Optimal. Leaf size=73 \[ \frac{(b c-a d)^3 \log (a+b x)}{b^4}+\frac{d x (b c-a d)^2}{b^3}+\frac{(c+d x)^2 (b c-a d)}{2 b^2}+\frac{(c+d x)^3}{3 b} \]
[Out]
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Rubi [A] time = 0.0984185, antiderivative size = 73, 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 \[ \frac{(b c-a d)^3 \log (a+b x)}{b^4}+\frac{d x (b c-a d)^2}{b^3}+\frac{(c+d x)^2 (b c-a d)}{2 b^2}+\frac{(c+d x)^3}{3 b} \]
Antiderivative was successfully verified.
[In] Int[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^4,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\left (c + d x\right )^{3}}{3 b} - \frac{\left (c + d x\right )^{2} \left (a d - b c\right )}{2 b^{2}} + \frac{\left (a d - b c\right )^{2} \int d\, dx}{b^{3}} - \frac{\left (a d - b c\right )^{3} \log{\left (a + b x \right )}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a*c+(a*d+b*c)*x+b*d*x**2)**3/(b*x+a)**4,x)
[Out]
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Mathematica [A] time = 0.0549696, size = 74, normalized size = 1.01 \[ \frac{b d x \left (6 a^2 d^2-3 a b d (6 c+d x)+b^2 \left (18 c^2+9 c d x+2 d^2 x^2\right )\right )+6 (b c-a d)^3 \log (a+b x)}{6 b^4} \]
Antiderivative was successfully verified.
[In] Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^4,x]
[Out]
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Maple [A] time = 0.006, size = 133, normalized size = 1.8 \[{\frac{{x}^{3}{d}^{3}}{3\,b}}-{\frac{{d}^{3}{x}^{2}a}{2\,{b}^{2}}}+{\frac{3\,c{d}^{2}{x}^{2}}{2\,b}}+{\frac{{a}^{2}{d}^{3}x}{{b}^{3}}}-3\,{\frac{ac{d}^{2}x}{{b}^{2}}}+3\,{\frac{x{c}^{2}d}{b}}-{\frac{\ln \left ( bx+a \right ){a}^{3}{d}^{3}}{{b}^{4}}}+3\,{\frac{\ln \left ( bx+a \right ){a}^{2}c{d}^{2}}{{b}^{3}}}-3\,{\frac{\ln \left ( bx+a \right ) a{c}^{2}d}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ){c}^{3}}{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a*c+(a*d+b*c)*x+x^2*b*d)^3/(b*x+a)^4,x)
[Out]
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Maxima [A] time = 0.752053, size = 154, normalized size = 2.11 \[ \frac{2 \, b^{2} d^{3} x^{3} + 3 \,{\left (3 \, b^{2} c d^{2} - a b d^{3}\right )} x^{2} + 6 \,{\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} x}{6 \, b^{3}} + \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^3/(b*x + a)^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.206661, size = 157, normalized size = 2.15 \[ \frac{2 \, b^{3} d^{3} x^{3} + 3 \,{\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 6 \,{\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x + 6 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{6 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^3/(b*x + a)^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.00857, size = 82, normalized size = 1.12 \[ \frac{d^{3} x^{3}}{3 b} - \frac{x^{2} \left (a d^{3} - 3 b c d^{2}\right )}{2 b^{2}} + \frac{x \left (a^{2} d^{3} - 3 a b c d^{2} + 3 b^{2} c^{2} d\right )}{b^{3}} - \frac{\left (a d - b c\right )^{3} \log{\left (a + b x \right )}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*c+(a*d+b*c)*x+b*d*x**2)**3/(b*x+a)**4,x)
[Out]
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GIAC/XCAS [A] time = 0.211686, size = 155, normalized size = 2.12 \[ \frac{2 \, b^{2} d^{3} x^{3} + 9 \, b^{2} c d^{2} x^{2} - 3 \, a b d^{3} x^{2} + 18 \, b^{2} c^{2} d x - 18 \, a b c d^{2} x + 6 \, a^{2} d^{3} x}{6 \, b^{3}} + \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^3/(b*x + a)^4,x, algorithm="giac")
[Out]