Optimal. Leaf size=156 \[ -\frac{a^2 (b c-a d)^3}{2 b^6 (a+b x)^2}+\frac{\left (10 a^2 d^2-8 a b c d+b^2 c^2\right ) (b c-a d) \log (a+b x)}{b^6}+\frac{a (2 b c-5 a d) (b c-a d)^2}{b^6 (a+b x)}+\frac{3 d x (b c-2 a d) (b c-a d)}{b^5}+\frac{3 d^2 x^2 (b c-a d)}{2 b^4}+\frac{d^3 x^3}{3 b^3} \]
[Out]
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Rubi [A] time = 0.368462, antiderivative size = 156, 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 \[ -\frac{a^2 (b c-a d)^3}{2 b^6 (a+b x)^2}+\frac{\left (10 a^2 d^2-8 a b c d+b^2 c^2\right ) (b c-a d) \log (a+b x)}{b^6}+\frac{a (2 b c-5 a d) (b c-a d)^2}{b^6 (a+b x)}+\frac{3 d x (b c-2 a d) (b c-a d)}{b^5}+\frac{3 d^2 x^2 (b c-a d)}{2 b^4}+\frac{d^3 x^3}{3 b^3} \]
Antiderivative was successfully verified.
[In] Int[(x^2*(c + d*x)^3)/(a + b*x)^3,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{a^{2} \left (a d - b c\right )^{3}}{2 b^{6} \left (a + b x\right )^{2}} - \frac{a \left (a d - b c\right )^{2} \left (5 a d - 2 b c\right )}{b^{6} \left (a + b x\right )} + \frac{d^{3} x^{3}}{3 b^{3}} - \frac{3 d^{2} \left (a d - b c\right ) \int x\, dx}{b^{4}} + \frac{3 d x \left (a d - b c\right ) \left (2 a d - b c\right )}{b^{5}} - \frac{\left (a d - b c\right ) \left (10 a^{2} d^{2} - 8 a b c d + b^{2} c^{2}\right ) \log{\left (a + b x \right )}}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(d*x+c)**3/(b*x+a)**3,x)
[Out]
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Mathematica [A] time = 0.145483, size = 160, normalized size = 1.03 \[ \frac{18 b d x \left (2 a^2 d^2-3 a b c d+b^2 c^2\right )+\frac{3 a^2 (a d-b c)^3}{(a+b x)^2}+6 \left (-10 a^3 d^3+18 a^2 b c d^2-9 a b^2 c^2 d+b^3 c^3\right ) \log (a+b x)+9 b^2 d^2 x^2 (b c-a d)-\frac{6 a (b c-a d)^2 (5 a d-2 b c)}{a+b x}+2 b^3 d^3 x^3}{6 b^6} \]
Antiderivative was successfully verified.
[In] Integrate[(x^2*(c + d*x)^3)/(a + b*x)^3,x]
[Out]
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Maple [A] time = 0.014, size = 280, normalized size = 1.8 \[{\frac{{d}^{3}{x}^{3}}{3\,{b}^{3}}}-{\frac{3\,{d}^{3}{x}^{2}a}{2\,{b}^{4}}}+{\frac{3\,{d}^{2}{x}^{2}c}{2\,{b}^{3}}}+6\,{\frac{{a}^{2}{d}^{3}x}{{b}^{5}}}-9\,{\frac{ac{d}^{2}x}{{b}^{4}}}+3\,{\frac{{c}^{2}dx}{{b}^{3}}}-10\,{\frac{\ln \left ( bx+a \right ){a}^{3}{d}^{3}}{{b}^{6}}}+18\,{\frac{\ln \left ( bx+a \right ){a}^{2}c{d}^{2}}{{b}^{5}}}-9\,{\frac{\ln \left ( bx+a \right ) a{c}^{2}d}{{b}^{4}}}+{\frac{\ln \left ( bx+a \right ){c}^{3}}{{b}^{3}}}-5\,{\frac{{a}^{4}{d}^{3}}{{b}^{6} \left ( bx+a \right ) }}+12\,{\frac{{a}^{3}c{d}^{2}}{{b}^{5} \left ( bx+a \right ) }}-9\,{\frac{{a}^{2}{c}^{2}d}{{b}^{4} \left ( bx+a \right ) }}+2\,{\frac{a{c}^{3}}{{b}^{3} \left ( bx+a \right ) }}+{\frac{{a}^{5}{d}^{3}}{2\,{b}^{6} \left ( bx+a \right ) ^{2}}}-{\frac{3\,{a}^{4}c{d}^{2}}{2\,{b}^{5} \left ( bx+a \right ) ^{2}}}+{\frac{3\,{a}^{3}{c}^{2}d}{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}-{\frac{{a}^{2}{c}^{3}}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(d*x+c)^3/(b*x+a)^3,x)
[Out]
