Optimal. Leaf size=78 \[ \frac{3 d^2 (b c-a d) \log (a+b x)}{b^4}-\frac{3 d (b c-a d)^2}{b^4 (a+b x)}-\frac{(b c-a d)^3}{2 b^4 (a+b x)^2}+\frac{d^3 x}{b^3} \]
[Out]
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Rubi [A] time = 0.152647, antiderivative size = 78, 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 \[ \frac{3 d^2 (b c-a d) \log (a+b x)}{b^4}-\frac{3 d (b c-a d)^2}{b^4 (a+b x)}-\frac{(b c-a d)^3}{2 b^4 (a+b x)^2}+\frac{d^3 x}{b^3} \]
Antiderivative was successfully verified.
[In] Int[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^6,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ d^{3} \int \frac{1}{b^{3}}\, dx - \frac{3 d^{2} \left (a d - b c\right ) \log{\left (a + b x \right )}}{b^{4}} - \frac{3 d \left (a d - b c\right )^{2}}{b^{4} \left (a + b x\right )} + \frac{\left (a d - b c\right )^{3}}{2 b^{4} \left (a + b x\right )^{2}} \]
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)**6,x)
[Out]
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Mathematica [A] time = 0.0765707, size = 114, normalized size = 1.46 \[ \frac{-5 a^3 d^3+a^2 b d^2 (9 c-4 d x)+a b^2 d \left (-3 c^2+12 c d x+4 d^2 x^2\right )-6 d^2 (a+b x)^2 (a d-b c) \log (a+b x)+b^3 \left (-\left (c^3+6 c^2 d x-2 d^3 x^3\right )\right )}{2 b^4 (a+b x)^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^6,x]
[Out]
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Maple [B] time = 0.011, size = 160, normalized size = 2.1 \[{\frac{{d}^{3}x}{{b}^{3}}}+{\frac{{a}^{3}{d}^{3}}{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}-{\frac{3\,{a}^{2}c{d}^{2}}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}}+{\frac{3\,a{c}^{2}d}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{{c}^{3}}{2\,b \left ( bx+a \right ) ^{2}}}-3\,{\frac{{d}^{3}\ln \left ( bx+a \right ) a}{{b}^{4}}}+3\,{\frac{{d}^{2}\ln \left ( bx+a \right ) c}{{b}^{3}}}-3\,{\frac{{a}^{2}{d}^{3}}{{b}^{4} \left ( bx+a \right ) }}+6\,{\frac{ac{d}^{2}}{{b}^{3} \left ( bx+a \right ) }}-3\,{\frac{{c}^{2}d}{{b}^{2} \left ( bx+a \right ) }} \]
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)^6,x)
[Out]
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Maxima [A] time = 0.763691, size = 169, normalized size = 2.17 \[ \frac{d^{3} x}{b^{3}} - \frac{b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 9 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3} + 6 \,{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{2 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} + \frac{3 \,{\left (b c d^{2} - a d^{3}\right )} \log \left (b x + a\right )}{b^{4}} \]
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)^6,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.219929, size = 254, normalized size = 3.26 \[ \frac{2 \, b^{3} d^{3} x^{3} + 4 \, a b^{2} d^{3} x^{2} - b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3} - 2 \,{\left (3 \, b^{3} c^{2} d - 6 \, a b^{2} c d^{2} + 2 \, a^{2} b d^{3}\right )} x + 6 \,{\left (a^{2} b c d^{2} - a^{3} d^{3} +{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 2 \,{\left (a b^{2} c d^{2} - a^{2} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{2 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} \]
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)^6,x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.2713, size = 128, normalized size = 1.64 \[ - \frac{5 a^{3} d^{3} - 9 a^{2} b c d^{2} + 3 a b^{2} c^{2} d + b^{3} c^{3} + x \left (6 a^{2} b d^{3} - 12 a b^{2} c d^{2} + 6 b^{3} c^{2} d\right )}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} + \frac{d^{3} x}{b^{3}} - \frac{3 d^{2} \left (a d - b c\right ) \log{\left (a + b x \right )}}{b^{4}} \]
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)**6,x)
[Out]
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GIAC/XCAS [A] time = 0.212182, size = 151, normalized size = 1.94 \[ \frac{d^{3} x}{b^{3}} + \frac{3 \,{\left (b c d^{2} - a d^{3}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{4}} - \frac{b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 9 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3} + 6 \,{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{2 \,{\left (b x + a\right )}^{2} b^{4}} \]
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)^6,x, algorithm="giac")
[Out]