3.261 \(\int (A+B x) (a+c x^2)^2 \, dx\)

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

[Out]

a^2*A*x + (2*a*A*c*x^3)/3 + (A*c^2*x^5)/5 + (B*(a + c*x^2)^3)/(6*c)

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Rubi [A]  time = 0.0148019, 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} \[ a^2 A x+\frac{2}{3} a A c x^3+\frac{B \left (a+c x^2\right )^3}{6 c}+\frac{1}{5} A c^2 x^5 \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(a + c*x^2)^2,x]

[Out]

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

Mathematica [A]  time = 0.0020613, size = 60, normalized size = 1.33 \[ a^2 A x+\frac{1}{2} a^2 B x^2+\frac{2}{3} a A c x^3+\frac{1}{2} a B c x^4+\frac{1}{5} A c^2 x^5+\frac{1}{6} B c^2 x^6 \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(a + c*x^2)^2,x]

[Out]

a^2*A*x + (a^2*B*x^2)/2 + (2*a*A*c*x^3)/3 + (a*B*c*x^4)/2 + (A*c^2*x^5)/5 + (B*c^2*x^6)/6

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Maple [A]  time = 0.001, size = 51, normalized size = 1.1 \begin{align*}{\frac{B{c}^{2}{x}^{6}}{6}}+{\frac{A{c}^{2}{x}^{5}}{5}}+{\frac{aBc{x}^{4}}{2}}+{\frac{2\,aAc{x}^{3}}{3}}+{\frac{{a}^{2}B{x}^{2}}{2}}+{a}^{2}Ax \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^2,x)

[Out]

1/6*B*c^2*x^6+1/5*A*c^2*x^5+1/2*a*B*c*x^4+2/3*a*A*c*x^3+1/2*a^2*B*x^2+a^2*A*x

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Maxima [A]  time = 1.03248, size = 68, normalized size = 1.51 \begin{align*} \frac{1}{6} \, B c^{2} x^{6} + \frac{1}{5} \, A c^{2} x^{5} + \frac{1}{2} \, B a c x^{4} + \frac{2}{3} \, A a c x^{3} + \frac{1}{2} \, B a^{2} x^{2} + A a^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*B*c^2*x^6 + 1/5*A*c^2*x^5 + 1/2*B*a*c*x^4 + 2/3*A*a*c*x^3 + 1/2*B*a^2*x^2 + A*a^2*x

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Fricas [A]  time = 1.28154, size = 120, normalized size = 2.67 \begin{align*} \frac{1}{6} x^{6} c^{2} B + \frac{1}{5} x^{5} c^{2} A + \frac{1}{2} x^{4} c a B + \frac{2}{3} x^{3} c a A + \frac{1}{2} x^{2} a^{2} B + x a^{2} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^6*c^2*B + 1/5*x^5*c^2*A + 1/2*x^4*c*a*B + 2/3*x^3*c*a*A + 1/2*x^2*a^2*B + x*a^2*A

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Sympy [A]  time = 0.12706, size = 58, normalized size = 1.29 \begin{align*} A a^{2} x + \frac{2 A a c x^{3}}{3} + \frac{A c^{2} x^{5}}{5} + \frac{B a^{2} x^{2}}{2} + \frac{B a c x^{4}}{2} + \frac{B c^{2} x^{6}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**2*x + 2*A*a*c*x**3/3 + A*c**2*x**5/5 + B*a**2*x**2/2 + B*a*c*x**4/2 + B*c**2*x**6/6

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Giac [A]  time = 1.10761, size = 68, normalized size = 1.51 \begin{align*} \frac{1}{6} \, B c^{2} x^{6} + \frac{1}{5} \, A c^{2} x^{5} + \frac{1}{2} \, B a c x^{4} + \frac{2}{3} \, A a c x^{3} + \frac{1}{2} \, B a^{2} x^{2} + A a^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*B*c^2*x^6 + 1/5*A*c^2*x^5 + 1/2*B*a*c*x^4 + 2/3*A*a*c*x^3 + 1/2*B*a^2*x^2 + A*a^2*x