Optimal. Leaf size=231 \[ -\frac{3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{4 e^7 (d+e x)^4}+\frac{(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{5 e^7 (d+e x)^5}-\frac{d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{2 e^7 (d+e x)^6}+\frac{c^2 (2 c d-b e)}{e^7 (d+e x)^3}-\frac{d^3 (c d-b e)^3}{8 e^7 (d+e x)^8}+\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{7 e^7 (d+e x)^7}-\frac{c^3}{2 e^7 (d+e x)^2} \]
[Out]
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Rubi [A] time = 0.529349, antiderivative size = 231, 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 \[ -\frac{3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{4 e^7 (d+e x)^4}+\frac{(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{5 e^7 (d+e x)^5}-\frac{d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{2 e^7 (d+e x)^6}+\frac{c^2 (2 c d-b e)}{e^7 (d+e x)^3}-\frac{d^3 (c d-b e)^3}{8 e^7 (d+e x)^8}+\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{7 e^7 (d+e x)^7}-\frac{c^3}{2 e^7 (d+e x)^2} \]
Antiderivative was successfully verified.
[In] Int[(b*x + c*x^2)^3/(d + e*x)^9,x]
[Out]
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Rubi in Sympy [A] time = 73.9436, size = 221, normalized size = 0.96 \[ - \frac{c^{3}}{2 e^{7} \left (d + e x\right )^{2}} - \frac{c^{2} \left (b e - 2 c d\right )}{e^{7} \left (d + e x\right )^{3}} - \frac{3 c \left (b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{4 e^{7} \left (d + e x\right )^{4}} + \frac{d^{3} \left (b e - c d\right )^{3}}{8 e^{7} \left (d + e x\right )^{8}} - \frac{3 d^{2} \left (b e - 2 c d\right ) \left (b e - c d\right )^{2}}{7 e^{7} \left (d + e x\right )^{7}} + \frac{d \left (b e - c d\right ) \left (b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{2 e^{7} \left (d + e x\right )^{6}} - \frac{\left (b e - 2 c d\right ) \left (b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{5 e^{7} \left (d + e x\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x)**3/(e*x+d)**9,x)
[Out]
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Mathematica [A] time = 0.15458, size = 221, normalized size = 0.96 \[ -\frac{b^3 e^3 \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+3 b^2 c e^2 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+5 b c^2 e \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )+5 c^3 \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )}{280 e^7 (d+e x)^8} \]
Antiderivative was successfully verified.
[In] Integrate[(b*x + c*x^2)^3/(d + e*x)^9,x]
[Out]
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Maple [A] time = 0.011, size = 274, normalized size = 1.2 \[{\frac{{d}^{3} \left ({b}^{3}{e}^{3}-3\,{b}^{2}cd{e}^{2}+3\,b{c}^{2}{d}^{2}e-{c}^{3}{d}^{3} \right ) }{8\,{e}^{7} \left ( ex+d \right ) ^{8}}}-{\frac{3\,{d}^{2} \left ({b}^{3}{e}^{3}-4\,{b}^{2}cd{e}^{2}+5\,b{c}^{2}{d}^{2}e-2\,{c}^{3}{d}^{3} \right ) }{7\,{e}^{7} \left ( ex+d \right ) ^{7}}}-{\frac{{b}^{3}{e}^{3}-12\,{b}^{2}cd{e}^{2}+30\,b{c}^{2}{d}^{2}e-20\,{c}^{3}{d}^{3}}{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}-{\frac{{c}^{3}}{2\,{e}^{7} \left ( ex+d \right ) ^{2}}}-{\frac{{c}^{2} \left ( be-2\,cd \right ) }{{e}^{7} \left ( ex+d \right ) ^{3}}}+{\frac{d \left ({b}^{3}{e}^{3}-6\,{b}^{2}cd{e}^{2}+10\,b{c}^{2}{d}^{2}e-5\,{c}^{3}{d}^{3} \right ) }{2\,{e}^{7} \left ( ex+d \right ) ^{6}}}-{\frac{3\,c \left ({b}^{2}{e}^{2}-5\,bcde+5\,{c}^{2}{d}^{2} \right ) }{4\,{e}^{7} \left ( ex+d \right ) ^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x)^3/(e*x+d)^9,x)
