3.338 \(\int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{12}} \, dx\)

Optimal. Leaf size=49 \[ \frac {F^a \Gamma \left (\frac {11}{2},-\frac {b \log (F)}{(c+d x)^2}\right )}{2 d (c+d x)^{11} \left (-\frac {b \log (F)}{(c+d x)^2}\right )^{11/2}} \]

[Out]

1/2*F^a*(1048576/61836869254970658257624840625*GAMMA(51/2,-b*ln(F)/(d*x+c)^2)-1048576/618368692549706582576248
40625*(-b*ln(F)/(d*x+c)^2)^(49/2)*exp(b*ln(F)/(d*x+c)^2)-524288/1261976923570829760359690625*(-b*ln(F)/(d*x+c)
^2)^(47/2)*exp(b*ln(F)/(d*x+c)^2)-262144/26850572841932548092759375*(-b*ln(F)/(d*x+c)^2)^(45/2)*exp(b*ln(F)/(d
*x+c)^2)-131072/596679396487389957616875*(-b*ln(F)/(d*x+c)^2)^(43/2)*exp(b*ln(F)/(d*x+c)^2)-65536/138762650345
90464130625*(-b*ln(F)/(d*x+c)^2)^(41/2)*exp(b*ln(F)/(d*x+c)^2)-32768/338445488648547905625*(-b*ln(F)/(d*x+c)^2
)^(39/2)*exp(b*ln(F)/(d*x+c)^2)-16384/8678089452526869375*(-b*ln(F)/(d*x+c)^2)^(37/2)*exp(b*ln(F)/(d*x+c)^2)-8
192/234542958176401875*(-b*ln(F)/(d*x+c)^2)^(35/2)*exp(b*ln(F)/(d*x+c)^2)-4096/6701227376468625*(-b*ln(F)/(d*x
+c)^2)^(33/2)*exp(b*ln(F)/(d*x+c)^2)-2048/203067496256625*(-b*ln(F)/(d*x+c)^2)^(31/2)*exp(b*ln(F)/(d*x+c)^2)-1
024/6550564395375*(-b*ln(F)/(d*x+c)^2)^(29/2)*exp(b*ln(F)/(d*x+c)^2)-512/225881530875*(-b*ln(F)/(d*x+c)^2)^(27
/2)*exp(b*ln(F)/(d*x+c)^2)-256/8365982625*(-b*ln(F)/(d*x+c)^2)^(25/2)*exp(b*ln(F)/(d*x+c)^2)-128/334639305*(-b
*ln(F)/(d*x+c)^2)^(23/2)*exp(b*ln(F)/(d*x+c)^2)-64/14549535*(-b*ln(F)/(d*x+c)^2)^(21/2)*exp(b*ln(F)/(d*x+c)^2)
-32/692835*(-b*ln(F)/(d*x+c)^2)^(19/2)*exp(b*ln(F)/(d*x+c)^2)-16/36465*(-b*ln(F)/(d*x+c)^2)^(17/2)*exp(b*ln(F)
/(d*x+c)^2)-8/2145*(-b*ln(F)/(d*x+c)^2)^(15/2)*exp(b*ln(F)/(d*x+c)^2)-4/143*(-b*ln(F)/(d*x+c)^2)^(13/2)*exp(b*
ln(F)/(d*x+c)^2)-2/11*(-b*ln(F)/(d*x+c)^2)^(11/2)*exp(b*ln(F)/(d*x+c)^2))/d/(d*x+c)^11/(-b*ln(F)/(d*x+c)^2)^(1
1/2)

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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} \[ \frac {F^a \text {Gamma}\left (\frac {11}{2},-\frac {b \log (F)}{(c+d x)^2}\right )}{2 d (c+d x)^{11} \left (-\frac {b \log (F)}{(c+d x)^2}\right )^{11/2}} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b/(c + d*x)^2)/(c + d*x)^12,x]

[Out]

(F^a*Gamma[11/2, -((b*Log[F])/(c + d*x)^2)])/(2*d*(c + d*x)^11*(-((b*Log[F])/(c + d*x)^2))^(11/2))

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 \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{12}} \, dx &=\frac {F^a \Gamma \left (\frac {11}{2},-\frac {b \log (F)}{(c+d x)^2}\right )}{2 d (c+d x)^{11} \left (-\frac {b \log (F)}{(c+d x)^2}\right )^{11/2}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 49, normalized size = 1.00 \[ \frac {F^a \Gamma \left (\frac {11}{2},-\frac {b \log (F)}{(c+d x)^2}\right )}{2 d (c+d x)^{11} \left (-\frac {b \log (F)}{(c+d x)^2}\right )^{11/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b/(c + d*x)^2)/(c + d*x)^12,x]

