Optimal. Leaf size=112 \[ \frac{3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{5/2}}-\frac{3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{64 c^2}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c} \]
[Out]
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Rubi [A] time = 0.0935915, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{5/2}}-\frac{3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{64 c^2}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 8.8518, size = 105, normalized size = 0.94 \[ \frac{\left (b + 2 c x\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{8 c} - \frac{3 \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}}{64 c^{2}} + \frac{3 \left (- 4 a c + b^{2}\right )^{2} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{128 c^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.112391, size = 99, normalized size = 0.88 \[ \frac{2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left (4 c \left (5 a+2 c x^2\right )-3 b^2+8 b c x\right )+3 \left (b^2-4 a c\right )^2 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{128 c^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^(3/2),x]
[Out]
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Maple [B] time = 0.005, size = 201, normalized size = 1.8 \[{\frac{2\,cx+b}{8\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,ax}{8}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,{b}^{2}x}{32\,c}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,ab}{16\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,{b}^{3}}{64\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,{a}^{2}}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}-{\frac{3\,a{b}^{2}}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{3\,{b}^{4}}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^(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((c*x^2 + b*x + a)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.245165, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (16 \, c^{3} x^{3} + 24 \, b c^{2} x^{2} - 3 \, b^{3} + 20 \, a b c + 2 \,{\left (b^{2} c + 20 \, a c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} + 3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \log \left (-4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{256 \, c^{\frac{5}{2}}}, \frac{2 \,{\left (16 \, c^{3} x^{3} + 24 \, b c^{2} x^{2} - 3 \, b^{3} + 20 \, a b c + 2 \,{\left (b^{2} c + 20 \, a c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} + 3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{128 \, \sqrt{-c} c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (a + b x + c x^{2}\right )^{\frac{3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.216883, size = 166, normalized size = 1.48 \[ \frac{1}{64} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \, c x + 3 \, b\right )} x + \frac{b^{2} c^{2} + 20 \, a c^{3}}{c^{3}}\right )} x - \frac{3 \, b^{3} c - 20 \, a b c^{2}}{c^{3}}\right )} - \frac{3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(3/2),x, algorithm="giac")
[Out]