3.463 \(\int (d+e x) (a+c x^2)^2 \, dx\)

Optimal. Leaf size=45 \[ a^2 d x+\frac{2}{3} a c d x^3+\frac{e \left (a+c x^2\right )^3}{6 c}+\frac{1}{5} c^2 d x^5 \]

[Out]

a^2*d*x + (2*a*c*d*x^3)/3 + (c^2*d*x^5)/5 + (e*(a + c*x^2)^3)/(6*c)

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Rubi [A]  time = 0.0152334, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {641, 194} \[ a^2 d x+\frac{2}{3} a c d x^3+\frac{e \left (a+c x^2\right )^3}{6 c}+\frac{1}{5} c^2 d x^5 \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)*(a + c*x^2)^2,x]

[Out]

a^2*d*x + (2*a*c*d*x^3)/3 + (c^2*d*x^5)/5 + (e*(a + c*x^2)^3)/(6*c)

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

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 (d+e x) \left (a+c x^2\right )^2 \, dx &=\frac{e \left (a+c x^2\right )^3}{6 c}+d \int \left (a+c x^2\right )^2 \, dx\\ &=\frac{e \left (a+c x^2\right )^3}{6 c}+d \int \left (a^2+2 a c x^2+c^2 x^4\right ) \, dx\\ &=a^2 d x+\frac{2}{3} a c d x^3+\frac{1}{5} c^2 d x^5+\frac{e \left (a+c x^2\right )^3}{6 c}\\ \end{align*}

Mathematica [A]  time = 0.0021371, size = 60, normalized size = 1.33 \[ a^2 d x+\frac{1}{2} a^2 e x^2+\frac{2}{3} a c d x^3+\frac{1}{2} a c e x^4+\frac{1}{5} c^2 d x^5+\frac{1}{6} c^2 e x^6 \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)*(a + c*x^2)^2,x]

[Out]

a^2*d*x + (a^2*e*x^2)/2 + (2*a*c*d*x^3)/3 + (a*c*e*x^4)/2 + (c^2*d*x^5)/5 + (c^2*e*x^6)/6

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Maple [A]  time = 0.041, size = 51, normalized size = 1.1 \begin{align*}{\frac{{c}^{2}e{x}^{6}}{6}}+{\frac{{c}^{2}d{x}^{5}}{5}}+{\frac{ace{x}^{4}}{2}}+{\frac{2\,acd{x}^{3}}{3}}+{\frac{{a}^{2}e{x}^{2}}{2}}+{a}^{2}dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(c*x^2+a)^2,x)

[Out]

1/6*c^2*e*x^6+1/5*c^2*d*x^5+1/2*a*c*e*x^4+2/3*a*c*d*x^3+1/2*a^2*e*x^2+a^2*d*x

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Maxima [A]  time = 1.13861, size = 68, normalized size = 1.51 \begin{align*} \frac{1}{6} \, c^{2} e x^{6} + \frac{1}{5} \, c^{2} d x^{5} + \frac{1}{2} \, a c e x^{4} + \frac{2}{3} \, a c d x^{3} + \frac{1}{2} \, a^{2} e x^{2} + a^{2} d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*c^2*e*x^6 + 1/5*c^2*d*x^5 + 1/2*a*c*e*x^4 + 2/3*a*c*d*x^3 + 1/2*a^2*e*x^2 + a^2*d*x

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Fricas [A]  time = 1.57441, size = 120, normalized size = 2.67 \begin{align*} \frac{1}{6} x^{6} e c^{2} + \frac{1}{5} x^{5} d c^{2} + \frac{1}{2} x^{4} e c a + \frac{2}{3} x^{3} d c a + \frac{1}{2} x^{2} e a^{2} + x d a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^6*e*c^2 + 1/5*x^5*d*c^2 + 1/2*x^4*e*c*a + 2/3*x^3*d*c*a + 1/2*x^2*e*a^2 + x*d*a^2

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Sympy [A]  time = 0.123217, size = 58, normalized size = 1.29 \begin{align*} a^{2} d x + \frac{a^{2} e x^{2}}{2} + \frac{2 a c d x^{3}}{3} + \frac{a c e x^{4}}{2} + \frac{c^{2} d x^{5}}{5} + \frac{c^{2} e x^{6}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*d*x + a**2*e*x**2/2 + 2*a*c*d*x**3/3 + a*c*e*x**4/2 + c**2*d*x**5/5 + c**2*e*x**6/6

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Giac [A]  time = 1.24112, size = 72, normalized size = 1.6 \begin{align*} \frac{1}{6} \, c^{2} x^{6} e + \frac{1}{5} \, c^{2} d x^{5} + \frac{1}{2} \, a c x^{4} e + \frac{2}{3} \, a c d x^{3} + \frac{1}{2} \, a^{2} x^{2} e + a^{2} d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*c^2*x^6*e + 1/5*c^2*d*x^5 + 1/2*a*c*x^4*e + 2/3*a*c*d*x^3 + 1/2*a^2*x^2*e + a^2*d*x