[Out]
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Maxima [A] time = 0.720866, size = 464, normalized size = 2.01 \[ -\frac{140 \, c^{3} e^{6} x^{6} + 5 \, c^{3} d^{6} + 5 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + b^{3} d^{3} e^{3} + 280 \,{\left (c^{3} d e^{5} + b c^{2} e^{6}\right )} x^{5} + 70 \,{\left (5 \, c^{3} d^{2} e^{4} + 5 \, b c^{2} d e^{5} + 3 \, b^{2} c e^{6}\right )} x^{4} + 56 \,{\left (5 \, c^{3} d^{3} e^{3} + 5 \, b c^{2} d^{2} e^{4} + 3 \, b^{2} c d e^{5} + b^{3} e^{6}\right )} x^{3} + 28 \,{\left (5 \, c^{3} d^{4} e^{2} + 5 \, b c^{2} d^{3} e^{3} + 3 \, b^{2} c d^{2} e^{4} + b^{3} d e^{5}\right )} x^{2} + 8 \,{\left (5 \, c^{3} d^{5} e + 5 \, b c^{2} d^{4} e^{2} + 3 \, b^{2} c d^{3} e^{3} + b^{3} d^{2} e^{4}\right )} x}{280 \,{\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} e^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^3/(e*x + d)^9,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.217087, size = 464, normalized size = 2.01 \[ -\frac{140 \, c^{3} e^{6} x^{6} + 5 \, c^{3} d^{6} + 5 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + b^{3} d^{3} e^{3} + 280 \,{\left (c^{3} d e^{5} + b c^{2} e^{6}\right )} x^{5} + 70 \,{\left (5 \, c^{3} d^{2} e^{4} + 5 \, b c^{2} d e^{5} + 3 \, b^{2} c e^{6}\right )} x^{4} + 56 \,{\left (5 \, c^{3} d^{3} e^{3} + 5 \, b c^{2} d^{2} e^{4} + 3 \, b^{2} c d e^{5} + b^{3} e^{6}\right )} x^{3} + 28 \,{\left (5 \, c^{3} d^{4} e^{2} + 5 \, b c^{2} d^{3} e^{3} + 3 \, b^{2} c d^{2} e^{4} + b^{3} d e^{5}\right )} x^{2} + 8 \,{\left (5 \, c^{3} d^{5} e + 5 \, b c^{2} d^{4} e^{2} + 3 \, b^{2} c d^{3} e^{3} + b^{3} d^{2} e^{4}\right )} x}{280 \,{\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} e^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^3/(e*x + d)^9,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x)**3/(e*x+d)**9,x)
[Out]
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GIAC/XCAS [A] time = 0.207438, size = 360, normalized size = 1.56 \[ -\frac{{\left (140 \, c^{3} x^{6} e^{6} + 280 \, c^{3} d x^{5} e^{5} + 350 \, c^{3} d^{2} x^{4} e^{4} + 280 \, c^{3} d^{3} x^{3} e^{3} + 140 \, c^{3} d^{4} x^{2} e^{2} + 40 \, c^{3} d^{5} x e + 5 \, c^{3} d^{6} + 280 \, b c^{2} x^{5} e^{6} + 350 \, b c^{2} d x^{4} e^{5} + 280 \, b c^{2} d^{2} x^{3} e^{4} + 140 \, b c^{2} d^{3} x^{2} e^{3} + 40 \, b c^{2} d^{4} x e^{2} + 5 \, b c^{2} d^{5} e + 210 \, b^{2} c x^{4} e^{6} + 168 \, b^{2} c d x^{3} e^{5} + 84 \, b^{2} c d^{2} x^{2} e^{4} + 24 \, b^{2} c d^{3} x e^{3} + 3 \, b^{2} c d^{4} e^{2} + 56 \, b^{3} x^{3} e^{6} + 28 \, b^{3} d x^{2} e^{5} + 8 \, b^{3} d^{2} x e^{4} + b^{3} d^{3} e^{3}\right )} e^{\left (-7\right )}}{280 \,{\left (x e + d\right )}^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^3/(e*x + d)^9,x, algorithm="giac")
[Out]