\(\int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^8 \, dx\) [328]

Optimal result

Integrand size = 21, antiderivative size = 49 \[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^8 \, dx=\frac {F^a (c+d x)^9 \Gamma \left (-\frac {9}{2},-\frac {b \log (F)}{(c+d x)^2}\right ) \left (-\frac {b \log (F)}{(c+d x)^2}\right )^{9/2}}{2 d} \]

[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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2250} \[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^8 \, dx=\frac {F^a (c+d x)^9 \left (-\frac {b \log (F)}{(c+d x)^2}\right )^{9/2} \Gamma \left (-\frac {9}{2},-\frac {b \log (F)}{(c+d x)^2}\right )}{2 d} \]

[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 2250

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

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00 \[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^8 \, dx=\frac {F^a (c+d x)^9 \Gamma \left (-\frac {9}{2},-\frac {b \log (F)}{(c+d x)^2}\right ) \left (-\frac {b \log (F)}{(c+d x)^2}\right )^{9/2}}{2 d} \]

[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)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(825\) vs. \(2(190)=380\).

Time = 2.62 (sec) , antiderivative size = 826, normalized size of antiderivative = 16.86

method	result	size
risch	\(F^{a} d^{7} F^{\frac {b}{\left (d x +c \right )^{2}}} c \,x^{8}+4 F^{a} d^{6} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{2} x^{7}+\frac {28 F^{a} d^{5} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{3} x^{6}}{3}+14 F^{a} d^{4} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{4} x^{5}+14 F^{a} d^{3} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{5} x^{4}+\frac {28 F^{a} d^{2} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{6} x^{3}}{3}+4 F^{a} d \,F^{\frac {b}{\left (d x +c \right )^{2}}} c^{7} x^{2}+\frac {16 F^{a} b^{4} \ln \left (F \right )^{4} F^{\frac {b}{\left (d x +c \right )^{2}}} x}{945}+\frac {2 F^{a} d^{6} b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} x^{7}}{63}+\frac {4 F^{a} d^{4} b^{2} \ln \left (F \right )^{2} F^{\frac {b}{\left (d x +c \right )^{2}}} x^{5}}{315}+\frac {8 F^{a} d^{2} b^{3} \ln \left (F \right )^{3} F^{\frac {b}{\left (d x +c \right )^{2}}} x^{3}}{945}+\frac {2 F^{a} b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} c^{7}}{63 d}+\frac {4 F^{a} b^{2} \ln \left (F \right )^{2} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{5}}{315 d}+\frac {8 F^{a} b^{3} \ln \left (F \right )^{3} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{3}}{945 d}+\frac {16 F^{a} b^{4} \ln \left (F \right )^{4} F^{\frac {b}{\left (d x +c \right )^{2}}} c}{945 d}+\frac {8 F^{a} b^{3} \ln \left (F \right )^{3} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{2} x}{315}+\frac {2 F^{a} b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} c^{6} x}{9}+\frac {4 F^{a} b^{2} \ln \left (F \right )^{2} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{4} x}{63}+F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{8} x +\frac {F^{a} d^{8} F^{\frac {b}{\left (d x +c \right )^{2}}} x^{9}}{9}+\frac {F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{9}}{9 d}-\frac {16 F^{a} b^{5} \ln \left (F \right )^{5} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-b \ln \left (F \right )}}{d x +c}\right )}{945 d \sqrt {-b \ln \left (F \right )}}+\frac {2 F^{a} d^{5} b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} c \,x^{6}}{9}+\frac {2 F^{a} d^{4} b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} c^{2} x^{5}}{3}+\frac {10 F^{a} d^{3} b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} c^{3} x^{4}}{9}+\frac {10 F^{a} d^{2} b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} c^{4} x^{3}}{9}+\frac {2 F^{a} d b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} c^{5} x^{2}}{3}+\frac {4 F^{a} d^{3} b^{2} \ln \left (F \right )^{2} F^{\frac {b}{\left (d x +c \right )^{2}}} c \,x^{4}}{63}+\frac {8 F^{a} d^{2} b^{2} \ln \left (F \right )^{2} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{2} x^{3}}{63}+\frac {8 F^{a} d \,b^{2} \ln \left (F \right )^{2} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{3} x^{2}}{63}+\frac {8 F^{a} d \,b^{3} \ln \left (F \right )^{3} F^{\frac {b}{\left (d x +c \right )^{2}}} c \,x^{2}}{315}\)	\(826\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 413 vs. \(2 (185) = 370\).

