3.327 \(\int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^{10} \, dx\)

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

[Out]

1/2*F^a*(d*x+c)^11*(64/10395*Pi^(1/2)*erfc((-b*ln(F)/(d*x+c)^2)^(1/2))-64/10395/(-b*ln(F)/(d*x+c)^2)^(1/2)*exp
(b*ln(F)/(d*x+c)^2)+32/10395/(-b*ln(F)/(d*x+c)^2)^(3/2)*exp(b*ln(F)/(d*x+c)^2)-16/3465/(-b*ln(F)/(d*x+c)^2)^(5
/2)*exp(b*ln(F)/(d*x+c)^2)+8/693/(-b*ln(F)/(d*x+c)^2)^(7/2)*exp(b*ln(F)/(d*x+c)^2)-4/99/(-b*ln(F)/(d*x+c)^2)^(
9/2)*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))*(-b*ln(F)/(d*x+c)^2)^(11/
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} \[ \frac {F^a (c+d x)^{11} \left (-\frac {b \log (F)}{(c+d x)^2}\right )^{11/2} \text {Gamma}\left (-\frac {11}{2},-\frac {b \log (F)}{(c+d x)^2}\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b/(c + d*x)^2)*(c + d*x)^10,x]

[Out]

(F^a*(c + d*x)^11*Gamma[-11/2, -((b*Log[F])/(c + d*x)^2)]*(-((b*Log[F])/(c + d*x)^2))^(11/2))/(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)^{10} \, dx &=\frac {F^a (c+d x)^{11} \Gamma \left (-\frac {11}{2},-\frac {b \log (F)}{(c+d x)^2}\right ) \left (-\frac {b \log (F)}{(c+d x)^2}\right )^{11/2}}{2 d}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 49, normalized size = 1.00 \[ \frac {F^a (c+d x)^{11} \left (-\frac {b \log (F)}{(c+d x)^2}\right )^{11/2} \Gamma \left (-\frac {11}{2},-\frac {b \log (F)}{(c+d x)^2}\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b/(c + d*x)^2)*(c + d*x)^10,x]

[Out]

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

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fricas [B]  time = 0.43, size = 561, normalized size = 11.45 \[ \frac {32 \, \sqrt {\pi } F^{a} b^{5} d \sqrt {-\frac {b \log \relax (F)}{d^{2}}} \operatorname {erf}\left (\frac {d \sqrt {-\frac {b \log \relax (F)}{d^{2}}}}{d x + c}\right ) \log \relax (F)^{5} + {\left (945 \, d^{11} x^{11} + 10395 \, c d^{10} x^{10} + 51975 \, c^{2} d^{9} x^{9} + 155925 \, c^{3} d^{8} x^{8} + 311850 \, c^{4} d^{7} x^{7} + 436590 \, c^{5} d^{6} x^{6} + 436590 \, c^{6} d^{5} x^{5} + 311850 \, c^{7} d^{4} x^{4} + 155925 \, c^{8} d^{3} x^{3} + 51975 \, c^{9} d^{2} x^{2} + 10395 \, c^{10} d x + 945 \, c^{11} + 32 \, {\left (b^{5} d x + b^{5} c\right )} \log \relax (F)^{5} + 16 \, {\left (b^{4} d^{3} x^{3} + 3 \, b^{4} c d^{2} x^{2} + 3 \, b^{4} c^{2} d x + b^{4} c^{3}\right )} \log \relax (F)^{4} + 24 \, {\left (b^{3} d^{5} x^{5} + 5 \, b^{3} c d^{4} x^{4} + 10 \, b^{3} c^{2} d^{3} x^{3} + 10 \, b^{3} c^{3} d^{2} x^{2} + 5 \, b^{3} c^{4} d x + b^{3} c^{5}\right )} \log \relax (F)^{3} + 60 \, {\left (b^{2} d^{7} x^{7} + 7 \, b^{2} c d^{6} x^{6} + 21 \, b^{2} c^{2} d^{5} x^{5} + 35 \, b^{2} c^{3} d^{4} x^{4} + 35 \, b^{2} c^{4} d^{3} x^{3} + 21 \, b^{2} c^{5} d^{2} x^{2} + 7 \, b^{2} c^{6} d x + b^{2} c^{7}\right )} \log \relax (F)^{2} + 210 \, {\left (b d^{9} x^{9} + 9 \, b c d^{8} x^{8} + 36 \, b c^{2} d^{7} x^{7} + 84 \, b c^{3} d^{6} x^{6} + 126 \, b c^{4} d^{5} x^{5} + 126 \, b c^{5} d^{4} x^{4} + 84 \, b c^{6} d^{3} x^{3} + 36 \, b c^{7} d^{2} x^{2} + 9 \, b c^{8} d x + b c^{9}\right )} \log \relax (F)\right )} F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{10395 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^2)*(d*x+c)^10,x, algorithm="fricas")

