3.158 \(\int \frac{(a+b x^2)^2 (c+d x^2)^2}{x^4} \, dx\)

Optimal. Leaf size=80 \[ x \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{a^2 c^2}{3 x^3}+\frac{2}{3} b d x^3 (a d+b c)-\frac{2 a c (a d+b c)}{x}+\frac{1}{5} b^2 d^2 x^5 \]

[Out]

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

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Rubi [A]  time = 0.0527717, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {448} \[ x \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{a^2 c^2}{3 x^3}+\frac{2}{3} b d x^3 (a d+b c)-\frac{2 a c (a d+b c)}{x}+\frac{1}{5} b^2 d^2 x^5 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^2*(c + d*x^2)^2)/x^4,x]

[Out]

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

Rule 448

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

Rubi steps

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

Mathematica [A]  time = 0.0409307, size = 80, normalized size = 1. \[ x \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{a^2 c^2}{3 x^3}+\frac{2}{3} b d x^3 (a d+b c)-\frac{2 a c (a d+b c)}{x}+\frac{1}{5} b^2 d^2 x^5 \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^2*(c + d*x^2)^2)/x^4,x]

[Out]

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

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Maple [A]  time = 0.005, size = 81, normalized size = 1. \begin{align*}{\frac{{b}^{2}{d}^{2}{x}^{5}}{5}}+{\frac{2\,{x}^{3}ab{d}^{2}}{3}}+{\frac{2\,{x}^{3}{b}^{2}cd}{3}}+{a}^{2}{d}^{2}x+4\,cabdx+{b}^{2}{c}^{2}x-{\frac{{a}^{2}{c}^{2}}{3\,{x}^{3}}}-2\,{\frac{ac \left ( ad+bc \right ) }{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2*(d*x^2+c)^2/x^4,x)

[Out]

1/5*b^2*d^2*x^5+2/3*x^3*a*b*d^2+2/3*x^3*b^2*c*d+a^2*d^2*x+4*c*a*b*d*x+b^2*c^2*x-1/3*a^2*c^2/x^3-2*a*c*(a*d+b*c
)/x

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Maxima [A]  time = 0.999957, size = 113, normalized size = 1.41 \begin{align*} \frac{1}{5} \, b^{2} d^{2} x^{5} + \frac{2}{3} \,{\left (b^{2} c d + a b d^{2}\right )} x^{3} +{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x - \frac{a^{2} c^{2} + 6 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^2/x^4,x, algorithm="maxima")

[Out]

1/5*b^2*d^2*x^5 + 2/3*(b^2*c*d + a*b*d^2)*x^3 + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x - 1/3*(a^2*c^2 + 6*(a*b*c^2
+ a^2*c*d)*x^2)/x^3

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Fricas [A]  time = 1.17608, size = 185, normalized size = 2.31 \begin{align*} \frac{3 \, b^{2} d^{2} x^{8} + 10 \,{\left (b^{2} c d + a b d^{2}\right )} x^{6} + 15 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} - 5 \, a^{2} c^{2} - 30 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2}}{15 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^2/x^4,x, algorithm="fricas")

[Out]

1/15*(3*b^2*d^2*x^8 + 10*(b^2*c*d + a*b*d^2)*x^6 + 15*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4 - 5*a^2*c^2 - 30*(a*
b*c^2 + a^2*c*d)*x^2)/x^3

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Sympy [A]  time = 0.426533, size = 90, normalized size = 1.12 \begin{align*} \frac{b^{2} d^{2} x^{5}}{5} + x^{3} \left (\frac{2 a b d^{2}}{3} + \frac{2 b^{2} c d}{3}\right ) + x \left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2}\right ) - \frac{a^{2} c^{2} + x^{2} \left (6 a^{2} c d + 6 a b c^{2}\right )}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2*(d*x**2+c)**2/x**4,x)

[Out]

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

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Giac [A]  time = 1.15307, size = 119, normalized size = 1.49 \begin{align*} \frac{1}{5} \, b^{2} d^{2} x^{5} + \frac{2}{3} \, b^{2} c d x^{3} + \frac{2}{3} \, a b d^{2} x^{3} + b^{2} c^{2} x + 4 \, a b c d x + a^{2} d^{2} x - \frac{6 \, a b c^{2} x^{2} + 6 \, a^{2} c d x^{2} + a^{2} c^{2}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^2/x^4,x, algorithm="giac")

[Out]

1/5*b^2*d^2*x^5 + 2/3*b^2*c*d*x^3 + 2/3*a*b*d^2*x^3 + b^2*c^2*x + 4*a*b*c*d*x + a^2*d^2*x - 1/3*(6*a*b*c^2*x^2
+ 6*a^2*c*d*x^2 + a^2*c^2)/x^3