Time = 0.33 (sec) , antiderivative size = 413, normalized size of antiderivative = 8.43 \[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^8 \, dx=\frac {16 \, \sqrt {\pi } F^{a} b^{4} d \sqrt {-\frac {b \log \left (F\right )}{d^{2}}} \operatorname {erf}\left (\frac {d \sqrt {-\frac {b \log \left (F\right )}{d^{2}}}}{d x + c}\right ) \log \left (F\right )^{4} + {\left (105 \, d^{9} x^{9} + 945 \, c d^{8} x^{8} + 3780 \, c^{2} d^{7} x^{7} + 8820 \, c^{3} d^{6} x^{6} + 13230 \, c^{4} d^{5} x^{5} + 13230 \, c^{5} d^{4} x^{4} + 8820 \, c^{6} d^{3} x^{3} + 3780 \, c^{7} d^{2} x^{2} + 945 \, c^{8} d x + 105 \, c^{9} + 16 \, {\left (b^{4} d x + b^{4} c\right )} \log \left (F\right )^{4} + 8 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \log \left (F\right )^{3} + 12 \, {\left (b^{2} d^{5} x^{5} + 5 \, b^{2} c d^{4} x^{4} + 10 \, b^{2} c^{2} d^{3} x^{3} + 10 \, b^{2} c^{3} d^{2} x^{2} + 5 \, b^{2} c^{4} d x + b^{2} c^{5}\right )} \log \left (F\right )^{2} + 30 \, {\left (b d^{7} x^{7} + 7 \, b c d^{6} x^{6} + 21 \, b c^{2} d^{5} x^{5} + 35 \, b c^{3} d^{4} x^{4} + 35 \, b c^{4} d^{3} x^{3} + 21 \, b c^{5} d^{2} x^{2} + 7 \, b c^{6} d x + b c^{7}\right )} \log \left (F\right )\right )} F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{945 \, d} \]

[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

Sympy [F]

\[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^8 \, dx=\int F^{a + \frac {b}{\left (c + d x\right )^{2}}} \left (c + d x\right )^{8}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^8 \, dx=\int { {\left (d x + c\right )}^{8} F^{a + \frac {b}{{\left (d x + c\right )}^{2}}} \,d x } \]

[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)

Giac [F]

\[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^8 \, dx=\int { {\left (d x + c\right )}^{8} F^{a + \frac {b}{{\left (d x + c\right )}^{2}}} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 0.82 (sec) , antiderivative size = 232, normalized size of antiderivative = 4.73 \[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^8 \, dx=\frac {F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,{\left (c+d\,x\right )}^9}{9\,d}+\frac {16\,F^a\,\sqrt {\pi }\,{\left (c+d\,x\right )}^9\,{\left (-\frac {b\,\ln \left (F\right )}{{\left (c+d\,x\right )}^2}\right )}^{9/2}}{945\,d}+\frac {4\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^2\,{\ln \left (F\right )}^2\,{\left (c+d\,x\right )}^5}{315\,d}+\frac {8\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^3\,{\ln \left (F\right )}^3\,{\left (c+d\,x\right )}^3}{945\,d}+\frac {2\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^7}{63\,d}+\frac {16\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^4\,{\ln \left (F\right )}^4\,\left (c+d\,x\right )}{945\,d}-\frac {16\,F^a\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-\frac {b\,\ln \left (F\right )}{{\left (c+d\,x\right )}^2}}\right )\,{\left (c+d\,x\right )}^9\,{\left (-\frac {b\,\ln \left (F\right )}{{\left (c+d\,x\right )}^2}\right )}^{9/2}}{945\,d} \]

[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)