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

Optimal. Leaf size=17 \[ \frac{a x^3}{3}+\frac{b x^4}{4} \]

[Out]

(a*x^3)/3 + (b*x^4)/4

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Rubi [A]  time = 0.0029354, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 0, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \frac{a x^3}{3}+\frac{b x^4}{4} \]

Antiderivative was successfully verified.

[In]

Int[a*x^2 + b*x^3,x]

[Out]

(a*x^3)/3 + (b*x^4)/4

Rubi steps

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

Mathematica [A]  time = 0.0000287, size = 17, normalized size = 1. \[ \frac{a x^3}{3}+\frac{b x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[a*x^2 + b*x^3,x]

[Out]

(a*x^3)/3 + (b*x^4)/4

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Maple [A]  time = 0., size = 14, normalized size = 0.8 \begin{align*}{\frac{a{x}^{3}}{3}}+{\frac{b{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*x^3+1/4*b*x^4

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Maxima [A]  time = 0.98957, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{4} \, b x^{4} + \frac{1}{3} \, a 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]

1/4*b*x^4 + 1/3*a*x^3

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Fricas [A]  time = 1.09005, size = 31, normalized size = 1.82 \begin{align*} \frac{1}{4} x^{4} b + \frac{1}{3} x^{3} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4*b + 1/3*x^3*a

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Sympy [A]  time = 0.217801, size = 12, normalized size = 0.71 \begin{align*} \frac{a x^{3}}{3} + \frac{b x^{4}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**3/3 + b*x**4/4

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Giac [A]  time = 1.15425, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{4} \, b x^{4} + \frac{1}{3} \, a 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]

1/4*b*x^4 + 1/3*a*x^3