Optimal. Leaf size=61 \[ \frac{a^3 x^{m+1}}{m+1}+\frac{3 a^2 b x^{m+2}}{m+2}+\frac{3 a b^2 x^{m+3}}{m+3}+\frac{b^3 x^{m+4}}{m+4} \]
[Out]
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Rubi [A] time = 0.0481728, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{a^3 x^{m+1}}{m+1}+\frac{3 a^2 b x^{m+2}}{m+2}+\frac{3 a b^2 x^{m+3}}{m+3}+\frac{b^3 x^{m+4}}{m+4} \]
Antiderivative was successfully verified.
[In] Int[x^m*(a + b*x)^3,x]
[Out]
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Rubi in Sympy [A] time = 9.38047, size = 53, normalized size = 0.87 \[ \frac{a^{3} x^{m + 1}}{m + 1} + \frac{3 a^{2} b x^{m + 2}}{m + 2} + \frac{3 a b^{2} x^{m + 3}}{m + 3} + \frac{b^{3} x^{m + 4}}{m + 4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**m*(b*x+a)**3,x)
[Out]
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Mathematica [A] time = 0.038294, size = 55, normalized size = 0.9 \[ x^m \left (\frac{a^3 x}{m+1}+\frac{3 a^2 b x^2}{m+2}+\frac{3 a b^2 x^3}{m+3}+\frac{b^3 x^4}{m+4}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^m*(a + b*x)^3,x]
[Out]
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Maple [B] time = 0.001, size = 170, normalized size = 2.8 \[{\frac{{x}^{1+m} \left ({b}^{3}{m}^{3}{x}^{3}+3\,a{b}^{2}{m}^{3}{x}^{2}+6\,{b}^{3}{m}^{2}{x}^{3}+3\,{a}^{2}b{m}^{3}x+21\,a{b}^{2}{m}^{2}{x}^{2}+11\,{b}^{3}m{x}^{3}+{a}^{3}{m}^{3}+24\,{a}^{2}b{m}^{2}x+42\,a{b}^{2}m{x}^{2}+6\,{b}^{3}{x}^{3}+9\,{a}^{3}{m}^{2}+57\,{a}^{2}bmx+24\,a{b}^{2}{x}^{2}+26\,{a}^{3}m+36\,{a}^{2}bx+24\,{a}^{3} \right ) }{ \left ( 4+m \right ) \left ( 3+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^m*(b*x+a)^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*x + a)^3*x^m,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.226682, size = 212, normalized size = 3.48 \[ \frac{{\left ({\left (b^{3} m^{3} + 6 \, b^{3} m^{2} + 11 \, b^{3} m + 6 \, b^{3}\right )} x^{4} + 3 \,{\left (a b^{2} m^{3} + 7 \, a b^{2} m^{2} + 14 \, a b^{2} m + 8 \, a b^{2}\right )} x^{3} + 3 \,{\left (a^{2} b m^{3} + 8 \, a^{2} b m^{2} + 19 \, a^{2} b m + 12 \, a^{2} b\right )} x^{2} +{\left (a^{3} m^{3} + 9 \, a^{3} m^{2} + 26 \, a^{3} m + 24 \, a^{3}\right )} x\right )} x^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^3*x^m,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.35971, size = 663, normalized size = 10.87 \[ \begin{cases} - \frac{a^{3}}{3 x^{3}} - \frac{3 a^{2} b}{2 x^{2}} - \frac{3 a b^{2}}{x} + b^{3} \log{\left (x \right )} & \text{for}\: m = -4 \\- \frac{a^{3}}{2 x^{2}} - \frac{3 a^{2} b}{x} + 3 a b^{2} \log{\left (x \right )} + b^{3} x & \text{for}\: m = -3 \\- \frac{a^{3}}{x} + 3 a^{2} b \log{\left (x \right )} + 3 a b^{2} x + \frac{b^{3} x^{2}}{2} & \text{for}\: m = -2 \\a^{3} \log{\left (x \right )} + 3 a^{2} b x + \frac{3 a b^{2} x^{2}}{2} + \frac{b^{3} x^{3}}{3} & \text{for}\: m = -1 \\\frac{a^{3} m^{3} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{9 a^{3} m^{2} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{26 a^{3} m x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{24 a^{3} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{3 a^{2} b m^{3} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{24 a^{2} b m^{2} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{57 a^{2} b m x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{36 a^{2} b x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{3 a b^{2} m^{3} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{21 a b^{2} m^{2} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{42 a b^{2} m x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{24 a b^{2} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{b^{3} m^{3} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{6 b^{3} m^{2} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{11 b^{3} m x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{6 b^{3} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**m*(b*x+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.210137, size = 346, normalized size = 5.67 \[ \frac{b^{3} m^{3} x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + 3 \, a b^{2} m^{3} x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 6 \, b^{3} m^{2} x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + 3 \, a^{2} b m^{3} x^{2} e^{\left (m{\rm ln}\left (x\right )\right )} + 21 \, a b^{2} m^{2} x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 11 \, b^{3} m x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + a^{3} m^{3} x e^{\left (m{\rm ln}\left (x\right )\right )} + 24 \, a^{2} b m^{2} x^{2} e^{\left (m{\rm ln}\left (x\right )\right )} + 42 \, a b^{2} m x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 6 \, b^{3} x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + 9 \, a^{3} m^{2} x e^{\left (m{\rm ln}\left (x\right )\right )} + 57 \, a^{2} b m x^{2} e^{\left (m{\rm ln}\left (x\right )\right )} + 24 \, a b^{2} x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 26 \, a^{3} m x e^{\left (m{\rm ln}\left (x\right )\right )} + 36 \, a^{2} b x^{2} e^{\left (m{\rm ln}\left (x\right )\right )} + 24 \, a^{3} x e^{\left (m{\rm ln}\left (x\right )\right )}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^3*x^m,x, algorithm="giac")
[Out]