[Out]

1/10395*(32*sqrt(pi)*F^a*b^5*d*sqrt(-b*log(F)/d^2)*erf(d*sqrt(-b*log(F)/d^2)/(d*x + c))*log(F)^5 + (945*d^11*x
^11 + 10395*c*d^10*x^10 + 51975*c^2*d^9*x^9 + 155925*c^3*d^8*x^8 + 311850*c^4*d^7*x^7 + 436590*c^5*d^6*x^6 + 4
36590*c^6*d^5*x^5 + 311850*c^7*d^4*x^4 + 155925*c^8*d^3*x^3 + 51975*c^9*d^2*x^2 + 10395*c^10*d*x + 945*c^11 +
32*(b^5*d*x + b^5*c)*log(F)^5 + 16*(b^4*d^3*x^3 + 3*b^4*c*d^2*x^2 + 3*b^4*c^2*d*x + b^4*c^3)*log(F)^4 + 24*(b^
3*d^5*x^5 + 5*b^3*c*d^4*x^4 + 10*b^3*c^2*d^3*x^3 + 10*b^3*c^3*d^2*x^2 + 5*b^3*c^4*d*x + b^3*c^5)*log(F)^3 + 60
*(b^2*d^7*x^7 + 7*b^2*c*d^6*x^6 + 21*b^2*c^2*d^5*x^5 + 35*b^2*c^3*d^4*x^4 + 35*b^2*c^4*d^3*x^3 + 21*b^2*c^5*d^
2*x^2 + 7*b^2*c^6*d*x + b^2*c^7)*log(F)^2 + 210*(b*d^9*x^9 + 9*b*c*d^8*x^8 + 36*b*c^2*d^7*x^7 + 84*b*c^3*d^6*x
^6 + 126*b*c^4*d^5*x^5 + 126*b*c^5*d^4*x^4 + 84*b*c^6*d^3*x^3 + 36*b*c^7*d^2*x^2 + 9*b*c^8*d*x + b*c^9)*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 \[ \int {\left (d x + c\right )}^{10} F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^2)*(d*x+c)^10,x, algorithm="giac")

[Out]

integrate((d*x + c)^10*F^(a + b/(d*x + c)^2), x)

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maple [B]  time = 0.18, size = 1173, normalized size = 23.94 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+1/(d*x+c)^2*b)*(d*x+c)^10,x)

[Out]