[Out]

(F^a*Gamma[11/2, -((b*Log[F])/(c + d*x)^2)])/(2*d*(c + d*x)^11*(-((b*Log[F])/(c + d*x)^2))^(11/2))

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fricas [A]  time = 0.48, size = 601, normalized size = 12.27 \[ -\frac {945 \, \sqrt {\pi } {\left (d^{10} x^{9} + 9 \, c d^{9} x^{8} + 36 \, c^{2} d^{8} x^{7} + 84 \, c^{3} d^{7} x^{6} + 126 \, c^{4} d^{6} x^{5} + 126 \, c^{5} d^{5} x^{4} + 84 \, c^{6} d^{4} x^{3} + 36 \, c^{7} d^{3} x^{2} + 9 \, c^{8} d^{2} x + c^{9} d\right )} F^{a} \sqrt {-\frac {b \log \relax (F)}{d^{2}}} \operatorname {erf}\left (\frac {d \sqrt {-\frac {b \log \relax (F)}{d^{2}}}}{d x + c}\right ) + 2 \, {\left (16 \, b^{5} \log \relax (F)^{5} - 72 \, {\left (b^{4} d^{2} x^{2} + 2 \, b^{4} c d x + b^{4} c^{2}\right )} \log \relax (F)^{4} + 252 \, {\left (b^{3} d^{4} x^{4} + 4 \, b^{3} c d^{3} x^{3} + 6 \, b^{3} c^{2} d^{2} x^{2} + 4 \, b^{3} c^{3} d x + b^{3} c^{4}\right )} \log \relax (F)^{3} - 630 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 15 \, b^{2} c^{4} d^{2} x^{2} + 6 \, b^{2} c^{5} d x + b^{2} c^{6}\right )} \log \relax (F)^{2} + 945 \, {\left (b d^{8} x^{8} + 8 \, b c d^{7} x^{7} + 28 \, b c^{2} d^{6} x^{6} + 56 \, b c^{3} d^{5} x^{5} + 70 \, b c^{4} d^{4} x^{4} + 56 \, b c^{5} d^{3} x^{3} + 28 \, b c^{6} d^{2} x^{2} + 8 \, b c^{7} d x + b c^{8}\right )} \log \relax (F)\right )} F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{64 \, {\left (b^{6} d^{10} x^{9} + 9 \, b^{6} c d^{9} x^{8} + 36 \, b^{6} c^{2} d^{8} x^{7} + 84 \, b^{6} c^{3} d^{7} x^{6} + 126 \, b^{6} c^{4} d^{6} x^{5} + 126 \, b^{6} c^{5} d^{5} x^{4} + 84 \, b^{6} c^{6} d^{4} x^{3} + 36 \, b^{6} c^{7} d^{3} x^{2} + 9 \, b^{6} c^{8} d^{2} x + b^{6} c^{9} d\right )} \log \relax (F)^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^2)/(d*x+c)^12,x, algorithm="fricas")

[Out]

-1/64*(945*sqrt(pi)*(d^10*x^9 + 9*c*d^9*x^8 + 36*c^2*d^8*x^7 + 84*c^3*d^7*x^6 + 126*c^4*d^6*x^5 + 126*c^5*d^5*
x^4 + 84*c^6*d^4*x^3 + 36*c^7*d^3*x^2 + 9*c^8*d^2*x + c^9*d)*F^a*sqrt(-b*log(F)/d^2)*erf(d*sqrt(-b*log(F)/d^2)
/(d*x + c)) + 2*(16*b^5*log(F)^5 - 72*(b^4*d^2*x^2 + 2*b^4*c*d*x + b^4*c^2)*log(F)^4 + 252*(b^3*d^4*x^4 + 4*b^
3*c*d^3*x^3 + 6*b^3*c^2*d^2*x^2 + 4*b^3*c^3*d*x + b^3*c^4)*log(F)^3 - 630*(b^2*d^6*x^6 + 6*b^2*c*d^5*x^5 + 15*
b^2*c^2*d^4*x^4 + 20*b^2*c^3*d^3*x^3 + 15*b^2*c^4*d^2*x^2 + 6*b^2*c^5*d*x + b^2*c^6)*log(F)^2 + 945*(b*d^8*x^8
+ 8*b*c*d^7*x^7 + 28*b*c^2*d^6*x^6 + 56*b*c^3*d^5*x^5 + 70*b*c^4*d^4*x^4 + 56*b*c^5*d^3*x^3 + 28*b*c^6*d^2*x^
2 + 8*b*c^7*d*x + b*c^8)*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^6*d^10
*x^9 + 9*b^6*c*d^9*x^8 + 36*b^6*c^2*d^8*x^7 + 84*b^6*c^3*d^7*x^6 + 126*b^6*c^4*d^6*x^5 + 126*b^6*c^5*d^5*x^4 +
84*b^6*c^6*d^4*x^3 + 36*b^6*c^7*d^3*x^2 + 9*b^6*c^8*d^2*x + b^6*c^9*d)*log(F)^6)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{12}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^2)/(d*x+c)^12,x, algorithm="giac")

