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

Optimal. Leaf size=106 \[ -\frac{b \left (c+d x^2\right )^6 (3 b c-2 a d)}{12 d^4}+\frac{\left (c+d x^2\right )^5 (b c-a d) (3 b c-a d)}{10 d^4}-\frac{c \left (c+d x^2\right )^4 (b c-a d)^2}{8 d^4}+\frac{b^2 \left (c+d x^2\right )^7}{14 d^4} \]

[Out]

-(c*(b*c - a*d)^2*(c + d*x^2)^4)/(8*d^4) + ((b*c - a*d)*(3*b*c - a*d)*(c + d*x^2)^5)/(10*d^4) - (b*(3*b*c - 2*
a*d)*(c + d*x^2)^6)/(12*d^4) + (b^2*(c + d*x^2)^7)/(14*d^4)

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Rubi [A]  time = 0.224857, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {446, 77} \[ -\frac{b \left (c+d x^2\right )^6 (3 b c-2 a d)}{12 d^4}+\frac{\left (c+d x^2\right )^5 (b c-a d) (3 b c-a d)}{10 d^4}-\frac{c \left (c+d x^2\right )^4 (b c-a d)^2}{8 d^4}+\frac{b^2 \left (c+d x^2\right )^7}{14 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c*(b*c - a*d)^2*(c + d*x^2)^4)/(8*d^4) + ((b*c - a*d)*(3*b*c - a*d)*(c + d*x^2)^5)/(10*d^4) - (b*(3*b*c - 2*
a*d)*(c + d*x^2)^6)/(12*d^4) + (b^2*(c + d*x^2)^7)/(14*d^4)

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(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] && IntegerQ[Simplify[(m + 1)/n]]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x (a+b x)^2 (c+d x)^3 \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{c (b c-a d)^2 (c+d x)^3}{d^3}+\frac{(b c-a d) (3 b c-a d) (c+d x)^4}{d^3}-\frac{b (3 b c-2 a d) (c+d x)^5}{d^3}+\frac{b^2 (c+d x)^6}{d^3}\right ) \, dx,x,x^2\right )\\ &=-\frac{c (b c-a d)^2 \left (c+d x^2\right )^4}{8 d^4}+\frac{(b c-a d) (3 b c-a d) \left (c+d x^2\right )^5}{10 d^4}-\frac{b (3 b c-2 a d) \left (c+d x^2\right )^6}{12 d^4}+\frac{b^2 \left (c+d x^2\right )^7}{14 d^4}\\ \end{align*}

Mathematica [A]  time = 0.0320621, size = 119, normalized size = 1.12 \[ \frac{1}{840} x^4 \left (84 d x^6 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+105 c x^4 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+210 a^2 c^3+140 a c^2 x^2 (3 a d+2 b c)+70 b d^2 x^8 (2 a d+3 b c)+60 b^2 d^3 x^{10}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^4*(210*a^2*c^3 + 140*a*c^2*(2*b*c + 3*a*d)*x^2 + 105*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^4 + 84*d*(3*b^2*
c^2 + 6*a*b*c*d + a^2*d^2)*x^6 + 70*b*d^2*(3*b*c + 2*a*d)*x^8 + 60*b^2*d^3*x^10))/840

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Maple [A]  time = 0., size = 128, normalized size = 1.2 \begin{align*}{\frac{{b}^{2}{d}^{3}{x}^{14}}{14}}+{\frac{ \left ( 2\,ab{d}^{3}+3\,{b}^{2}c{d}^{2} \right ){x}^{12}}{12}}+{\frac{ \left ({a}^{2}{d}^{3}+6\,abc{d}^{2}+3\,{b}^{2}{c}^{2}d \right ){x}^{10}}{10}}+{\frac{ \left ( 3\,{a}^{2}c{d}^{2}+6\,ab{c}^{2}d+{b}^{2}{c}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( 3\,{a}^{2}{c}^{2}d+2\,ab{c}^{3} \right ){x}^{6}}{6}}+{\frac{{a}^{2}{c}^{3}{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x^2+a)^2*(d*x^2+c)^3,x)

[Out]

1/14*b^2*d^3*x^14+1/12*(2*a*b*d^3+3*b^2*c*d^2)*x^12+1/10*(a^2*d^3+6*a*b*c*d^2+3*b^2*c^2*d)*x^10+1/8*(3*a^2*c*d
^2+6*a*b*c^2*d+b^2*c^3)*x^8+1/6*(3*a^2*c^2*d+2*a*b*c^3)*x^6+1/4*a^2*c^3*x^4

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Maxima [A]  time = 1.0068, size = 171, normalized size = 1.61 \begin{align*} \frac{1}{14} \, b^{2} d^{3} x^{14} + \frac{1}{12} \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{12} + \frac{1}{10} \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{10} + \frac{1}{4} \, a^{2} c^{3} x^{4} + \frac{1}{8} \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{8} + \frac{1}{6} \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^2+a)^2*(d*x^2+c)^3,x, algorithm="maxima")