F^a*F^(1/(d*x+c)^2*b)*c^10*x+1/11*F^a*d^10*F^(1/(d*x+c)^2*b)*x^11+1/11*F^a/d*F^(1/(d*x+c)^2*b)*c^11+4/693*F^a/
d*b^2*ln(F)^2*F^(1/(d*x+c)^2*b)*c^7+8/3465*F^a/d*b^3*ln(F)^3*F^(1/(d*x+c)^2*b)*c^5+16/10395*F^a/d*b^4*ln(F)^4*
F^(1/(d*x+c)^2*b)*c^3+32/10395*F^a/d*b^5*ln(F)^5*F^(1/(d*x+c)^2*b)*c+2/99*F^a/d*b*ln(F)*F^(1/(d*x+c)^2*b)*c^9+
4/693*F^a*d^6*b^2*ln(F)^2*F^(1/(d*x+c)^2*b)*x^7+8/3465*F^a*d^4*b^3*ln(F)^3*F^(1/(d*x+c)^2*b)*x^5+16/10395*F^a*
d^2*b^4*ln(F)^4*F^(1/(d*x+c)^2*b)*x^3+2/99*F^a*d^8*b*ln(F)*F^(1/(d*x+c)^2*b)*x^9+2/11*F^a*b*ln(F)*F^(1/(d*x+c)
^2*b)*c^8*x+4/99*F^a*b^2*ln(F)^2*F^(1/(d*x+c)^2*b)*c^6*x+8/693*F^a*b^3*ln(F)^3*F^(1/(d*x+c)^2*b)*c^4*x+16/3465
*F^a*b^4*ln(F)^4*F^(1/(d*x+c)^2*b)*c^2*x+32/10395*F^a*b^5*ln(F)^5*F^(1/(d*x+c)^2*b)*x+F^a*d^9*F^(1/(d*x+c)^2*b
)*c*x^10+5*F^a*d^8*F^(1/(d*x+c)^2*b)*c^2*x^9+15*F^a*d^7*F^(1/(d*x+c)^2*b)*c^3*x^8+30*F^a*d^6*F^(1/(d*x+c)^2*b)
*c^4*x^7+42*F^a*d^5*F^(1/(d*x+c)^2*b)*c^5*x^6+42*F^a*d^4*F^(1/(d*x+c)^2*b)*c^6*x^5+30*F^a*d^3*F^(1/(d*x+c)^2*b
)*c^7*x^4+15*F^a*d^2*F^(1/(d*x+c)^2*b)*c^8*x^3+5*F^a*d*F^(1/(d*x+c)^2*b)*c^9*x^2+2/11*F^a*d^7*b*ln(F)*F^(1/(d*
x+c)^2*b)*c*x^8+8/11*F^a*d^6*b*ln(F)*F^(1/(d*x+c)^2*b)*c^2*x^7+56/33*F^a*d^5*b*ln(F)*F^(1/(d*x+c)^2*b)*c^3*x^6
+28/11*F^a*d^4*b*ln(F)*F^(1/(d*x+c)^2*b)*c^4*x^5+28/11*F^a*d^3*b*ln(F)*F^(1/(d*x+c)^2*b)*c^5*x^4+56/33*F^a*d^2
*b*ln(F)*F^(1/(d*x+c)^2*b)*c^6*x^3+8/11*F^a*d*b*ln(F)*F^(1/(d*x+c)^2*b)*c^7*x^2+4/99*F^a*d^5*b^2*ln(F)^2*F^(1/
(d*x+c)^2*b)*c*x^6+4/33*F^a*d^4*b^2*ln(F)^2*F^(1/(d*x+c)^2*b)*c^2*x^5+20/99*F^a*d^3*b^2*ln(F)^2*F^(1/(d*x+c)^2
*b)*c^3*x^4+20/99*F^a*d^2*b^2*ln(F)^2*F^(1/(d*x+c)^2*b)*c^4*x^3+4/33*F^a*d*b^2*ln(F)^2*F^(1/(d*x+c)^2*b)*c^5*x
^2+8/693*F^a*d^3*b^3*ln(F)^3*F^(1/(d*x+c)^2*b)*c*x^4+16/693*F^a*d^2*b^3*ln(F)^3*F^(1/(d*x+c)^2*b)*c^2*x^3+16/6
93*F^a*d*b^3*ln(F)^3*F^(1/(d*x+c)^2*b)*c^3*x^2+16/3465*F^a*d*b^4*ln(F)^4*F^(1/(d*x+c)^2*b)*c*x^2-32/10395*F^a/