[Out]

integrate(F^(a + b/(d*x + c)^2)/(d*x + c)^12, x)

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maple [A]  time = 0.27, size = 208, normalized size = 4.24 \[ -\frac {F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{2 \left (d x +c \right )^{9} b d \ln \relax (F )}+\frac {9 F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{4 \left (d x +c \right )^{7} b^{2} d \ln \relax (F )^{2}}-\frac {63 F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{8 \left (d x +c \right )^{5} b^{3} d \ln \relax (F )^{3}}+\frac {315 F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{16 \left (d x +c \right )^{3} b^{4} d \ln \relax (F )^{4}}-\frac {945 F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{32 \left (d x +c \right ) b^{5} d \ln \relax (F )^{5}}+\frac {945 \sqrt {\pi }\, F^{a} \erf \left (\frac {\sqrt {-b \ln \relax (F )}}{d x +c}\right )}{64 \sqrt {-b \ln \relax (F )}\, b^{5} d \ln \relax (F )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+1/(d*x+c)^2*b)/(d*x+c)^12,x)

[Out]

-1/2*F^a/d*F^(1/(d*x+c)^2*b)/(d*x+c)^9/b/ln(F)+9/4*F^a/d/b^2/ln(F)^2*F^(1/(d*x+c)^2*b)/(d*x+c)^7-63/8*F^a/d/b^
3/ln(F)^3*F^(1/(d*x+c)^2*b)/(d*x+c)^5+315/16*F^a/d/b^4/ln(F)^4*F^(1/(d*x+c)^2*b)/(d*x+c)^3-945/32*F^a/d/b^5/ln
(F)^5*F^(1/(d*x+c)^2*b)/(d*x+c)+945/64*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
))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{12}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^2)/(d*x+c)^12,x, algorithm="maxima")

[Out]

integrate(F^(a + b/(d*x + c)^2)/(d*x + c)^12, x)

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mupad [B]  time = 4.92, size = 189, normalized size = 3.86 \[ \frac {\frac {F^a\,\left (945\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,\ln \relax (F)}{\sqrt {b\,\ln \relax (F)}\,\left (c+d\,x\right )}\right )-\frac {1890\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,\sqrt {b\,\ln \relax (F)}}{c+d\,x}\right )}{64\,\sqrt {b\,\ln \relax (F)}}-\frac {63\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^2\,{\ln \relax (F)}^2}{8\,{\left (c+d\,x\right )}^5}+\frac {9\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^3\,{\ln \relax (F)}^3}{4\,{\left (c+d\,x\right )}^7}-\frac {F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^4\,{\ln \relax (F)}^4}{2\,{\left (c+d\,x\right )}^9}+\frac {315\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b\,\ln \relax (F)}{16\,{\left (c+d\,x\right )}^3}}{b^5\,d\,{\ln \relax (F)}^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a + b/(c + d*x)^2)/(c + d*x)^12,x)

[Out]

((F^a*(945*pi^(1/2)*erfi((b*log(F))/((b*log(F))^(1/2)*(c + d*x))) - (1890*F^(b/(c + d*x)^2)*(b*log(F))^(1/2))/
(c + d*x)))/(64*(b*log(F))^(1/2)) - (63*F^a*F^(b/(c + d*x)^2)*b^2*log(F)^2)/(8*(c + d*x)^5) + (9*F^a*F^(b/(c +
d*x)^2)*b^3*log(F)^3)/(4*(c + d*x)^7) - (F^a*F^(b/(c + d*x)^2)*b^4*log(F)^4)/(2*(c + d*x)^9) + (315*F^a*F^(b/
(c + d*x)^2)*b*log(F))/(16*(c + d*x)^3))/(b^5*d*log(F)^5)

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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)**12,x)

[Out]

Timed out

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