3.144 \(\int x (a+b x^2)^2 (c+d x^2) \, dx\)

Optimal. Leaf size=42 \[ \frac{\left (a+b x^2\right )^3 (b c-a d)}{6 b^2}+\frac{d \left (a+b x^2\right )^4}{8 b^2} \]

[Out]

((b*c - a*d)*(a + b*x^2)^3)/(6*b^2) + (d*(a + b*x^2)^4)/(8*b^2)

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Rubi [A]  time = 0.0592418, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {444, 43} \[ \frac{\left (a+b x^2\right )^3 (b c-a d)}{6 b^2}+\frac{d \left (a+b x^2\right )^4}{8 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^2)^2*(c + d*x^2),x]

[Out]

((b*c - a*d)*(a + b*x^2)^3)/(6*b^2) + (d*(a + b*x^2)^4)/(8*b^2)

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int (a+b x)^2 (c+d x) \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{(b c-a d) (a+b x)^2}{b}+\frac{d (a+b x)^3}{b}\right ) \, dx,x,x^2\right )\\ &=\frac{(b c-a d) \left (a+b x^2\right )^3}{6 b^2}+\frac{d \left (a+b x^2\right )^4}{8 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0123594, size = 51, normalized size = 1.21 \[ \frac{1}{24} x^2 \left (12 a^2 c+4 b x^4 (2 a d+b c)+6 a x^2 (a d+2 b c)+3 b^2 d x^6\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^2)^2*(c + d*x^2),x]

[Out]

(x^2*(12*a^2*c + 6*a*(2*b*c + a*d)*x^2 + 4*b*(b*c + 2*a*d)*x^4 + 3*b^2*d*x^6))/24

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Maple [A]  time = 0.001, size = 52, normalized size = 1.2 \begin{align*}{\frac{{b}^{2}d{x}^{8}}{8}}+{\frac{ \left ( 2\,abd+{b}^{2}c \right ){x}^{6}}{6}}+{\frac{ \left ({a}^{2}d+2\,abc \right ){x}^{4}}{4}}+{\frac{{a}^{2}c{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b^2*d*x^8+1/6*(2*a*b*d+b^2*c)*x^6+1/4*(a^2*d+2*a*b*c)*x^4+1/2*a^2*c*x^2

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Maxima [A]  time = 1.00039, size = 69, normalized size = 1.64 \begin{align*} \frac{1}{8} \, b^{2} d x^{8} + \frac{1}{6} \,{\left (b^{2} c + 2 \, a b d\right )} x^{6} + \frac{1}{2} \, a^{2} c x^{2} + \frac{1}{4} \,{\left (2 \, a b c + a^{2} d\right )} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b^2*d*x^8 + 1/6*(b^2*c + 2*a*b*d)*x^6 + 1/2*a^2*c*x^2 + 1/4*(2*a*b*c + a^2*d)*x^4

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Fricas [A]  time = 1.03948, size = 128, normalized size = 3.05 \begin{align*} \frac{1}{8} x^{8} d b^{2} + \frac{1}{6} x^{6} c b^{2} + \frac{1}{3} x^{6} d b a + \frac{1}{2} x^{4} c b a + \frac{1}{4} x^{4} d a^{2} + \frac{1}{2} x^{2} c a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x^8*d*b^2 + 1/6*x^6*c*b^2 + 1/3*x^6*d*b*a + 1/2*x^4*c*b*a + 1/4*x^4*d*a^2 + 1/2*x^2*c*a^2

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Sympy [A]  time = 0.066984, size = 53, normalized size = 1.26 \begin{align*} \frac{a^{2} c x^{2}}{2} + \frac{b^{2} d x^{8}}{8} + x^{6} \left (\frac{a b d}{3} + \frac{b^{2} c}{6}\right ) + x^{4} \left (\frac{a^{2} d}{4} + \frac{a b c}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*c*x**2/2 + b**2*d*x**8/8 + x**6*(a*b*d/3 + b**2*c/6) + x**4*(a**2*d/4 + a*b*c/2)

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Giac [A]  time = 1.12865, size = 72, normalized size = 1.71 \begin{align*} \frac{1}{8} \, b^{2} d x^{8} + \frac{1}{6} \, b^{2} c x^{6} + \frac{1}{3} \, a b d x^{6} + \frac{1}{2} \, a b c x^{4} + \frac{1}{4} \, a^{2} d x^{4} + \frac{1}{2} \, a^{2} c x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b^2*d*x^8 + 1/6*b^2*c*x^6 + 1/3*a*b*d*x^6 + 1/2*a*b*c*x^4 + 1/4*a^2*d*x^4 + 1/2*a^2*c*x^2