3.2112 \(\int (a+b \sqrt{x}) x^3 \, dx\)

Optimal. Leaf size=19 \[ \frac{a x^4}{4}+\frac{2}{9} b x^{9/2} \]

[Out]

(a*x^4)/4 + (2*b*x^(9/2))/9

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Rubi [A]  time = 0.0049615, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {14} \[ \frac{a x^4}{4}+\frac{2}{9} b x^{9/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])*x^3,x]

[Out]

(a*x^4)/4 + (2*b*x^(9/2))/9

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \left (a+b \sqrt{x}\right ) x^3 \, dx &=\int \left (a x^3+b x^{7/2}\right ) \, dx\\ &=\frac{a x^4}{4}+\frac{2}{9} b x^{9/2}\\ \end{align*}

Mathematica [A]  time = 0.0053905, size = 19, normalized size = 1. \[ \frac{a x^4}{4}+\frac{2}{9} b x^{9/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])*x^3,x]

[Out]

(a*x^4)/4 + (2*b*x^(9/2))/9

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Maple [A]  time = 0.001, size = 14, normalized size = 0.7 \begin{align*}{\frac{a{x}^{4}}{4}}+{\frac{2\,b}{9}{x}^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*x^(1/2)),x)

[Out]

1/4*a*x^4+2/9*b*x^(9/2)

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Maxima [B]  time = 0.957458, size = 178, normalized size = 9.37 \begin{align*} \frac{2 \,{\left (b \sqrt{x} + a\right )}^{9}}{9 \, b^{8}} - \frac{7 \,{\left (b \sqrt{x} + a\right )}^{8} a}{4 \, b^{8}} + \frac{6 \,{\left (b \sqrt{x} + a\right )}^{7} a^{2}}{b^{8}} - \frac{35 \,{\left (b \sqrt{x} + a\right )}^{6} a^{3}}{3 \, b^{8}} + \frac{14 \,{\left (b \sqrt{x} + a\right )}^{5} a^{4}}{b^{8}} - \frac{21 \,{\left (b \sqrt{x} + a\right )}^{4} a^{5}}{2 \, b^{8}} + \frac{14 \,{\left (b \sqrt{x} + a\right )}^{3} a^{6}}{3 \, b^{8}} - \frac{{\left (b \sqrt{x} + a\right )}^{2} a^{7}}{b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*x^(1/2)),x, algorithm="maxima")

[Out]

2/9*(b*sqrt(x) + a)^9/b^8 - 7/4*(b*sqrt(x) + a)^8*a/b^8 + 6*(b*sqrt(x) + a)^7*a^2/b^8 - 35/3*(b*sqrt(x) + a)^6
*a^3/b^8 + 14*(b*sqrt(x) + a)^5*a^4/b^8 - 21/2*(b*sqrt(x) + a)^4*a^5/b^8 + 14/3*(b*sqrt(x) + a)^3*a^6/b^8 - (b
*sqrt(x) + a)^2*a^7/b^8

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Fricas [A]  time = 1.48778, size = 36, normalized size = 1.89 \begin{align*} \frac{2}{9} \, b x^{\frac{9}{2}} + \frac{1}{4} \, a x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*x^(1/2)),x, algorithm="fricas")

[Out]

2/9*b*x^(9/2) + 1/4*a*x^4

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Sympy [A]  time = 1.07947, size = 15, normalized size = 0.79 \begin{align*} \frac{a x^{4}}{4} + \frac{2 b x^{\frac{9}{2}}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*x**(1/2)),x)

[Out]

a*x**4/4 + 2*b*x**(9/2)/9

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Giac [A]  time = 1.09918, size = 18, normalized size = 0.95 \begin{align*} \frac{2}{9} \, b x^{\frac{9}{2}} + \frac{1}{4} \, a x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*x^(1/2)),x, algorithm="giac")

[Out]

2/9*b*x^(9/2) + 1/4*a*x^4