3.1.51 \(\int (a+c x^2)^2 \, dx\) [51]

Optimal. Leaf size=25 \[ a^2 x+\frac {2}{3} a c x^3+\frac {c^2 x^5}{5} \]

[Out]

a^2*x+2/3*a*c*x^3+1/5*c^2*x^5

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Rubi [A]
time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {200} \begin {gather*} a^2 x+\frac {2}{3} a c x^3+\frac {c^2 x^5}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^2,x]

[Out]

a^2*x + (2*a*c*x^3)/3 + (c^2*x^5)/5

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 25, normalized size = 1.00 \begin {gather*} a^2 x+\frac {2}{3} a c x^3+\frac {c^2 x^5}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^2,x]

[Out]

a^2*x + (2*a*c*x^3)/3 + (c^2*x^5)/5

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Maple [A]
time = 0.36, size = 22, normalized size = 0.88

method	result	size
gosper	\(a^{2} x +\frac {2}{3} a c \,x^{3}+\frac {1}{5} c^{2} x^{5}\)	\(22\)
default	\(a^{2} x +\frac {2}{3} a c \,x^{3}+\frac {1}{5} c^{2} x^{5}\)	\(22\)
norman	\(a^{2} x +\frac {2}{3} a c \,x^{3}+\frac {1}{5} c^{2} x^{5}\)	\(22\)
risch	\(a^{2} x +\frac {2}{3} a c \,x^{3}+\frac {1}{5} c^{2} x^{5}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*x+2/3*a*c*x^3+1/5*c^2*x^5

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Maxima [A]
time = 0.26, size = 21, normalized size = 0.84 \begin {gather*} \frac {1}{5} \, c^{2} x^{5} + \frac {2}{3} \, a c x^{3} + a^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c^2*x^5 + 2/3*a*c*x^3 + a^2*x

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Fricas [A]
time = 1.49, size = 21, normalized size = 0.84 \begin {gather*} \frac {1}{5} \, c^{2} x^{5} + \frac {2}{3} \, a c x^{3} + a^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c^2*x^5 + 2/3*a*c*x^3 + a^2*x

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Sympy [A]
time = 0.01, size = 22, normalized size = 0.88 \begin {gather*} a^{2} x + \frac {2 a c x^{3}}{3} + \frac {c^{2} x^{5}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x + 2*a*c*x**3/3 + c**2*x**5/5

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Giac [A]
time = 0.87, size = 21, normalized size = 0.84 \begin {gather*} \frac {1}{5} \, c^{2} x^{5} + \frac {2}{3} \, a c x^{3} + a^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c^2*x^5 + 2/3*a*c*x^3 + a^2*x

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Mupad [B]
time = 0.03, size = 21, normalized size = 0.84 \begin {gather*} a^2\,x+\frac {2\,a\,c\,x^3}{3}+\frac {c^2\,x^5}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^2,x)

[Out]

a^2*x + (c^2*x^5)/5 + (2*a*c*x^3)/3

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