Optimal. Leaf size=58 \[ \frac{a^2 (c x)^{m+1}}{c (m+1)}+\frac{2 a b (c x)^{m+4}}{c^4 (m+4)}+\frac{b^2 (c x)^{m+7}}{c^7 (m+7)} \]
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Rubi [A] time = 0.0777792, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{a^2 (c x)^{m+1}}{c (m+1)}+\frac{2 a b (c x)^{m+4}}{c^4 (m+4)}+\frac{b^2 (c x)^{m+7}}{c^7 (m+7)} \]
Antiderivative was successfully verified.
[In] Int[(c*x)^m*(a + b*x^3)^2,x]
[Out]
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Rubi in Sympy [A] time = 13.0846, size = 49, normalized size = 0.84 \[ \frac{a^{2} \left (c x\right )^{m + 1}}{c \left (m + 1\right )} + \frac{2 a b \left (c x\right )^{m + 4}}{c^{4} \left (m + 4\right )} + \frac{b^{2} \left (c x\right )^{m + 7}}{c^{7} \left (m + 7\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x)**m*(b*x**3+a)**2,x)
[Out]
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Mathematica [A] time = 0.0289169, size = 41, normalized size = 0.71 \[ (c x)^m \left (\frac{a^2 x}{m+1}+\frac{2 a b x^4}{m+4}+\frac{b^2 x^7}{m+7}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(c*x)^m*(a + b*x^3)^2,x]
[Out]
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Maple [A] time = 0.014, size = 94, normalized size = 1.6 \[{\frac{ \left ({b}^{2}{m}^{2}{x}^{6}+5\,{b}^{2}m{x}^{6}+4\,{b}^{2}{x}^{6}+2\,ab{m}^{2}{x}^{3}+16\,abm{x}^{3}+14\,ab{x}^{3}+{a}^{2}{m}^{2}+11\,m{a}^{2}+28\,{a}^{2} \right ) x \left ( cx \right ) ^{m}}{ \left ( 7+m \right ) \left ( 4+m \right ) \left ( 1+m \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x)^m*(b*x^3+a)^2,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^3 + a)^2*(c*x)^m,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.227715, size = 117, normalized size = 2.02 \[ \frac{{\left ({\left (b^{2} m^{2} + 5 \, b^{2} m + 4 \, b^{2}\right )} x^{7} + 2 \,{\left (a b m^{2} + 8 \, a b m + 7 \, a b\right )} x^{4} +{\left (a^{2} m^{2} + 11 \, a^{2} m + 28 \, a^{2}\right )} x\right )} \left (c x\right )^{m}}{m^{3} + 12 \, m^{2} + 39 \, m + 28} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)^2*(c*x)^m,x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.31937, size = 352, normalized size = 6.07 \[ \begin{cases} \frac{- \frac{a^{2}}{6 x^{6}} - \frac{2 a b}{3 x^{3}} + b^{2} \log{\left (x \right )}}{c^{7}} & \text{for}\: m = -7 \\\frac{- \frac{a^{2}}{3 x^{3}} + 2 a b \log{\left (x \right )} + \frac{b^{2} x^{3}}{3}}{c^{4}} & \text{for}\: m = -4 \\\frac{a^{2} \log{\left (x \right )} + \frac{2 a b x^{3}}{3} + \frac{b^{2} x^{6}}{6}}{c} & \text{for}\: m = -1 \\\frac{a^{2} c^{m} m^{2} x x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{11 a^{2} c^{m} m x x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{28 a^{2} c^{m} x x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{2 a b c^{m} m^{2} x^{4} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{16 a b c^{m} m x^{4} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{14 a b c^{m} x^{4} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{b^{2} c^{m} m^{2} x^{7} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{5 b^{2} c^{m} m x^{7} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{4 b^{2} c^{m} x^{7} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x)**m*(b*x**3+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.220322, size = 207, normalized size = 3.57 \[ \frac{b^{2} m^{2} x^{7} e^{\left (m{\rm ln}\left (c x\right )\right )} + 5 \, b^{2} m x^{7} e^{\left (m{\rm ln}\left (c x\right )\right )} + 4 \, b^{2} x^{7} e^{\left (m{\rm ln}\left (c x\right )\right )} + 2 \, a b m^{2} x^{4} e^{\left (m{\rm ln}\left (c x\right )\right )} + 16 \, a b m x^{4} e^{\left (m{\rm ln}\left (c x\right )\right )} + 14 \, a b x^{4} e^{\left (m{\rm ln}\left (c x\right )\right )} + a^{2} m^{2} x e^{\left (m{\rm ln}\left (c x\right )\right )} + 11 \, a^{2} m x e^{\left (m{\rm ln}\left (c x\right )\right )} + 28 \, a^{2} x e^{\left (m{\rm ln}\left (c x\right )\right )}}{m^{3} + 12 \, m^{2} + 39 \, m + 28} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)^2*(c*x)^m,x, algorithm="giac")
[Out]