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

Optimal. Leaf size=183 \[ -\frac{105 \sqrt{\pi } F^a \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (F)}}{c+d x}\right )}{32 b^{9/2} d \log ^{\frac{9}{2}}(F)}+\frac{7 F^{a+\frac{b}{(c+d x)^2}}}{4 b^2 d \log ^2(F) (c+d x)^5}-\frac{35 F^{a+\frac{b}{(c+d x)^2}}}{8 b^3 d \log ^3(F) (c+d x)^3}+\frac{105 F^{a+\frac{b}{(c+d x)^2}}}{16 b^4 d \log ^4(F) (c+d x)}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)^7} \]

[Out]

(-105*F^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[F]])/(c + d*x)])/(32*b^(9/2)*d*Log[F]^(9/2)) + (105*F^(a + b/(c + d*
x)^2))/(16*b^4*d*(c + d*x)*Log[F]^4) - (35*F^(a + b/(c + d*x)^2))/(8*b^3*d*(c + d*x)^3*Log[F]^3) + (7*F^(a + b
/(c + d*x)^2))/(4*b^2*d*(c + d*x)^5*Log[F]^2) - F^(a + b/(c + d*x)^2)/(2*b*d*(c + d*x)^7*Log[F])

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Rubi [A]  time = 0.264987, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2212, 2211, 2204} \[ -\frac{105 \sqrt{\pi } F^a \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (F)}}{c+d x}\right )}{32 b^{9/2} d \log ^{\frac{9}{2}}(F)}+\frac{7 F^{a+\frac{b}{(c+d x)^2}}}{4 b^2 d \log ^2(F) (c+d x)^5}-\frac{35 F^{a+\frac{b}{(c+d x)^2}}}{8 b^3 d \log ^3(F) (c+d x)^3}+\frac{105 F^{a+\frac{b}{(c+d x)^2}}}{16 b^4 d \log ^4(F) (c+d x)}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-105*F^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[F]])/(c + d*x)])/(32*b^(9/2)*d*Log[F]^(9/2)) + (105*F^(a + b/(c + d*
x)^2))/(16*b^4*d*(c + d*x)*Log[F]^4) - (35*F^(a + b/(c + d*x)^2))/(8*b^3*d*(c + d*x)^3*Log[F]^3) + (7*F^(a + b
/(c + d*x)^2))/(4*b^2*d*(c + d*x)^5*Log[F]^2) - F^(a + b/(c + d*x)^2)/(2*b*d*(c + d*x)^7*Log[F])

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])

Rule 2211

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dist[1/(d*(m + 1))
, Subst[Int[F^(a + b*x^2), x], x, (c + d*x)^(m + 1)], x] /; FreeQ[{F, a, b, c, d, m, n}, x] && EqQ[n, 2*(m + 1
)]

