Optimal. Leaf size=113 \[ -\frac{3 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{32 c^{5/2} d^2}+\frac{3 (b+2 c x) \sqrt{a+b x+c x^2}}{16 c^2 d^2}-\frac{\left (a+b x+c x^2\right )^{3/2}}{2 c d^2 (b+2 c x)} \]
[Out]
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Rubi [A] time = 0.126603, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{3 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{32 c^{5/2} d^2}+\frac{3 (b+2 c x) \sqrt{a+b x+c x^2}}{16 c^2 d^2}-\frac{\left (a+b x+c x^2\right )^{3/2}}{2 c d^2 (b+2 c x)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^2,x]
[Out]
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Rubi in Sympy [A] time = 20.3321, size = 105, normalized size = 0.93 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{2 c d^{2} \left (b + 2 c x\right )} + \frac{3 \left (b + 2 c x\right ) \sqrt{a + b x + c x^{2}}}{16 c^{2} d^{2}} - \frac{3 \left (- 4 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{32 c^{\frac{5}{2}} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)**(3/2)/(2*c*d*x+b*d)**2,x)
[Out]
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Mathematica [A] time = 0.217512, size = 100, normalized size = 0.88 \[ \frac{\frac{\sqrt{a+x (b+c x)} \left (4 c \left (c x^2-2 a\right )+3 b^2+4 b c x\right )}{16 c^2 (b+2 c x)}-\frac{3 \left (b^2-4 a c\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{32 c^{5/2}}}{d^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^2,x]
[Out]
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Maple [B] time = 0.017, size = 570, normalized size = 5. \[ -{\frac{1}{c{d}^{2} \left ( 4\,ac-{b}^{2} \right ) } \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}}+{\frac{x}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) } \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{3}{2}}}}+{\frac{b}{2\,c{d}^{2} \left ( 4\,ac-{b}^{2} \right ) } \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{3}{2}}}}+{\frac{3\,ax}{2\,{d}^{2} \left ( 4\,ac-{b}^{2} \right ) }\sqrt{ \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}}}}-{\frac{3\,{b}^{2}x}{8\,c{d}^{2} \left ( 4\,ac-{b}^{2} \right ) }\sqrt{ \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}}}}+{\frac{3\,ab}{4\,c{d}^{2} \left ( 4\,ac-{b}^{2} \right ) }\sqrt{ \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}}}}-{\frac{3\,{b}^{3}}{16\,{c}^{2}{d}^{2} \left ( 4\,ac-{b}^{2} \right ) }\sqrt{ \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}}}}+{\frac{3\,{a}^{2}}{2\,{d}^{2} \left ( 4\,ac-{b}^{2} \right ) }\ln \left ( \sqrt{c} \left ( x+{\frac{b}{2\,c}} \right ) +\sqrt{ \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}}} \right ){\frac{1}{\sqrt{c}}}}-{\frac{3\,a{b}^{2}}{4\,{d}^{2} \left ( 4\,ac-{b}^{2} \right ) }\ln \left ( \sqrt{c} \left ( x+{\frac{b}{2\,c}} \right ) +\sqrt{ \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}}} \right ){c}^{-{\frac{3}{2}}}}+{\frac{3\,{b}^{4}}{32\,{d}^{2} \left ( 4\,ac-{b}^{2} \right ) }\ln \left ( \sqrt{c} \left ( x+{\frac{b}{2\,c}} \right ) +\sqrt{ \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}}} \right ){c}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^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)/(2*c*d*x + b*d)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.314655, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (4 \, c^{2} x^{2} + 4 \, b c x + 3 \, b^{2} - 8 \, a c\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} - 3 \,{\left (b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x\right )} \log \left (-4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{64 \,{\left (2 \, c^{3} d^{2} x + b c^{2} d^{2}\right )} \sqrt{c}}, \frac{2 \,{\left (4 \, c^{2} x^{2} + 4 \, b c x + 3 \, b^{2} - 8 \, a c\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} - 3 \,{\left (b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{32 \,{\left (2 \, c^{3} d^{2} x + b c^{2} d^{2}\right )} \sqrt{-c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(3/2)/(2*c*d*x + b*d)^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{a \sqrt{a + b x + c x^{2}}}{b^{2} + 4 b c x + 4 c^{2} x^{2}}\, dx + \int \frac{b x \sqrt{a + b x + c x^{2}}}{b^{2} + 4 b c x + 4 c^{2} x^{2}}\, dx + \int \frac{c x^{2} \sqrt{a + b x + c x^{2}}}{b^{2} + 4 b c x + 4 c^{2} x^{2}}\, dx}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)**(3/2)/(2*c*d*x+b*d)**2,x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(3/2)/(2*c*d*x + b*d)^2,x, algorithm="giac")
[Out]