3.37 \(\int \frac{F^{a+b x}}{x^{7/2}} \, dx\)

Optimal. Leaf size=100 \[ \frac{8}{15} \sqrt{\pi } b^{5/2} F^a \log ^{\frac{5}{2}}(F) \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )-\frac{8 b^2 \log ^2(F) F^{a+b x}}{15 \sqrt{x}}-\frac{2 F^{a+b x}}{5 x^{5/2}}-\frac{4 b \log (F) F^{a+b x}}{15 x^{3/2}} \]

[Out]

(-2*F^(a + b*x))/(5*x^(5/2)) - (4*b*F^(a + b*x)*Log[F])/(15*x^(3/2)) - (8*b^2*F^
(a + b*x)*Log[F]^2)/(15*Sqrt[x]) + (8*b^(5/2)*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*
Sqrt[Log[F]]]*Log[F]^(5/2))/15

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Rubi [A]  time = 0.139042, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{8}{15} \sqrt{\pi } b^{5/2} F^a \log ^{\frac{5}{2}}(F) \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )-\frac{8 b^2 \log ^2(F) F^{a+b x}}{15 \sqrt{x}}-\frac{2 F^{a+b x}}{5 x^{5/2}}-\frac{4 b \log (F) F^{a+b x}}{15 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b*x)/x^(7/2),x]

[Out]

(-2*F^(a + b*x))/(5*x^(5/2)) - (4*b*F^(a + b*x)*Log[F])/(15*x^(3/2)) - (8*b^2*F^
(a + b*x)*Log[F]^2)/(15*Sqrt[x]) + (8*b^(5/2)*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*
Sqrt[Log[F]]]*Log[F]^(5/2))/15

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Rubi in Sympy [A]  time = 15.7628, size = 100, normalized size = 1. \[ \frac{8 \sqrt{\pi } F^{a} b^{\frac{5}{2}} \log{\left (F \right )}^{\frac{5}{2}} \operatorname{erfi}{\left (\sqrt{b} \sqrt{x} \sqrt{\log{\left (F \right )}} \right )}}{15} - \frac{8 F^{a + b x} b^{2} \log{\left (F \right )}^{2}}{15 \sqrt{x}} - \frac{4 F^{a + b x} b \log{\left (F \right )}}{15 x^{\frac{3}{2}}} - \frac{2 F^{a + b x}}{5 x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(b*x+a)/x**(7/2),x)

[Out]

8*sqrt(pi)*F**a*b**(5/2)*log(F)**(5/2)*erfi(sqrt(b)*sqrt(x)*sqrt(log(F)))/15 - 8
*F**(a + b*x)*b**2*log(F)**2/(15*sqrt(x)) - 4*F**(a + b*x)*b*log(F)/(15*x**(3/2)
) - 2*F**(a + b*x)/(5*x**(5/2))

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Mathematica [A]  time = 0.079998, size = 76, normalized size = 0.76 \[ \frac{2}{15} F^a \left (4 \sqrt{\pi } b^{5/2} \log ^{\frac{5}{2}}(F) \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )-\frac{F^{b x} \left (4 b^2 x^2 \log ^2(F)+2 b x \log (F)+3\right )}{x^{5/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[F^(a + b*x)/x^(7/2),x]

[Out]

(2*F^a*(4*b^(5/2)*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]]*Log[F]^(5/2) - (F^
(b*x)*(3 + 2*b*x*Log[F] + 4*b^2*x^2*Log[F]^2))/x^(5/2)))/15

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Maple [A]  time = 0.02, size = 84, normalized size = 0.8 \[ -{\frac{{F}^{a}}{b} \left ( -b \right ) ^{{\frac{7}{2}}} \left ( \ln \left ( F \right ) \right ) ^{{\frac{5}{2}}} \left ( -{\frac{2\,{{\rm e}^{b\ln \left ( F \right ) x}}}{5} \left ({\frac{4\,{b}^{2}{x}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}}{3}}+{\frac{2\,b\ln \left ( F \right ) x}{3}}+1 \right ){x}^{-{\frac{5}{2}}} \left ( -b \right ) ^{-{\frac{5}{2}}} \left ( \ln \left ( F \right ) \right ) ^{-{\frac{5}{2}}}}+{\frac{8\,\sqrt{\pi }}{15}{b}^{{\frac{5}{2}}}{\it erfi} \left ( \sqrt{b}\sqrt{x}\sqrt{\ln \left ( F \right ) } \right ) \left ( -b \right ) ^{-{\frac{5}{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(b*x+a)/x^(7/2),x)

[Out]

-F^a*(-b)^(7/2)*ln(F)^(5/2)/b*(-2/5/x^(5/2)/(-b)^(5/2)/ln(F)^(5/2)*(4/3*b^2*x^2*
ln(F)^2+2/3*b*ln(F)*x+1)*exp(b*ln(F)*x)+8/15/(-b)^(5/2)*b^(5/2)*Pi^(1/2)*erfi(b^
(1/2)*x^(1/2)*ln(F)^(1/2)))

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Maxima [A]  time = 0.836354, size = 32, normalized size = 0.32 \[ -\frac{\left (-b x \log \left (F\right )\right )^{\frac{5}{2}} F^{a} \Gamma \left (-\frac{5}{2}, -b x \log \left (F\right )\right )}{x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(b*x + a)/x^(7/2),x, algorithm="maxima")

[Out]

-(-b*x*log(F))^(5/2)*F^a*gamma(-5/2, -b*x*log(F))/x^(5/2)

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Fricas [A]  time = 0.270217, size = 107, normalized size = 1.07 \[ \frac{2 \,{\left (4 \, \sqrt{\pi } F^{a} b^{3} x^{\frac{5}{2}} \operatorname{erf}\left (\sqrt{-b \log \left (F\right )} \sqrt{x}\right ) \log \left (F\right )^{3} -{\left (4 \, b^{2} x^{2} \log \left (F\right )^{2} + 2 \, b x \log \left (F\right ) + 3\right )} \sqrt{-b \log \left (F\right )} F^{b x + a}\right )}}{15 \, \sqrt{-b \log \left (F\right )} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(b*x + a)/x^(7/2),x, algorithm="fricas")

[Out]

2/15*(4*sqrt(pi)*F^a*b^3*x^(5/2)*erf(sqrt(-b*log(F))*sqrt(x))*log(F)^3 - (4*b^2*
x^2*log(F)^2 + 2*b*x*log(F) + 3)*sqrt(-b*log(F))*F^(b*x + a))/(sqrt(-b*log(F))*x
^(5/2))

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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**(b*x+a)/x**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{b x + a}}{x^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(b*x + a)/x^(7/2),x, algorithm="giac")

[Out]

integrate(F^(b*x + a)/x^(7/2), x)