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rubi steps

\begin{align*} \int \frac{F^{a+\frac{b}{(c+d x)^2}}}{(c+d x)^{10}} \, dx &=-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d (c+d x)^7 \log (F)}-\frac{7 \int \frac{F^{a+\frac{b}{(c+d x)^2}}}{(c+d x)^8} \, dx}{2 b \log (F)}\\ &=\frac{7 F^{a+\frac{b}{(c+d x)^2}}}{4 b^2 d (c+d x)^5 \log ^2(F)}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d (c+d x)^7 \log (F)}+\frac{35 \int \frac{F^{a+\frac{b}{(c+d x)^2}}}{(c+d x)^6} \, dx}{4 b^2 \log ^2(F)}\\ &=-\frac{35 F^{a+\frac{b}{(c+d x)^2}}}{8 b^3 d (c+d x)^3 \log ^3(F)}+\frac{7 F^{a+\frac{b}{(c+d x)^2}}}{4 b^2 d (c+d x)^5 \log ^2(F)}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d (c+d x)^7 \log (F)}-\frac{105 \int \frac{F^{a+\frac{b}{(c+d x)^2}}}{(c+d x)^4} \, dx}{8 b^3 \log ^3(F)}\\ &=\frac{105 F^{a+\frac{b}{(c+d x)^2}}}{16 b^4 d (c+d x) \log ^4(F)}-\frac{35 F^{a+\frac{b}{(c+d x)^2}}}{8 b^3 d (c+d x)^3 \log ^3(F)}+\frac{7 F^{a+\frac{b}{(c+d x)^2}}}{4 b^2 d (c+d x)^5 \log ^2(F)}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d (c+d x)^7 \log (F)}+\frac{105 \int \frac{F^{a+\frac{b}{(c+d x)^2}}}{(c+d x)^2} \, dx}{16 b^4 \log ^4(F)}\\ &=\frac{105 F^{a+\frac{b}{(c+d x)^2}}}{16 b^4 d (c+d x) \log ^4(F)}-\frac{35 F^{a+\frac{b}{(c+d x)^2}}}{8 b^3 d (c+d x)^3 \log ^3(F)}+\frac{7 F^{a+\frac{b}{(c+d x)^2}}}{4 b^2 d (c+d x)^5 \log ^2(F)}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d (c+d x)^7 \log (F)}-\frac{105 \operatorname{Subst}\left (\int F^{a+b x^2} \, dx,x,\frac{1}{c+d x}\right )}{16 b^4 d \log ^4(F)}\\ &=-\frac{105 F^a \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{b} \sqrt{\log (F)}}{c+d x}\right )}{32 b^{9/2} d \log ^{\frac{9}{2}}(F)}+\frac{105 F^{a+\frac{b}{(c+d x)^2}}}{16 b^4 d (c+d x) \log ^4(F)}-\frac{35 F^{a+\frac{b}{(c+d x)^2}}}{8 b^3 d (c+d x)^3 \log ^3(F)}+\frac{7 F^{a+\frac{b}{(c+d x)^2}}}{4 b^2 d (c+d x)^5 \log ^2(F)}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d (c+d x)^7 \log (F)}\\ \end{align*}

Mathematica [A]  time = 0.154358, size = 127, normalized size = 0.69 \[ \frac{F^a \left (\frac{2 \sqrt{b} \sqrt{\log (F)} F^{\frac{b}{(c+d x)^2}} \left (28 b^2 \log ^2(F) (c+d x)^2-8 b^3 \log ^3(F)-70 b \log (F) (c+d x)^4+105 (c+d x)^6\right )}{(c+d x)^7}-105 \sqrt{\pi } \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (F)}}{c+d x}\right )\right )}{32 b^{9/2} d \log ^{\frac{9}{2}}(F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(F^a*(-105*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[F]])/(c + d*x)] + (2*Sqrt[b]*F^(b/(c + d*x)^2)*Sqrt[Log[F]]*(105*(c
+ d*x)^6 - 70*b*(c + d*x)^4*Log[F] + 28*b^2*(c + d*x)^2*Log[F]^2 - 8*b^3*Log[F]^3))/(c + d*x)^7))/(32*b^(9/2)
*d*Log[F]^(9/2))

