Optimal. Leaf size=85 \[ -\frac{(b c-a d)^3 \log (a+b x)}{a^3 b}+\frac{c^2 (b c-3 a d)}{a^2 x}+\frac{c \log (x) \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{a^3}-\frac{c^3}{2 a x^2} \]
[Out]
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Rubi [A] time = 0.136631, antiderivative size = 85, 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 \[ -\frac{(b c-a d)^3 \log (a+b x)}{a^3 b}+\frac{c^2 (b c-3 a d)}{a^2 x}+\frac{c \log (x) \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{a^3}-\frac{c^3}{2 a x^2} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)^3/(x^3*(a + b*x)),x]
[Out]
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Rubi in Sympy [A] time = 36.0037, size = 78, normalized size = 0.92 \[ - \frac{c^{3}}{2 a x^{2}} - \frac{c^{2} \left (3 a d - b c\right )}{a^{2} x} + \frac{c \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right ) \log{\left (x \right )}}{a^{3}} + \frac{\left (a d - b c\right )^{3} \log{\left (a + b x \right )}}{a^{3} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**3/x**3/(b*x+a),x)
[Out]
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Mathematica [A] time = 0.142552, size = 78, normalized size = 0.92 \[ -\frac{-2 c \log (x) \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )+\frac{a c^2 (a (c+6 d x)-2 b c x)}{x^2}+\frac{2 (b c-a d)^3 \log (a+b x)}{b}}{2 a^3} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)^3/(x^3*(a + b*x)),x]
[Out]
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Maple [A] time = 0.014, size = 132, normalized size = 1.6 \[ -{\frac{{c}^{3}}{2\,a{x}^{2}}}+3\,{\frac{c\ln \left ( x \right ){d}^{2}}{a}}-3\,{\frac{{c}^{2}\ln \left ( x \right ) bd}{{a}^{2}}}+{\frac{{c}^{3}\ln \left ( x \right ){b}^{2}}{{a}^{3}}}-3\,{\frac{{c}^{2}d}{ax}}+{\frac{b{c}^{3}}{x{a}^{2}}}+{\frac{\ln \left ( bx+a \right ){d}^{3}}{b}}-3\,{\frac{\ln \left ( bx+a \right ) c{d}^{2}}{a}}+3\,{\frac{b\ln \left ( bx+a \right ){c}^{2}d}{{a}^{2}}}-{\frac{{b}^{2}\ln \left ( bx+a \right ){c}^{3}}{{a}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^3/x^3/(b*x+a),x)
[Out]
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Maxima [A] time = 1.35875, size = 151, normalized size = 1.78 \[ \frac{{\left (b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} \log \left (x\right )}{a^{3}} - \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{a^{3} b} - \frac{a c^{3} - 2 \,{\left (b c^{3} - 3 \, a c^{2} d\right )} x}{2 \, a^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^3/((b*x + a)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.223112, size = 167, normalized size = 1.96 \[ -\frac{a^{2} b c^{3} + 2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2} \log \left (b x + a\right ) - 2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2}\right )} x^{2} \log \left (x\right ) - 2 \,{\left (a b^{2} c^{3} - 3 \, a^{2} b c^{2} d\right )} x}{2 \, a^{3} b x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^3/((b*x + a)*x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.34295, size = 257, normalized size = 3.02 \[ - \frac{a c^{3} + x \left (6 a c^{2} d - 2 b c^{3}\right )}{2 a^{2} x^{2}} + \frac{c \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right ) \log{\left (x + \frac{- 3 a^{3} c d^{2} + 3 a^{2} b c^{2} d - a b^{2} c^{3} + a c \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right )}{a^{3} d^{3} - 6 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{a^{3}} + \frac{\left (a d - b c\right )^{3} \log{\left (x + \frac{- 3 a^{3} c d^{2} + 3 a^{2} b c^{2} d - a b^{2} c^{3} + \frac{a \left (a d - b c\right )^{3}}{b}}{a^{3} d^{3} - 6 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{a^{3} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)**3/x**3/(b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.263854, size = 161, normalized size = 1.89 \[ \frac{{\left (b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} - \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{3} b} - \frac{a^{2} c^{3} - 2 \,{\left (a b c^{3} - 3 \, a^{2} c^{2} d\right )} x}{2 \, a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^3/((b*x + a)*x^3),x, algorithm="giac")
[Out]