Optimal. Leaf size=94 \[ \frac{a^4 c^3 (e x)^{m+1}}{e (m+1)}-\frac{2 a^3 b c^3 (e x)^{m+2}}{e^2 (m+2)}+\frac{2 a b^3 c^3 (e x)^{m+4}}{e^4 (m+4)}-\frac{b^4 c^3 (e x)^{m+5}}{e^5 (m+5)} \]
[Out]
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Rubi [A] time = 0.142111, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{a^4 c^3 (e x)^{m+1}}{e (m+1)}-\frac{2 a^3 b c^3 (e x)^{m+2}}{e^2 (m+2)}+\frac{2 a b^3 c^3 (e x)^{m+4}}{e^4 (m+4)}-\frac{b^4 c^3 (e x)^{m+5}}{e^5 (m+5)} \]
Antiderivative was successfully verified.
[In] Int[(e*x)^m*(a + b*x)*(a*c - b*c*x)^3,x]
[Out]
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Rubi in Sympy [A] time = 31.3657, size = 85, normalized size = 0.9 \[ \frac{a^{4} c^{3} \left (e x\right )^{m + 1}}{e \left (m + 1\right )} - \frac{2 a^{3} b c^{3} \left (e x\right )^{m + 2}}{e^{2} \left (m + 2\right )} + \frac{2 a b^{3} c^{3} \left (e x\right )^{m + 4}}{e^{4} \left (m + 4\right )} - \frac{b^{4} c^{3} \left (e x\right )^{m + 5}}{e^{5} \left (m + 5\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**m*(b*x+a)*(-b*c*x+a*c)**3,x)
[Out]
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Mathematica [A] time = 0.0844646, size = 112, normalized size = 1.19 \[ -\frac{c^3 x (e x)^m \left (a^4 \left (-\left (m^3+11 m^2+38 m+40\right )\right )+2 a^3 b \left (m^3+10 m^2+29 m+20\right ) x-2 a b^3 \left (m^3+8 m^2+17 m+10\right ) x^3+b^4 \left (m^3+7 m^2+14 m+8\right ) x^4\right )}{(m+1) (m+2) (m+4) (m+5)} \]
Antiderivative was successfully verified.
[In] Integrate[(e*x)^m*(a + b*x)*(a*c - b*c*x)^3,x]
[Out]
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Maple [A] time = 0.008, size = 175, normalized size = 1.9 \[{\frac{{c}^{3} \left ( ex \right ) ^{m} \left ( -{b}^{4}{m}^{3}{x}^{4}+2\,a{b}^{3}{m}^{3}{x}^{3}-7\,{b}^{4}{m}^{2}{x}^{4}+16\,a{b}^{3}{m}^{2}{x}^{3}-14\,{b}^{4}m{x}^{4}-2\,{a}^{3}b{m}^{3}x+34\,a{b}^{3}m{x}^{3}-8\,{b}^{4}{x}^{4}+{a}^{4}{m}^{3}-20\,{a}^{3}b{m}^{2}x+20\,a{b}^{3}{x}^{3}+11\,{a}^{4}{m}^{2}-58\,{a}^{3}bmx+38\,{a}^{4}m-40\,{a}^{3}bx+40\,{a}^{4} \right ) x}{ \left ( 5+m \right ) \left ( 4+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^m*(b*x+a)*(-b*c*x+a*c)^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(b*c*x - a*c)^3*(b*x + a)*(e*x)^m,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.223986, size = 282, normalized size = 3. \[ -\frac{{\left ({\left (b^{4} c^{3} m^{3} + 7 \, b^{4} c^{3} m^{2} + 14 \, b^{4} c^{3} m + 8 \, b^{4} c^{3}\right )} x^{5} - 2 \,{\left (a b^{3} c^{3} m^{3} + 8 \, a b^{3} c^{3} m^{2} + 17 \, a b^{3} c^{3} m + 10 \, a b^{3} c^{3}\right )} x^{4} + 2 \,{\left (a^{3} b c^{3} m^{3} + 10 \, a^{3} b c^{3} m^{2} + 29 \, a^{3} b c^{3} m + 20 \, a^{3} b c^{3}\right )} x^{2} -{\left (a^{4} c^{3} m^{3} + 11 \, a^{4} c^{3} m^{2} + 38 \, a^{4} c^{3} m + 40 \, a^{4} c^{3}\right )} x\right )} \left (e x\right )^{m}}{m^{4} + 12 \, m^{3} + 49 \, m^{2} + 78 \, m + 40} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(b*c*x - a*c)^3*(b*x + a)*(e*x)^m,x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.53939, size = 838, normalized size = 8.91 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)**m*(b*x+a)*(-b*c*x+a*c)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.214333, size = 456, normalized size = 4.85 \[ -\frac{b^{4} c^{3} m^{3} x^{5} e^{\left (m{\rm ln}\left (x\right ) + m\right )} - 2 \, a b^{3} c^{3} m^{3} x^{4} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 7 \, b^{4} c^{3} m^{2} x^{5} e^{\left (m{\rm ln}\left (x\right ) + m\right )} - 16 \, a b^{3} c^{3} m^{2} x^{4} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 14 \, b^{4} c^{3} m x^{5} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 2 \, a^{3} b c^{3} m^{3} x^{2} e^{\left (m{\rm ln}\left (x\right ) + m\right )} - 34 \, a b^{3} c^{3} m x^{4} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 8 \, b^{4} c^{3} x^{5} e^{\left (m{\rm ln}\left (x\right ) + m\right )} - a^{4} c^{3} m^{3} x e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 20 \, a^{3} b c^{3} m^{2} x^{2} e^{\left (m{\rm ln}\left (x\right ) + m\right )} - 20 \, a b^{3} c^{3} x^{4} e^{\left (m{\rm ln}\left (x\right ) + m\right )} - 11 \, a^{4} c^{3} m^{2} x e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 58 \, a^{3} b c^{3} m x^{2} e^{\left (m{\rm ln}\left (x\right ) + m\right )} - 38 \, a^{4} c^{3} m x e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 40 \, a^{3} b c^{3} x^{2} e^{\left (m{\rm ln}\left (x\right ) + m\right )} - 40 \, a^{4} c^{3} x e^{\left (m{\rm ln}\left (x\right ) + m\right )}}{m^{4} + 12 \, m^{3} + 49 \, m^{2} + 78 \, m + 40} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(b*c*x - a*c)^3*(b*x + a)*(e*x)^m,x, algorithm="giac")
[Out]