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

Optimal. Leaf size=81 \[ \frac{1}{3} x^3 \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{a^2 c^2}{x}+\frac{2}{5} b d x^5 (a d+b c)+2 a c x (a d+b c)+\frac{1}{7} b^2 d^2 x^7 \]

[Out]

-((a^2*c^2)/x) + 2*a*c*(b*c + a*d)*x + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^3)/3 + (2*b*d*(b*c + a*d)*x^5)/5 + (
b^2*d^2*x^7)/7

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Rubi [A]  time = 0.0426554, antiderivative size = 81, 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} \[ \frac{1}{3} x^3 \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{a^2 c^2}{x}+\frac{2}{5} b d x^5 (a d+b c)+2 a c x (a d+b c)+\frac{1}{7} b^2 d^2 x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^2*c^2)/x) + 2*a*c*(b*c + a*d)*x + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^3)/3 + (2*b*d*(b*c + a*d)*x^5)/5 + (
b^2*d^2*x^7)/7

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

Mathematica [A]  time = 0.0377618, size = 81, normalized size = 1. \[ \frac{1}{3} x^3 \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{a^2 c^2}{x}+\frac{2}{5} b d x^5 (a d+b c)+2 a c x (a d+b c)+\frac{1}{7} b^2 d^2 x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^2*c^2)/x) + 2*a*c*(b*c + a*d)*x + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^3)/3 + (2*b*d*(b*c + a*d)*x^5)/5 + (
b^2*d^2*x^7)/7

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Maple [A]  time = 0.003, size = 91, normalized size = 1.1 \begin{align*}{\frac{{b}^{2}{d}^{2}{x}^{7}}{7}}+{\frac{2\,{x}^{5}ab{d}^{2}}{5}}+{\frac{2\,{x}^{5}{b}^{2}cd}{5}}+{\frac{{x}^{3}{a}^{2}{d}^{2}}{3}}+{\frac{4\,{x}^{3}abcd}{3}}+{\frac{{x}^{3}{b}^{2}{c}^{2}}{3}}+2\,{a}^{2}cdx+2\,ab{c}^{2}x-{\frac{{a}^{2}{c}^{2}}{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^2,x)

[Out]

1/7*b^2*d^2*x^7+2/5*x^5*a*b*d^2+2/5*x^5*b^2*c*d+1/3*x^3*a^2*d^2+4/3*x^3*a*b*c*d+1/3*x^3*b^2*c^2+2*a^2*c*d*x+2*
a*b*c^2*x-a^2*c^2/x

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Maxima [A]  time = 1.00465, size = 112, normalized size = 1.38 \begin{align*} \frac{1}{7} \, b^{2} d^{2} x^{7} + \frac{2}{5} \,{\left (b^{2} c d + a b d^{2}\right )} x^{5} + \frac{1}{3} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{3} - \frac{a^{2} c^{2}}{x} + 2 \,{\left (a b c^{2} + a^{2} c d\right )} x \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^2,x, algorithm="maxima")

[Out]

1/7*b^2*d^2*x^7 + 2/5*(b^2*c*d + a*b*d^2)*x^5 + 1/3*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^3 - a^2*c^2/x + 2*(a*b*c
^2 + a^2*c*d)*x

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Fricas [A]  time = 1.28043, size = 189, normalized size = 2.33 \begin{align*} \frac{15 \, b^{2} d^{2} x^{8} + 42 \,{\left (b^{2} c d + a b d^{2}\right )} x^{6} + 35 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} - 105 \, a^{2} c^{2} + 210 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2}}{105 \, x} \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^2,x, algorithm="fricas")

[Out]

1/105*(15*b^2*d^2*x^8 + 42*(b^2*c*d + a*b*d^2)*x^6 + 35*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4 - 105*a^2*c^2 + 21
0*(a*b*c^2 + a^2*c*d)*x^2)/x

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Sympy [A]  time = 0.315012, size = 92, normalized size = 1.14 \begin{align*} - \frac{a^{2} c^{2}}{x} + \frac{b^{2} d^{2} x^{7}}{7} + x^{5} \left (\frac{2 a b d^{2}}{5} + \frac{2 b^{2} c d}{5}\right ) + x^{3} \left (\frac{a^{2} d^{2}}{3} + \frac{4 a b c d}{3} + \frac{b^{2} c^{2}}{3}\right ) + x \left (2 a^{2} c d + 2 a b c^{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)**2/x**2,x)

[Out]

-a**2*c**2/x + b**2*d**2*x**7/7 + x**5*(2*a*b*d**2/5 + 2*b**2*c*d/5) + x**3*(a**2*d**2/3 + 4*a*b*c*d/3 + b**2*
c**2/3) + x*(2*a**2*c*d + 2*a*b*c**2)

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Giac [A]  time = 1.3286, size = 122, normalized size = 1.51 \begin{align*} \frac{1}{7} \, b^{2} d^{2} x^{7} + \frac{2}{5} \, b^{2} c d x^{5} + \frac{2}{5} \, a b d^{2} x^{5} + \frac{1}{3} \, b^{2} c^{2} x^{3} + \frac{4}{3} \, a b c d x^{3} + \frac{1}{3} \, a^{2} d^{2} x^{3} + 2 \, a b c^{2} x + 2 \, a^{2} c d x - \frac{a^{2} c^{2}}{x} \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^2,x, algorithm="giac")

[Out]

1/7*b^2*d^2*x^7 + 2/5*b^2*c*d*x^5 + 2/5*a*b*d^2*x^5 + 1/3*b^2*c^2*x^3 + 4/3*a*b*c*d*x^3 + 1/3*a^2*d^2*x^3 + 2*
a*b*c^2*x + 2*a^2*c*d*x - a^2*c^2/x