Optimal. Leaf size=64 \[ -\frac{(b c-a d)^3 \log (a+b x)}{a b^3}+\frac{d^2 x (3 b c-a d)}{b^2}+\frac{c^3 \log (x)}{a}+\frac{d^3 x^2}{2 b} \]
[Out]
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Rubi [A] time = 0.110603, antiderivative size = 64, 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 \[ -\frac{(b c-a d)^3 \log (a+b x)}{a b^3}+\frac{d^2 x (3 b c-a d)}{b^2}+\frac{c^3 \log (x)}{a}+\frac{d^3 x^2}{2 b} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)^3/(x*(a + b*x)),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - d^{2} \left (a d - 3 b c\right ) \int \frac{1}{b^{2}}\, dx + \frac{d^{3} \int x\, dx}{b} + \frac{c^{3} \log{\left (x \right )}}{a} + \frac{\left (a d - b c\right )^{3} \log{\left (a + b x \right )}}{a b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**3/x/(b*x+a),x)
[Out]
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Mathematica [A] time = 0.044101, size = 59, normalized size = 0.92 \[ \frac{a b d^2 x (-2 a d+6 b c+b d x)-2 (b c-a d)^3 \log (a+b x)+2 b^3 c^3 \log (x)}{2 a b^3} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)^3/(x*(a + b*x)),x]
[Out]
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Maple [A] time = 0.009, size = 103, normalized size = 1.6 \[{\frac{{d}^{3}{x}^{2}}{2\,b}}-{\frac{{d}^{3}ax}{{b}^{2}}}+3\,{\frac{{d}^{2}xc}{b}}+{\frac{{c}^{3}\ln \left ( x \right ) }{a}}+{\frac{{a}^{2}\ln \left ( bx+a \right ){d}^{3}}{{b}^{3}}}-3\,{\frac{a\ln \left ( bx+a \right ) c{d}^{2}}{{b}^{2}}}+3\,{\frac{\ln \left ( bx+a \right ){c}^{2}d}{b}}-{\frac{\ln \left ( bx+a \right ){c}^{3}}{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^3/x/(b*x+a),x)
[Out]
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Maxima [A] time = 1.32911, size = 123, normalized size = 1.92 \[ \frac{c^{3} \log \left (x\right )}{a} + \frac{b d^{3} x^{2} + 2 \,{\left (3 \, b c d^{2} - a d^{3}\right )} x}{2 \, b^{2}} - \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{a b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^3/((b*x + a)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.210761, size = 131, normalized size = 2.05 \[ \frac{a b^{2} d^{3} x^{2} + 2 \, b^{3} c^{3} \log \left (x\right ) + 2 \,{\left (3 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x - 2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{2 \, a b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^3/((b*x + a)*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 7.53387, size = 112, normalized size = 1.75 \[ \frac{d^{3} x^{2}}{2 b} - \frac{x \left (a d^{3} - 3 b c d^{2}\right )}{b^{2}} + \frac{c^{3} \log{\left (x \right )}}{a} + \frac{\left (a d - b c\right )^{3} \log{\left (x + \frac{- a b^{2} c^{3} + \frac{a \left (a d - b c\right )^{3}}{b}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{a b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)**3/x/(b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.281084, size = 123, normalized size = 1.92 \[ \frac{c^{3}{\rm ln}\left ({\left | x \right |}\right )}{a} + \frac{b d^{3} x^{2} + 6 \, b c d^{2} x - 2 \, a d^{3} x}{2 \, b^{2}} - \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{a b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^3/((b*x + a)*x),x, algorithm="giac")
[Out]