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

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

[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*Log[F]^3) - (7*F^(a + b*x)*x^(5/2))/(2*b^2*Log[F]^2) + (F^(a
+ b*x)*x^(7/2))/(b*Log[F])

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Rubi [A]  time = 0.147164, 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, Rules used = {2176, 2180, 2204} \[ \frac{105 \sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )}{16 b^{9/2} \log ^{\frac{9}{2}}(F)}-\frac{7 x^{5/2} F^{a+b x}}{2 b^2 \log ^2(F)}+\frac{35 x^{3/2} F^{a+b x}}{4 b^3 \log ^3(F)}-\frac{105 \sqrt{x} F^{a+b x}}{8 b^4 \log ^4(F)}+\frac{x^{7/2} F^{a+b x}}{b \log (F)} \]

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*Log[F]^3) - (7*F^(a + b*x)*x^(5/2))/(2*b^2*Log[F]^2) + (F^(a
+ b*x)*x^(7/2))/(b*Log[F])

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2180

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - (c*
f)/d) + (f*g*x^2)/d), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rubi steps

\begin{align*} \int F^{a+b x} x^{7/2} \, dx &=\frac{F^{a+b x} x^{7/2}}{b \log (F)}-\frac{7 \int F^{a+b x} x^{5/2} \, dx}{2 b \log (F)}\\ &=-\frac{7 F^{a+b x} x^{5/2}}{2 b^2 \log ^2(F)}+\frac{F^{a+b x} x^{7/2}}{b \log (F)}+\frac{35 \int F^{a+b x} x^{3/2} \, dx}{4 b^2 \log ^2(F)}\\ &=\frac{35 F^{a+b x} x^{3/2}}{4 b^3 \log ^3(F)}-\frac{7 F^{a+b x} x^{5/2}}{2 b^2 \log ^2(F)}+\frac{F^{a+b x} x^{7/2}}{b \log (F)}-\frac{105 \int F^{a+b x} \sqrt{x} \, dx}{8 b^3 \log ^3(F)}\\ &=-\frac{105 F^{a+b x} \sqrt{x}}{8 b^4 \log ^4(F)}+\frac{35 F^{a+b x} x^{3/2}}{4 b^3 \log ^3(F)}-\frac{7 F^{a+b x} x^{5/2}}{2 b^2 \log ^2(F)}+\frac{F^{a+b x} x^{7/2}}{b \log (F)}+\frac{105 \int \frac{F^{a+b x}}{\sqrt{x}} \, dx}{16 b^4 \log ^4(F)}\\ &=-\frac{105 F^{a+b x} \sqrt{x}}{8 b^4 \log ^4(F)}+\frac{35 F^{a+b x} x^{3/2}}{4 b^3 \log ^3(F)}-\frac{7 F^{a+b x} x^{5/2}}{2 b^2 \log ^2(F)}+\frac{F^{a+b x} x^{7/2}}{b \log (F)}+\frac{105 \operatorname{Subst}\left (\int F^{a+b x^2} \, dx,x,\sqrt{x}\right )}{8 b^4 \log ^4(F)}\\ &=\frac{105 F^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )}{16 b^{9/2} \log ^{\frac{9}{2}}(F)}-\frac{105 F^{a+b x} \sqrt{x}}{8 b^4 \log ^4(F)}+\frac{35 F^{a+b x} x^{3/2}}{4 b^3 \log ^3(F)}-\frac{7 F^{a+b x} x^{5/2}}{2 b^2 \log ^2(F)}+\frac{F^{a+b x} x^{7/2}}{b \log (F)}\\ \end{align*}

Mathematica [A]  time = 0.0062582, size = 36, normalized size = 0.27 \[ \frac{F^a \sqrt{-b x \log (F)} \text{Gamma}\left (\frac{9}{2},-b x \log (F)\right )}{b^5 \sqrt{x} \log ^5(F)} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*x)*x^(7/2),x]

[Out]

(F^a*Gamma[9/2, -(b*x*Log[F])]*Sqrt[-(b*x*Log[F])])/(b^5*Sqrt[x]*Log[F]^5)

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Maple [A]  time = 0.022, size = 99, normalized size = 0.8 \begin{align*} -{\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}}}} \end{align*}

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)^2-6
30*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 = 1.22927, size = 32, normalized size = 0.24 \begin{align*} -\frac{F^{a} x^{\frac{9}{2}} \Gamma \left (\frac{9}{2}, -b x \log \left (F\right )\right )}{\left (-b x \log \left (F\right )\right )^{\frac{9}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-F^a*x^(9/2)*gamma(9/2, -b*x*log(F))/(-b*x*log(F))^(9/2)

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Fricas [A]  time = 1.53279, size = 254, normalized size = 1.94 \begin{align*} -\frac{105 \, \sqrt{\pi } \sqrt{-b \log \left (F\right )} F^{a} \operatorname{erf}\left (\sqrt{-b \log \left (F\right )} \sqrt{x}\right ) - 2 \,{\left (8 \, b^{4} x^{3} \log \left (F\right )^{4} - 28 \, b^{3} x^{2} \log \left (F\right )^{3} + 70 \, b^{2} x \log \left (F\right )^{2} - 105 \, b \log \left (F\right )\right )} F^{b x + a} \sqrt{x}}{16 \, b^{5} \log \left (F\right )^{5}} \end{align*}

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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 [A]  time = 1.2203, size = 127, normalized size = 0.97 \begin{align*} -\frac{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}} + \frac{{\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 )} e^{\left (b x \log \left (F\right ) + a \log \left (F\right )\right )}}{8 \, b^{4} \log \left (F\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-105/16*sqrt(pi)*F^a*erf(-sqrt(-b*log(F))*sqrt(x))/(sqrt(-b*log(F))*b^4*log(F)^4) + 1/8*(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))*e^(b*x*log(F) + a*log(F))/(b^4*log(F)^4)