3.50 \(\int (a+c x^2)^3 \, dx\)

Optimal. Leaf size=35 \[ a^3 x+a^2 c x^3+\frac {3}{5} a c^2 x^5+\frac {c^3 x^7}{7} \]

[Out]

a^3*x+a^2*c*x^3+3/5*a*c^2*x^5+1/7*c^3*x^7

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Rubi [A]  time = 0.01, antiderivative size = 35, 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 = {194} \[ a^2 c x^3+a^3 x+\frac {3}{5} a c^2 x^5+\frac {c^3 x^7}{7} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^3,x]

[Out]

a^3*x + a^2*c*x^3 + (3*a*c^2*x^5)/5 + (c^3*x^7)/7

Rule 194

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 )^3 \, dx &=\int \left (a^3+3 a^2 c x^2+3 a c^2 x^4+c^3 x^6\right ) \, dx\\ &=a^3 x+a^2 c x^3+\frac {3}{5} a c^2 x^5+\frac {c^3 x^7}{7}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 35, normalized size = 1.00 \[ a^3 x+a^2 c x^3+\frac {3}{5} a c^2 x^5+\frac {c^3 x^7}{7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^3,x]

[Out]

a^3*x + a^2*c*x^3 + (3*a*c^2*x^5)/5 + (c^3*x^7)/7

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fricas [A]  time = 0.75, size = 31, normalized size = 0.89 \[ \frac {1}{7} x^{7} c^{3} + \frac {3}{5} x^{5} c^{2} a + x^{3} c a^{2} + x a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*c^3 + 3/5*x^5*c^2*a + x^3*c*a^2 + x*a^3

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giac [A]  time = 0.53, size = 31, normalized size = 0.89 \[ \frac {1}{7} \, c^{3} x^{7} + \frac {3}{5} \, a c^{2} x^{5} + a^{2} c x^{3} + a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^3,x, algorithm="giac")

[Out]

1/7*c^3*x^7 + 3/5*a*c^2*x^5 + a^2*c*x^3 + a^3*x

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maple [A]  time = 0.04, size = 32, normalized size = 0.91 \[ \frac {1}{7} c^{3} x^{7}+\frac {3}{5} a \,c^{2} x^{5}+a^{2} c \,x^{3}+a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^3*x+a^2*c*x^3+3/5*a*c^2*x^5+1/7*c^3*x^7

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maxima [A]  time = 1.35, size = 31, normalized size = 0.89 \[ \frac {1}{7} \, c^{3} x^{7} + \frac {3}{5} \, a c^{2} x^{5} + a^{2} c x^{3} + a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^3,x, algorithm="maxima")

[Out]

1/7*c^3*x^7 + 3/5*a*c^2*x^5 + a^2*c*x^3 + a^3*x

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mupad [B]  time = 0.04, size = 31, normalized size = 0.89 \[ a^3\,x+a^2\,c\,x^3+\frac {3\,a\,c^2\,x^5}{5}+\frac {c^3\,x^7}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^3,x)

[Out]

a^3*x + (c^3*x^7)/7 + a^2*c*x^3 + (3*a*c^2*x^5)/5

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sympy [A]  time = 0.07, size = 32, normalized size = 0.91 \[ a^{3} x + a^{2} c x^{3} + \frac {3 a c^{2} x^{5}}{5} + \frac {c^{3} x^{7}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x + a**2*c*x**3 + 3*a*c**2*x**5/5 + c**3*x**7/7

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