Optimal. Leaf size=58 \[ \frac{a^2 (c x)^{m+1}}{c (m+1)}+\frac{2 a b (c x)^{m+3}}{c^3 (m+3)}+\frac{b^2 (c x)^{m+5}}{c^5 (m+5)} \]
[Out]
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Rubi [A] time = 0.0749419, 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 \[ \frac{a^2 (c x)^{m+1}}{c (m+1)}+\frac{2 a b (c x)^{m+3}}{c^3 (m+3)}+\frac{b^2 (c x)^{m+5}}{c^5 (m+5)} \]
Antiderivative was successfully verified.
[In] Int[(c*x)^m*(a + b*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 13.4871, size = 49, normalized size = 0.84 \[ \frac{a^{2} \left (c x\right )^{m + 1}}{c \left (m + 1\right )} + \frac{2 a b \left (c x\right )^{m + 3}}{c^{3} \left (m + 3\right )} + \frac{b^{2} \left (c x\right )^{m + 5}}{c^{5} \left (m + 5\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x)**m*(b*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.0265579, size = 41, normalized size = 0.71 \[ (c x)^m \left (\frac{a^2 x}{m+1}+\frac{2 a b x^3}{m+3}+\frac{b^2 x^5}{m+5}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(c*x)^m*(a + b*x^2)^2,x]
[Out]
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Maple [A] time = 0.008, size = 94, normalized size = 1.6 \[{\frac{ \left ({b}^{2}{m}^{2}{x}^{4}+4\,{b}^{2}m{x}^{4}+2\,ab{m}^{2}{x}^{2}+3\,{b}^{2}{x}^{4}+12\,abm{x}^{2}+{a}^{2}{m}^{2}+10\,ab{x}^{2}+8\,m{a}^{2}+15\,{a}^{2} \right ) x \left ( cx \right ) ^{m}}{ \left ( 5+m \right ) \left ( 3+m \right ) \left ( 1+m \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x)^m*(b*x^2+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^2 + a)^2*(c*x)^m,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.229408, size = 117, normalized size = 2.02 \[ \frac{{\left ({\left (b^{2} m^{2} + 4 \, b^{2} m + 3 \, b^{2}\right )} x^{5} + 2 \,{\left (a b m^{2} + 6 \, a b m + 5 \, a b\right )} x^{3} +{\left (a^{2} m^{2} + 8 \, a^{2} m + 15 \, a^{2}\right )} x\right )} \left (c x\right )^{m}}{m^{3} + 9 \, m^{2} + 23 \, m + 15} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(c*x)^m,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.65615, size = 345, normalized size = 5.95 \[ \begin{cases} \frac{- \frac{a^{2}}{4 x^{4}} - \frac{a b}{x^{2}} + b^{2} \log{\left (x \right )}}{c^{5}} & \text{for}\: m = -5 \\\frac{- \frac{a^{2}}{2 x^{2}} + 2 a b \log{\left (x \right )} + \frac{b^{2} x^{2}}{2}}{c^{3}} & \text{for}\: m = -3 \\\frac{a^{2} \log{\left (x \right )} + a b x^{2} + \frac{b^{2} x^{4}}{4}}{c} & \text{for}\: m = -1 \\\frac{a^{2} c^{m} m^{2} x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{8 a^{2} c^{m} m x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{15 a^{2} c^{m} x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{2 a b c^{m} m^{2} x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{12 a b c^{m} m x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{10 a b c^{m} x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{b^{2} c^{m} m^{2} x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{4 b^{2} c^{m} m x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{3 b^{2} c^{m} x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x)**m*(b*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.218646, size = 207, normalized size = 3.57 \[ \frac{b^{2} m^{2} x^{5} e^{\left (m{\rm ln}\left (c x\right )\right )} + 4 \, b^{2} m x^{5} e^{\left (m{\rm ln}\left (c x\right )\right )} + 2 \, a b m^{2} x^{3} e^{\left (m{\rm ln}\left (c x\right )\right )} + 3 \, b^{2} x^{5} e^{\left (m{\rm ln}\left (c x\right )\right )} + 12 \, a b m x^{3} e^{\left (m{\rm ln}\left (c x\right )\right )} + a^{2} m^{2} x e^{\left (m{\rm ln}\left (c x\right )\right )} + 10 \, a b x^{3} e^{\left (m{\rm ln}\left (c x\right )\right )} + 8 \, a^{2} m x e^{\left (m{\rm ln}\left (c x\right )\right )} + 15 \, a^{2} x e^{\left (m{\rm ln}\left (c x\right )\right )}}{m^{3} + 9 \, m^{2} + 23 \, m + 15} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(c*x)^m,x, algorithm="giac")
[Out]