Optimal. Leaf size=228 \[ -\frac{3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{2 e^7 (d+e x)^2}+\frac{(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{3 e^7 (d+e x)^3}-\frac{3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{4 e^7 (d+e x)^4}+\frac{3 c^2 (2 c d-b e)}{e^7 (d+e x)}-\frac{d^3 (c d-b e)^3}{6 e^7 (d+e x)^6}+\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{5 e^7 (d+e x)^5}+\frac{c^3 \log (d+e x)}{e^7} \]
[Out]
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Rubi [A] time = 0.514409, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{2 e^7 (d+e x)^2}+\frac{(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{3 e^7 (d+e x)^3}-\frac{3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{4 e^7 (d+e x)^4}+\frac{3 c^2 (2 c d-b e)}{e^7 (d+e x)}-\frac{d^3 (c d-b e)^3}{6 e^7 (d+e x)^6}+\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{5 e^7 (d+e x)^5}+\frac{c^3 \log (d+e x)}{e^7} \]
Antiderivative was successfully verified.
[In] Int[(b*x + c*x^2)^3/(d + e*x)^7,x]
[Out]
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Rubi in Sympy [A] time = 68.6195, size = 221, normalized size = 0.97 \[ \frac{c^{3} \log{\left (d + e x \right )}}{e^{7}} - \frac{3 c^{2} \left (b e - 2 c d\right )}{e^{7} \left (d + e x\right )} - \frac{3 c \left (b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{2 e^{7} \left (d + e x\right )^{2}} + \frac{d^{3} \left (b e - c d\right )^{3}}{6 e^{7} \left (d + e x\right )^{6}} - \frac{3 d^{2} \left (b e - 2 c d\right ) \left (b e - c d\right )^{2}}{5 e^{7} \left (d + e x\right )^{5}} + \frac{3 d \left (b e - c d\right ) \left (b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{4 e^{7} \left (d + e x\right )^{4}} - \frac{\left (b e - 2 c d\right ) \left (b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{3 e^{7} \left (d + e x\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x)**3/(e*x+d)**7,x)
[Out]
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Mathematica [A] time = 0.154949, size = 231, normalized size = 1.01 \[ \frac{-b^3 e^3 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )-6 b^2 c e^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )-30 b c^2 e \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )+c^3 d \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )+60 c^3 (d+e x)^6 \log (d+e x)}{60 e^7 (d+e x)^6} \]
Antiderivative was successfully verified.
[In] Integrate[(b*x + c*x^2)^3/(d + e*x)^7,x]
[Out]
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Maple [A] time = 0.013, size = 387, normalized size = 1.7 \[ -{\frac{3\,{b}^{3}{d}^{2}}{5\,{e}^{4} \left ( ex+d \right ) ^{5}}}+{\frac{12\,{d}^{3}{b}^{2}c}{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}-3\,{\frac{{d}^{4}b{c}^{2}}{{e}^{6} \left ( ex+d \right ) ^{5}}}+{\frac{6\,{c}^{3}{d}^{5}}{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}-{\frac{3\,{b}^{2}c}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}+{\frac{15\,db{c}^{2}}{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}-{\frac{15\,{d}^{2}{c}^{3}}{2\,{e}^{7} \left ( ex+d \right ) ^{2}}}+{\frac{{c}^{3}\ln \left ( ex+d \right ) }{{e}^{7}}}-{\frac{{b}^{3}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}+4\,{\frac{d{b}^{2}c}{{e}^{5} \left ( ex+d \right ) ^{3}}}-10\,{\frac{{d}^{2}b{c}^{2}}{{e}^{6} \left ( ex+d \right ) ^{3}}}+{\frac{20\,{c}^{3}{d}^{3}}{3\,{e}^{7} \left ( ex+d \right ) ^{3}}}+{\frac{{b}^{3}{d}^{3}}{6\,{e}^{4} \left ( ex+d \right ) ^{6}}}-{\frac{{d}^{4}{b}^{2}c}{2\,{e}^{5} \left ( ex+d \right ) ^{6}}}+{\frac{b{c}^{2}{d}^{5}}{2\,{e}^{6} \left ( ex+d \right ) ^{6}}}-{\frac{{c}^{3}{d}^{6}}{6\,{e}^{7} \left ( ex+d \right ) ^{6}}}-3\,{\frac{b{c}^{2}}{{e}^{6} \left ( ex+d \right ) }}+6\,{\frac{d{c}^{3}}{{e}^{7} \left ( ex+d \right ) }}+{\frac{3\,{b}^{3}d}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{9\,{d}^{2}{b}^{2}c}{2\,{e}^{5} \left ( ex+d \right ) ^{4}}}+{\frac{15\,{d}^{3}b{c}^{2}}{2\,{e}^{6} \left ( ex+d \right ) ^{4}}}-{\frac{15\,{d}^{4}{c}^{3}}{4\,{e}^{7} \left ( ex+d \right ) ^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x)^3/(e*x+d)^7,x)
