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

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

[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 = 0.438499, 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 \[ -\frac{105 \sqrt{\pi } F^a \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (F)}}{c+d x}\right )}{32 b^{9/2} d \log ^{\frac{9}{2}}(F)}+\frac{105 F^{a+\frac{b}{(c+d x)^2}}}{16 b^4 d \log ^4(F) (c+d x)}-\frac{35 F^{a+\frac{b}{(c+d x)^2}}}{8 b^3 d \log ^3(F) (c+d x)^3}+\frac{7 F^{a+\frac{b}{(c+d x)^2}}}{4 b^2 d \log ^2(F) (c+d x)^5}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)^7} \]

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])

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Rubi in Sympy [A]  time = 44.2207, size = 165, normalized size = 0.9 \[ - \frac{105 \sqrt{\pi } F^{a} \operatorname{erfi}{\left (\frac{\sqrt{b} \sqrt{\log{\left (F \right )}}}{c + d x} \right )}}{32 b^{\frac{9}{2}} d \log{\left (F \right )}^{\frac{9}{2}}} - \frac{F^{a + \frac{b}{\left (c + d x\right )^{2}}}}{2 b d \left (c + d x\right )^{7} \log{\left (F \right )}} + \frac{7 F^{a + \frac{b}{\left (c + d x\right )^{2}}}}{4 b^{2} d \left (c + d x\right )^{5} \log{\left (F \right )}^{2}} - \frac{35 F^{a + \frac{b}{\left (c + d x\right )^{2}}}}{8 b^{3} d \left (c + d x\right )^{3} \log{\left (F \right )}^{3}} + \frac{105 F^{a + \frac{b}{\left (c + d x\right )^{2}}}}{16 b^{4} d \left (c + d x\right ) \log{\left (F \right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-105*sqrt(pi)*F**a*erfi(sqrt(b)*sqrt(log(F))/(c + d*x))/(32*b**(9/2)*d*log(F)**(
9/2)) - F**(a + b/(c + d*x)**2)/(2*b*d*(c + d*x)**7*log(F)) + 7*F**(a + b/(c + d
*x)**2)/(4*b**2*d*(c + d*x)**5*log(F)**2) - 35*F**(a + b/(c + d*x)**2)/(8*b**3*d
*(c + d*x)**3*log(F)**3) + 105*F**(a + b/(c + d*x)**2)/(16*b**4*d*(c + d*x)*log(
F)**4)

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Mathematica [A]  time = 0.185695, size = 127, normalized size = 0.69 \[ \frac{F^a \left (\frac{2 \sqrt{b} \sqrt{\log (F)} F^{\frac{b}{(c+d x)^2}} \left (-8 b^3 \log ^3(F)+28 b^2 \log ^2(F) (c+d x)^2-70 b \log (F) (c+d x)^4+105 (c+d x)^6\right )}{(c+d x)^7}-105 \sqrt{\pi } \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (F)}}{c+d x}\right )\right )}{32 b^{9/2} d \log ^{\frac{9}{2}}(F)} \]

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 = 0.197, size = 175, normalized size = 1. \[ -{\frac{{F}^{a}}{2\,d \left ( dx+c \right ) ^{7}b\ln \left ( F \right ) }{F}^{{\frac{b}{ \left ( dx+c \right ) ^{2}}}}}+{\frac{7\,{F}^{a}}{4\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}d \left ( dx+c \right ) ^{5}}{F}^{{\frac{b}{ \left ( dx+c \right ) ^{2}}}}}-{\frac{35\,{F}^{a}}{8\,d{b}^{3} \left ( \ln \left ( F \right ) \right ) ^{3} \left ( dx+c \right ) ^{3}}{F}^{{\frac{b}{ \left ( dx+c \right ) ^{2}}}}}+{\frac{105\,{F}^{a}}{16\,d{b}^{4} \left ( \ln \left ( F \right ) \right ) ^{4} \left ( dx+c \right ) }{F}^{{\frac{b}{ \left ( dx+c \right ) ^{2}}}}}-{\frac{105\,{F}^{a}\sqrt{\pi }}{32\,d{b}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}}{\it Erf} \left ({\frac{1}{dx+c}\sqrt{-b\ln \left ( F \right ) }} \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}} \]

