Optimal. Leaf size=103 \[ -\frac{b^3 x (3 b c-4 a d)}{d^4}+\frac{6 b^2 (b c-a d)^2 \log (c+d x)}{d^5}+\frac{4 b (b c-a d)^3}{d^5 (c+d x)}-\frac{(b c-a d)^4}{2 d^5 (c+d x)^2}+\frac{b^4 x^2}{2 d^3} \]
[Out]
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Rubi [A] time = 0.230864, antiderivative size = 103, 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 \[ -\frac{b^3 x (3 b c-4 a d)}{d^4}+\frac{6 b^2 (b c-a d)^2 \log (c+d x)}{d^5}+\frac{4 b (b c-a d)^3}{d^5 (c+d x)}-\frac{(b c-a d)^4}{2 d^5 (c+d x)^2}+\frac{b^4 x^2}{2 d^3} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^7/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{b^{4} \int x\, dx}{d^{3}} + \frac{6 b^{2} \left (a d - b c\right )^{2} \log{\left (c + d x \right )}}{d^{5}} - \frac{4 b \left (a d - b c\right )^{3}}{d^{5} \left (c + d x\right )} + \frac{\left (4 a d - 3 b c\right ) \int b^{3}\, dx}{d^{4}} - \frac{\left (a d - b c\right )^{4}}{2 d^{5} \left (c + d x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**7/(a*c+(a*d+b*c)*x+b*d*x**2)**3,x)
[Out]
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Mathematica [A] time = 0.106027, size = 167, normalized size = 1.62 \[ \frac{-a^4 d^4-4 a^3 b d^3 (c+2 d x)+6 a^2 b^2 c d^2 (3 c+4 d x)+4 a b^3 d \left (-5 c^3-4 c^2 d x+4 c d^2 x^2+2 d^3 x^3\right )+12 b^2 (c+d x)^2 (b c-a d)^2 \log (c+d x)+b^4 \left (7 c^4+2 c^3 d x-11 c^2 d^2 x^2-4 c d^3 x^3+d^4 x^4\right )}{2 d^5 (c+d x)^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^7/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]
[Out]
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Maple [B] time = 0.012, size = 245, normalized size = 2.4 \[{\frac{{b}^{4}{x}^{2}}{2\,{d}^{3}}}+4\,{\frac{a{b}^{3}x}{{d}^{3}}}-3\,{\frac{{b}^{4}cx}{{d}^{4}}}-4\,{\frac{{a}^{3}b}{{d}^{2} \left ( dx+c \right ) }}+12\,{\frac{{a}^{2}{b}^{2}c}{{d}^{3} \left ( dx+c \right ) }}-12\,{\frac{a{b}^{3}{c}^{2}}{{d}^{4} \left ( dx+c \right ) }}+4\,{\frac{{b}^{4}{c}^{3}}{{d}^{5} \left ( dx+c \right ) }}+6\,{\frac{{b}^{2}\ln \left ( dx+c \right ){a}^{2}}{{d}^{3}}}-12\,{\frac{{b}^{3}\ln \left ( dx+c \right ) ca}{{d}^{4}}}+6\,{\frac{{b}^{4}\ln \left ( dx+c \right ){c}^{2}}{{d}^{5}}}-{\frac{{a}^{4}}{2\,d \left ( dx+c \right ) ^{2}}}+2\,{\frac{{a}^{3}bc}{{d}^{2} \left ( dx+c \right ) ^{2}}}-3\,{\frac{{a}^{2}{b}^{2}{c}^{2}}{{d}^{3} \left ( dx+c \right ) ^{2}}}+2\,{\frac{a{b}^{3}{c}^{3}}{{d}^{4} \left ( dx+c \right ) ^{2}}}-{\frac{{b}^{4}{c}^{4}}{2\,{d}^{5} \left ( dx+c \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^7/(a*c+(a*d+b*c)*x+x^2*b*d)^3,x)
[Out]
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Maxima [A] time = 0.774883, size = 258, normalized size = 2.5 \[ \frac{7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4} + 8 \,{\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x}{2 \,{\left (d^{7} x^{2} + 2 \, c d^{6} x + c^{2} d^{5}\right )}} + \frac{b^{4} d x^{2} - 2 \,{\left (3 \, b^{4} c - 4 \, a b^{3} d\right )} x}{2 \, d^{4}} + \frac{6 \,{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \log \left (d x + c\right )}{d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^7/(b*d*x^2 + a*c + (b*c + a*d)*x)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.20456, size = 393, normalized size = 3.82 \[ \frac{b^{4} d^{4} x^{4} + 7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4} - 4 \,{\left (b^{4} c d^{3} - 2 \, a b^{3} d^{4}\right )} x^{3} -{\left (11 \, b^{4} c^{2} d^{2} - 16 \, a b^{3} c d^{3}\right )} x^{2} + 2 \,{\left (b^{4} c^{3} d - 8 \, a b^{3} c^{2} d^{2} + 12 \, a^{2} b^{2} c d^{3} - 4 \, a^{3} b d^{4}\right )} x + 12 \,{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + a^{2} b^{2} c^{2} d^{2} +{\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} + 2 \,{\left (b^{4} c^{3} d - 2 \, a b^{3} c^{2} d^{2} + a^{2} b^{2} c d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \,{\left (d^{7} x^{2} + 2 \, c d^{6} x + c^{2} d^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^7/(b*d*x^2 + a*c + (b*c + a*d)*x)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.98582, size = 184, normalized size = 1.79 \[ \frac{b^{4} x^{2}}{2 d^{3}} + \frac{6 b^{2} \left (a d - b c\right )^{2} \log{\left (c + d x \right )}}{d^{5}} - \frac{a^{4} d^{4} + 4 a^{3} b c d^{3} - 18 a^{2} b^{2} c^{2} d^{2} + 20 a b^{3} c^{3} d - 7 b^{4} c^{4} + x \left (8 a^{3} b d^{4} - 24 a^{2} b^{2} c d^{3} + 24 a b^{3} c^{2} d^{2} - 8 b^{4} c^{3} d\right )}{2 c^{2} d^{5} + 4 c d^{6} x + 2 d^{7} x^{2}} + \frac{x \left (4 a b^{3} d - 3 b^{4} c\right )}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**7/(a*c+(a*d+b*c)*x+b*d*x**2)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.216525, size = 247, normalized size = 2.4 \[ \frac{6 \,{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )}{\rm ln}\left ({\left | d x + c \right |}\right )}{d^{5}} + \frac{b^{4} d^{3} x^{2} - 6 \, b^{4} c d^{2} x + 8 \, a b^{3} d^{3} x}{2 \, d^{6}} + \frac{7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4} + 8 \,{\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x}{2 \,{\left (d x + c\right )}^{2} d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^7/(b*d*x^2 + a*c + (b*c + a*d)*x)^3,x, algorithm="giac")
[Out]