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

Optimal. Leaf size=51 \[ -\frac{a^2 c}{2 x^2}+\frac{1}{2} b x^2 (2 a d+b c)+a \log (x) (a d+2 b c)+\frac{1}{4} b^2 d x^4 \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.125256, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{a^2 c}{2 x^2}+\frac{1}{2} b x^2 (2 a d+b c)+a \log (x) (a d+2 b c)+\frac{1}{4} b^2 d x^4 \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0418176, size = 49, normalized size = 0.96 \[ \frac{1}{4} \left (-\frac{2 a^2 c}{x^2}+2 b x^2 (2 a d+b c)+4 a \log (x) (a d+2 b c)+b^2 d x^4\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 50, normalized size = 1. \[{\frac{{b}^{2}d{x}^{4}}{4}}+{x}^{2}abd+{\frac{{b}^{2}c{x}^{2}}{2}}+\ln \left ( x \right ){a}^{2}d+2\,\ln \left ( x \right ) abc-{\frac{{a}^{2}c}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.35337, size = 70, normalized size = 1.37 \[ \frac{1}{4} \, b^{2} d x^{4} + \frac{1}{2} \,{\left (b^{2} c + 2 \, a b d\right )} x^{2} + \frac{1}{2} \,{\left (2 \, a b c + a^{2} d\right )} \log \left (x^{2}\right ) - \frac{a^{2} c}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.22059, size = 73, normalized size = 1.43 \[ \frac{b^{2} d x^{6} + 2 \,{\left (b^{2} c + 2 \, a b d\right )} x^{4} + 4 \,{\left (2 \, a b c + a^{2} d\right )} x^{2} \log \left (x\right ) - 2 \, a^{2} c}{4 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 1.61829, size = 48, normalized size = 0.94 \[ - \frac{a^{2} c}{2 x^{2}} + a \left (a d + 2 b c\right ) \log{\left (x \right )} + \frac{b^{2} d x^{4}}{4} + x^{2} \left (a b d + \frac{b^{2} c}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.223446, size = 95, normalized size = 1.86 \[ \frac{1}{4} \, b^{2} d x^{4} + \frac{1}{2} \, b^{2} c x^{2} + a b d x^{2} + \frac{1}{2} \,{\left (2 \, a b c + a^{2} d\right )}{\rm ln}\left (x^{2}\right ) - \frac{2 \, a b c x^{2} + a^{2} d x^{2} + a^{2} c}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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