[Out]

1/14*b^2*d^3*x^14 + 1/12*(3*b^2*c*d^2 + 2*a*b*d^3)*x^12 + 1/10*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^10 + 1/
4*a^2*c^3*x^4 + 1/8*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^8 + 1/6*(2*a*b*c^3 + 3*a^2*c^2*d)*x^6

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Fricas [A]  time = 1.13755, size = 319, normalized size = 3.01 \begin{align*} \frac{1}{14} x^{14} d^{3} b^{2} + \frac{1}{4} x^{12} d^{2} c b^{2} + \frac{1}{6} x^{12} d^{3} b a + \frac{3}{10} x^{10} d c^{2} b^{2} + \frac{3}{5} x^{10} d^{2} c b a + \frac{1}{10} x^{10} d^{3} a^{2} + \frac{1}{8} x^{8} c^{3} b^{2} + \frac{3}{4} x^{8} d c^{2} b a + \frac{3}{8} x^{8} d^{2} c a^{2} + \frac{1}{3} x^{6} c^{3} b a + \frac{1}{2} x^{6} d c^{2} a^{2} + \frac{1}{4} x^{4} c^{3} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^2+a)^2*(d*x^2+c)^3,x, algorithm="fricas")

[Out]

1/14*x^14*d^3*b^2 + 1/4*x^12*d^2*c*b^2 + 1/6*x^12*d^3*b*a + 3/10*x^10*d*c^2*b^2 + 3/5*x^10*d^2*c*b*a + 1/10*x^
10*d^3*a^2 + 1/8*x^8*c^3*b^2 + 3/4*x^8*d*c^2*b*a + 3/8*x^8*d^2*c*a^2 + 1/3*x^6*c^3*b*a + 1/2*x^6*d*c^2*a^2 + 1
/4*x^4*c^3*a^2

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Sympy [A]  time = 0.084389, size = 138, normalized size = 1.3 \begin{align*} \frac{a^{2} c^{3} x^{4}}{4} + \frac{b^{2} d^{3} x^{14}}{14} + x^{12} \left (\frac{a b d^{3}}{6} + \frac{b^{2} c d^{2}}{4}\right ) + x^{10} \left (\frac{a^{2} d^{3}}{10} + \frac{3 a b c d^{2}}{5} + \frac{3 b^{2} c^{2} d}{10}\right ) + x^{8} \left (\frac{3 a^{2} c d^{2}}{8} + \frac{3 a b c^{2} d}{4} + \frac{b^{2} c^{3}}{8}\right ) + x^{6} \left (\frac{a^{2} c^{2} d}{2} + \frac{a b c^{3}}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x**2+a)**2*(d*x**2+c)**3,x)

[Out]

a**2*c**3*x**4/4 + b**2*d**3*x**14/14 + x**12*(a*b*d**3/6 + b**2*c*d**2/4) + x**10*(a**2*d**3/10 + 3*a*b*c*d**
2/5 + 3*b**2*c**2*d/10) + x**8*(3*a**2*c*d**2/8 + 3*a*b*c**2*d/4 + b**2*c**3/8) + x**6*(a**2*c**2*d/2 + a*b*c*
*3/3)

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Giac [A]  time = 1.13849, size = 182, normalized size = 1.72 \begin{align*} \frac{1}{14} \, b^{2} d^{3} x^{14} + \frac{1}{4} \, b^{2} c d^{2} x^{12} + \frac{1}{6} \, a b d^{3} x^{12} + \frac{3}{10} \, b^{2} c^{2} d x^{10} + \frac{3}{5} \, a b c d^{2} x^{10} + \frac{1}{10} \, a^{2} d^{3} x^{10} + \frac{1}{8} \, b^{2} c^{3} x^{8} + \frac{3}{4} \, a b c^{2} d x^{8} + \frac{3}{8} \, a^{2} c d^{2} x^{8} + \frac{1}{3} \, a b c^{3} x^{6} + \frac{1}{2} \, a^{2} c^{2} d x^{6} + \frac{1}{4} \, a^{2} c^{3} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^2+a)^2*(d*x^2+c)^3,x, algorithm="giac")

[Out]

1/14*b^2*d^3*x^14 + 1/4*b^2*c*d^2*x^12 + 1/6*a*b*d^3*x^12 + 3/10*b^2*c^2*d*x^10 + 3/5*a*b*c*d^2*x^10 + 1/10*a^
2*d^3*x^10 + 1/8*b^2*c^3*x^8 + 3/4*a*b*c^2*d*x^8 + 3/8*a^2*c*d^2*x^8 + 1/3*a*b*c^3*x^6 + 1/2*a^2*c^2*d*x^6 + 1
/4*a^2*c^3*x^4