3.1148 \(\int \frac{(b d+2 c d x)^3}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=36 \[ d^3 \left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )+d^3 (b+2 c x)^2 \]

[Out]

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

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Rubi [A]  time = 0.057868, antiderivative size = 36, 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 \[ d^3 \left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )+d^3 (b+2 c x)^2 \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.7539, size = 34, normalized size = 0.94 \[ d^{3} \left (b + 2 c x\right )^{2} + d^{3} \left (- 4 a c + b^{2}\right ) \log{\left (a + b x + c x^{2} \right )} \]

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),x)

[Out]

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

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Mathematica [A]  time = 0.0274497, size = 33, normalized size = 0.92 \[ d^3 \left (\left (b^2-4 a c\right ) \log (a+x (b+c x))+4 c x (b+c x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 57, normalized size = 1.6 \[ 4\,{x}^{2}{c}^{2}{d}^{3}-4\,\ln \left ( c{x}^{2}+bx+a \right ) ac{d}^{3}+\ln \left ( c{x}^{2}+bx+a \right ){b}^{2}{d}^{3}+4\,xbc{d}^{3} \]

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),x)

[Out]

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

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Maxima [A]  time = 0.694869, size = 58, normalized size = 1.61 \[ 4 \, c^{2} d^{3} x^{2} + 4 \, b c d^{3} x +{\left (b^{2} - 4 \, a c\right )} d^{3} \log \left (c x^{2} + b x + a\right ) \]

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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.203616, size = 58, normalized size = 1.61 \[ 4 \, c^{2} d^{3} x^{2} + 4 \, b c d^{3} x +{\left (b^{2} - 4 \, a c\right )} d^{3} \log \left (c x^{2} + b x + a\right ) \]

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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.16274, size = 44, normalized size = 1.22 \[ 4 b c d^{3} x + 4 c^{2} d^{3} x^{2} - d^{3} \left (4 a c - b^{2}\right ) \log{\left (a + b x + c x^{2} \right )} \]

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),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.215881, size = 72, normalized size = 2. \[{\left (b^{2} d^{3} - 4 \, a c d^{3}\right )}{\rm ln}\left (c x^{2} + b x + a\right ) + \frac{4 \,{\left (c^{4} d^{3} x^{2} + b c^{3} d^{3} x\right )}}{c^{2}} \]

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),x, algorithm="giac")

[Out]

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