3.162 \(\int x (a+b x^2)^2 (c+d x^2)^3 \, dx\)

Optimal. Leaf size=71 \[ -\frac{b \left (c+d x^2\right )^5 (b c-a d)}{5 d^3}+\frac{\left (c+d x^2\right )^4 (b c-a d)^2}{8 d^3}+\frac{b^2 \left (c+d x^2\right )^6}{12 d^3} \]

[Out]

((b*c - a*d)^2*(c + d*x^2)^4)/(8*d^3) - (b*(b*c - a*d)*(c + d*x^2)^5)/(5*d^3) + (b^2*(c + d*x^2)^6)/(12*d^3)

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Rubi [A]  time = 0.118306, 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} \[ -\frac{b \left (c+d x^2\right )^5 (b c-a d)}{5 d^3}+\frac{\left (c+d x^2\right )^4 (b c-a d)^2}{8 d^3}+\frac{b^2 \left (c+d x^2\right )^6}{12 d^3} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^2)^2*(c + d*x^2)^3,x]

[Out]

((b*c - a*d)^2*(c + d*x^2)^4)/(8*d^3) - (b*(b*c - a*d)*(c + d*x^2)^5)/(5*d^3) + (b^2*(c + d*x^2)^6)/(12*d^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 )^3 \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int (a+b x)^2 (c+d x)^3 \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{(-b c+a d)^2 (c+d x)^3}{d^2}-\frac{2 b (b c-a d) (c+d x)^4}{d^2}+\frac{b^2 (c+d x)^5}{d^2}\right ) \, dx,x,x^2\right )\\ &=\frac{(b c-a d)^2 \left (c+d x^2\right )^4}{8 d^3}-\frac{b (b c-a d) \left (c+d x^2\right )^5}{5 d^3}+\frac{b^2 \left (c+d x^2\right )^6}{12 d^3}\\ \end{align*}

Mathematica [A]  time = 0.0283017, size = 119, normalized size = 1.68 \[ \frac{1}{120} x^2 \left (15 d x^6 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+20 c x^4 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+60 a^2 c^3+30 a c^2 x^2 (3 a d+2 b c)+12 b d^2 x^8 (2 a d+3 b c)+10 b^2 d^3 x^{10}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^2)^2*(c + d*x^2)^3,x]

[Out]

(x^2*(60*a^2*c^3 + 30*a*c^2*(2*b*c + 3*a*d)*x^2 + 20*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^4 + 15*d*(3*b^2*c^2
+ 6*a*b*c*d + a^2*d^2)*x^6 + 12*b*d^2*(3*b*c + 2*a*d)*x^8 + 10*b^2*d^3*x^10))/120

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Maple [A]  time = 0.001, size = 128, normalized size = 1.8 \begin{align*}{\frac{{b}^{2}{d}^{3}{x}^{12}}{12}}+{\frac{ \left ( 2\,ab{d}^{3}+3\,{b}^{2}c{d}^{2} \right ){x}^{10}}{10}}+{\frac{ \left ({a}^{2}{d}^{3}+6\,abc{d}^{2}+3\,{b}^{2}{c}^{2}d \right ){x}^{8}}{8}}+{\frac{ \left ( 3\,{a}^{2}c{d}^{2}+6\,ab{c}^{2}d+{b}^{2}{c}^{3} \right ){x}^{6}}{6}}+{\frac{ \left ( 3\,{a}^{2}{c}^{2}d+2\,ab{c}^{3} \right ){x}^{4}}{4}}+{\frac{{a}^{2}{c}^{3}{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)^3,x)

[Out]

1/12*b^2*d^3*x^12+1/10*(2*a*b*d^3+3*b^2*c*d^2)*x^10+1/8*(a^2*d^3+6*a*b*c*d^2+3*b^2*c^2*d)*x^8+1/6*(3*a^2*c*d^2
+6*a*b*c^2*d+b^2*c^3)*x^6+1/4*(3*a^2*c^2*d+2*a*b*c^3)*x^4+1/2*a^2*c^3*x^2

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Maxima [A]  time = 1.006, size = 171, normalized size = 2.41 \begin{align*} \frac{1}{12} \, b^{2} d^{3} x^{12} + \frac{1}{10} \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{10} + \frac{1}{8} \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{8} + \frac{1}{2} \, a^{2} c^{3} x^{2} + \frac{1}{6} \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{6} + \frac{1}{4} \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{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)^3,x, algorithm="maxima")

[Out]

