3.236 \(\int \frac{1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx\)

Optimal. Leaf size=87 \[ \frac{b^2 \log \left (a+b x^2\right )}{2 a^2 (b c-a d)}-\frac{\log (x) (a d+b c)}{a^2 c^2}-\frac{d^2 \log \left (c+d x^2\right )}{2 c^2 (b c-a d)}-\frac{1}{2 a c x^2} \]

[Out]

-1/(2*a*c*x^2) - ((b*c + a*d)*Log[x])/(a^2*c^2) + (b^2*Log[a + b*x^2])/(2*a^2*(b
*c - a*d)) - (d^2*Log[c + d*x^2])/(2*c^2*(b*c - a*d))

_______________________________________________________________________________________

Rubi [A]  time = 0.226018, antiderivative size = 87, 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 \[ \frac{b^2 \log \left (a+b x^2\right )}{2 a^2 (b c-a d)}-\frac{\log (x) (a d+b c)}{a^2 c^2}-\frac{d^2 \log \left (c+d x^2\right )}{2 c^2 (b c-a d)}-\frac{1}{2 a c x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + b*x^2)*(c + d*x^2)),x]

[Out]

-1/(2*a*c*x^2) - ((b*c + a*d)*Log[x])/(a^2*c^2) + (b^2*Log[a + b*x^2])/(2*a^2*(b
*c - a*d)) - (d^2*Log[c + d*x^2])/(2*c^2*(b*c - a*d))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 33.7355, size = 76, normalized size = 0.87 \[ \frac{d^{2} \log{\left (c + d x^{2} \right )}}{2 c^{2} \left (a d - b c\right )} - \frac{1}{2 a c x^{2}} - \frac{b^{2} \log{\left (a + b x^{2} \right )}}{2 a^{2} \left (a d - b c\right )} - \frac{\left (a d + b c\right ) \log{\left (x^{2} \right )}}{2 a^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(b*x**2+a)/(d*x**2+c),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0658263, size = 88, normalized size = 1.01 \[ -\frac{b^2 \log \left (a+b x^2\right )}{2 a^2 (a d-b c)}+\frac{\log (x) (-a d-b c)}{a^2 c^2}-\frac{d^2 \log \left (c+d x^2\right )}{2 c^2 (b c-a d)}-\frac{1}{2 a c x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(a + b*x^2)*(c + d*x^2)),x]

[Out]

-1/(2*a*c*x^2) + ((-(b*c) - a*d)*Log[x])/(a^2*c^2) - (b^2*Log[a + b*x^2])/(2*a^2
*(-(b*c) + a*d)) - (d^2*Log[c + d*x^2])/(2*c^2*(b*c - a*d))

_______________________________________________________________________________________

Maple [A]  time = 0.016, size = 87, normalized size = 1. \[ -{\frac{1}{2\,ac{x}^{2}}}-{\frac{\ln \left ( x \right ) d}{a{c}^{2}}}-{\frac{b\ln \left ( x \right ) }{{a}^{2}c}}+{\frac{{d}^{2}\ln \left ( d{x}^{2}+c \right ) }{2\,{c}^{2} \left ( ad-bc \right ) }}-{\frac{{b}^{2}\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{2} \left ( ad-bc \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x^2+a)/(d*x^2+c),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.38471, size = 117, normalized size = 1.34 \[ \frac{b^{2} \log \left (b x^{2} + a\right )}{2 \,{\left (a^{2} b c - a^{3} d\right )}} - \frac{d^{2} \log \left (d x^{2} + c\right )}{2 \,{\left (b c^{3} - a c^{2} d\right )}} - \frac{{\left (b c + a d\right )} \log \left (x^{2}\right )}{2 \, a^{2} c^{2}} - \frac{1}{2 \, a c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*(d*x^2 + c)*x^3),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.439202, size = 134, normalized size = 1.54 \[ \frac{b^{2} c^{2} x^{2} \log \left (b x^{2} + a\right ) - a^{2} d^{2} x^{2} \log \left (d x^{2} + c\right ) - a b c^{2} + a^{2} c d - 2 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{2} \log \left (x\right )}{2 \,{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*(d*x^2 + c)*x^3),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(b*x**2+a)/(d*x**2+c),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*(d*x^2 + c)*x^3),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError