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

Optimal. Leaf size=102 \[ -\frac{2 \sqrt{\pi } b^{3/2} F^a \log ^{\frac{3}{2}}(F) \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (F)}}{c+d x}\right )}{3 d}+\frac{(c+d x)^3 F^{a+\frac{b}{(c+d x)^2}}}{3 d}+\frac{2 b \log (F) (c+d x) F^{a+\frac{b}{(c+d x)^2}}}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.191996, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ -\frac{2 \sqrt{\pi } b^{3/2} F^a \log ^{\frac{3}{2}}(F) \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (F)}}{c+d x}\right )}{3 d}+\frac{(c+d x)^3 F^{a+\frac{b}{(c+d x)^2}}}{3 d}+\frac{2 b \log (F) (c+d x) F^{a+\frac{b}{(c+d x)^2}}}{3 d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.6631, size = 92, normalized size = 0.9 \[ - \frac{2 \sqrt{\pi } F^{a} b^{\frac{3}{2}} \log{\left (F \right )}^{\frac{3}{2}} \operatorname{erfi}{\left (\frac{\sqrt{b} \sqrt{\log{\left (F \right )}}}{c + d x} \right )}}{3 d} + \frac{2 F^{a + \frac{b}{\left (c + d x\right )^{2}}} b \left (c + d x\right ) \log{\left (F \right )}}{3 d} + \frac{F^{a + \frac{b}{\left (c + d x\right )^{2}}} \left (c + d x\right )^{3}}{3 d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.105439, size = 79, normalized size = 0.77 \[ \frac{F^a \left ((c+d x) F^{\frac{b}{(c+d x)^2}} \left (2 b \log (F)+(c+d x)^2\right )-2 \sqrt{\pi } b^{3/2} \log ^{\frac{3}{2}}(F) \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (F)}}{c+d x}\right )\right )}{3 d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.041, size = 169, normalized size = 1.7 \[{\frac{{F}^{a}{d}^{2}{x}^{3}}{3}{F}^{{\frac{b}{ \left ( dx+c \right ) ^{2}}}}}+{F}^{a}d{F}^{{\frac{b}{ \left ( dx+c \right ) ^{2}}}}c{x}^{2}+{F}^{a}{F}^{{\frac{b}{ \left ( dx+c \right ) ^{2}}}}{c}^{2}x+{\frac{{F}^{a}{c}^{3}}{3\,d}{F}^{{\frac{b}{ \left ( dx+c \right ) ^{2}}}}}+{\frac{2\,{F}^{a}b\ln \left ( F \right ) x}{3}{F}^{{\frac{b}{ \left ( dx+c \right ) ^{2}}}}}+{\frac{2\,{F}^{a}b\ln \left ( F \right ) c}{3\,d}{F}^{{\frac{b}{ \left ( dx+c \right ) ^{2}}}}}-{\frac{2\,{F}^{a}{b}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}\sqrt{\pi }}{3\,d}{\it Erf} \left ({\frac{1}{dx+c}\sqrt{-b\ln \left ( F \right ) }} \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{1}{3} \,{\left (F^{a} d^{2} x^{3} + 3 \, F^{a} c d x^{2} +{\left (3 \, F^{a} c^{2} + 2 \, F^{a} b \log \left (F\right )\right )} x\right )} F^{\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}} + \int \frac{2 \,{\left (2 \, F^{a} b^{2} d x \log \left (F\right )^{2} - F^{a} b c^{3} \log \left (F\right )\right )} F^{\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{3 \,{\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(F^a*d^2*x^3 + 3*F^a*c*d*x^2 + (3*F^a*c^2 + 2*F^a*b*log(F))*x)*F^(b/(d^2*x^2
+ 2*c*d*x + c^2)) + integrate(2/3*(2*F^a*b^2*d*x*log(F)^2 - F^a*b*c^3*log(F))*F
^(b/(d^2*x^2 + 2*c*d*x + c^2))/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3), x)

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Fricas [A]  time = 0.278927, size = 204, normalized size = 2. \[ -\frac{2 \, \sqrt{\pi } F^{a} b^{2} \operatorname{erf}\left (\frac{d \sqrt{-\frac{b \log \left (F\right )}{d^{2}}}}{d x + c}\right ) \log \left (F\right )^{2} -{\left (d^{4} x^{3} + 3 \, c d^{3} x^{2} + 3 \, c^{2} d^{2} x + c^{3} d + 2 \,{\left (b d^{2} x + b c d\right )} \log \left (F\right )\right )} F^{\frac{a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}} \sqrt{-\frac{b \log \left (F\right )}{d^{2}}}}{3 \, d^{2} \sqrt{-\frac{b \log \left (F\right )}{d^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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