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*F^a/d*F^(b/(d*x+c)^2)/(d*x+c)^7/b/ln(F)+7/4*F^a/d/b^2/ln(F)^2*F^(b/(d*x+c)^
2)/(d*x+c)^5-35/8*F^a/d/b^3/ln(F)^3*F^(b/(d*x+c)^2)/(d*x+c)^3+105/16*F^a/d/b^4/l
n(F)^4*F^(b/(d*x+c)^2)/(d*x+c)-105/32*F^a/d/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. \[ \int \frac{F^{a + \frac{b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{10}}\,{d x} \]

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]

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

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Fricas [A]  time = 0.286014, size = 587, normalized size = 3.21 \[ -\frac{105 \, \sqrt{\pi }{\left (d^{7} x^{7} + 7 \, c d^{6} x^{6} + 21 \, c^{2} d^{5} x^{5} + 35 \, c^{3} d^{4} x^{4} + 35 \, c^{4} d^{3} x^{3} + 21 \, c^{5} d^{2} x^{2} + 7 \, c^{6} d x + c^{7}\right )} F^{a} \operatorname{erf}\left (\frac{d \sqrt{-\frac{b \log \left (F\right )}{d^{2}}}}{d x + c}\right ) - 2 \,{\left (105 \, d^{7} x^{6} + 630 \, c d^{6} x^{5} + 1575 \, c^{2} d^{5} x^{4} + 2100 \, c^{3} d^{4} x^{3} + 1575 \, c^{4} d^{3} x^{2} + 630 \, c^{5} d^{2} x + 105 \, c^{6} d - 8 \, b^{3} d \log \left (F\right )^{3} + 28 \,{\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d\right )} \log \left (F\right )^{2} - 70 \,{\left (b d^{5} x^{4} + 4 \, b c d^{4} x^{3} + 6 \, b c^{2} d^{3} x^{2} + 4 \, b c^{3} d^{2} x + b c^{4} d\right )} \log \left (F\right )\right )} F^{\frac{a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}} \sqrt{-\frac{b \log \left (F\right )}{d^{2}}}}{32 \,{\left (b^{4} d^{9} x^{7} + 7 \, b^{4} c d^{8} x^{6} + 21 \, b^{4} c^{2} d^{7} x^{5} + 35 \, b^{4} c^{3} d^{6} x^{4} + 35 \, b^{4} c^{4} d^{5} x^{3} + 21 \, b^{4} c^{5} d^{4} x^{2} + 7 \, b^{4} c^{6} d^{3} x + b^{4} c^{7} d^{2}\right )} \sqrt{-\frac{b \log \left (F\right )}{d^{2}}} \log \left (F\right )^{4}} \]

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^7*x^7 + 7*c*d^6*x^6 + 21*c^2*d^5*x^5 + 35*c^3*d^4*x^4 + 3
5*c^4*d^3*x^3 + 21*c^5*d^2*x^2 + 7*c^6*d*x + c^7)*F^a*erf(d*sqrt(-b*log(F)/d^2)/
(d*x + c)) - 2*(105*d^7*x^6 + 630*c*d^6*x^5 + 1575*c^2*d^5*x^4 + 2100*c^3*d^4*x^
3 + 1575*c^4*d^3*x^2 + 630*c^5*d^2*x + 105*c^6*d - 8*b^3*d*log(F)^3 + 28*(b^2*d^
3*x^2 + 2*b^2*c*d^2*x + b^2*c^2*d)*log(F)^2 - 70*(b*d^5*x^4 + 4*b*c*d^4*x^3 + 6*
b*c^2*d^3*x^2 + 4*b*c^3*d^2*x + b*c^4*d)*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))*sqrt(-b*log(F)/d^2))/((b^4*d^9*x^7 + 7*b^4*c*
d^8*x^6 + 21*b^4*c^2*d^7*x^5 + 35*b^4*c^3*d^6*x^4 + 35*b^4*c^4*d^5*x^3 + 21*b^4*
c^5*d^4*x^2 + 7*b^4*c^6*d^3*x + b^4*c^7*d^2)*sqrt(-b*log(F)/d^2)*log(F)^4)

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

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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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{a + \frac{b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{10}}\,{d x} \]

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