3.1802 \(\int \frac{(a+b x)^3}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx\)

Optimal. Leaf size=31 \[ \frac{b c-a d}{d^2 (c+d x)}+\frac{b \log (c+d x)}{d^2} \]

[Out]

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

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Rubi [A]  time = 0.0698133, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{b c-a d}{d^2 (c+d x)}+\frac{b \log (c+d x)}{d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.1453, size = 26, normalized size = 0.84 \[ \frac{b \log{\left (c + d x \right )}}{d^{2}} - \frac{a d - b c}{d^{2} \left (c + d x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0195983, size = 31, normalized size = 1. \[ \frac{b c-a d}{d^2 (c+d x)}+\frac{b \log (c+d x)}{d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 39, normalized size = 1.3 \[ -{\frac{a}{d \left ( dx+c \right ) }}+{\frac{bc}{{d}^{2} \left ( dx+c \right ) }}+{\frac{b\ln \left ( dx+c \right ) }{{d}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.730316, size = 46, normalized size = 1.48 \[ \frac{b c - a d}{d^{3} x + c d^{2}} + \frac{b \log \left (d x + c\right )}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.237424, size = 50, normalized size = 1.61 \[ \frac{b c - a d +{\left (b d x + b c\right )} \log \left (d x + c\right )}{d^{3} x + c d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.49113, size = 27, normalized size = 0.87 \[ \frac{b \log{\left (c + d x \right )}}{d^{2}} - \frac{a d - b c}{c d^{2} + d^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.213626, size = 43, normalized size = 1.39 \[ \frac{b{\rm ln}\left ({\left | d x + c \right |}\right )}{d^{2}} + \frac{b c - a d}{{\left (d x + c\right )} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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