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

Optimal. Leaf size=84 \[ \frac{1}{2} x^2 \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{a^2 c^2}{2 x^2}+\frac{1}{2} b d x^4 (a d+b c)+2 a c \log (x) (a d+b c)+\frac{1}{6} b^2 d^2 x^6 \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.213321, antiderivative size = 84, 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 \[ \frac{1}{2} x^2 \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{a^2 c^2}{2 x^2}+\frac{1}{2} b d x^4 (a d+b c)+2 a c \log (x) (a d+b c)+\frac{1}{6} b^2 d^2 x^6 \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2} c^{2}}{2 x^{2}} + a c \left (a d + b c\right ) \log{\left (x^{2} \right )} + \frac{b^{2} d^{2} x^{6}}{6} + b d \left (a d + b c\right ) \int ^{x^{2}} x\, dx + \frac{\left (a^{2} d^{2} + b c \left (4 a d + b c\right )\right ) \int ^{x^{2}} a^{2}\, dx}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2*(d*x**2+c)**2/x**3,x)

[Out]

-a**2*c**2/(2*x**2) + a*c*(a*d + b*c)*log(x**2) + b**2*d**2*x**6/6 + b*d*(a*d +
b*c)*Integral(x, (x, x**2)) + (a**2*d**2 + b*c*(4*a*d + b*c))*Integral(a**2, (x,
x**2))/(2*a**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0786349, size = 83, normalized size = 0.99 \[ \frac{1}{6} \left (\frac{3 a^2 \left (d^2 x^4-c^2\right )}{x^2}+3 a b d x^2 \left (4 c+d x^2\right )+12 a c \log (x) (a d+b c)+b^2 x^2 \left (3 c^2+3 c d x^2+d^2 x^4\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(3*a*b*d*x^2*(4*c + d*x^2) + (3*a^2*(-c^2 + d^2*x^4))/x^2 + b^2*x^2*(3*c^2 + 3*c
*d*x^2 + d^2*x^4) + 12*a*c*(b*c + a*d)*Log[x])/6

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 93, normalized size = 1.1 \[{\frac{{b}^{2}{d}^{2}{x}^{6}}{6}}+{\frac{{x}^{4}ab{d}^{2}}{2}}+{\frac{{x}^{4}{b}^{2}cd}{2}}+{\frac{{x}^{2}{a}^{2}{d}^{2}}{2}}+2\,{x}^{2}abcd+{\frac{{x}^{2}{b}^{2}{c}^{2}}{2}}+2\,\ln \left ( x \right ){a}^{2}cd+2\,\ln \left ( x \right ) ab{c}^{2}-{\frac{{a}^{2}{c}^{2}}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.35796, size = 115, normalized size = 1.37 \[ \frac{1}{6} \, b^{2} d^{2} x^{6} + \frac{1}{2} \,{\left (b^{2} c d + a b d^{2}\right )} x^{4} + \frac{1}{2} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{2} - \frac{a^{2} c^{2}}{2 \, x^{2}} +{\left (a b c^{2} + a^{2} c d\right )} \log \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.211374, size = 119, normalized size = 1.42 \[ \frac{b^{2} d^{2} x^{8} + 3 \,{\left (b^{2} c d + a b d^{2}\right )} x^{6} + 3 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} - 3 \, a^{2} c^{2} + 12 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2} \log \left (x\right )}{6 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(b^2*d^2*x^8 + 3*(b^2*c*d + a*b*d^2)*x^6 + 3*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)
*x^4 - 3*a^2*c^2 + 12*(a*b*c^2 + a^2*c*d)*x^2*log(x))/x^2

_______________________________________________________________________________________

Sympy [A]  time = 1.74269, size = 87, normalized size = 1.04 \[ - \frac{a^{2} c^{2}}{2 x^{2}} + 2 a c \left (a d + b c\right ) \log{\left (x \right )} + \frac{b^{2} d^{2} x^{6}}{6} + x^{4} \left (\frac{a b d^{2}}{2} + \frac{b^{2} c d}{2}\right ) + x^{2} \left (\frac{a^{2} d^{2}}{2} + 2 a b c d + \frac{b^{2} c^{2}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.227304, size = 154, normalized size = 1.83 \[ \frac{1}{6} \, b^{2} d^{2} x^{6} + \frac{1}{2} \, b^{2} c d x^{4} + \frac{1}{2} \, a b d^{2} x^{4} + \frac{1}{2} \, b^{2} c^{2} x^{2} + 2 \, a b c d x^{2} + \frac{1}{2} \, a^{2} d^{2} x^{2} +{\left (a b c^{2} + a^{2} c d\right )}{\rm ln}\left (x^{2}\right ) - \frac{2 \, a b c^{2} x^{2} + 2 \, a^{2} c d x^{2} + a^{2} c^{2}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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