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

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 = 0.127475, 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 \[ \frac{1}{3} x^3 \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{a^2 c^2}{x}+\frac{2}{5} b d x^5 (a d+b c)+2 a c x (a d+b c)+\frac{1}{7} b^2 d^2 x^7 \]

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

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Rubi in Sympy [A]  time = 25.3281, size = 80, normalized size = 0.99 \[ - \frac{a^{2} c^{2}}{x} + 2 a c x \left (a d + b c\right ) + \frac{b^{2} d^{2} x^{7}}{7} + \frac{2 b d x^{5} \left (a d + b c\right )}{5} + x^{3} \left (\frac{a^{2} d^{2}}{3} + \frac{4 a b c d}{3} + \frac{b^{2} c^{2}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2*(d*x**2+c)**2/x**2,x)

[Out]

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

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Mathematica [A]  time = 0.0637022, 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 = 0.006, size = 91, normalized size = 1.1 \[{\frac{{b}^{2}{d}^{2}{x}^{7}}{7}}+{\frac{2\,{x}^{5}ab{d}^{2}}{5}}+{\frac{2\,{x}^{5}{b}^{2}cd}{5}}+{\frac{{x}^{3}{a}^{2}{d}^{2}}{3}}+{\frac{4\,{x}^{3}abcd}{3}}+{\frac{{x}^{3}{b}^{2}{c}^{2}}{3}}+2\,x{a}^{2}cd+2\,xab{c}^{2}-{\frac{{a}^{2}{c}^{2}}{x}} \]

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*x*a^2*c*d+2*x*a*b*c^2-a^2*c^2/x

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Maxima [A]  time = 1.35274, size = 112, normalized size = 1.38 \[ \frac{1}{7} \, b^{2} d^{2} x^{7} + \frac{2}{5} \,{\left (b^{2} c d + a b d^{2}\right )} x^{5} + \frac{1}{3} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{3} - \frac{a^{2} c^{2}}{x} + 2 \,{\left (a b c^{2} + a^{2} c d\right )} x \]

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 = 0.215754, size = 117, normalized size = 1.44 \[ \frac{15 \, b^{2} d^{2} x^{8} + 42 \,{\left (b^{2} c d + a b d^{2}\right )} x^{6} + 35 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} - 105 \, a^{2} c^{2} + 210 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2}}{105 \, x} \]

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 + 210*(a*b*c^2 + a^2*c*d)*x^2)/x

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Sympy [A]  time = 1.38349, size = 92, normalized size = 1.14 \[ - \frac{a^{2} c^{2}}{x} + \frac{b^{2} d^{2} x^{7}}{7} + x^{5} \left (\frac{2 a b d^{2}}{5} + \frac{2 b^{2} c d}{5}\right ) + x^{3} \left (\frac{a^{2} d^{2}}{3} + \frac{4 a b c d}{3} + \frac{b^{2} c^{2}}{3}\right ) + x \left (2 a^{2} c d + 2 a b c^{2}\right ) \]

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/XCAS [A]  time = 0.227856, size = 122, normalized size = 1.51 \[ \frac{1}{7} \, b^{2} d^{2} x^{7} + \frac{2}{5} \, b^{2} c d x^{5} + \frac{2}{5} \, a b d^{2} x^{5} + \frac{1}{3} \, b^{2} c^{2} x^{3} + \frac{4}{3} \, a b c d x^{3} + \frac{1}{3} \, a^{2} d^{2} x^{3} + 2 \, a b c^{2} x + 2 \, a^{2} c d x - \frac{a^{2} c^{2}}{x} \]

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