[Out]
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Maxima [A] time = 0.719314, size = 447, normalized size = 1.96 \[ \frac{147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} + 180 \,{\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 90 \,{\left (15 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} - b^{2} c e^{6}\right )} x^{4} + 20 \,{\left (110 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} - 6 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} x^{3} + 15 \,{\left (125 \, c^{3} d^{4} e^{2} - 30 \, b c^{2} d^{3} e^{3} - 6 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 6 \,{\left (137 \, c^{3} d^{5} e - 30 \, b c^{2} d^{4} e^{2} - 6 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x}{60 \,{\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} + \frac{c^{3} \log \left (e x + d\right )}{e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^3/(e*x + d)^7,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.221694, size = 549, normalized size = 2.41 \[ \frac{147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} + 180 \,{\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 90 \,{\left (15 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} - b^{2} c e^{6}\right )} x^{4} + 20 \,{\left (110 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} - 6 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} x^{3} + 15 \,{\left (125 \, c^{3} d^{4} e^{2} - 30 \, b c^{2} d^{3} e^{3} - 6 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 6 \,{\left (137 \, c^{3} d^{5} e - 30 \, b c^{2} d^{4} e^{2} - 6 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x + 60 \,{\left (c^{3} e^{6} x^{6} + 6 \, c^{3} d e^{5} x^{5} + 15 \, c^{3} d^{2} e^{4} x^{4} + 20 \, c^{3} d^{3} e^{3} x^{3} + 15 \, c^{3} d^{4} e^{2} x^{2} + 6 \, c^{3} d^{5} e x + c^{3} d^{6}\right )} \log \left (e x + d\right )}{60 \,{\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^3/(e*x + d)^7,x, algorithm="fricas")
[Out]
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Sympy [A] time = 88.5041, size = 343, normalized size = 1.5 \[ \frac{c^{3} \log{\left (d + e x \right )}}{e^{7}} - \frac{b^{3} d^{3} e^{3} + 6 b^{2} c d^{4} e^{2} + 30 b c^{2} d^{5} e - 147 c^{3} d^{6} + x^{5} \left (180 b c^{2} e^{6} - 360 c^{3} d e^{5}\right ) + x^{4} \left (90 b^{2} c e^{6} + 450 b c^{2} d e^{5} - 1350 c^{3} d^{2} e^{4}\right ) + x^{3} \left (20 b^{3} e^{6} + 120 b^{2} c d e^{5} + 600 b c^{2} d^{2} e^{4} - 2200 c^{3} d^{3} e^{3}\right ) + x^{2} \left (15 b^{3} d e^{5} + 90 b^{2} c d^{2} e^{4} + 450 b c^{2} d^{3} e^{3} - 1875 c^{3} d^{4} e^{2}\right ) + x \left (6 b^{3} d^{2} e^{4} + 36 b^{2} c d^{3} e^{3} + 180 b c^{2} d^{4} e^{2} - 822 c^{3} d^{5} e\right )}{60 d^{6} e^{7} + 360 d^{5} e^{8} x + 900 d^{4} e^{9} x^{2} + 1200 d^{3} e^{10} x^{3} + 900 d^{2} e^{11} x^{4} + 360 d e^{12} x^{5} + 60 e^{13} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x)**3/(e*x+d)**7,x)
[Out]
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GIAC/XCAS [A] time = 0.209338, size = 351, normalized size = 1.54 \[ c^{3} e^{\left (-7\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{{\left (180 \,{\left (2 \, c^{3} d e^{4} - b c^{2} e^{5}\right )} x^{5} + 90 \,{\left (15 \, c^{3} d^{2} e^{3} - 5 \, b c^{2} d e^{4} - b^{2} c e^{5}\right )} x^{4} + 20 \,{\left (110 \, c^{3} d^{3} e^{2} - 30 \, b c^{2} d^{2} e^{3} - 6 \, b^{2} c d e^{4} - b^{3} e^{5}\right )} x^{3} + 15 \,{\left (125 \, c^{3} d^{4} e - 30 \, b c^{2} d^{3} e^{2} - 6 \, b^{2} c d^{2} e^{3} - b^{3} d e^{4}\right )} x^{2} + 6 \,{\left (137 \, c^{3} d^{5} - 30 \, b c^{2} d^{4} e - 6 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )} x +{\left (147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} e^{\left (-1\right )}\right )} e^{\left (-6\right )}}{60 \,{\left (x e + d\right )}^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^3/(e*x + d)^7,x, algorithm="giac")
[Out]