Optimal. Leaf size=104 \[ -\frac{2 b^3 (c+d x)^2 (b c-a d)}{d^5}+\frac{6 b^2 x (b c-a d)^2}{d^4}-\frac{(b c-a d)^4}{d^5 (c+d x)}-\frac{4 b (b c-a d)^3 \log (c+d x)}{d^5}+\frac{b^4 (c+d x)^3}{3 d^5} \]
[Out]
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Rubi [A] time = 0.237281, antiderivative size = 104, 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 \[ -\frac{2 b^3 (c+d x)^2 (b c-a d)}{d^5}+\frac{6 b^2 x (b c-a d)^2}{d^4}-\frac{(b c-a d)^4}{d^5 (c+d x)}-\frac{4 b (b c-a d)^3 \log (c+d x)}{d^5}+\frac{b^4 (c+d x)^3}{3 d^5} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^6/(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 37.3211, size = 94, normalized size = 0.9 \[ \frac{b^{4} \left (c + d x\right )^{3}}{3 d^{5}} + \frac{2 b^{3} \left (c + d x\right )^{2} \left (a d - b c\right )}{d^{5}} + \frac{6 b^{2} x \left (a d - b c\right )^{2}}{d^{4}} + \frac{4 b \left (a d - b c\right )^{3} \log{\left (c + d x \right )}}{d^{5}} - \frac{\left (a d - b c\right )^{4}}{d^{5} \left (c + d x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**6/(a*c+(a*d+b*c)*x+b*d*x**2)**2,x)
[Out]
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Mathematica [A] time = 0.112073, size = 165, normalized size = 1.59 \[ \frac{-3 a^4 d^4+12 a^3 b c d^3+18 a^2 b^2 d^2 \left (-c^2+c d x+d^2 x^2\right )+6 a b^3 d \left (2 c^3-4 c^2 d x-3 c d^2 x^2+d^3 x^3\right )-12 b (c+d x) (b c-a d)^3 \log (c+d x)+b^4 \left (-3 c^4+9 c^3 d x+6 c^2 d^2 x^2-2 c d^3 x^3+d^4 x^4\right )}{3 d^5 (c+d x)} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^6/(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]
[Out]
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Maple [B] time = 0.012, size = 230, normalized size = 2.2 \[{\frac{{b}^{4}{x}^{3}}{3\,{d}^{2}}}+2\,{\frac{{b}^{3}{x}^{2}a}{{d}^{2}}}-{\frac{{b}^{4}{x}^{2}c}{{d}^{3}}}+6\,{\frac{{a}^{2}{b}^{2}x}{{d}^{2}}}-8\,{\frac{ac{b}^{3}x}{{d}^{3}}}+3\,{\frac{{b}^{4}{c}^{2}x}{{d}^{4}}}-{\frac{{a}^{4}}{d \left ( dx+c \right ) }}+4\,{\frac{{a}^{3}bc}{{d}^{2} \left ( dx+c \right ) }}-6\,{\frac{{a}^{2}{b}^{2}{c}^{2}}{{d}^{3} \left ( dx+c \right ) }}+4\,{\frac{a{b}^{3}{c}^{3}}{{d}^{4} \left ( dx+c \right ) }}-{\frac{{b}^{4}{c}^{4}}{{d}^{5} \left ( dx+c \right ) }}+4\,{\frac{b\ln \left ( dx+c \right ){a}^{3}}{{d}^{2}}}-12\,{\frac{{b}^{2}\ln \left ( dx+c \right ){a}^{2}c}{{d}^{3}}}+12\,{\frac{{b}^{3}\ln \left ( dx+c \right ) a{c}^{2}}{{d}^{4}}}-4\,{\frac{{b}^{4}\ln \left ( dx+c \right ){c}^{3}}{{d}^{5}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^6/(a*c+(a*d+b*c)*x+x^2*b*d)^2,x)
[Out]
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Maxima [A] time = 0.752315, size = 247, normalized size = 2.38 \[ -\frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{d^{6} x + c d^{5}} + \frac{b^{4} d^{2} x^{3} - 3 \,{\left (b^{4} c d - 2 \, a b^{3} d^{2}\right )} x^{2} + 3 \,{\left (3 \, b^{4} c^{2} - 8 \, a b^{3} c d + 6 \, a^{2} b^{2} d^{2}\right )} x}{3 \, d^{4}} - \frac{4 \,{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \log \left (d x + c\right )}{d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^6/(b*d*x^2 + a*c + (b*c + a*d)*x)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.202796, size = 360, normalized size = 3.46 \[ \frac{b^{4} d^{4} x^{4} - 3 \, b^{4} c^{4} + 12 \, a b^{3} c^{3} d - 18 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} - 3 \, a^{4} d^{4} - 2 \,{\left (b^{4} c d^{3} - 3 \, a b^{3} d^{4}\right )} x^{3} + 6 \,{\left (b^{4} c^{2} d^{2} - 3 \, a b^{3} c d^{3} + 3 \, a^{2} b^{2} d^{4}\right )} x^{2} + 3 \,{\left (3 \, b^{4} c^{3} d - 8 \, a b^{3} c^{2} d^{2} + 6 \, a^{2} b^{2} c d^{3}\right )} x - 12 \,{\left (b^{4} c^{4} - 3 \, a b^{3} c^{3} d + 3 \, a^{2} b^{2} c^{2} d^{2} - a^{3} b c d^{3} +{\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x\right )} \log \left (d x + c\right )}{3 \,{\left (d^{6} x + c d^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^6/(b*d*x^2 + a*c + (b*c + a*d)*x)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.77771, size = 151, normalized size = 1.45 \[ \frac{b^{4} x^{3}}{3 d^{2}} + \frac{4 b \left (a d - b c\right )^{3} \log{\left (c + d x \right )}}{d^{5}} - \frac{a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}}{c d^{5} + d^{6} x} + \frac{x^{2} \left (2 a b^{3} d - b^{4} c\right )}{d^{3}} + \frac{x \left (6 a^{2} b^{2} d^{2} - 8 a b^{3} c d + 3 b^{4} c^{2}\right )}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**6/(a*c+(a*d+b*c)*x+b*d*x**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.213154, size = 254, normalized size = 2.44 \[ -\frac{4 \,{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )}{\rm ln}\left ({\left | d x + c \right |}\right )}{d^{5}} + \frac{b^{4} d^{4} x^{3} - 3 \, b^{4} c d^{3} x^{2} + 6 \, a b^{3} d^{4} x^{2} + 9 \, b^{4} c^{2} d^{2} x - 24 \, a b^{3} c d^{3} x + 18 \, a^{2} b^{2} d^{4} x}{3 \, d^{6}} - \frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{{\left (d x + c\right )} d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^6/(b*d*x^2 + a*c + (b*c + a*d)*x)^2,x, algorithm="giac")
[Out]