3.228 \(\int \frac{x^5}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx\)

Optimal. Leaf size=70 \[ \frac{a^2 \log \left (a+b x^2\right )}{2 b^2 (b c-a d)}-\frac{c^2 \log \left (c+d x^2\right )}{2 d^2 (b c-a d)}+\frac{x^2}{2 b d} \]

[Out]

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

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Rubi [A]  time = 0.17453, antiderivative size = 70, 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 \[ \frac{a^2 \log \left (a+b x^2\right )}{2 b^2 (b c-a d)}-\frac{c^2 \log \left (c+d x^2\right )}{2 d^2 (b c-a d)}+\frac{x^2}{2 b d} \]

Antiderivative was successfully verified.

[In]  Int[x^5/((a + b*x^2)*(c + d*x^2)),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2} \log{\left (a + b x^{2} \right )}}{2 b^{2} \left (a d - b c\right )} + \frac{c^{2} \log{\left (c + d x^{2} \right )}}{2 d^{2} \left (a d - b c\right )} + \frac{\int ^{x^{2}} \frac{1}{b}\, dx}{2 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b*x**2+a)/(d*x**2+c),x)

[Out]

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

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Mathematica [A]  time = 0.0500274, size = 66, normalized size = 0.94 \[ \frac{a^2 d^2 \log \left (a+b x^2\right )-b \left (d x^2 (a d-b c)+b c^2 \log \left (c+d x^2\right )\right )}{2 b^2 d^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/((a + b*x^2)*(c + d*x^2)),x]

[Out]

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

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Maple [A]  time = 0.01, size = 65, normalized size = 0.9 \[{\frac{{x}^{2}}{2\,bd}}+{\frac{{c}^{2}\ln \left ( d{x}^{2}+c \right ) }{ \left ( 2\,ad-2\,bc \right ){d}^{2}}}-{\frac{{a}^{2}\ln \left ( b{x}^{2}+a \right ) }{ \left ( 2\,ad-2\,bc \right ){b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x^2+a)/(d*x^2+c),x)

[Out]

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

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Maxima [A]  time = 1.33316, size = 92, normalized size = 1.31 \[ \frac{a^{2} \log \left (b x^{2} + a\right )}{2 \,{\left (b^{3} c - a b^{2} d\right )}} - \frac{c^{2} \log \left (d x^{2} + c\right )}{2 \,{\left (b c d^{2} - a d^{3}\right )}} + \frac{x^{2}}{2 \, b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((b*x^2 + a)*(d*x^2 + c)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.247015, size = 97, normalized size = 1.39 \[ \frac{a^{2} d^{2} \log \left (b x^{2} + a\right ) - b^{2} c^{2} \log \left (d x^{2} + c\right ) +{\left (b^{2} c d - a b d^{2}\right )} x^{2}}{2 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((b*x^2 + a)*(d*x^2 + c)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 11.7071, size = 201, normalized size = 2.87 \[ - \frac{a^{2} \log{\left (x^{2} + \frac{\frac{a^{4} d^{3}}{b \left (a d - b c\right )} - \frac{2 a^{3} c d^{2}}{a d - b c} + \frac{a^{2} b c^{2} d}{a d - b c} + a^{2} c d + a b c^{2}}{a^{2} d^{2} + b^{2} c^{2}} \right )}}{2 b^{2} \left (a d - b c\right )} + \frac{c^{2} \log{\left (x^{2} + \frac{- \frac{a^{2} b c^{2} d}{a d - b c} + a^{2} c d + \frac{2 a b^{2} c^{3}}{a d - b c} + a b c^{2} - \frac{b^{3} c^{4}}{d \left (a d - b c\right )}}{a^{2} d^{2} + b^{2} c^{2}} \right )}}{2 d^{2} \left (a d - b c\right )} + \frac{x^{2}}{2 b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b*x**2+a)/(d*x**2+c),x)

[Out]

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

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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^5/((b*x^2 + a)*(d*x^2 + c)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError