} d^{2} \log \relax (F)^{3}\right )} x^{3} + 6 \, {\left (630 \, F^{a} c^{7} d + 105 \, F^{a} b c^{5} d \log \relax (F) + 20 \, F^{a} b^{2} c^{3} d \log \relax (F)^{2} + 4 \, F^{a} b^{3} c d \log \relax (F)^{3}\right )} x^{2} + {\left (945 \, F^{a} c^{8} + 210 \, F^{a} b c^{6} \log \relax (F) + 60 \, F^{a} b^{2} c^{4} \log \relax (F)^{2} + 24 \, F^{a} b^{3} c^{2} \log \relax (F)^{3} + 16 \, F^{a} b^{4} \log \relax (F)^{4}\right )} x\right )} F^{\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}} + \int \frac {2 \, {\left (16 \, F^{a} b^{5} d x \log \relax (F)^{5} - 105 \, F^{a} b c^{9} \log \relax (F) - 30 \, F^{a} b^{2} c^{7} \log \relax (F)^{2} - 12 \, F^{a} b^{3} c^{5} \log \relax (F)^{3} - 8 \, F^{a} b^{4} c^{3} \log \relax (F)^{4}\right )} F^{\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{945 \, {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )}}\,{d x} \]

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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mupad [B]  time = 4.17, size = 232, normalized size = 4.73 \[ \frac {F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,{\left (c+d\,x\right )}^9}{9\,d}+\frac {16\,F^a\,\sqrt {\pi }\,{\left (c+d\,x\right )}^9\,{\left (-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^2}\right )}^{9/2}}{945\,d}+\frac {4\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^2\,{\ln \relax (F)}^2\,{\left (c+d\,x\right )}^5}{315\,d}+\frac {8\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^3\,{\ln \relax (F)}^3\,{\left (c+d\,x\right )}^3}{945\,d}+\frac {2\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b\,\ln \relax (F)\,{\left (c+d\,x\right )}^7}{63\,d}+\frac {16\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^4\,{\ln \relax (F)}^4\,\left (c+d\,x\right )}{945\,d}-\frac {16\,F^a\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^2}}\right )\,{\left (c+d\,x\right )}^9\,{\left (-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^2}\right )}^{9/2}}{945\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(F^a*F^(b/(c + d*x)^2)*(c + d*x)^9)/(9*d) + (16*F^a*pi^(1/2)*(c + d*x)^9*(-(b*log(F))/(c + d*x)^2)^(9/2))/(945
*d) + (4*F^a*F^(b/(c + d*x)^2)*b^2*log(F)^2*(c + d*x)^5)/(315*d) + (8*F^a*F^(b/(c + d*x)^2)*b^3*log(F)^3*(c +
d*x)^3)/(945*d) + (2*F^a*F^(b/(c + d*x)^2)*b*log(F)*(c + d*x)^7)/(63*d) + (16*F^a*F^(b/(c + d*x)^2)*b^4*log(F)
^4*(c + d*x))/(945*d) - (16*F^a*pi^(1/2)*erfc((-(b*log(F))/(c + d*x)^2)^(1/2))*(c + d*x)^9*(-(b*log(F))/(c + d
*x)^2)^(9/2))/(945*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)**8,x)

[Out]

Timed out

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