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Maxima [A] time = 1.36273, size = 306, normalized size = 1.96 \[ \frac{3 \, a^{2} b^{3} c^{3} - 15 \, a^{3} b^{2} c^{2} d + 21 \, a^{4} b c d^{2} - 9 \, a^{5} d^{3} + 2 \,{\left (2 \, a b^{4} c^{3} - 9 \, a^{2} b^{3} c^{2} d + 12 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x}{2 \,{\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} + \frac{2 \, b^{2} d^{3} x^{3} + 9 \,{\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2} + 18 \,{\left (b^{2} c^{2} d - 3 \, a b c d^{2} + 2 \, a^{2} d^{3}\right )} x}{6 \, b^{5}} + \frac{{\left (b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 10 \, a^{3} d^{3}\right )} \log \left (b x + a\right )}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^3*x^2/(b*x + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.209433, size = 487, normalized size = 3.12 \[ \frac{2 \, b^{5} d^{3} x^{5} + 9 \, a^{2} b^{3} c^{3} - 45 \, a^{3} b^{2} c^{2} d + 63 \, a^{4} b c d^{2} - 27 \, a^{5} d^{3} +{\left (9 \, b^{5} c d^{2} - 5 \, a b^{4} d^{3}\right )} x^{4} + 2 \,{\left (9 \, b^{5} c^{2} d - 18 \, a b^{4} c d^{2} + 10 \, a^{2} b^{3} d^{3}\right )} x^{3} + 9 \,{\left (4 \, a b^{4} c^{2} d - 11 \, a^{2} b^{3} c d^{2} + 7 \, a^{3} b^{2} d^{3}\right )} x^{2} + 6 \,{\left (2 \, a b^{4} c^{3} - 6 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} + a^{4} b d^{3}\right )} x + 6 \,{\left (a^{2} b^{3} c^{3} - 9 \, a^{3} b^{2} c^{2} d + 18 \, a^{4} b c d^{2} - 10 \, a^{5} d^{3} +{\left (b^{5} c^{3} - 9 \, a b^{4} c^{2} d + 18 \, a^{2} b^{3} c d^{2} - 10 \, a^{3} b^{2} d^{3}\right )} x^{2} + 2 \,{\left (a b^{4} c^{3} - 9 \, a^{2} b^{3} c^{2} d + 18 \, a^{3} b^{2} c d^{2} - 10 \, a^{4} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{6 \,{\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^3*x^2/(b*x + a)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 9.42366, size = 230, normalized size = 1.47 \[ - \frac{9 a^{5} d^{3} - 21 a^{4} b c d^{2} + 15 a^{3} b^{2} c^{2} d - 3 a^{2} b^{3} c^{3} + x \left (10 a^{4} b d^{3} - 24 a^{3} b^{2} c d^{2} + 18 a^{2} b^{3} c^{2} d - 4 a b^{4} c^{3}\right )}{2 a^{2} b^{6} + 4 a b^{7} x + 2 b^{8} x^{2}} + \frac{d^{3} x^{3}}{3 b^{3}} - \frac{x^{2} \left (3 a d^{3} - 3 b c d^{2}\right )}{2 b^{4}} + \frac{x \left (6 a^{2} d^{3} - 9 a b c d^{2} + 3 b^{2} c^{2} d\right )}{b^{5}} - \frac{\left (a d - b c\right ) \left (10 a^{2} d^{2} - 8 a b c d + b^{2} c^{2}\right ) \log{\left (a + b x \right )}}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(d*x+c)**3/(b*x+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.276668, size = 300, normalized size = 1.92 \[ \frac{{\left (b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 10 \, a^{3} d^{3}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{6}} + \frac{3 \, a^{2} b^{3} c^{3} - 15 \, a^{3} b^{2} c^{2} d + 21 \, a^{4} b c d^{2} - 9 \, a^{5} d^{3} + 2 \,{\left (2 \, a b^{4} c^{3} - 9 \, a^{2} b^{3} c^{2} d + 12 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x}{2 \,{\left (b x + a\right )}^{2} b^{6}} + \frac{2 \, b^{6} d^{3} x^{3} + 9 \, b^{6} c d^{2} x^{2} - 9 \, a b^{5} d^{3} x^{2} + 18 \, b^{6} c^{2} d x - 54 \, a b^{5} c d^{2} x + 36 \, a^{2} b^{4} d^{3} x}{6 \, b^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^3*x^2/(b*x + a)^3,x, algorithm="giac")
[Out]