3.3.66 \(\int \frac {F^{a+b (c+d x)^2}}{(c+d x)^{11}} \, dx\) [266]

Optimal. Leaf size=31 \[ \frac {b^5 F^a \Gamma \left (-5,-b (c+d x)^2 \log (F)\right ) \log ^5(F)}{2 d} \]

[Out]

-1/2*F^a/(d*x+c)^10*Ei(6,-b*(d*x+c)^2*ln(F))/d

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Rubi [A]
time = 0.04, antiderivative size = 31, 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 = {2250} \begin {gather*} \frac {b^5 F^a \log ^5(F) \text {Gamma}\left (-5,-b \log (F) (c+d x)^2\right )}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b^5*F^a*Gamma[-5, -(b*(c + d*x)^2*Log[F])]*Log[F]^5)/(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*} \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^{11}} \, dx &=\frac {b^5 F^a \Gamma \left (-5,-b (c+d x)^2 \log (F)\right ) \log ^5(F)}{2 d}\\ \end {align*}

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Mathematica [A]
time = 0.19, size = 31, normalized size = 1.00 \begin {gather*} \frac {b^5 F^a \Gamma \left (-5,-b (c+d x)^2 \log (F)\right ) \log ^5(F)}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b^5*F^a*Gamma[-5, -(b*(c + d*x)^2*Log[F])]*Log[F]^5)/(2*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(184\) vs. \(2(29)=58\).
time = 0.12, size = 185, normalized size = 5.97

method	result	size
risch	\(-\frac {F^{b \left (d x +c \right )^{2}} F^{a}}{10 d \left (d x +c \right )^{10}}-\frac {b \ln \left (F \right ) F^{b \left (d x +c \right )^{2}} F^{a}}{40 d \left (d x +c \right )^{8}}-\frac {b^{2} \ln \left (F \right )^{2} F^{b \left (d x +c \right )^{2}} F^{a}}{120 d \left (d x +c \right )^{6}}-\frac {b^{3} \ln \left (F \right )^{3} F^{b \left (d x +c \right )^{2}} F^{a}}{240 d \left (d x +c \right )^{4}}-\frac {b^{4} \ln \left (F \right )^{4} F^{b \left (d x +c \right )^{2}} F^{a}}{240 d \left (d x +c \right )^{2}}-\frac {b^{5} \ln \left (F \right )^{5} F^{a} \expIntegral \left (1, -b \left (d x +c \right )^{2} \ln \left (F \right )\right )}{240 d}\)	\(185\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b*(d*x+c)^2)/(d*x+c)^11,x,method=_RETURNVERBOSE)

[Out]

-1/10/d/(d*x+c)^10*F^(b*(d*x+c)^2)*F^a-1/40/d*b*ln(F)/(d*x+c)^8*F^(b*(d*x+c)^2)*F^a-1/120/d*b^2*ln(F)^2/(d*x+c
)^6*F^(b*(d*x+c)^2)*F^a-1/240/d*b^3*ln(F)^3/(d*x+c)^4*F^(b*(d*x+c)^2)*F^a-1/240/d*b^4*ln(F)^4/(d*x+c)^2*F^(b*(
d*x+c)^2)*F^a-1/240/d*b^5*ln(F)^5*F^a*Ei(1,-b*(d*x+c)^2*ln(F))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^2)/(d*x+c)^11,x, algorithm="maxima")

[Out]

integrate(F^((d*x + c)^2*b + a)/(d*x + c)^11, x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 596 vs. \(2 (29) = 58\).
time = 0.10, size = 596, normalized size = 19.23 \begin {gather*} \frac {{\left (b^{5} d^{10} x^{10} + 10 \, b^{5} c d^{9} x^{9} + 45 \, b^{5} c^{2} d^{8} x^{8} + 120 \, b^{5} c^{3} d^{7} x^{7} + 210 \, b^{5} c^{4} d^{6} x^{6} + 252 \, b^{5} c^{5} d^{5} x^{5} + 210 \, b^{5} c^{6} d^{4} x^{4} + 120 \, b^{5} c^{7} d^{3} x^{3} + 45 \, b^{5} c^{8} d^{2} x^{2} + 10 \, b^{5} c^{9} d x + b^{5} c^{10}\right )} F^{a} {\rm Ei}\left ({\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \left (F\right )\right ) \log \left (F\right )^{5} - {\left ({\left (b^{4} d^{8} x^{8} + 8 \, b^{4} c d^{7} x^{7} + 28 \, b^{4} c^{2} d^{6} x^{6} + 56 \, b^{4} c^{3} d^{5} x^{5} + 70 \, b^{4} c^{4} d^{4} x^{4} + 56 \, b^{4} c^{5} d^{3} x^{3} + 28 \, b^{4} c^{6} d^{2} x^{2} + 8 \, b^{4} c^{7} d x + b^{4} c^{8}\right )} \log \left (F\right )^{4} + {\left (b^{3} d^{6} x^{6} + 6 \, b^{3} c d^{5} x^{5} + 15 \, b^{3} c^{2} d^{4} x^{4} + 20 \, b^{3} c^{3} d^{3} x^{3} + 15 \, b^{3} c^{4} d^{2} x^{2} + 6 \, b^{3} c^{5} d x + b^{3} c^{6}\right )} \log \left (F\right )^{3} + 2 \, {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \left (F\right )^{2} + 6 \, {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \left (F\right ) + 24\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{240 \, {\left (d^{11} x^{10} + 10 \, c d^{10} x^{9} + 45 \, c^{2} d^{9} x^{8} + 120 \, c^{3} d^{8} x^{7} + 210 \, c^{4} d^{7} x^{6} + 252 \, c^{5} d^{6} x^{5} + 210 \, c^{6} d^{5} x^{4} + 120 \, c^{7} d^{4} x^{3} + 45 \, c^{8} d^{3} x^{2} + 10 \, c^{9} d^{2} x + c^{10} d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^2)/(d*x+c)^11,x, algorithm="fricas")

