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

Optimal. Leaf size=71 \[ \frac{d \left (a+b x^2\right )^4 (b c-a d)}{4 b^3}+\frac{\left (a+b x^2\right )^3 (b c-a d)^2}{6 b^3}+\frac{d^2 \left (a+b x^2\right )^5}{10 b^3} \]

[Out]

((b*c - a*d)^2*(a + b*x^2)^3)/(6*b^3) + (d*(b*c - a*d)*(a + b*x^2)^4)/(4*b^3) + (d^2*(a + b*x^2)^5)/(10*b^3)

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Rubi [A]  time = 0.105496, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {444, 43} \[ \frac{d \left (a+b x^2\right )^4 (b c-a d)}{4 b^3}+\frac{\left (a+b x^2\right )^3 (b c-a d)^2}{6 b^3}+\frac{d^2 \left (a+b x^2\right )^5}{10 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^2)^2*(c + d*x^2)^2,x]

[Out]

((b*c - a*d)^2*(a + b*x^2)^3)/(6*b^3) + (d*(b*c - a*d)*(a + b*x^2)^4)/(4*b^3) + (d^2*(a + b*x^2)^5)/(10*b^3)

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 )^2 \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int (a+b x)^2 (c+d x)^2 \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{(b c-a d)^2 (a+b x)^2}{b^2}+\frac{2 d (b c-a d) (a+b x)^3}{b^2}+\frac{d^2 (a+b x)^4}{b^2}\right ) \, dx,x,x^2\right )\\ &=\frac{(b c-a d)^2 \left (a+b x^2\right )^3}{6 b^3}+\frac{d (b c-a d) \left (a+b x^2\right )^4}{4 b^3}+\frac{d^2 \left (a+b x^2\right )^5}{10 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0218552, size = 81, normalized size = 1.14 \[ \frac{1}{60} x^2 \left (10 x^4 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+30 a^2 c^2+15 b d x^6 (a d+b c)+30 a c x^2 (a d+b c)+6 b^2 d^2 x^8\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(30*a^2*c^2 + 30*a*c*(b*c + a*d)*x^2 + 10*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4 + 15*b*d*(b*c + a*d)*x^6 +
6*b^2*d^2*x^8))/60

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Maple [A]  time = 0.001, size = 90, normalized size = 1.3 \begin{align*}{\frac{{b}^{2}{d}^{2}{x}^{10}}{10}}+{\frac{ \left ( 2\,ab{d}^{2}+2\,{b}^{2}cd \right ){x}^{8}}{8}}+{\frac{ \left ({a}^{2}{d}^{2}+4\,cabd+{b}^{2}{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 2\,{a}^{2}cd+2\,ab{c}^{2} \right ){x}^{4}}{4}}+{\frac{{a}^{2}{c}^{2}{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)^2,x)

[Out]

1/10*b^2*d^2*x^10+1/8*(2*a*b*d^2+2*b^2*c*d)*x^8+1/6*(a^2*d^2+4*a*b*c*d+b^2*c^2)*x^6+1/4*(2*a^2*c*d+2*a*b*c^2)*
x^4+1/2*a^2*c^2*x^2

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Maxima [A]  time = 1.0141, size = 115, normalized size = 1.62 \begin{align*} \frac{1}{10} \, b^{2} d^{2} x^{10} + \frac{1}{4} \,{\left (b^{2} c d + a b d^{2}\right )} x^{8} + \frac{1}{6} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{6} + \frac{1}{2} \, a^{2} c^{2} x^{2} + \frac{1}{2} \,{\left (a b c^{2} + a^{2} c 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)^2,x, algorithm="maxima")

[Out]

1/10*b^2*d^2*x^10 + 1/4*(b^2*c*d + a*b*d^2)*x^8 + 1/6*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^6 + 1/2*a^2*c^2*x^2 +
1/2*(a*b*c^2 + a^2*c*d)*x^4

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Fricas [A]  time = 1.0144, size = 220, normalized size = 3.1 \begin{align*} \frac{1}{10} x^{10} d^{2} b^{2} + \frac{1}{4} x^{8} d c b^{2} + \frac{1}{4} x^{8} d^{2} b a + \frac{1}{6} x^{6} c^{2} b^{2} + \frac{2}{3} x^{6} d c b a + \frac{1}{6} x^{6} d^{2} a^{2} + \frac{1}{2} x^{4} c^{2} b a + \frac{1}{2} x^{4} d c a^{2} + \frac{1}{2} x^{2} c^{2} 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)^2,x, algorithm="fricas")

[Out]

1/10*x^10*d^2*b^2 + 1/4*x^8*d*c*b^2 + 1/4*x^8*d^2*b*a + 1/6*x^6*c^2*b^2 + 2/3*x^6*d*c*b*a + 1/6*x^6*d^2*a^2 +
1/2*x^4*c^2*b*a + 1/2*x^4*d*c*a^2 + 1/2*x^2*c^2*a^2

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Sympy [A]  time = 0.07623, size = 94, normalized size = 1.32 \begin{align*} \frac{a^{2} c^{2} x^{2}}{2} + \frac{b^{2} d^{2} x^{10}}{10} + x^{8} \left (\frac{a b d^{2}}{4} + \frac{b^{2} c d}{4}\right ) + x^{6} \left (\frac{a^{2} d^{2}}{6} + \frac{2 a b c d}{3} + \frac{b^{2} c^{2}}{6}\right ) + x^{4} \left (\frac{a^{2} c d}{2} + \frac{a b c^{2}}{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)**2,x)

[Out]

a**2*c**2*x**2/2 + b**2*d**2*x**10/10 + x**8*(a*b*d**2/4 + b**2*c*d/4) + x**6*(a**2*d**2/6 + 2*a*b*c*d/3 + b**
2*c**2/6) + x**4*(a**2*c*d/2 + a*b*c**2/2)

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Giac [A]  time = 1.14301, size = 127, normalized size = 1.79 \begin{align*} \frac{1}{10} \, b^{2} d^{2} x^{10} + \frac{1}{4} \, b^{2} c d x^{8} + \frac{1}{4} \, a b d^{2} x^{8} + \frac{1}{6} \, b^{2} c^{2} x^{6} + \frac{2}{3} \, a b c d x^{6} + \frac{1}{6} \, a^{2} d^{2} x^{6} + \frac{1}{2} \, a b c^{2} x^{4} + \frac{1}{2} \, a^{2} c d x^{4} + \frac{1}{2} \, a^{2} c^{2} 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)^2,x, algorithm="giac")

[Out]

1/10*b^2*d^2*x^10 + 1/4*b^2*c*d*x^8 + 1/4*a*b*d^2*x^8 + 1/6*b^2*c^2*x^6 + 2/3*a*b*c*d*x^6 + 1/6*a^2*d^2*x^6 +
1/2*a*b*c^2*x^4 + 1/2*a^2*c*d*x^4 + 1/2*a^2*c^2*x^2