3.1337 \(\int \frac{(a+b x)^2}{c+d x} \, dx\)

Optimal. Leaf size=50 \[ \frac{(b c-a d)^2 \log (c+d x)}{d^3}-\frac{b x (b c-a d)}{d^2}+\frac{(a+b x)^2}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.0507067, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{(b c-a d)^2 \log (c+d x)}{d^3}-\frac{b x (b c-a d)}{d^2}+\frac{(a+b x)^2}{2 d} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^2/(c + d*x),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\left (a + b x\right )^{2}}{2 d} + \frac{\left (a d - b c\right ) \int b\, dx}{d^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (c + d x \right )}}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0268914, size = 43, normalized size = 0.86 \[ \frac{b d x (4 a d-2 b c+b d x)+2 (b c-a d)^2 \log (c+d x)}{2 d^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 74, normalized size = 1.5 \[{\frac{{b}^{2}{x}^{2}}{2\,d}}+2\,{\frac{abx}{d}}-{\frac{{b}^{2}xc}{{d}^{2}}}+{\frac{\ln \left ( dx+c \right ){a}^{2}}{d}}-2\,{\frac{\ln \left ( dx+c \right ) abc}{{d}^{2}}}+{\frac{\ln \left ( dx+c \right ){b}^{2}{c}^{2}}{{d}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2/(d*x+c),x)

[Out]

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

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Maxima [A]  time = 1.33422, size = 81, normalized size = 1.62 \[ \frac{b^{2} d x^{2} - 2 \,{\left (b^{2} c - 2 \, a b d\right )} x}{2 \, d^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (d x + c\right )}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.207573, size = 84, normalized size = 1.68 \[ \frac{b^{2} d^{2} x^{2} - 2 \,{\left (b^{2} c d - 2 \, a b d^{2}\right )} x + 2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (d x + c\right )}{2 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.687509, size = 44, normalized size = 0.88 \[ \frac{b^{2} x^{2}}{2 d} + \frac{x \left (2 a b d - b^{2} c\right )}{d^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (c + d x \right )}}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**2/(d*x+c),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.222066, size = 81, normalized size = 1.62 \[ \frac{b^{2} d x^{2} - 2 \, b^{2} c x + 4 \, a b d x}{2 \, d^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\rm ln}\left ({\left | d x + c \right |}\right )}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b^2*d*x^2 - 2*b^2*c*x + 4*a*b*d*x)/d^2 + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*ln
(abs(d*x + c))/d^3