3.324 \(\int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^9} \, dx\)

Optimal. Leaf size=126 \[ \frac {3 F^{a+\frac {b}{(c+d x)^2}}}{b^4 d \log ^4(F)}-\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{b^3 d \log ^3(F) (c+d x)^2}+\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{2 b^2 d \log ^2(F) (c+d x)^4}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)^6} \]

[Out]

3*F^(a+b/(d*x+c)^2)/b^4/d/ln(F)^4-3*F^(a+b/(d*x+c)^2)/b^3/d/(d*x+c)^2/ln(F)^3+3/2*F^(a+b/(d*x+c)^2)/b^2/d/(d*x
+c)^4/ln(F)^2-1/2*F^(a+b/(d*x+c)^2)/b/d/(d*x+c)^6/ln(F)

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Rubi [A]  time = 0.19, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2212, 2209} \[ \frac {3 F^{a+\frac {b}{(c+d x)^2}}}{2 b^2 d \log ^2(F) (c+d x)^4}-\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{b^3 d \log ^3(F) (c+d x)^2}+\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{b^4 d \log ^4(F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*F^(a + b/(c + d*x)^2))/(b^4*d*Log[F]^4) - (3*F^(a + b/(c + d*x)^2))/(b^3*d*(c + d*x)^2*Log[F]^3) + (3*F^(a
+ b/(c + d*x)^2))/(2*b^2*d*(c + d*x)^4*Log[F]^2) - F^(a + b/(c + d*x)^2)/(2*b*d*(c + d*x)^6*Log[F])

Rule 2209

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

Rule 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rubi steps

\begin {align*} \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^9} \, dx &=-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d (c+d x)^6 \log (F)}-\frac {3 \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^7} \, dx}{b \log (F)}\\ &=\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{2 b^2 d (c+d x)^4 \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d (c+d x)^6 \log (F)}+\frac {6 \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^5} \, dx}{b^2 \log ^2(F)}\\ &=-\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{b^3 d (c+d x)^2 \log ^3(F)}+\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{2 b^2 d (c+d x)^4 \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d (c+d x)^6 \log (F)}-\frac {6 \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^3} \, dx}{b^3 \log ^3(F)}\\ &=\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{b^4 d \log ^4(F)}-\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{b^3 d (c+d x)^2 \log ^3(F)}+\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{2 b^2 d (c+d x)^4 \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d (c+d x)^6 \log (F)}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 81, normalized size = 0.64 \[ \frac {F^{a+\frac {b}{(c+d x)^2}} \left (-b^3 \log ^3(F)+3 b^2 \log ^2(F) (c+d x)^2-6 b \log (F) (c+d x)^4+6 (c+d x)^6\right )}{2 b^4 d \log ^4(F) (c+d x)^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(F^(a + b/(c + d*x)^2)*(6*(c + d*x)^6 - 6*b*(c + d*x)^4*Log[F] + 3*b^2*(c + d*x)^2*Log[F]^2 - b^3*Log[F]^3))/(
2*b^4*d*(c + d*x)^6*Log[F]^4)

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fricas [B]  time = 0.42, size = 287, normalized size = 2.28 \[ \frac {{\left (6 \, d^{6} x^{6} + 36 \, c d^{5} x^{5} + 90 \, c^{2} d^{4} x^{4} + 120 \, c^{3} d^{3} x^{3} + 90 \, c^{4} d^{2} x^{2} + 36 \, c^{5} d x + 6 \, c^{6} - b^{3} \log \relax (F)^{3} + 3 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \relax (F)^{2} - 6 \, {\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4}\right )} \log \relax (F)\right )} F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \, {\left (b^{4} d^{7} x^{6} + 6 \, b^{4} c d^{6} x^{5} + 15 \, b^{4} c^{2} d^{5} x^{4} + 20 \, b^{4} c^{3} d^{4} x^{3} + 15 \, b^{4} c^{4} d^{3} x^{2} + 6 \, b^{4} c^{5} d^{2} x + b^{4} c^{6} d\right )} \log \relax (F)^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^2)/(d*x+c)^9,x, algorithm="fricas")

[Out]

