3.204 \(\int \frac{x^2}{(a+b x) (c+d x)} \, dx\)

Optimal. Leaf size=56 \[ \frac{a^2 \log (a+b x)}{b^2 (b c-a d)}-\frac{c^2 \log (c+d x)}{d^2 (b c-a d)}+\frac{x}{b d} \]

[Out]

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

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Rubi [A]  time = 0.0923564, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{a^2 \log (a+b x)}{b^2 (b c-a d)}-\frac{c^2 \log (c+d x)}{d^2 (b c-a d)}+\frac{x}{b d} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0419386, size = 56, normalized size = 1. \[ \frac{a^2 \log (a+b x)}{b^2 (b c-a d)}-\frac{c^2 \log (c+d x)}{d^2 (b c-a d)}+\frac{x}{b d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 57, normalized size = 1. \[{\frac{x}{bd}}+{\frac{{c}^{2}\ln \left ( dx+c \right ) }{{d}^{2} \left ( ad-bc \right ) }}-{\frac{{a}^{2}\ln \left ( bx+a \right ) }{{b}^{2} \left ( ad-bc \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x+a)/(d*x+c),x)

[Out]

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

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Maxima [A]  time = 1.35613, size = 81, normalized size = 1.45 \[ \frac{a^{2} \log \left (b x + a\right )}{b^{3} c - a b^{2} d} - \frac{c^{2} \log \left (d x + c\right )}{b c d^{2} - a d^{3}} + \frac{x}{b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.22171, size = 88, normalized size = 1.57 \[ \frac{a^{2} d^{2} \log \left (b x + a\right ) - b^{2} c^{2} \log \left (d x + c\right ) +{\left (b^{2} c d - a b d^{2}\right )} x}{b^{3} c d^{2} - a b^{2} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.21615, size = 190, normalized size = 3.39 \[ - \frac{a^{2} \log{\left (x + \frac{\frac{a^{4} d^{3}}{b \left (a d - b c\right )} - \frac{2 a^{3} c d^{2}}{a d - b c} + \frac{a^{2} b c^{2} d}{a d - b c} + a^{2} c d + a b c^{2}}{a^{2} d^{2} + b^{2} c^{2}} \right )}}{b^{2} \left (a d - b c\right )} + \frac{c^{2} \log{\left (x + \frac{- \frac{a^{2} b c^{2} d}{a d - b c} + a^{2} c d + \frac{2 a b^{2} c^{3}}{a d - b c} + a b c^{2} - \frac{b^{3} c^{4}}{d \left (a d - b c\right )}}{a^{2} d^{2} + b^{2} c^{2}} \right )}}{d^{2} \left (a d - b c\right )} + \frac{x}{b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: NotImplementedError