3.3.5 \(\int (a x^2+b x^3) \, dx\) [205]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \frac {a x^3}{3}+\frac {b x^4}{4} \end {gather*}

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*}

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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {a x^3}{3}+\frac {b x^4}{4} \end {gather*}

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 = 0.02, size = 14, normalized size = 0.82

method	result	size
gosper	\(\frac {x^{3} \left (3 b x +4 a \right )}{12}\)	\(14\)
default	\(\frac {1}{3} a \,x^{3}+\frac {1}{4} b \,x^{4}\)	\(14\)
norman	\(\frac {1}{3} a \,x^{3}+\frac {1}{4} b \,x^{4}\)	\(14\)
risch	\(\frac {1}{3} a \,x^{3}+\frac {1}{4} b \,x^{4}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{4} \, b x^{4} + \frac {1}{3} \, a x^{3} \end {gather*}

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 = 3.56, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{4} \, b x^{4} + \frac {1}{3} \, a x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b*x^4 + 1/3*a*x^3

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Sympy [A]
time = 0.01, size = 12, normalized size = 0.71 \begin {gather*} \frac {a x^{3}}{3} + \frac {b x^{4}}{4} \end {gather*}

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 = 1.46, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{4} \, b x^{4} + \frac {1}{3} \, a x^{3} \end {gather*}

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

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Mupad [B]
time = 0.02, size = 13, normalized size = 0.76 \begin {gather*} \frac {x^3\,\left (4\,a+3\,b\,x\right )}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a*x^2 + b*x^3,x)

[Out]

(x^3*(4*a + 3*b*x))/12

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