Optimal. Leaf size=28 \[ \frac{c (a+b x)^3 \sqrt{c (a+b x)^2}}{4 b} \]
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Rubi [A] time = 0.0286164, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{c (a+b x)^3 \sqrt{c (a+b x)^2}}{4 b} \]
Antiderivative was successfully verified.
[In] Int[(c*(a + b*x)^2)^(3/2),x]
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Rubi in Sympy [A] time = 2.7658, size = 36, normalized size = 1.29 \[ \frac{\left (2 a + 2 b x\right ) \left (a^{2} c + 2 a b c x + b^{2} c x^{2}\right )^{\frac{3}{2}}}{8 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*(b*x+a)**2)**(3/2),x)
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Mathematica [A] time = 0.0186441, size = 25, normalized size = 0.89 \[ \frac{(a+b x) \left (c (a+b x)^2\right )^{3/2}}{4 b} \]
Antiderivative was successfully verified.
[In] Integrate[(c*(a + b*x)^2)^(3/2),x]
[Out]
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Maple [B] time = 0.005, size = 51, normalized size = 1.8 \[{\frac{x \left ({b}^{3}{x}^{3}+4\,a{b}^{2}{x}^{2}+6\,{a}^{2}bx+4\,{a}^{3} \right ) }{4\, \left ( bx+a \right ) ^{3}} \left ( c \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*(b*x+a)^2)^(3/2),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 + a)^2*c)^(3/2),x, algorithm="maxima")
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Fricas [A] time = 0.212155, size = 90, normalized size = 3.21 \[ \frac{{\left (b^{3} c x^{4} + 4 \, a b^{2} c x^{3} + 6 \, a^{2} b c x^{2} + 4 \, a^{3} c x\right )} \sqrt{b^{2} c x^{2} + 2 \, a b c x + a^{2} c}}{4 \,{\left (b x + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((b*x + a)^2*c)^(3/2),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (c \left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*(b*x+a)**2)**(3/2),x)
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GIAC/XCAS [A] time = 0.217388, size = 28, normalized size = 1. \[ \frac{{\left (b x + a\right )}^{4} c^{\frac{3}{2}}{\rm sign}\left (b x + a\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((b*x + a)^2*c)^(3/2),x, algorithm="giac")
[Out]