1/2*(6*d^6*x^6 + 36*c*d^5*x^5 + 90*c^2*d^4*x^4 + 120*c^3*d^3*x^3 + 90*c^4*d^2*x^2 + 36*c^5*d*x + 6*c^6 - b^3*l
og(F)^3 + 3*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*log(F)^2 - 6*(b*d^4*x^4 + 4*b*c*d^3*x^3 + 6*b*c^2*d^2*x^2 +
4*b*c^3*d*x + b*c^4)*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))/((b^4*d^7*x^6 +
6*b^4*c*d^6*x^5 + 15*b^4*c^2*d^5*x^4 + 20*b^4*c^3*d^4*x^3 + 15*b^4*c^4*d^3*x^2 + 6*b^4*c^5*d^2*x + b^4*c^6*d)
*log(F)^4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{9}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^2)/(d*x+c)^9,x, algorithm="giac")

[Out]

integrate(F^(a + b/(d*x + c)^2)/(d*x + c)^9, x)

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maple [B]  time = 0.12, size = 444, normalized size = 3.52 \[ \frac {\frac {3 d^{7} x^{8} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{b^{4} \ln \relax (F )^{4}}+\frac {24 c \,d^{6} x^{7} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{b^{4} \ln \relax (F )^{4}}-\frac {3 \left (b \ln \relax (F )-28 c^{2}\right ) d^{5} x^{6} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{b^{4} \ln \relax (F )^{4}}-\frac {6 \left (3 b \ln \relax (F )-28 c^{2}\right ) c \,d^{4} x^{5} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{b^{4} \ln \relax (F )^{4}}+\frac {3 \left (b^{2} \ln \relax (F )^{2}-30 b \,c^{2} \ln \relax (F )+140 c^{4}\right ) d^{3} x^{4} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{2 b^{4} \ln \relax (F )^{4}}+\frac {6 \left (b^{2} \ln \relax (F )^{2}-10 b \,c^{2} \ln \relax (F )+28 c^{4}\right ) c \,d^{2} x^{3} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{b^{4} \ln \relax (F )^{4}}-\frac {\left (b^{3} \ln \relax (F )^{3}-18 b^{2} c^{2} \ln \relax (F )^{2}+90 b \,c^{4} \ln \relax (F )-168 c^{6}\right ) d \,x^{2} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{2 b^{4} \ln \relax (F )^{4}}-\frac {\left (b^{3} \ln \relax (F )^{3}-6 b^{2} c^{2} \ln \relax (F )^{2}+18 b \,c^{4} \ln \relax (F )-24 c^{6}\right ) c x \,{\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{b^{4} \ln \relax (F )^{4}}-\frac {\left (b^{3} \ln \relax (F )^{3}-3 b^{2} c^{2} \ln \relax (F )^{2}+6 b \,c^{4} \ln \relax (F )-6 c^{6}\right ) c^{2} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}}{2 b^{4} d \ln \relax (F )^{4}}}{\left (d x +c \right )^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+1/(d*x+c)^2*b)/(d*x+c)^9,x)

[Out]

(3*d^7/ln(F)^4/b^4*x^8*exp((a+1/(d*x+c)^2*b)*ln(F))-c*(b^3*ln(F)^3-6*b^2*c^2*ln(F)^2+18*b*c^4*ln(F)-24*c^6)/b^
4/ln(F)^4*x*exp((a+1/(d*x+c)^2*b)*ln(F))-1/2*d*(b^3*ln(F)^3-18*b^2*c^2*ln(F)^2+90*b*c^4*ln(F)-168*c^6)/ln(F)^4
/b^4*x^2*exp((a+1/(d*x+c)^2*b)*ln(F))+3/2*d^3*(b^2*ln(F)^2-30*b*c^2*ln(F)+140*c^4)/ln(F)^4/b^4*x^4*exp((a+1/(d
*x+c)^2*b)*ln(F))-3*d^5*(b*ln(F)-28*c^2)/ln(F)^4/b^4*x^6*exp((a+1/(d*x+c)^2*b)*ln(F))+24*d^6*c/ln(F)^4/b^4*x^7
*exp((a+1/(d*x+c)^2*b)*ln(F))-1/2*(b^3*ln(F)^3-3*b^2*c^2*ln(F)^2+6*b*c^4*ln(F)-6*c^6)*c^2/b^4/ln(F)^4/d*exp((a
+1/(d*x+c)^2*b)*ln(F))+6*c*d^2*(b^2*ln(F)^2-10*b*c^2*ln(F)+28*c^4)/ln(F)^4/b^4*x^3*exp((a+1/(d*x+c)^2*b)*ln(F)
)-6*c*d^4*(3*b*ln(F)-28*c^2)/ln(F)^4/b^4*x^5*exp((a+1/(d*x+c)^2*b)*ln(F)))/(d*x+c)^8

