3.38 \(\int \frac{F^{a+b x}}{x^{9/2}} \, dx\)

Optimal. Leaf size=123 \[ \frac{16}{105} \sqrt{\pi } b^{7/2} F^a \log ^{\frac{7}{2}}(F) \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )-\frac{16 b^3 \log ^3(F) F^{a+b x}}{105 \sqrt{x}}-\frac{8 b^2 \log ^2(F) F^{a+b x}}{105 x^{3/2}}-\frac{2 F^{a+b x}}{7 x^{7/2}}-\frac{4 b \log (F) F^{a+b x}}{35 x^{5/2}} \]

[Out]

(-2*F^(a + b*x))/(7*x^(7/2)) - (4*b*F^(a + b*x)*Log[F])/(35*x^(5/2)) - (8*b^2*F^
(a + b*x)*Log[F]^2)/(105*x^(3/2)) - (16*b^3*F^(a + b*x)*Log[F]^3)/(105*Sqrt[x])
+ (16*b^(7/2)*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]]*Log[F]^(7/2))/105

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Rubi [A]  time = 0.175157, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{16}{105} \sqrt{\pi } b^{7/2} F^a \log ^{\frac{7}{2}}(F) \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )-\frac{16 b^3 \log ^3(F) F^{a+b x}}{105 \sqrt{x}}-\frac{8 b^2 \log ^2(F) F^{a+b x}}{105 x^{3/2}}-\frac{2 F^{a+b x}}{7 x^{7/2}}-\frac{4 b \log (F) F^{a+b x}}{35 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b*x)/x^(9/2),x]

[Out]

(-2*F^(a + b*x))/(7*x^(7/2)) - (4*b*F^(a + b*x)*Log[F])/(35*x^(5/2)) - (8*b^2*F^
(a + b*x)*Log[F]^2)/(105*x^(3/2)) - (16*b^3*F^(a + b*x)*Log[F]^3)/(105*Sqrt[x])
+ (16*b^(7/2)*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]]*Log[F]^(7/2))/105

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Rubi in Sympy [A]  time = 20.7114, size = 124, normalized size = 1.01 \[ \frac{16 \sqrt{\pi } F^{a} b^{\frac{7}{2}} \log{\left (F \right )}^{\frac{7}{2}} \operatorname{erfi}{\left (\sqrt{b} \sqrt{x} \sqrt{\log{\left (F \right )}} \right )}}{105} - \frac{16 F^{a + b x} b^{3} \log{\left (F \right )}^{3}}{105 \sqrt{x}} - \frac{8 F^{a + b x} b^{2} \log{\left (F \right )}^{2}}{105 x^{\frac{3}{2}}} - \frac{4 F^{a + b x} b \log{\left (F \right )}}{35 x^{\frac{5}{2}}} - \frac{2 F^{a + b x}}{7 x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(b*x+a)/x**(9/2),x)

[Out]

16*sqrt(pi)*F**a*b**(7/2)*log(F)**(7/2)*erfi(sqrt(b)*sqrt(x)*sqrt(log(F)))/105 -
16*F**(a + b*x)*b**3*log(F)**3/(105*sqrt(x)) - 8*F**(a + b*x)*b**2*log(F)**2/(1
05*x**(3/2)) - 4*F**(a + b*x)*b*log(F)/(35*x**(5/2)) - 2*F**(a + b*x)/(7*x**(7/2
))

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Mathematica [A]  time = 0.053314, size = 92, normalized size = 0.75 \[ -\frac{2 F^a \left (F^{b x} \left (8 b^3 x^3 \log ^3(F)+4 b^2 x^2 \log ^2(F)+6 b x \log (F)+15\right )-8 \sqrt{\pi } b^{7/2} x^{7/2} \log ^{\frac{7}{2}}(F) \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )\right )}{105 x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(a + b*x)/x^(9/2),x]

[Out]

(-2*F^a*(-8*b^(7/2)*Sqrt[Pi]*x^(7/2)*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]]*Log[F]^(
7/2) + F^(b*x)*(15 + 6*b*x*Log[F] + 4*b^2*x^2*Log[F]^2 + 8*b^3*x^3*Log[F]^3)))/(
105*x^(7/2))

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Maple [A]  time = 0.02, size = 96, normalized size = 0.8 \[ -{\frac{{F}^{a}}{b} \left ( -b \right ) ^{{\frac{9}{2}}} \left ( \ln \left ( F \right ) \right ) ^{{\frac{7}{2}}} \left ( -{\frac{2\,{{\rm e}^{b\ln \left ( F \right ) x}}}{7} \left ({\frac{8\,{b}^{3}{x}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}}{15}}+{\frac{4\,{b}^{2}{x}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}}{15}}+{\frac{2\,b\ln \left ( F \right ) x}{5}}+1 \right ){x}^{-{\frac{7}{2}}} \left ( -b \right ) ^{-{\frac{7}{2}}} \left ( \ln \left ( F \right ) \right ) ^{-{\frac{7}{2}}}}+{\frac{16\,\sqrt{\pi }}{105}{b}^{{\frac{7}{2}}}{\it erfi} \left ( \sqrt{b}\sqrt{x}\sqrt{\ln \left ( F \right ) } \right ) \left ( -b \right ) ^{-{\frac{7}{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(b*x+a)/x^(9/2),x)

[Out]

-F^a*(-b)^(9/2)*ln(F)^(7/2)/b*(-2/7/x^(7/2)/(-b)^(7/2)/ln(F)^(7/2)*(8/15*b^3*x^3
*ln(F)^3+4/15*b^2*x^2*ln(F)^2+2/5*b*ln(F)*x+1)*exp(b*ln(F)*x)+16/105/(-b)^(7/2)*
b^(7/2)*Pi^(1/2)*erfi(b^(1/2)*x^(1/2)*ln(F)^(1/2)))

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Maxima [A]  time = 0.837909, size = 32, normalized size = 0.26 \[ -\frac{\left (-b x \log \left (F\right )\right )^{\frac{7}{2}} F^{a} \Gamma \left (-\frac{7}{2}, -b x \log \left (F\right )\right )}{x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(b*x + a)/x^(9/2),x, algorithm="maxima")

[Out]

-(-b*x*log(F))^(7/2)*F^a*gamma(-7/2, -b*x*log(F))/x^(7/2)

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Fricas [A]  time = 0.269327, size = 123, normalized size = 1. \[ \frac{2 \,{\left (8 \, \sqrt{\pi } F^{a} b^{4} x^{\frac{7}{2}} \operatorname{erf}\left (\sqrt{-b \log \left (F\right )} \sqrt{x}\right ) \log \left (F\right )^{4} -{\left (8 \, b^{3} x^{3} \log \left (F\right )^{3} + 4 \, b^{2} x^{2} \log \left (F\right )^{2} + 6 \, b x \log \left (F\right ) + 15\right )} \sqrt{-b \log \left (F\right )} F^{b x + a}\right )}}{105 \, \sqrt{-b \log \left (F\right )} x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(b*x + a)/x^(9/2),x, algorithm="fricas")

[Out]

2/105*(8*sqrt(pi)*F^a*b^4*x^(7/2)*erf(sqrt(-b*log(F))*sqrt(x))*log(F)^4 - (8*b^3
*x^3*log(F)^3 + 4*b^2*x^2*log(F)^2 + 6*b*x*log(F) + 15)*sqrt(-b*log(F))*F^(b*x +
a))/(sqrt(-b*log(F))*x^(7/2))

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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**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{b x + a}}{x^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(b*x + a)/x^(9/2),x, algorithm="giac")

[Out]

integrate(F^(b*x + a)/x^(9/2), x)