3.380 \(\int F^{c (a+b x)^n} (a+b x)^{-1+\frac{n}{2}} \, dx\)

Optimal. Leaf size=47 \[ \frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{c} \sqrt{\log (F)} (a+b x)^{n/2}\right )}{b \sqrt{c} n \sqrt{\log (F)}} \]

[Out]

(Sqrt[Pi]*Erfi[Sqrt[c]*(a + b*x)^(n/2)*Sqrt[Log[F]]])/(b*Sqrt[c]*n*Sqrt[Log[F]])

_______________________________________________________________________________________

Rubi [A]  time = 0.0784218, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{c} \sqrt{\log (F)} (a+b x)^{n/2}\right )}{b \sqrt{c} n \sqrt{\log (F)}} \]

Antiderivative was successfully verified.

[In]  Int[F^(c*(a + b*x)^n)*(a + b*x)^(-1 + n/2),x]

[Out]

(Sqrt[Pi]*Erfi[Sqrt[c]*(a + b*x)^(n/2)*Sqrt[Log[F]]])/(b*Sqrt[c]*n*Sqrt[Log[F]])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.31634, size = 41, normalized size = 0.87 \[ \frac{\sqrt{\pi } \operatorname{erfi}{\left (\sqrt{c} \left (a + b x\right )^{\frac{n}{2}} \sqrt{\log{\left (F \right )}} \right )}}{b \sqrt{c} n \sqrt{\log{\left (F \right )}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(c*(b*x+a)**n)*(b*x+a)**(-1+1/2*n),x)

[Out]

sqrt(pi)*erfi(sqrt(c)*(a + b*x)**(n/2)*sqrt(log(F)))/(b*sqrt(c)*n*sqrt(log(F)))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0570414, size = 0, normalized size = 0. \[ \int F^{c (a+b x)^n} (a+b x)^{-1+\frac{n}{2}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[F^(c*(a + b*x)^n)*(a + b*x)^(-1 + n/2),x]

[Out]

Integrate[F^(c*(a + b*x)^n)*(a + b*x)^(-1 + n/2), x]

_______________________________________________________________________________________

Maple [A]  time = 0.111, size = 36, normalized size = 0.8 \[{\frac{\sqrt{\pi }}{bn}{\it Erf} \left ( \sqrt{-c\ln \left ( F \right ) } \left ( bx+a \right ) ^{{\frac{n}{2}}} \right ){\frac{1}{\sqrt{-c\ln \left ( F \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(c*(b*x+a)^n)*(b*x+a)^(-1+1/2*n),x)

[Out]

1/n/b*Pi^(1/2)/(-c*ln(F))^(1/2)*erf((-c*ln(F))^(1/2)*(b*x+a)^(1/2*n))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{\frac{1}{2} \, n - 1} F^{{\left (b x + a\right )}^{n} c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/2*n - 1)*F^((b*x + a)^n*c),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(1/2*n - 1)*F^((b*x + a)^n*c), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.262004, size = 57, normalized size = 1.21 \[ \frac{\sqrt{\pi } \operatorname{erf}\left ({\left (b x + a\right )} \sqrt{-c \log \left (F\right )}{\left (b x + a\right )}^{\frac{1}{2} \, n - 1}\right )}{\sqrt{-c \log \left (F\right )} b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/2*n - 1)*F^((b*x + a)^n*c),x, algorithm="fricas")

[Out]

sqrt(pi)*erf((b*x + a)*sqrt(-c*log(F))*(b*x + a)^(1/2*n - 1))/(sqrt(-c*log(F))*b
*n)

_______________________________________________________________________________________

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**(c*(b*x+a)**n)*(b*x+a)**(-1+1/2*n),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{\frac{1}{2} \, n - 1} F^{{\left (b x + a\right )}^{n} c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/2*n - 1)*F^((b*x + a)^n*c),x, algorithm="giac")

[Out]

integrate((b*x + a)^(1/2*n - 1)*F^((b*x + a)^n*c), x)