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maxima [B]  time = 1.08, size = 349, normalized size = 2.77 \[ \frac {{\left (6 \, F^{a} d^{6} x^{6} + 36 \, F^{a} c d^{5} x^{5} + 6 \, F^{a} c^{6} - 6 \, F^{a} b c^{4} \log \relax (F) + 3 \, F^{a} b^{2} c^{2} \log \relax (F)^{2} - F^{a} b^{3} \log \relax (F)^{3} + 6 \, {\left (15 \, F^{a} c^{2} d^{4} - F^{a} b d^{4} \log \relax (F)\right )} x^{4} + 24 \, {\left (5 \, F^{a} c^{3} d^{3} - F^{a} b c d^{3} \log \relax (F)\right )} x^{3} + 3 \, {\left (30 \, F^{a} c^{4} d^{2} - 12 \, F^{a} b c^{2} d^{2} \log \relax (F) + F^{a} b^{2} d^{2} \log \relax (F)^{2}\right )} x^{2} + 6 \, {\left (6 \, F^{a} c^{5} d - 4 \, F^{a} b c^{3} d \log \relax (F) + F^{a} b^{2} c d \log \relax (F)^{2}\right )} x\right )} F^{\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \, {\left (b^{4} d^{7} x^{6} \log \relax (F)^{4} + 6 \, b^{4} c d^{6} x^{5} \log \relax (F)^{4} + 15 \, b^{4} c^{2} d^{5} x^{4} \log \relax (F)^{4} + 20 \, b^{4} c^{3} d^{4} x^{3} \log \relax (F)^{4} + 15 \, b^{4} c^{4} d^{3} x^{2} \log \relax (F)^{4} + 6 \, b^{4} c^{5} d^{2} x \log \relax (F)^{4} + b^{4} c^{6} d \log \relax (F)^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^2)/(d*x+c)^9,x, algorithm="maxima")

[Out]

1/2*(6*F^a*d^6*x^6 + 36*F^a*c*d^5*x^5 + 6*F^a*c^6 - 6*F^a*b*c^4*log(F) + 3*F^a*b^2*c^2*log(F)^2 - F^a*b^3*log(
F)^3 + 6*(15*F^a*c^2*d^4 - F^a*b*d^4*log(F))*x^4 + 24*(5*F^a*c^3*d^3 - F^a*b*c*d^3*log(F))*x^3 + 3*(30*F^a*c^4
*d^2 - 12*F^a*b*c^2*d^2*log(F) + F^a*b^2*d^2*log(F)^2)*x^2 + 6*(6*F^a*c^5*d - 4*F^a*b*c^3*d*log(F) + F^a*b^2*c
*d*log(F)^2)*x)*F^(b/(d^2*x^2 + 2*c*d*x + c^2))/(b^4*d^7*x^6*log(F)^4 + 6*b^4*c*d^6*x^5*log(F)^4 + 15*b^4*c^2*
d^5*x^4*log(F)^4 + 20*b^4*c^3*d^4*x^3*log(F)^4 + 15*b^4*c^4*d^3*x^2*log(F)^4 + 6*b^4*c^5*d^2*x*log(F)^4 + b^4*
c^6*d*log(F)^4)

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mupad [B]  time = 4.25, size = 292, normalized size = 2.32 \[ \frac {F^a\,F^{\frac {b}{c^2+2\,c\,d\,x+d^2\,x^2}}\,\left (\frac {3\,x^6}{b^4\,d\,{\ln \relax (F)}^4}-\frac {b^3\,{\ln \relax (F)}^3-3\,b^2\,c^2\,{\ln \relax (F)}^2+6\,b\,c^4\,\ln \relax (F)-6\,c^6}{2\,b^4\,d^7\,{\ln \relax (F)}^4}+\frac {18\,c\,x^5}{b^4\,d^2\,{\ln \relax (F)}^4}+\frac {3\,x^2\,\left (b^2\,{\ln \relax (F)}^2-12\,b\,c^2\,\ln \relax (F)+30\,c^4\right )}{2\,b^4\,d^5\,{\ln \relax (F)}^4}-\frac {3\,x^4\,\left (b\,\ln \relax (F)-15\,c^2\right )}{b^4\,d^3\,{\ln \relax (F)}^4}-\frac {12\,c\,x^3\,\left (b\,\ln \relax (F)-5\,c^2\right )}{b^4\,d^4\,{\ln \relax (F)}^4}+\frac {3\,c\,x\,\left (b^2\,{\ln \relax (F)}^2-4\,b\,c^2\,\ln \relax (F)+6\,c^4\right )}{b^4\,d^6\,{\ln \relax (F)}^4}\right )}{x^6+\frac {c^6}{d^6}+\frac {6\,c\,x^5}{d}+\frac {6\,c^5\,x}{d^5}+\frac {15\,c^2\,x^4}{d^2}+\frac {20\,c^3\,x^3}{d^3}+\frac {15\,c^4\,x^2}{d^4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a + b/(c + d*x)^2)/(c + d*x)^9,x)