d*b^6*ln(F)^6*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 \[ \frac {1}{10395} \, {\left (945 \, F^{a} d^{10} x^{11} + 10395 \, F^{a} c d^{9} x^{10} + 105 \, {\left (495 \, F^{a} c^{2} d^{8} + 2 \, F^{a} b d^{8} \log \relax (F)\right )} x^{9} + 945 \, {\left (165 \, F^{a} c^{3} d^{7} + 2 \, F^{a} b c d^{7} \log \relax (F)\right )} x^{8} + 30 \, {\left (10395 \, F^{a} c^{4} d^{6} + 252 \, F^{a} b c^{2} d^{6} \log \relax (F) + 2 \, F^{a} b^{2} d^{6} \log \relax (F)^{2}\right )} x^{7} + 210 \, {\left (2079 \, F^{a} c^{5} d^{5} + 84 \, F^{a} b c^{3} d^{5} \log \relax (F) + 2 \, F^{a} b^{2} c d^{5} \log \relax (F)^{2}\right )} x^{6} + 6 \, {\left (72765 \, F^{a} c^{6} d^{4} + 4410 \, F^{a} b c^{4} d^{4} \log \relax (F) + 210 \, F^{a} b^{2} c^{2} d^{4} \log \relax (F)^{2} + 4 \, F^{a} b^{3} d^{4} \log \relax (F)^{3}\right )} x^{5} + 30 \, {\left (10395 \, F^{a} c^{7} d^{3} + 882 \, F^{a} b c^{5} d^{3} \log \relax (F) + 70 \, F^{a} b^{2} c^{3} d^{3} \log \relax (F)^{2} + 4 \, F^{a} b^{3} c d^{3} \log \relax (F)^{3}\right )} x^{4} + {\left (155925 \, F^{a} c^{8} d^{2} + 17640 \, F^{a} b c^{6} d^{2} \log \relax (F) + 2100 \, F^{a} b^{2} c^{4} d^{2} \log \relax (F)^{2} + 240 \, F^{a} b^{3} c^{2} d^{2} \log \relax (F)^{3} + 16 \, F^{a} b^{4} d^{2} \log \relax (F)^{4}\right )} x^{3} + 3 \, {\left (17325 \, F^{a} c^{9} d + 2520 \, F^{a} b c^{7} d \log \relax (F) + 420 \, F^{a} b^{2} c^{5} d \log \relax (F)^{2} + 80 \, F^{a} b^{3} c^{3} d \log \relax (F)^{3} + 16 \, F^{a} b^{4} c d \log \relax (F)^{4}\right )} x^{2} + {\left (10395 \, F^{a} c^{10} + 1890 \, F^{a} b c^{8} \log \relax (F) + 420 \, F^{a} b^{2} c^{6} \log \relax (F)^{2} + 120 \, F^{a} b^{3} c^{4} \log \relax (F)^{3} + 48 \, F^{a} b^{4} c^{2} \log \relax (F)^{4} + 32 \, F^{a} b^{5} \log \relax (F)^{5}\right )} x\right )} F^{\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}} + \int \frac {2 \, {\left (32 \, F^{a} b^{6} d x \log \relax (F)^{6} - 945 \, F^{a} b c^{11} \log \relax (F) - 210 \, F^{a} b^{2} c^{9} \log \relax (F)^{2} - 60 \, F^{a} b^{3} c^{7} \log \relax (F)^{3} - 24 \, F^{a} b^{4} c^{5} \log \relax (F)^{4} - 16 \, F^{a} b^{5} c^{3} \log \relax (F)^{5}\right )} F^{\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{10395 \, {\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)^10,x, algorithm="maxima")