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Maple [A]  time = 0.155, size = 175, normalized size = 1. \begin{align*} -{\frac{{F}^{a}}{2\,d \left ( dx+c \right ) ^{7}b\ln \left ( F \right ) }{F}^{{\frac{b}{ \left ( dx+c \right ) ^{2}}}}}+{\frac{7\,{F}^{a}}{4\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}d \left ( dx+c \right ) ^{5}}{F}^{{\frac{b}{ \left ( dx+c \right ) ^{2}}}}}-{\frac{35\,{F}^{a}}{8\,d{b}^{3} \left ( \ln \left ( F \right ) \right ) ^{3} \left ( dx+c \right ) ^{3}}{F}^{{\frac{b}{ \left ( dx+c \right ) ^{2}}}}}+{\frac{105\,{F}^{a}}{16\,d{b}^{4} \left ( \ln \left ( F \right ) \right ) ^{4} \left ( dx+c \right ) }{F}^{{\frac{b}{ \left ( dx+c \right ) ^{2}}}}}-{\frac{105\,{F}^{a}\sqrt{\pi }}{32\,d{b}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}}{\it Erf} \left ({\frac{1}{dx+c}\sqrt{-b\ln \left ( F \right ) }} \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/d*F^a*F^(b/(d*x+c)^2)/(d*x+c)^7/b/ln(F)+7/4/d*F^a/b^2/ln(F)^2*F^(b/(d*x+c)^2)/(d*x+c)^5-35/8/d*F^a/b^3/ln
(F)^3*F^(b/(d*x+c)^2)/(d*x+c)^3+105/16/d*F^a/b^4/ln(F)^4*F^(b/(d*x+c)^2)/(d*x+c)-105/32/d*F^a/b^4/ln(F)^4*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., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{a + \frac{b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{10}}\,{d x} \end{align*}

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]

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

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Fricas [B]  time = 1.79248, size = 950, normalized size = 5.19 \begin{align*} \frac{105 \, \sqrt{\pi }{\left (d^{8} x^{7} + 7 \, c d^{7} x^{6} + 21 \, c^{2} d^{6} x^{5} + 35 \, c^{3} d^{5} x^{4} + 35 \, c^{4} d^{4} x^{3} + 21 \, c^{5} d^{3} x^{2} + 7 \, c^{6} d^{2} x + c^{7} d\right )} F^{a} \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 ) - 2 \,{\left (8 \, b^{4} \log \left (F\right )^{4} - 28 \,{\left (b^{3} d^{2} x^{2} + 2 \, b^{3} c d x + b^{3} c^{2}\right )} \log \left (F\right )^{3} + 70 \,{\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} - 105 \,{\left (b d^{6} x^{6} + 6 \, b c d^{5} x^{5} + 15 \, b c^{2} d^{4} x^{4} + 20 \, b c^{3} d^{3} x^{3} + 15 \, b c^{4} d^{2} x^{2} + 6 \, b c^{5} d x + b c^{6}\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}}}}{32 \,{\left (b^{5} d^{8} x^{7} + 7 \, b^{5} c d^{7} x^{6} + 21 \, b^{5} c^{2} d^{6} x^{5} + 35 \, b^{5} c^{3} d^{5} x^{4} + 35 \, b^{5} c^{4} d^{4} x^{3} + 21 \, b^{5} c^{5} d^{3} x^{2} + 7 \, b^{5} c^{6} d^{2} x + b^{5} c^{7} d\right )} \log \left (F\right )^{5}} \end{align*}

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]

1/32*(105*sqrt(pi)*(d^8*x^7 + 7*c*d^7*x^6 + 21*c^2*d^6*x^5 + 35*c^3*d^5*x^4 + 35*c^4*d^4*x^3 + 21*c^5*d^3*x^2
+ 7*c^6*d^2*x + c^7*d)*F^a*sqrt(-b*log(F)/d^2)*erf(d*sqrt(-b*log(F)/d^2)/(d*x + c)) - 2*(8*b^4*log(F)^4 - 28*(
b^3*d^2*x^2 + 2*b^3*c*d*x + b^3*c^2)*log(F)^3 + 70*(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 - 105*(b*d^6*x^6 + 6*b*c*d^5*x^5 + 15*b*c^2*d^4*x^4 + 20*b*c^3*d^3*x^3 + 15*b*c^4*
d^2*x^2 + 6*b*c^5*d*x + b*c^6)*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^
5*d^8*x^7 + 7*b^5*c*d^7*x^6 + 21*b^5*c^2*d^6*x^5 + 35*b^5*c^3*d^5*x^4 + 35*b^5*c^4*d^4*x^3 + 21*b^5*c^5*d^3*x^
2 + 7*b^5*c^6*d^2*x + b^5*c^7*d)*log(F)^5)

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

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{a + \frac{b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{10}}\,{d x} \end{align*}

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