3.75 \(\int (a x+b x^3+c x^5)^2 \, dx\)

Optimal. Leaf size=54 \[ \frac{a^2 x^3}{3}+\frac{1}{7} x^7 \left (2 a c+b^2\right )+\frac{2}{5} a b x^5+\frac{2}{9} b c x^9+\frac{c^2 x^{11}}{11} \]

[Out]

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

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Rubi [A]  time = 0.0263768, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1594, 1108} \[ \frac{a^2 x^3}{3}+\frac{1}{7} x^7 \left (2 a c+b^2\right )+\frac{2}{5} a b x^5+\frac{2}{9} b c x^9+\frac{c^2 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

Int[(a*x + b*x^3 + c*x^5)^2,x]

[Out]

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

Rule 1594

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

Rule 1108

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^m*(a
+ b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] &&  !IntegerQ[(m + 1)/2]

Rubi steps

\begin{align*} \int \left (a x+b x^3+c x^5\right )^2 \, dx &=\int x^2 \left (a+b x^2+c x^4\right )^2 \, dx\\ &=\int \left (a^2 x^2+2 a b x^4+\left (b^2+2 a c\right ) x^6+2 b c x^8+c^2 x^{10}\right ) \, dx\\ &=\frac{a^2 x^3}{3}+\frac{2}{5} a b x^5+\frac{1}{7} \left (b^2+2 a c\right ) x^7+\frac{2}{9} b c x^9+\frac{c^2 x^{11}}{11}\\ \end{align*}

Mathematica [A]  time = 0.0075766, size = 54, normalized size = 1. \[ \frac{a^2 x^3}{3}+\frac{1}{7} x^7 \left (2 a c+b^2\right )+\frac{2}{5} a b x^5+\frac{2}{9} b c x^9+\frac{c^2 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x + b*x^3 + c*x^5)^2,x]

[Out]

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

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Maple [A]  time = 0.002, size = 45, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}{x}^{3}}{3}}+{\frac{2\,ab{x}^{5}}{5}}+{\frac{ \left ( 2\,ac+{b}^{2} \right ){x}^{7}}{7}}+{\frac{2\,bc{x}^{9}}{9}}+{\frac{{c}^{2}{x}^{11}}{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^5+b*x^3+a*x)^2,x)

[Out]

1/3*a^2*x^3+2/5*a*b*x^5+1/7*(2*a*c+b^2)*x^7+2/9*b*c*x^9+1/11*c^2*x^11

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Maxima [A]  time = 1.12043, size = 65, normalized size = 1.2 \begin{align*} \frac{1}{11} \, c^{2} x^{11} + \frac{2}{9} \, b c x^{9} + \frac{1}{7} \, b^{2} x^{7} + \frac{1}{3} \, a^{2} x^{3} + \frac{2}{35} \,{\left (5 \, c x^{7} + 7 \, b x^{5}\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^5+b*x^3+a*x)^2,x, algorithm="maxima")

[Out]

1/11*c^2*x^11 + 2/9*b*c*x^9 + 1/7*b^2*x^7 + 1/3*a^2*x^3 + 2/35*(5*c*x^7 + 7*b*x^5)*a

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Fricas [A]  time = 1.11846, size = 115, normalized size = 2.13 \begin{align*} \frac{1}{11} x^{11} c^{2} + \frac{2}{9} x^{9} c b + \frac{1}{7} x^{7} b^{2} + \frac{2}{7} x^{7} c a + \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((c*x^5+b*x^3+a*x)^2,x, algorithm="fricas")

[Out]

1/11*x^11*c^2 + 2/9*x^9*c*b + 1/7*x^7*b^2 + 2/7*x^7*c*a + 2/5*x^5*b*a + 1/3*x^3*a^2

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Sympy [A]  time = 0.069614, size = 51, normalized size = 0.94 \begin{align*} \frac{a^{2} x^{3}}{3} + \frac{2 a b x^{5}}{5} + \frac{2 b c x^{9}}{9} + \frac{c^{2} x^{11}}{11} + x^{7} \left (\frac{2 a c}{7} + \frac{b^{2}}{7}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**5+b*x**3+a*x)**2,x)

[Out]

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

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Giac [A]  time = 1.09849, size = 62, normalized size = 1.15 \begin{align*} \frac{1}{11} \, c^{2} x^{11} + \frac{2}{9} \, b c x^{9} + \frac{1}{7} \, b^{2} x^{7} + \frac{2}{7} \, a c 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((c*x^5+b*x^3+a*x)^2,x, algorithm="giac")

[Out]

1/11*c^2*x^11 + 2/9*b*c*x^9 + 1/7*b^2*x^7 + 2/7*a*c*x^7 + 2/5*a*b*x^5 + 1/3*a^2*x^3