3.430 \(\int x^{3/2} (a+b x) \, dx\)

Optimal. Leaf size=21 \[ \frac{2}{5} a x^{5/2}+\frac{2}{7} b x^{7/2} \]

[Out]

(2*a*x^(5/2))/5 + (2*b*x^(7/2))/7

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Rubi [A]  time = 0.0036704, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ \frac{2}{5} a x^{5/2}+\frac{2}{7} b x^{7/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)*(a + b*x),x]

[Out]

(2*a*x^(5/2))/5 + (2*b*x^(7/2))/7

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^{3/2} (a+b x) \, dx &=\int \left (a x^{3/2}+b x^{5/2}\right ) \, dx\\ &=\frac{2}{5} a x^{5/2}+\frac{2}{7} b x^{7/2}\\ \end{align*}

Mathematica [A]  time = 0.0041991, size = 17, normalized size = 0.81 \[ \frac{2}{35} x^{5/2} (7 a+5 b x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)*(a + b*x),x]

[Out]

(2*x^(5/2)*(7*a + 5*b*x))/35

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Maple [A]  time = 0.002, size = 14, normalized size = 0.7 \begin{align*}{\frac{10\,bx+14\,a}{35}{x}^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*x^(5/2)*(5*b*x+7*a)

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Maxima [A]  time = 1.05742, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{7} \, b x^{\frac{7}{2}} + \frac{2}{5} \, a x^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*b*x^(7/2) + 2/5*a*x^(5/2)

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Fricas [A]  time = 1.59495, size = 46, normalized size = 2.19 \begin{align*} \frac{2}{35} \,{\left (5 \, b x^{3} + 7 \, a x^{2}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(5*b*x^3 + 7*a*x^2)*sqrt(x)

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Sympy [A]  time = 0.846176, size = 19, normalized size = 0.9 \begin{align*} \frac{2 a x^{\frac{5}{2}}}{5} + \frac{2 b x^{\frac{7}{2}}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*x**(5/2)/5 + 2*b*x**(7/2)/7

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Giac [A]  time = 1.18716, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{7} \, b x^{\frac{7}{2}} + \frac{2}{5} \, a x^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*b*x^(7/2) + 2/5*a*x^(5/2)