3.1160 \(\int \frac{(b d+2 c d x)^3}{\left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=43 \[ 4 c d^3 \log \left (a+b x+c x^2\right )-\frac{d^3 (b+2 c x)^2}{a+b x+c x^2} \]

[Out]

-((d^3*(b + 2*c*x)^2)/(a + b*x + c*x^2)) + 4*c*d^3*Log[a + b*x + c*x^2]

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Rubi [A]  time = 0.0555548, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ 4 c d^3 \log \left (a+b x+c x^2\right )-\frac{d^3 (b+2 c x)^2}{a+b x+c x^2} \]

Antiderivative was successfully verified.

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

[Out]

-((d^3*(b + 2*c*x)^2)/(a + b*x + c*x^2)) + 4*c*d^3*Log[a + b*x + c*x^2]

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Rubi in Sympy [A]  time = 18.1164, size = 39, normalized size = 0.91 \[ 4 c d^{3} \log{\left (a + b x + c x^{2} \right )} - \frac{d^{3} \left (b + 2 c x\right )^{2}}{a + b x + c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0443778, size = 42, normalized size = 0.98 \[ d^3 \left (\frac{4 a c-b^2}{a+b x+c x^2}+4 c \log \left (a+b x+c x^2\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

d^3*((-b^2 + 4*a*c)/(a + b*x + c*x^2) + 4*c*Log[a + b*x + c*x^2])

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Maple [A]  time = 0.012, size = 58, normalized size = 1.4 \[ 4\,{\frac{{d}^{3}ac}{c{x}^{2}+bx+a}}-{\frac{{d}^{3}{b}^{2}}{c{x}^{2}+bx+a}}+4\,c{d}^{3}\ln \left ( c{x}^{2}+bx+a \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.672262, size = 58, normalized size = 1.35 \[ 4 \, c d^{3} \log \left (c x^{2} + b x + a\right ) - \frac{{\left (b^{2} - 4 \, a c\right )} d^{3}}{c x^{2} + b x + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.20853, size = 86, normalized size = 2. \[ -\frac{{\left (b^{2} - 4 \, a c\right )} d^{3} - 4 \,{\left (c^{2} d^{3} x^{2} + b c d^{3} x + a c d^{3}\right )} \log \left (c x^{2} + b x + a\right )}{c x^{2} + b x + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.11369, size = 42, normalized size = 0.98 \[ 4 c d^{3} \log{\left (a + b x + c x^{2} \right )} + \frac{4 a c d^{3} - b^{2} d^{3}}{a + b x + c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.217561, size = 63, normalized size = 1.47 \[ 4 \, c d^{3}{\rm ln}\left (c x^{2} + b x + a\right ) - \frac{b^{2} d^{3} - 4 \, a c d^{3}}{c x^{2} + b x + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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