3.750 \(\int (\frac{x^2 (5 e^x+3 x^2)}{5 \sqrt{5 e^x+x^3}}+\frac{4}{5} x \sqrt{5 e^x+x^3}) \, dx\)

Optimal. Leaf size=20 \[ \frac{2}{5} x^2 \sqrt{x^3+5 e^x} \]

[Out]

(2*x^2*Sqrt[5*E^x + x^3])/5

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Rubi [A]  time = 0.597482, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 50, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {6742, 2262} \[ \frac{2}{5} x^2 \sqrt{x^3+5 e^x} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*(5*E^x + 3*x^2))/(5*Sqrt[5*E^x + x^3]) + (4*x*Sqrt[5*E^x + x^3])/5,x]

[Out]

(2*x^2*Sqrt[5*E^x + x^3])/5

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 2262

Int[(F_)^((e_.)*((c_.) + (d_.)*(x_)))*(x_)^(m_.)*((b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))) + (a_.)*(x_)^(n_.))^
(p_.), x_Symbol] :> Simp[(x^m*(a*x^n + b*F^(e*(c + d*x)))^(p + 1))/(b*d*e*(p + 1)*Log[F]), x] + (-Dist[m/(b*d*
e*(p + 1)*Log[F]), Int[x^(m - 1)*(a*x^n + b*F^(e*(c + d*x)))^(p + 1), x], x] - Dist[(a*n)/(b*d*e*Log[F]), Int[
x^(m + n - 1)*(a*x^n + b*F^(e*(c + d*x)))^p, x], x]) /; FreeQ[{F, a, b, c, d, e, m, n, p}, x] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \left (\frac{x^2 \left (5 e^x+3 x^2\right )}{5 \sqrt{5 e^x+x^3}}+\frac{4}{5} x \sqrt{5 e^x+x^3}\right ) \, dx &=\frac{1}{5} \int \frac{x^2 \left (5 e^x+3 x^2\right )}{\sqrt{5 e^x+x^3}} \, dx+\frac{4}{5} \int x \sqrt{5 e^x+x^3} \, dx\\ &=\frac{1}{5} \int \left (\frac{5 e^x x^2}{\sqrt{5 e^x+x^3}}+\frac{3 x^4}{\sqrt{5 e^x+x^3}}\right ) \, dx+\frac{4}{5} \int x \sqrt{5 e^x+x^3} \, dx\\ &=\frac{3}{5} \int \frac{x^4}{\sqrt{5 e^x+x^3}} \, dx+\frac{4}{5} \int x \sqrt{5 e^x+x^3} \, dx+\int \frac{e^x x^2}{\sqrt{5 e^x+x^3}} \, dx\\ &=\frac{2}{5} x^2 \sqrt{5 e^x+x^3}\\ \end{align*}

Mathematica [A]  time = 0.161067, size = 20, normalized size = 1. \[ \frac{2}{5} x^2 \sqrt{x^3+5 e^x} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(5*E^x + 3*x^2))/(5*Sqrt[5*E^x + x^3]) + (4*x*Sqrt[5*E^x + x^3])/5,x]

[Out]

(2*x^2*Sqrt[5*E^x + x^3])/5

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Maple [A]  time = 0.085, size = 16, normalized size = 0.8 \begin{align*}{\frac{2\,{x}^{2}}{5}\sqrt{5\,{{\rm e}^{x}}+{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5*x^2*(5*exp(x)+3*x^2)/(5*exp(x)+x^3)^(1/2)+4/5*x*(5*exp(x)+x^3)^(1/2),x)

[Out]

2/5*x^2*(5*exp(x)+x^3)^(1/2)

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Maxima [A]  time = 1.13466, size = 31, normalized size = 1.55 \begin{align*} \frac{2 \,{\left (x^{5} + 5 \, x^{2} e^{x}\right )}}{5 \, \sqrt{x^{3} + 5 \, e^{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*x^2*(5*exp(x)+3*x^2)/(5*exp(x)+x^3)^(1/2)+4/5*x*(5*exp(x)+x^3)^(1/2),x, algorithm="maxima")

[Out]

2/5*(x^5 + 5*x^2*e^x)/sqrt(x^3 + 5*e^x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*x^2*(5*exp(x)+3*x^2)/(5*exp(x)+x^3)^(1/2)+4/5*x*(5*exp(x)+x^3)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{7 x^{4}}{\sqrt{x^{3} + 5 e^{x}}}\, dx + \int \frac{20 x e^{x}}{\sqrt{x^{3} + 5 e^{x}}}\, dx + \int \frac{5 x^{2} e^{x}}{\sqrt{x^{3} + 5 e^{x}}}\, dx}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*x**2*(5*exp(x)+3*x**2)/(5*exp(x)+x**3)**(1/2)+4/5*x*(5*exp(x)+x**3)**(1/2),x)

[Out]

(Integral(7*x**4/sqrt(x**3 + 5*exp(x)), x) + Integral(20*x*exp(x)/sqrt(x**3 + 5*exp(x)), x) + Integral(5*x**2*
exp(x)/sqrt(x**3 + 5*exp(x)), x))/5

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (3 \, x^{2} + 5 \, e^{x}\right )} x^{2}}{5 \, \sqrt{x^{3} + 5 \, e^{x}}} + \frac{4}{5} \, \sqrt{x^{3} + 5 \, e^{x}} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*x^2*(5*exp(x)+3*x^2)/(5*exp(x)+x^3)^(1/2)+4/5*x*(5*exp(x)+x^3)^(1/2),x, algorithm="giac")

[Out]

integrate(1/5*(3*x^2 + 5*e^x)*x^2/sqrt(x^3 + 5*e^x) + 4/5*sqrt(x^3 + 5*e^x)*x, x)