3.335 \(\int \frac{F^{a+\frac{b}{(c+d x)^2}}}{(c+d x)^6} \, dx\)

Optimal. Leaf size=115 \[ -\frac{3 \sqrt{\pi } F^a \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (F)}}{c+d x}\right )}{8 b^{5/2} d \log ^{\frac{5}{2}}(F)}+\frac{3 F^{a+\frac{b}{(c+d x)^2}}}{4 b^2 d \log ^2(F) (c+d x)}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)^3} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.253556, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{3 \sqrt{\pi } F^a \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (F)}}{c+d x}\right )}{8 b^{5/2} d \log ^{\frac{5}{2}}(F)}+\frac{3 F^{a+\frac{b}{(c+d x)^2}}}{4 b^2 d \log ^2(F) (c+d x)}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)^3} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b/(c + d*x)^2)/(c + d*x)^6,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 22.11, size = 100, normalized size = 0.87 \[ - \frac{3 \sqrt{\pi } F^{a} \operatorname{erfi}{\left (\frac{\sqrt{b} \sqrt{\log{\left (F \right )}}}{c + d x} \right )}}{8 b^{\frac{5}{2}} d \log{\left (F \right )}^{\frac{5}{2}}} - \frac{F^{a + \frac{b}{\left (c + d x\right )^{2}}}}{2 b d \left (c + d x\right )^{3} \log{\left (F \right )}} + \frac{3 F^{a + \frac{b}{\left (c + d x\right )^{2}}}}{4 b^{2} d \left (c + d x\right ) \log{\left (F \right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(a+b/(d*x+c)**2)/(d*x+c)**6,x)

[Out]

-3*sqrt(pi)*F**a*erfi(sqrt(b)*sqrt(log(F))/(c + d*x))/(8*b**(5/2)*d*log(F)**(5/2
)) - F**(a + b/(c + d*x)**2)/(2*b*d*(c + d*x)**3*log(F)) + 3*F**(a + b/(c + d*x)
**2)/(4*b**2*d*(c + d*x)*log(F)**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.216159, size = 93, normalized size = 0.81 \[ \frac{\frac{2 F^{a+\frac{b}{(c+d x)^2}} \left (3 (c+d x)^2-2 b \log (F)\right )}{b^2 \log ^2(F) (c+d x)^3}-\frac{3 \sqrt{\pi } F^a \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (F)}}{c+d x}\right )}{b^{5/2} \log ^{\frac{5}{2}}(F)}}{8 d} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(a + b/(c + d*x)^2)/(c + d*x)^6,x]

[Out]

((-3*F^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[F]])/(c + d*x)])/(b^(5/2)*Log[F]^(5/2))
+ (2*F^(a + b/(c + d*x)^2)*(3*(c + d*x)^2 - 2*b*Log[F]))/(b^2*(c + d*x)^3*Log[F
]^2))/(8*d)

_______________________________________________________________________________________

Maple [A]  time = 0.101, size = 109, normalized size = 1. \[ -{\frac{{F}^{a}}{2\,d \left ( dx+c \right ) ^{3}b\ln \left ( F \right ) }{F}^{{\frac{b}{ \left ( dx+c \right ) ^{2}}}}}+{\frac{3\,{F}^{a}}{4\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}d \left ( dx+c \right ) }{F}^{{\frac{b}{ \left ( dx+c \right ) ^{2}}}}}-{\frac{3\,{F}^{a}\sqrt{\pi }}{8\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}d}{\it Erf} \left ({\frac{1}{dx+c}\sqrt{-b\ln \left ( F \right ) }} \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b/(d*x+c)^2)/(d*x+c)^6,x)

[Out]

-1/2*F^a/d*F^(b/(d*x+c)^2)/(d*x+c)^3/b/ln(F)+3/4*F^a/d/b^2/ln(F)^2*F^(b/(d*x+c)^
2)/(d*x+c)-3/8*F^a/d/b^2/ln(F)^2*Pi^(1/2)/(-b*ln(F))^(1/2)*erf((-b*ln(F))^(1/2)/
(d*x+c))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{a + \frac{b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(a + b/(d*x + c)^2)/(d*x + c)^6,x, algorithm="maxima")

[Out]

integrate(F^(a + b/(d*x + c)^2)/(d*x + c)^6, x)

_______________________________________________________________________________________

Fricas [A]  time = 0.286386, size = 271, normalized size = 2.36 \[ -\frac{3 \, \sqrt{\pi }{\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )} F^{a} \operatorname{erf}\left (\frac{d \sqrt{-\frac{b \log \left (F\right )}{d^{2}}}}{d x + c}\right ) - 2 \,{\left (3 \, d^{3} x^{2} + 6 \, c d^{2} x + 3 \, c^{2} d - 2 \, b d \log \left (F\right )\right )} F^{\frac{a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}} \sqrt{-\frac{b \log \left (F\right )}{d^{2}}}}{8 \,{\left (b^{2} d^{5} x^{3} + 3 \, b^{2} c d^{4} x^{2} + 3 \, b^{2} c^{2} d^{3} x + b^{2} c^{3} d^{2}\right )} \sqrt{-\frac{b \log \left (F\right )}{d^{2}}} \log \left (F\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(a + b/(d*x + c)^2)/(d*x + c)^6,x, algorithm="fricas")

[Out]

-1/8*(3*sqrt(pi)*(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)*F^a*erf(d*sqrt(-b*log
(F)/d^2)/(d*x + c)) - 2*(3*d^3*x^2 + 6*c*d^2*x + 3*c^2*d - 2*b*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^2*d^5*x^3 + 3*b^2*c*d^4*x^2 + 3*b^2*c^2*d^3*x + b^2*c^3*d^2)*sqrt(-b*log(F)/
d^2)*log(F)^2)

_______________________________________________________________________________________

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)**6,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{a + \frac{b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(a + b/(d*x + c)^2)/(d*x + c)^6,x, algorithm="giac")

[Out]

integrate(F^(a + b/(d*x + c)^2)/(d*x + c)^6, x)