3.8 \(\int (a x+b x^3)^2 \, dx\)

Optimal. Leaf size=30 \[ \frac{a^2 x^3}{3}+\frac{2}{5} a b x^5+\frac{b^2 x^7}{7} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0106576, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {1593, 270} \[ \frac{a^2 x^3}{3}+\frac{2}{5} a b x^5+\frac{b^2 x^7}{7} \]

Antiderivative was successfully verified.

[In]

Int[(a*x + b*x^3)^2,x]

[Out]

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

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \left (a x+b x^3\right )^2 \, dx &=\int x^2 \left (a+b x^2\right )^2 \, dx\\ &=\int \left (a^2 x^2+2 a b x^4+b^2 x^6\right ) \, dx\\ &=\frac{a^2 x^3}{3}+\frac{2}{5} a b x^5+\frac{b^2 x^7}{7}\\ \end{align*}

Mathematica [A]  time = 0.001557, size = 30, normalized size = 1. \[ \frac{a^2 x^3}{3}+\frac{2}{5} a b x^5+\frac{b^2 x^7}{7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x + b*x^3)^2,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 25, normalized size = 0.8 \begin{align*}{\frac{{x}^{3}{a}^{2}}{3}}+{\frac{2\,ab{x}^{5}}{5}}+{\frac{{b}^{2}{x}^{7}}{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*a^2+2/5*a*b*x^5+1/7*b^2*x^7

________________________________________________________________________________________

Maxima [A]  time = 1.08874, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{7} \, b^{2} x^{7} + \frac{2}{5} \, a b x^{5} + \frac{1}{3} \, a^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^2*x^7 + 2/5*a*b*x^5 + 1/3*a^2*x^3

________________________________________________________________________________________

Fricas [A]  time = 1.20089, size = 55, normalized size = 1.83 \begin{align*} \frac{1}{7} x^{7} b^{2} + \frac{2}{5} x^{5} b a + \frac{1}{3} x^{3} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*b^2 + 2/5*x^5*b*a + 1/3*x^3*a^2

________________________________________________________________________________________

Sympy [A]  time = 0.059754, size = 26, normalized size = 0.87 \begin{align*} \frac{a^{2} x^{3}}{3} + \frac{2 a b x^{5}}{5} + \frac{b^{2} x^{7}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x**3/3 + 2*a*b*x**5/5 + b**2*x**7/7

________________________________________________________________________________________

Giac [A]  time = 1.20869, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{7} \, b^{2} x^{7} + \frac{2}{5} \, a b x^{5} + \frac{1}{3} \, a^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^2*x^7 + 2/5*a*b*x^5 + 1/3*a^2*x^3