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

Optimal. Leaf size=131 \[ \frac{105 \sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )}{16 b^{9/2} \log ^{\frac{9}{2}}(F)}-\frac{105 \sqrt{x} F^{a+b x}}{8 b^4 \log ^4(F)}+\frac{35 x^{3/2} F^{a+b x}}{4 b^3 \log ^3(F)}-\frac{7 x^{5/2} F^{a+b x}}{2 b^2 \log ^2(F)}+\frac{x^{7/2} F^{a+b x}}{b \log (F)} \]

[Out]

(105*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]])/(16*b^(9/2)*Log[F]^(9/2))
- (105*F^(a + b*x)*Sqrt[x])/(8*b^4*Log[F]^4) + (35*F^(a + b*x)*x^(3/2))/(4*b^3*L
og[F]^3) - (7*F^(a + b*x)*x^(5/2))/(2*b^2*Log[F]^2) + (F^(a + b*x)*x^(7/2))/(b*L
og[F])

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Rubi [A]  time = 0.242796, antiderivative size = 131, 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 \[ \frac{105 \sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )}{16 b^{9/2} \log ^{\frac{9}{2}}(F)}-\frac{105 \sqrt{x} F^{a+b x}}{8 b^4 \log ^4(F)}+\frac{35 x^{3/2} F^{a+b x}}{4 b^3 \log ^3(F)}-\frac{7 x^{5/2} F^{a+b x}}{2 b^2 \log ^2(F)}+\frac{x^{7/2} F^{a+b x}}{b \log (F)} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b*x)*x^(7/2),x]

[Out]

(105*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]])/(16*b^(9/2)*Log[F]^(9/2))
- (105*F^(a + b*x)*Sqrt[x])/(8*b^4*Log[F]^4) + (35*F^(a + b*x)*x^(3/2))/(4*b^3*L
og[F]^3) - (7*F^(a + b*x)*x^(5/2))/(2*b^2*Log[F]^2) + (F^(a + b*x)*x^(7/2))/(b*L
og[F])

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Rubi in Sympy [A]  time = 26.9223, size = 129, normalized size = 0.98 \[ \frac{105 \sqrt{\pi } F^{a} \operatorname{erfi}{\left (\sqrt{b} \sqrt{x} \sqrt{\log{\left (F \right )}} \right )}}{16 b^{\frac{9}{2}} \log{\left (F \right )}^{\frac{9}{2}}} + \frac{F^{a + b x} x^{\frac{7}{2}}}{b \log{\left (F \right )}} - \frac{7 F^{a + b x} x^{\frac{5}{2}}}{2 b^{2} \log{\left (F \right )}^{2}} + \frac{35 F^{a + b x} x^{\frac{3}{2}}}{4 b^{3} \log{\left (F \right )}^{3}} - \frac{105 F^{a + b x} \sqrt{x}}{8 b^{4} \log{\left (F \right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

105*sqrt(pi)*F**a*erfi(sqrt(b)*sqrt(x)*sqrt(log(F)))/(16*b**(9/2)*log(F)**(9/2))
+ F**(a + b*x)*x**(7/2)/(b*log(F)) - 7*F**(a + b*x)*x**(5/2)/(2*b**2*log(F)**2)
+ 35*F**(a + b*x)*x**(3/2)/(4*b**3*log(F)**3) - 105*F**(a + b*x)*sqrt(x)/(8*b**
4*log(F)**4)

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Mathematica [A]  time = 0.0971548, size = 99, normalized size = 0.76 \[ \frac{F^a \left (2 \sqrt{b} \sqrt{x} \sqrt{\log (F)} F^{b x} \left (8 b^3 x^3 \log ^3(F)-28 b^2 x^2 \log ^2(F)+70 b x \log (F)-105\right )+105 \sqrt{\pi } \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )\right )}{16 b^{9/2} \log ^{\frac{9}{2}}(F)} \]

Antiderivative was successfully verified.

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

[Out]

(F^a*(105*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]] + 2*Sqrt[b]*F^(b*x)*Sqrt[x
]*Sqrt[Log[F]]*(-105 + 70*b*x*Log[F] - 28*b^2*x^2*Log[F]^2 + 8*b^3*x^3*Log[F]^3)
))/(16*b^(9/2)*Log[F]^(9/2))

