3.659 \(\int x^{4/3} (a+b x)^2 \, dx\)

Optimal. Leaf size=36 \[ \frac{3}{7} a^2 x^{7/3}+\frac{3}{5} a b x^{10/3}+\frac{3}{13} b^2 x^{13/3} \]

[Out]

(3*a^2*x^(7/3))/7 + (3*a*b*x^(10/3))/5 + (3*b^2*x^(13/3))/13

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Rubi [A]  time = 0.0070806, antiderivative size = 36, 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} \[ \frac{3}{7} a^2 x^{7/3}+\frac{3}{5} a b x^{10/3}+\frac{3}{13} b^2 x^{13/3} \]

Antiderivative was successfully verified.

[In]

Int[x^(4/3)*(a + b*x)^2,x]

[Out]

(3*a^2*x^(7/3))/7 + (3*a*b*x^(10/3))/5 + (3*b^2*x^(13/3))/13

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^{4/3} (a+b x)^2 \, dx &=\int \left (a^2 x^{4/3}+2 a b x^{7/3}+b^2 x^{10/3}\right ) \, dx\\ &=\frac{3}{7} a^2 x^{7/3}+\frac{3}{5} a b x^{10/3}+\frac{3}{13} b^2 x^{13/3}\\ \end{align*}

Mathematica [A]  time = 0.0076244, size = 28, normalized size = 0.78 \[ \frac{3}{455} x^{7/3} \left (65 a^2+91 a b x+35 b^2 x^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^(4/3)*(a + b*x)^2,x]

[Out]

(3*x^(7/3)*(65*a^2 + 91*a*b*x + 35*b^2*x^2))/455

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Maple [A]  time = 0.005, size = 25, normalized size = 0.7 \begin{align*}{\frac{105\,{b}^{2}{x}^{2}+273\,abx+195\,{a}^{2}}{455}{x}^{{\frac{7}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(4/3)*(b*x+a)^2,x)

[Out]

3/455*x^(7/3)*(35*b^2*x^2+91*a*b*x+65*a^2)

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Maxima [A]  time = 1.03683, size = 32, normalized size = 0.89 \begin{align*} \frac{3}{13} \, b^{2} x^{\frac{13}{3}} + \frac{3}{5} \, a b x^{\frac{10}{3}} + \frac{3}{7} \, a^{2} x^{\frac{7}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(4/3)*(b*x+a)^2,x, algorithm="maxima")

[Out]

3/13*b^2*x^(13/3) + 3/5*a*b*x^(10/3) + 3/7*a^2*x^(7/3)

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Fricas [A]  time = 1.46162, size = 73, normalized size = 2.03 \begin{align*} \frac{3}{455} \,{\left (35 \, b^{2} x^{4} + 91 \, a b x^{3} + 65 \, a^{2} x^{2}\right )} x^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(4/3)*(b*x+a)^2,x, algorithm="fricas")

[Out]

3/455*(35*b^2*x^4 + 91*a*b*x^3 + 65*a^2*x^2)*x^(1/3)

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Sympy [A]  time = 3.16779, size = 34, normalized size = 0.94 \begin{align*} \frac{3 a^{2} x^{\frac{7}{3}}}{7} + \frac{3 a b x^{\frac{10}{3}}}{5} + \frac{3 b^{2} x^{\frac{13}{3}}}{13} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(4/3)*(b*x+a)**2,x)

[Out]

3*a**2*x**(7/3)/7 + 3*a*b*x**(10/3)/5 + 3*b**2*x**(13/3)/13

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Giac [A]  time = 1.05798, size = 32, normalized size = 0.89 \begin{align*} \frac{3}{13} \, b^{2} x^{\frac{13}{3}} + \frac{3}{5} \, a b x^{\frac{10}{3}} + \frac{3}{7} \, a^{2} x^{\frac{7}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(4/3)*(b*x+a)^2,x, algorithm="giac")

[Out]

3/13*b^2*x^(13/3) + 3/5*a*b*x^(10/3) + 3/7*a^2*x^(7/3)