3.3.13 \(\int (b x+c x^2)^2 \, dx\)

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

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Rubi [A]  time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {b^2 x^3}{3}+\frac {1}{2} b c x^4+\frac {c^2 x^5}{5} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 30, normalized size = 1.00 \begin {gather*} \frac {b^2 x^3}{3}+\frac {1}{2} b c x^4+\frac {c^2 x^5}{5} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b x+c x^2\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(b*x + c*x^2)^2,x]

[Out]

IntegrateAlgebraic[(b*x + c*x^2)^2, x]

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fricas [A]  time = 0.36, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{5} x^{5} c^{2} + \frac {1}{2} x^{4} c b + \frac {1}{3} x^{3} b^{2} \end {gather*}

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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giac [A]  time = 0.17, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{5} \, c^{2} x^{5} + \frac {1}{2} \, b c x^{4} + \frac {1}{3} \, b^{2} x^{3} \end {gather*}

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

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maple [A]  time = 0.04, size = 25, normalized size = 0.83 \begin {gather*} \frac {1}{5} c^{2} x^{5}+\frac {1}{2} b c \,x^{4}+\frac {1}{3} b^{2} x^{3} \end {gather*}

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 = 1.30, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{5} \, c^{2} x^{5} + \frac {1}{2} \, b c x^{4} + \frac {1}{3} \, b^{2} x^{3} \end {gather*}

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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mupad [B]  time = 0.03, size = 24, normalized size = 0.80 \begin {gather*} \frac {b^2\,x^3}{3}+\frac {b\,c\,x^4}{2}+\frac {c^2\,x^5}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^2,x)

[Out]

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

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sympy [A]  time = 0.07, size = 24, normalized size = 0.80 \begin {gather*} \frac {b^{2} x^{3}}{3} + \frac {b c x^{4}}{2} + \frac {c^{2} x^{5}}{5} \end {gather*}

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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