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

Optimal. Leaf size=53 \[ \frac{c \log \left (c+d x^2\right )}{2 d (b c-a d)}-\frac{a \log \left (a+b x^2\right )}{2 b (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.129768, antiderivative size = 53, 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 \[ \frac{c \log \left (c+d x^2\right )}{2 d (b c-a d)}-\frac{a \log \left (a+b x^2\right )}{2 b (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.8976, size = 39, normalized size = 0.74 \[ \frac{a \log{\left (a + b x^{2} \right )}}{2 b \left (a d - b c\right )} - \frac{c \log{\left (c + d x^{2} \right )}}{2 d \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0312515, size = 43, normalized size = 0.81 \[ -\frac{a d \log \left (a+b x^2\right )-b c \log \left (c+d x^2\right )}{2 b^2 c d-2 a b d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 50, normalized size = 0.9 \[ -{\frac{c\ln \left ( d{x}^{2}+c \right ) }{ \left ( 2\,ad-2\,bc \right ) d}}+{\frac{a\ln \left ( b{x}^{2}+a \right ) }{ \left ( 2\,ad-2\,bc \right ) b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x^2+a)/(d*x^2+c),x)

[Out]

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

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Maxima [A]  time = 1.34683, size = 66, normalized size = 1.25 \[ -\frac{a \log \left (b x^{2} + a\right )}{2 \,{\left (b^{2} c - a b d\right )}} + \frac{c \log \left (d x^{2} + c\right )}{2 \,{\left (b c d - a d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.231717, size = 57, normalized size = 1.08 \[ -\frac{a d \log \left (b x^{2} + a\right ) - b c \log \left (d x^{2} + c\right )}{2 \,{\left (b^{2} c d - a b d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.39369, size = 144, normalized size = 2.72 \[ \frac{a \log{\left (x^{2} + \frac{\frac{a^{3} d^{2}}{b \left (a d - b c\right )} - \frac{2 a^{2} c d}{a d - b c} + \frac{a b c^{2}}{a d - b c} + 2 a c}{a d + b c} \right )}}{2 b \left (a d - b c\right )} - \frac{c \log{\left (x^{2} + \frac{- \frac{a^{2} c d}{a d - b c} + \frac{2 a b c^{2}}{a d - b c} + 2 a c - \frac{b^{2} c^{3}}{d \left (a d - b c\right )}}{a d + b c} \right )}}{2 d \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: NotImplementedError