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

Optimal. Leaf size=80 \[ \frac{1}{4} x^4 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 \log (x)+\frac{1}{3} b d x^6 (a d+b c)+a c x^2 (a d+b c)+\frac{1}{8} b^2 d^2 x^8 \]

[Out]

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

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Rubi [A]  time = 0.168724, antiderivative size = 80, 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 \[ \frac{1}{4} x^4 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 \log (x)+\frac{1}{3} b d x^6 (a d+b c)+a c x^2 (a d+b c)+\frac{1}{8} b^2 d^2 x^8 \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{2} c^{2} \log{\left (x^{2} \right )}}{2} + a c x^{2} \left (a d + b c\right ) + \frac{b^{2} d^{2} x^{8}}{8} + \frac{b d x^{6} \left (a d + b c\right )}{3} + \left (\frac{a^{2} d^{2}}{2} + 2 a b c d + \frac{b^{2} c^{2}}{2}\right ) \int ^{x^{2}} x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*c**2*log(x**2)/2 + a*c*x**2*(a*d + b*c) + b**2*d**2*x**8/8 + b*d*x**6*(a*d
+ b*c)/3 + (a**2*d**2/2 + 2*a*b*c*d + b**2*c**2/2)*Integral(x, (x, x**2))

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Mathematica [A]  time = 0.0431123, size = 80, normalized size = 1. \[ \frac{1}{4} x^4 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 \log (x)+\frac{1}{3} b d x^6 (a d+b c)+a c x^2 (a d+b c)+\frac{1}{8} b^2 d^2 x^8 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 90, normalized size = 1.1 \[{\frac{{b}^{2}{d}^{2}{x}^{8}}{8}}+{\frac{{x}^{6}ab{d}^{2}}{3}}+{\frac{{x}^{6}{b}^{2}cd}{3}}+{\frac{{x}^{4}{a}^{2}{d}^{2}}{4}}+{x}^{4}abcd+{\frac{{x}^{4}{b}^{2}{c}^{2}}{4}}+{x}^{2}{a}^{2}cd+a{c}^{2}b{x}^{2}+{a}^{2}{c}^{2}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.35938, size = 115, normalized size = 1.44 \[ \frac{1}{8} \, b^{2} d^{2} x^{8} + \frac{1}{3} \,{\left (b^{2} c d + a b d^{2}\right )} x^{6} + \frac{1}{4} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} + \frac{1}{2} \, a^{2} c^{2} \log \left (x^{2}\right ) +{\left (a b c^{2} + a^{2} c d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.228289, size = 111, normalized size = 1.39 \[ \frac{1}{8} \, b^{2} d^{2} x^{8} + \frac{1}{3} \,{\left (b^{2} c d + a b d^{2}\right )} x^{6} + \frac{1}{4} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} \log \left (x\right ) +{\left (a b c^{2} + a^{2} c d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.41373, size = 85, normalized size = 1.06 \[ a^{2} c^{2} \log{\left (x \right )} + \frac{b^{2} d^{2} x^{8}}{8} + x^{6} \left (\frac{a b d^{2}}{3} + \frac{b^{2} c d}{3}\right ) + x^{4} \left (\frac{a^{2} d^{2}}{4} + a b c d + \frac{b^{2} c^{2}}{4}\right ) + x^{2} \left (a^{2} c d + a b c^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.228196, size = 124, normalized size = 1.55 \[ \frac{1}{8} \, b^{2} d^{2} x^{8} + \frac{1}{3} \, b^{2} c d x^{6} + \frac{1}{3} \, a b d^{2} x^{6} + \frac{1}{4} \, b^{2} c^{2} x^{4} + a b c d x^{4} + \frac{1}{4} \, a^{2} d^{2} x^{4} + a b c^{2} x^{2} + a^{2} c d x^{2} + \frac{1}{2} \, a^{2} c^{2}{\rm ln}\left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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