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

Optimal. Leaf size=51 \[ \frac{6}{7} a^2 b x^{7/2}+\frac{2}{5} a^3 x^{5/2}+\frac{2}{3} a b^2 x^{9/2}+\frac{2}{11} b^3 x^{11/2} \]

[Out]

(2*a^3*x^(5/2))/5 + (6*a^2*b*x^(7/2))/7 + (2*a*b^2*x^(9/2))/3 + (2*b^3*x^(11/2))/11

________________________________________________________________________________________

Rubi [A]  time = 0.0115775, antiderivative size = 51, 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 = {43} \[ \frac{6}{7} a^2 b x^{7/2}+\frac{2}{5} a^3 x^{5/2}+\frac{2}{3} a b^2 x^{9/2}+\frac{2}{11} b^3 x^{11/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)*(a + b*x)^3,x]

[Out]

(2*a^3*x^(5/2))/5 + (6*a^2*b*x^(7/2))/7 + (2*a*b^2*x^(9/2))/3 + (2*b^3*x^(11/2))/11

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)^3 \, dx &=\int \left (a^3 x^{3/2}+3 a^2 b x^{5/2}+3 a b^2 x^{7/2}+b^3 x^{9/2}\right ) \, dx\\ &=\frac{2}{5} a^3 x^{5/2}+\frac{6}{7} a^2 b x^{7/2}+\frac{2}{3} a b^2 x^{9/2}+\frac{2}{11} b^3 x^{11/2}\\ \end{align*}

Mathematica [A]  time = 0.0099204, size = 39, normalized size = 0.76 \[ \frac{2 x^{5/2} \left (495 a^2 b x+231 a^3+385 a b^2 x^2+105 b^3 x^3\right )}{1155} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)*(a + b*x)^3,x]

[Out]

(2*x^(5/2)*(231*a^3 + 495*a^2*b*x + 385*a*b^2*x^2 + 105*b^3*x^3))/1155

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 36, normalized size = 0.7 \begin{align*}{\frac{210\,{b}^{3}{x}^{3}+770\,a{b}^{2}{x}^{2}+990\,{a}^{2}bx+462\,{a}^{3}}{1155}{x}^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1155*x^(5/2)*(105*b^3*x^3+385*a*b^2*x^2+495*a^2*b*x+231*a^3)

________________________________________________________________________________________

Maxima [A]  time = 1.04363, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{11} \, b^{3} x^{\frac{11}{2}} + \frac{2}{3} \, a b^{2} x^{\frac{9}{2}} + \frac{6}{7} \, a^{2} b x^{\frac{7}{2}} + \frac{2}{5} \, a^{3} x^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*b^3*x^(11/2) + 2/3*a*b^2*x^(9/2) + 6/7*a^2*b*x^(7/2) + 2/5*a^3*x^(5/2)

________________________________________________________________________________________

Fricas [A]  time = 1.64924, size = 103, normalized size = 2.02 \begin{align*} \frac{2}{1155} \,{\left (105 \, b^{3} x^{5} + 385 \, a b^{2} x^{4} + 495 \, a^{2} b x^{3} + 231 \, a^{3} 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)^3,x, algorithm="fricas")

[Out]

2/1155*(105*b^3*x^5 + 385*a*b^2*x^4 + 495*a^2*b*x^3 + 231*a^3*x^2)*sqrt(x)

________________________________________________________________________________________

Sympy [A]  time = 2.27367, size = 49, normalized size = 0.96 \begin{align*} \frac{2 a^{3} x^{\frac{5}{2}}}{5} + \frac{6 a^{2} b x^{\frac{7}{2}}}{7} + \frac{2 a b^{2} x^{\frac{9}{2}}}{3} + \frac{2 b^{3} x^{\frac{11}{2}}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**3*x**(5/2)/5 + 6*a**2*b*x**(7/2)/7 + 2*a*b**2*x**(9/2)/3 + 2*b**3*x**(11/2)/11

________________________________________________________________________________________

Giac [A]  time = 1.21508, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{11} \, b^{3} x^{\frac{11}{2}} + \frac{2}{3} \, a b^{2} x^{\frac{9}{2}} + \frac{6}{7} \, a^{2} b x^{\frac{7}{2}} + \frac{2}{5} \, a^{3} x^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*b^3*x^(11/2) + 2/3*a*b^2*x^(9/2) + 6/7*a^2*b*x^(7/2) + 2/5*a^3*x^(5/2)