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Maple [A]  time = 0.158, size = 99, normalized size = 0.8 \[ -{\frac{{F}^{a}}{b} \left ( -{\frac{ \left ( -72\,{b}^{3}{x}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}+252\,{b}^{2}{x}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}-630\,b\ln \left ( F \right ) x+945 \right ){{\rm e}^{b\ln \left ( F \right ) x}}}{72\,{b}^{4}}\sqrt{x} \left ( -b \right ) ^{{\frac{9}{2}}}\sqrt{\ln \left ( F \right ) }}+{\frac{105\,\sqrt{\pi }}{16} \left ( -b \right ) ^{{\frac{9}{2}}}{\it erfi} \left ( \sqrt{b}\sqrt{x}\sqrt{\ln \left ( F \right ) } \right ){b}^{-{\frac{9}{2}}}} \right ) \left ( -b \right ) ^{-{\frac{7}{2}}} \left ( \ln \left ( F \right ) \right ) ^{-{\frac{9}{2}}}} \]

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)^(9/2)/b*(-1/72*x^(1/2)*(-b)^(9/2)*ln(F)^(1/2)*(-72*b^3*x^3
*ln(F)^3+252*b^2*x^2*ln(F)^2-630*b*ln(F)*x+945)/b^4*exp(b*ln(F)*x)+105/16*(-b)^(
9/2)/b^(9/2)*Pi^(1/2)*erfi(b^(1/2)*x^(1/2)*ln(F)^(1/2)))

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Maxima [A]  time = 0.819579, size = 120, normalized size = 0.92 \[ \frac{1}{16} \, F^{a}{\left (\frac{2 \,{\left (8 \, b^{3} x^{\frac{7}{2}} \log \left (F\right )^{3} - 28 \, b^{2} x^{\frac{5}{2}} \log \left (F\right )^{2} + 70 \, b x^{\frac{3}{2}} \log \left (F\right ) - 105 \, \sqrt{x}\right )} F^{b x}}{b^{4} \log \left (F\right )^{4}} + \frac{105 \, \sqrt{\pi } \operatorname{erf}\left (\sqrt{-b \log \left (F\right )} \sqrt{x}\right )}{\sqrt{-b \log \left (F\right )} b^{4} \log \left (F\right )^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*F^a*(2*(8*b^3*x^(7/2)*log(F)^3 - 28*b^2*x^(5/2)*log(F)^2 + 70*b*x^(3/2)*log
(F) - 105*sqrt(x))*F^(b*x)/(b^4*log(F)^4) + 105*sqrt(pi)*erf(sqrt(-b*log(F))*sqr
t(x))/(sqrt(-b*log(F))*b^4*log(F)^4))

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Fricas [A]  time = 0.274976, size = 119, normalized size = 0.91 \[ \frac{2 \,{\left (8 \, b^{3} x^{3} \log \left (F\right )^{3} - 28 \, b^{2} x^{2} \log \left (F\right )^{2} + 70 \, b x \log \left (F\right ) - 105\right )} \sqrt{-b \log \left (F\right )} F^{b x + a} \sqrt{x} + 105 \, \sqrt{\pi } F^{a} \operatorname{erf}\left (\sqrt{-b \log \left (F\right )} \sqrt{x}\right )}{16 \, \sqrt{-b \log \left (F\right )} b^{4} \log \left (F\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*(2*(8*b^3*x^3*log(F)^3 - 28*b^2*x^2*log(F)^2 + 70*b*x*log(F) - 105)*sqrt(-b
*log(F))*F^(b*x + a)*sqrt(x) + 105*sqrt(pi)*F^a*erf(sqrt(-b*log(F))*sqrt(x)))/(s
qrt(-b*log(F))*b^4*log(F)^4)

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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 [A]  time = 0.257172, size = 130, normalized size = 0.99 \[ \frac{{\left (8 \, b^{3} x^{\frac{7}{2}}{\rm ln}\left (F\right )^{3} - 28 \, b^{2} x^{\frac{5}{2}}{\rm ln}\left (F\right )^{2} + 70 \, b x^{\frac{3}{2}}{\rm ln}\left (F\right ) - 105 \, \sqrt{x}\right )} e^{\left (b x{\rm ln}\left (F\right ) + a{\rm ln}\left (F\right )\right )}}{8 \, b^{4}{\rm ln}\left (F\right )^{4}} - \frac{105 \, \sqrt{\pi } \operatorname{erf}\left (-\sqrt{-b{\rm ln}\left (F\right )} \sqrt{x}\right ) e^{\left (a{\rm ln}\left (F\right )\right )}}{16 \, \sqrt{-b{\rm ln}\left (F\right )} b^{4}{\rm ln}\left (F\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(8*b^3*x^(7/2)*ln(F)^3 - 28*b^2*x^(5/2)*ln(F)^2 + 70*b*x^(3/2)*ln(F) - 105*s
qrt(x))*e^(b*x*ln(F) + a*ln(F))/(b^4*ln(F)^4) - 105/16*sqrt(pi)*erf(-sqrt(-b*ln(
F))*sqrt(x))*e^(a*ln(F))/(sqrt(-b*ln(F))*b^4*ln(F)^4)