### 3.235 $$\int (b x+c x^2)^2 \, dx$$

Optimal. Leaf size=30 $\frac{b^2 x^3}{3}+\frac{1}{2} b c x^4+\frac{c^2 x^5}{5}$

[Out]

(b^2*x^3)/3 + (b*c*x^4)/2 + (c^2*x^5)/5

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Rubi [A]  time = 0.0093515, antiderivative size = 30, 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 = {611} $\frac{b^2 x^3}{3}+\frac{1}{2} b c x^4+\frac{c^2 x^5}{5}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*x + c*x^2)^2,x]

[Out]

(b^2*x^3)/3 + (b*c*x^4)/2 + (c^2*x^5)/5

Rule 611

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x + c*x^2)^p, x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && (EqQ[a, 0] ||  !PerfectSquareQ[b^2 - 4*a*c])

Rubi steps

\begin{align*} \int \left (b x+c x^2\right )^2 \, dx &=\int \left (b^2 x^2+2 b c x^3+c^2 x^4\right ) \, dx\\ &=\frac{b^2 x^3}{3}+\frac{1}{2} b c x^4+\frac{c^2 x^5}{5}\\ \end{align*}

Mathematica [A]  time = 0.0016168, size = 30, normalized size = 1. $\frac{b^2 x^3}{3}+\frac{1}{2} b c x^4+\frac{c^2 x^5}{5}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(b*x + c*x^2)^2,x]

[Out]

(b^2*x^3)/3 + (b*c*x^4)/2 + (c^2*x^5)/5

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Maple [A]  time = 0.04, size = 25, normalized size = 0.8 \begin{align*}{\frac{{b}^{2}{x}^{3}}{3}}+{\frac{bc{x}^{4}}{2}}+{\frac{{c}^{2}{x}^{5}}{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^2,x)

[Out]

1/3*b^2*x^3+1/2*b*c*x^4+1/5*c^2*x^5

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Maxima [A]  time = 1.15469, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{5} \, c^{2} x^{5} + \frac{1}{2} \, b c x^{4} + \frac{1}{3} \, b^{2} x^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c^2*x^5 + 1/2*b*c*x^4 + 1/3*b^2*x^3

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Fricas [A]  time = 1.22747, size = 55, normalized size = 1.83 \begin{align*} \frac{1}{5} x^{5} c^{2} + \frac{1}{2} x^{4} c b + \frac{1}{3} x^{3} b^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5*c^2 + 1/2*x^4*c*b + 1/3*x^3*b^2

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Sympy [A]  time = 0.252658, size = 24, normalized size = 0.8 \begin{align*} \frac{b^{2} x^{3}}{3} + \frac{b c x^{4}}{2} + \frac{c^{2} x^{5}}{5} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x)**2,x)

[Out]

b**2*x**3/3 + b*c*x**4/2 + c**2*x**5/5

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Giac [A]  time = 1.31254, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{5} \, c^{2} x^{5} + \frac{1}{2} \, b c x^{4} + \frac{1}{3} \, b^{2} x^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c^2*x^5 + 1/2*b*c*x^4 + 1/3*b^2*x^3