[Out]

(F^a*F^(b/(c^2 + d^2*x^2 + 2*c*d*x))*((3*x^6)/(b^4*d*log(F)^4) - (b^3*log(F)^3 - 6*c^6 + 6*b*c^4*log(F) - 3*b^
2*c^2*log(F)^2)/(2*b^4*d^7*log(F)^4) + (18*c*x^5)/(b^4*d^2*log(F)^4) + (3*x^2*(b^2*log(F)^2 + 30*c^4 - 12*b*c^
2*log(F)))/(2*b^4*d^5*log(F)^4) - (3*x^4*(b*log(F) - 15*c^2))/(b^4*d^3*log(F)^4) - (12*c*x^3*(b*log(F) - 5*c^2
))/(b^4*d^4*log(F)^4) + (3*c*x*(b^2*log(F)^2 + 6*c^4 - 4*b*c^2*log(F)))/(b^4*d^6*log(F)^4)))/(x^6 + c^6/d^6 +
(6*c*x^5)/d + (6*c^5*x)/d^5 + (15*c^2*x^4)/d^2 + (20*c^3*x^3)/d^3 + (15*c^4*x^2)/d^4)

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sympy [B]  time = 0.48, size = 333, normalized size = 2.64 \[ \frac {F^{a + \frac {b}{\left (c + d x\right )^{2}}} \left (- b^{3} \log {\relax (F )}^{3} + 3 b^{2} c^{2} \log {\relax (F )}^{2} + 6 b^{2} c d x \log {\relax (F )}^{2} + 3 b^{2} d^{2} x^{2} \log {\relax (F )}^{2} - 6 b c^{4} \log {\relax (F )} - 24 b c^{3} d x \log {\relax (F )} - 36 b c^{2} d^{2} x^{2} \log {\relax (F )} - 24 b c d^{3} x^{3} \log {\relax (F )} - 6 b d^{4} x^{4} \log {\relax (F )} + 6 c^{6} + 36 c^{5} d x + 90 c^{4} d^{2} x^{2} + 120 c^{3} d^{3} x^{3} + 90 c^{2} d^{4} x^{4} + 36 c d^{5} x^{5} + 6 d^{6} x^{6}\right )}{2 b^{4} c^{6} d \log {\relax (F )}^{4} + 12 b^{4} c^{5} d^{2} x \log {\relax (F )}^{4} + 30 b^{4} c^{4} d^{3} x^{2} \log {\relax (F )}^{4} + 40 b^{4} c^{3} d^{4} x^{3} \log {\relax (F )}^{4} + 30 b^{4} c^{2} d^{5} x^{4} \log {\relax (F )}^{4} + 12 b^{4} c d^{6} x^{5} \log {\relax (F )}^{4} + 2 b^{4} d^{7} x^{6} \log {\relax (F )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b/(d*x+c)**2)/(d*x+c)**9,x)

[Out]

F**(a + b/(c + d*x)**2)*(-b**3*log(F)**3 + 3*b**2*c**2*log(F)**2 + 6*b**2*c*d*x*log(F)**2 + 3*b**2*d**2*x**2*l
og(F)**2 - 6*b*c**4*log(F) - 24*b*c**3*d*x*log(F) - 36*b*c**2*d**2*x**2*log(F) - 24*b*c*d**3*x**3*log(F) - 6*b
*d**4*x**4*log(F) + 6*c**6 + 36*c**5*d*x + 90*c**4*d**2*x**2 + 120*c**3*d**3*x**3 + 90*c**2*d**4*x**4 + 36*c*d
**5*x**5 + 6*d**6*x**6)/(2*b**4*c**6*d*log(F)**4 + 12*b**4*c**5*d**2*x*log(F)**4 + 30*b**4*c**4*d**3*x**2*log(
F)**4 + 40*b**4*c**3*d**4*x**3*log(F)**4 + 30*b**4*c**2*d**5*x**4*log(F)**4 + 12*b**4*c*d**6*x**5*log(F)**4 +
2*b**4*d**7*x**6*log(F)**4)

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