1/12*b^2*d^3*x^12 + 1/10*(3*b^2*c*d^2 + 2*a*b*d^3)*x^10 + 1/8*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^8 + 1/2*
a^2*c^3*x^2 + 1/6*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^6 + 1/4*(2*a*b*c^3 + 3*a^2*c^2*d)*x^4

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Fricas [B]  time = 1.054, size = 308, normalized size = 4.34 \begin{align*} \frac{1}{12} x^{12} d^{3} b^{2} + \frac{3}{10} x^{10} d^{2} c b^{2} + \frac{1}{5} x^{10} d^{3} b a + \frac{3}{8} x^{8} d c^{2} b^{2} + \frac{3}{4} x^{8} d^{2} c b a + \frac{1}{8} x^{8} d^{3} a^{2} + \frac{1}{6} x^{6} c^{3} b^{2} + x^{6} d c^{2} b a + \frac{1}{2} x^{6} d^{2} c a^{2} + \frac{1}{2} x^{4} c^{3} b a + \frac{3}{4} x^{4} d c^{2} a^{2} + \frac{1}{2} x^{2} c^{3} 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)^3,x, algorithm="fricas")

[Out]

1/12*x^12*d^3*b^2 + 3/10*x^10*d^2*c*b^2 + 1/5*x^10*d^3*b*a + 3/8*x^8*d*c^2*b^2 + 3/4*x^8*d^2*c*b*a + 1/8*x^8*d
^3*a^2 + 1/6*x^6*c^3*b^2 + x^6*d*c^2*b*a + 1/2*x^6*d^2*c*a^2 + 1/2*x^4*c^3*b*a + 3/4*x^4*d*c^2*a^2 + 1/2*x^2*c
^3*a^2

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Sympy [B]  time = 0.083065, size = 136, normalized size = 1.92 \begin{align*} \frac{a^{2} c^{3} x^{2}}{2} + \frac{b^{2} d^{3} x^{12}}{12} + x^{10} \left (\frac{a b d^{3}}{5} + \frac{3 b^{2} c d^{2}}{10}\right ) + x^{8} \left (\frac{a^{2} d^{3}}{8} + \frac{3 a b c d^{2}}{4} + \frac{3 b^{2} c^{2} d}{8}\right ) + x^{6} \left (\frac{a^{2} c d^{2}}{2} + a b c^{2} d + \frac{b^{2} c^{3}}{6}\right ) + x^{4} \left (\frac{3 a^{2} c^{2} d}{4} + \frac{a b c^{3}}{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)**3,x)

[Out]

a**2*c**3*x**2/2 + b**2*d**3*x**12/12 + x**10*(a*b*d**3/5 + 3*b**2*c*d**2/10) + x**8*(a**2*d**3/8 + 3*a*b*c*d*
*2/4 + 3*b**2*c**2*d/8) + x**6*(a**2*c*d**2/2 + a*b*c**2*d + b**2*c**3/6) + x**4*(3*a**2*c**2*d/4 + a*b*c**3/2
)

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Giac [B]  time = 1.16478, size = 181, normalized size = 2.55 \begin{align*} \frac{1}{12} \, b^{2} d^{3} x^{12} + \frac{3}{10} \, b^{2} c d^{2} x^{10} + \frac{1}{5} \, a b d^{3} x^{10} + \frac{3}{8} \, b^{2} c^{2} d x^{8} + \frac{3}{4} \, a b c d^{2} x^{8} + \frac{1}{8} \, a^{2} d^{3} x^{8} + \frac{1}{6} \, b^{2} c^{3} x^{6} + a b c^{2} d x^{6} + \frac{1}{2} \, a^{2} c d^{2} x^{6} + \frac{1}{2} \, a b c^{3} x^{4} + \frac{3}{4} \, a^{2} c^{2} d x^{4} + \frac{1}{2} \, a^{2} c^{3} 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)^3,x, algorithm="giac")

[Out]

1/12*b^2*d^3*x^12 + 3/10*b^2*c*d^2*x^10 + 1/5*a*b*d^3*x^10 + 3/8*b^2*c^2*d*x^8 + 3/4*a*b*c*d^2*x^8 + 1/8*a^2*d
^3*x^8 + 1/6*b^2*c^3*x^6 + a*b*c^2*d*x^6 + 1/2*a^2*c*d^2*x^6 + 1/2*a*b*c^3*x^4 + 3/4*a^2*c^2*d*x^4 + 1/2*a^2*c
^3*x^2