[Out]

1/240*((b^5*d^10*x^10 + 10*b^5*c*d^9*x^9 + 45*b^5*c^2*d^8*x^8 + 120*b^5*c^3*d^7*x^7 + 210*b^5*c^4*d^6*x^6 + 25
2*b^5*c^5*d^5*x^5 + 210*b^5*c^6*d^4*x^4 + 120*b^5*c^7*d^3*x^3 + 45*b^5*c^8*d^2*x^2 + 10*b^5*c^9*d*x + b^5*c^10
)*F^a*Ei((b*d^2*x^2 + 2*b*c*d*x + b*c^2)*log(F))*log(F)^5 - ((b^4*d^8*x^8 + 8*b^4*c*d^7*x^7 + 28*b^4*c^2*d^6*x
^6 + 56*b^4*c^3*d^5*x^5 + 70*b^4*c^4*d^4*x^4 + 56*b^4*c^5*d^3*x^3 + 28*b^4*c^6*d^2*x^2 + 8*b^4*c^7*d*x + b^4*c
^8)*log(F)^4 + (b^3*d^6*x^6 + 6*b^3*c*d^5*x^5 + 15*b^3*c^2*d^4*x^4 + 20*b^3*c^3*d^3*x^3 + 15*b^3*c^4*d^2*x^2 +
6*b^3*c^5*d*x + b^3*c^6)*log(F)^3 + 2*(b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b^
2*c^4)*log(F)^2 + 6*(b*d^2*x^2 + 2*b*c*d*x + b*c^2)*log(F) + 24)*F^(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a))/(d^11*
x^10 + 10*c*d^10*x^9 + 45*c^2*d^9*x^8 + 120*c^3*d^8*x^7 + 210*c^4*d^7*x^6 + 252*c^5*d^6*x^5 + 210*c^6*d^5*x^4
+ 120*c^7*d^4*x^3 + 45*c^8*d^3*x^2 + 10*c^9*d^2*x + c^10*d)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b*(d*x+c)**2)/(d*x+c)**11,x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^2)/(d*x+c)^11,x, algorithm="giac")

[Out]

integrate(F^((d*x + c)^2*b + a)/(d*x + c)^11, x)

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Mupad [B]
time = 3.91, size = 136, normalized size = 4.39 \begin {gather*} -\frac {F^a\,b^5\,{\ln \left (F\right )}^5\,\mathrm {expint}\left (-b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^2\right )}{240\,d}-\frac {F^a\,F^{b\,{\left (c+d\,x\right )}^2}\,b^5\,{\ln \left (F\right )}^5\,\left (\frac {1}{120\,b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^2}+\frac {1}{120\,b^2\,{\ln \left (F\right )}^2\,{\left (c+d\,x\right )}^4}+\frac {1}{60\,b^3\,{\ln \left (F\right )}^3\,{\left (c+d\,x\right )}^6}+\frac {1}{20\,b^4\,{\ln \left (F\right )}^4\,{\left (c+d\,x\right )}^8}+\frac {1}{5\,b^5\,{\ln \left (F\right )}^5\,{\left (c+d\,x\right )}^{10}}\right )}{2\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a + b*(c + d*x)^2)/(c + d*x)^11,x)

[Out]

- (F^a*b^5*log(F)^5*expint(-b*log(F)*(c + d*x)^2))/(240*d) - (F^a*F^(b*(c + d*x)^2)*b^5*log(F)^5*(1/(120*b*log
(F)*(c + d*x)^2) + 1/(120*b^2*log(F)^2*(c + d*x)^4) + 1/(60*b^3*log(F)^3*(c + d*x)^6) + 1/(20*b^4*log(F)^4*(c
+ d*x)^8) + 1/(5*b^5*log(F)^5*(c + d*x)^10)))/(2*d)

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