[Out]

1/10395*(945*F^a*d^10*x^11 + 10395*F^a*c*d^9*x^10 + 105*(495*F^a*c^2*d^8 + 2*F^a*b*d^8*log(F))*x^9 + 945*(165*
F^a*c^3*d^7 + 2*F^a*b*c*d^7*log(F))*x^8 + 30*(10395*F^a*c^4*d^6 + 252*F^a*b*c^2*d^6*log(F) + 2*F^a*b^2*d^6*log
(F)^2)*x^7 + 210*(2079*F^a*c^5*d^5 + 84*F^a*b*c^3*d^5*log(F) + 2*F^a*b^2*c*d^5*log(F)^2)*x^6 + 6*(72765*F^a*c^
6*d^4 + 4410*F^a*b*c^4*d^4*log(F) + 210*F^a*b^2*c^2*d^4*log(F)^2 + 4*F^a*b^3*d^4*log(F)^3)*x^5 + 30*(10395*F^a
*c^7*d^3 + 882*F^a*b*c^5*d^3*log(F) + 70*F^a*b^2*c^3*d^3*log(F)^2 + 4*F^a*b^3*c*d^3*log(F)^3)*x^4 + (155925*F^
a*c^8*d^2 + 17640*F^a*b*c^6*d^2*log(F) + 2100*F^a*b^2*c^4*d^2*log(F)^2 + 240*F^a*b^3*c^2*d^2*log(F)^3 + 16*F^a
*b^4*d^2*log(F)^4)*x^3 + 3*(17325*F^a*c^9*d + 2520*F^a*b*c^7*d*log(F) + 420*F^a*b^2*c^5*d*log(F)^2 + 80*F^a*b^
3*c^3*d*log(F)^3 + 16*F^a*b^4*c*d*log(F)^4)*x^2 + (10395*F^a*c^10 + 1890*F^a*b*c^8*log(F) + 420*F^a*b^2*c^6*lo
g(F)^2 + 120*F^a*b^3*c^4*log(F)^3 + 48*F^a*b^4*c^2*log(F)^4 + 32*F^a*b^5*log(F)^5)*x)*F^(b/(d^2*x^2 + 2*c*d*x
+ c^2)) + integrate(2/10395*(32*F^a*b^6*d*x*log(F)^6 - 945*F^a*b*c^11*log(F) - 210*F^a*b^2*c^9*log(F)^2 - 60*F
^a*b^3*c^7*log(F)^3 - 24*F^a*b^4*c^5*log(F)^4 - 16*F^a*b^5*c^3*log(F)^5)*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 = 4.35, size = 265, normalized size = 5.41 \[ \frac {F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,{\left (c+d\,x\right )}^{11}}{11\,d}-\frac {32\,F^a\,\sqrt {\pi }\,{\left (c+d\,x\right )}^{11}\,{\left (-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^2}\right )}^{11/2}}{10395\,d}+\frac {4\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^2\,{\ln \relax (F)}^2\,{\left (c+d\,x\right )}^7}{693\,d}+\frac {8\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^3\,{\ln \relax (F)}^3\,{\left (c+d\,x\right )}^5}{3465\,d}+\frac {16\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^4\,{\ln \relax (F)}^4\,{\left (c+d\,x\right )}^3}{10395\,d}+\frac {2\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b\,\ln \relax (F)\,{\left (c+d\,x\right )}^9}{99\,d}+\frac {32\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^5\,{\ln \relax (F)}^5\,\left (c+d\,x\right )}{10395\,d}+\frac {32\,F^a\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^2}}\right )\,{\left (c+d\,x\right )}^{11}\,{\left (-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^2}\right )}^{11/2}}{10395\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a + b/(c + d*x)^2)*(c + d*x)^10,x)

[Out]

(F^a*F^(b/(c + d*x)^2)*(c + d*x)^11)/(11*d) - (32*F^a*pi^(1/2)*(c + d*x)^11*(-(b*log(F))/(c + d*x)^2)^(11/2))/
(10395*d) + (4*F^a*F^(b/(c + d*x)^2)*b^2*log(F)^2*(c + d*x)^7)/(693*d) + (8*F^a*F^(b/(c + d*x)^2)*b^3*log(F)^3
*(c + d*x)^5)/(3465*d) + (16*F^a*F^(b/(c + d*x)^2)*b^4*log(F)^4*(c + d*x)^3)/(10395*d) + (2*F^a*F^(b/(c + d*x)
^2)*b*log(F)*(c + d*x)^9)/(99*d) + (32*F^a*F^(b/(c + d*x)^2)*b^5*log(F)^5*(c + d*x))/(10395*d) + (32*F^a*pi^(1
/2)*erfc((-(b*log(F))/(c + d*x)^2)^(1/2))*(c + d*x)^11*(-(b*log(F))/(c + d*x)^2)^(11/2))/(10395*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)**10,x)

[Out]

Timed out

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