3.164 \(\int \frac{(a+b x^2)^2 (c+d x^2)^3}{x} \, dx\)

Optimal. Leaf size=123 \[ \frac{1}{6} d x^6 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{4} c x^4 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a^2 c^3 \log (x)+\frac{1}{2} a c^2 x^2 (3 a d+2 b c)+\frac{1}{8} b d^2 x^8 (2 a d+3 b c)+\frac{1}{10} b^2 d^3 x^{10} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.103883, antiderivative size = 123, 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, 88} \[ \frac{1}{6} d x^6 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{4} c x^4 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a^2 c^3 \log (x)+\frac{1}{2} a c^2 x^2 (3 a d+2 b c)+\frac{1}{8} b d^2 x^8 (2 a d+3 b c)+\frac{1}{10} b^2 d^3 x^{10} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

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

Mathematica [A]  time = 0.0298954, size = 123, normalized size = 1. \[ \frac{1}{6} d x^6 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{4} c x^4 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a^2 c^3 \log (x)+\frac{1}{2} a c^2 x^2 (3 a d+2 b c)+\frac{1}{8} b d^2 x^8 (2 a d+3 b c)+\frac{1}{10} b^2 d^3 x^{10} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 132, normalized size = 1.1 \begin{align*}{\frac{{b}^{2}{d}^{3}{x}^{10}}{10}}+{\frac{{x}^{8}ab{d}^{3}}{4}}+{\frac{3\,{x}^{8}{b}^{2}c{d}^{2}}{8}}+{\frac{{x}^{6}{a}^{2}{d}^{3}}{6}}+{x}^{6}abc{d}^{2}+{\frac{{x}^{6}{b}^{2}{c}^{2}d}{2}}+{\frac{3\,{x}^{4}{a}^{2}c{d}^{2}}{4}}+{\frac{3\,{x}^{4}ab{c}^{2}d}{2}}+{\frac{{x}^{4}{b}^{2}{c}^{3}}{4}}+{\frac{3\,{x}^{2}{a}^{2}{c}^{2}d}{2}}+{x}^{2}ab{c}^{3}+{a}^{2}{c}^{3}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.977729, size = 173, normalized size = 1.41 \begin{align*} \frac{1}{10} \, b^{2} d^{3} x^{10} + \frac{1}{8} \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{8} + \frac{1}{6} \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} + \frac{1}{2} \, a^{2} c^{3} \log \left (x^{2}\right ) + \frac{1}{4} \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} + \frac{1}{2} \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.20148, size = 275, normalized size = 2.24 \begin{align*} \frac{1}{10} \, b^{2} d^{3} x^{10} + \frac{1}{8} \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{8} + \frac{1}{6} \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} + a^{2} c^{3} \log \left (x\right ) + \frac{1}{4} \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} + \frac{1}{2} \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.36256, size = 133, normalized size = 1.08 \begin{align*} a^{2} c^{3} \log{\left (x \right )} + \frac{b^{2} d^{3} x^{10}}{10} + x^{8} \left (\frac{a b d^{3}}{4} + \frac{3 b^{2} c d^{2}}{8}\right ) + x^{6} \left (\frac{a^{2} d^{3}}{6} + a b c d^{2} + \frac{b^{2} c^{2} d}{2}\right ) + x^{4} \left (\frac{3 a^{2} c d^{2}}{4} + \frac{3 a b c^{2} d}{2} + \frac{b^{2} c^{3}}{4}\right ) + x^{2} \left (\frac{3 a^{2} c^{2} d}{2} + a b c^{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.13317, size = 181, normalized size = 1.47 \begin{align*} \frac{1}{10} \, b^{2} d^{3} x^{10} + \frac{3}{8} \, b^{2} c d^{2} x^{8} + \frac{1}{4} \, a b d^{3} x^{8} + \frac{1}{2} \, b^{2} c^{2} d x^{6} + a b c d^{2} x^{6} + \frac{1}{6} \, a^{2} d^{3} x^{6} + \frac{1}{4} \, b^{2} c^{3} x^{4} + \frac{3}{2} \, a b c^{2} d x^{4} + \frac{3}{4} \, a^{2} c d^{2} x^{4} + a b c^{3} x^{2} + \frac{3}{2} \, a^{2} c^{2} d x^{2} + \frac{1}{2} \, a^{2} c^{3} \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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