Optimal. Leaf size=165 \[ \frac{d^2 \left (b^2-4 a c\right )^3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{5/2}}+\frac{d^2 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2}}{128 c^2}-\frac{d^2 \left (b^2-4 a c\right ) (b+2 c x)^3 \sqrt{a+b x+c x^2}}{64 c^2}+\frac{d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{12 c} \]
[Out]
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Rubi [A] time = 0.248576, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{d^2 \left (b^2-4 a c\right )^3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{5/2}}+\frac{d^2 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2}}{128 c^2}-\frac{d^2 \left (b^2-4 a c\right ) (b+2 c x)^3 \sqrt{a+b x+c x^2}}{64 c^2}+\frac{d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{12 c} \]
Antiderivative was successfully verified.
[In] Int[(b*d + 2*c*d*x)^2*(a + b*x + c*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 49.9997, size = 155, normalized size = 0.94 \[ \frac{d^{2} \left (b + 2 c x\right )^{3} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{12 c} - \frac{d^{2} \left (b + 2 c x\right )^{3} \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}}{64 c^{2}} + \frac{d^{2} \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}}}{128 c^{2}} + \frac{d^{2} \left (- 4 a c + b^{2}\right )^{3} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{256 c^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*d*x+b*d)**2*(c*x**2+b*x+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.20763, size = 150, normalized size = 0.91 \[ \frac{d^2 \left (3 \left (b^2-4 a c\right )^3 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )-2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left (-16 c^2 \left (3 a^2+14 a c x^2+8 c^2 x^4\right )-8 b^2 c \left (4 a+17 c x^2\right )-32 b c^2 x \left (7 a+8 c x^2\right )+3 b^4-8 b^3 c x\right )\right )}{768 c^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(b*d + 2*c*d*x)^2*(a + b*x + c*x^2)^(3/2),x]
[Out]
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Maple [B] time = 0.013, size = 406, normalized size = 2.5 \[{\frac{{d}^{2}b}{3} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{2}}}}+{\frac{2\,c{d}^{2}x}{3} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{2}}}}-{\frac{a{d}^{2}b}{12} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}{d}^{2}b}{8}\sqrt{c{x}^{2}+bx+a}}-{\frac{{d}^{2}{b}^{4}x}{64\,c}\sqrt{c{x}^{2}+bx+a}}+{\frac{{b}^{3}{d}^{2}a}{16\,c}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,{b}^{2}{d}^{2}{a}^{2}}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}-{\frac{3\,{d}^{2}{b}^{4}a}{64}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{{b}^{2}{d}^{2}x}{24} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{3}{d}^{2}}{48\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{{d}^{2}{b}^{5}}{128\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{{d}^{2}{b}^{6}}{256}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{{b}^{2}{d}^{2}ax}{8}\sqrt{c{x}^{2}+bx+a}}-{\frac{c{d}^{2}ax}{6} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}c{d}^{2}x}{4}\sqrt{c{x}^{2}+bx+a}}-{\frac{{a}^{3}{d}^{2}}{4}\sqrt{c}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*d*x+b*d)^2*(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((2*c*d*x + b*d)^2*(c*x^2 + b*x + a)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.258352, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{2} \log \left (4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right ) - 4 \,{\left (256 \, c^{5} d^{2} x^{5} + 640 \, b c^{4} d^{2} x^{4} + 16 \,{\left (33 \, b^{2} c^{3} + 28 \, a c^{4}\right )} d^{2} x^{3} + 8 \,{\left (19 \, b^{3} c^{2} + 84 \, a b c^{3}\right )} d^{2} x^{2} + 2 \,{\left (b^{4} c + 144 \, a b^{2} c^{2} + 48 \, a^{2} c^{3}\right )} d^{2} x -{\left (3 \, b^{5} - 32 \, a b^{3} c - 48 \, a^{2} b c^{2}\right )} d^{2}\right )} \sqrt{c x^{2} + b x + a} \sqrt{c}}{1536 \, c^{\frac{5}{2}}}, \frac{3 \,{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{2} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right ) + 2 \,{\left (256 \, c^{5} d^{2} x^{5} + 640 \, b c^{4} d^{2} x^{4} + 16 \,{\left (33 \, b^{2} c^{3} + 28 \, a c^{4}\right )} d^{2} x^{3} + 8 \,{\left (19 \, b^{3} c^{2} + 84 \, a b c^{3}\right )} d^{2} x^{2} + 2 \,{\left (b^{4} c + 144 \, a b^{2} c^{2} + 48 \, a^{2} c^{3}\right )} d^{2} x -{\left (3 \, b^{5} - 32 \, a b^{3} c - 48 \, a^{2} b c^{2}\right )} d^{2}\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c}}{768 \, \sqrt{-c} c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^2*(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. \[ d^{2} \left (\int a b^{2} \sqrt{a + b x + c x^{2}}\, dx + \int b^{3} x \sqrt{a + b x + c x^{2}}\, dx + \int 4 c^{3} x^{4} \sqrt{a + b x + c x^{2}}\, dx + \int 4 a c^{2} x^{2} \sqrt{a + b x + c x^{2}}\, dx + \int 8 b c^{2} x^{3} \sqrt{a + b x + c x^{2}}\, dx + \int 5 b^{2} c x^{2} \sqrt{a + b x + c x^{2}}\, dx + \int 4 a b c x \sqrt{a + b x + c x^{2}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x+b*d)**2*(c*x**2+b*x+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.229692, size = 350, normalized size = 2.12 \[ \frac{1}{384} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \, c^{3} d^{2} x + 5 \, b c^{2} d^{2}\right )} x + \frac{33 \, b^{2} c^{6} d^{2} + 28 \, a c^{7} d^{2}}{c^{5}}\right )} x + \frac{19 \, b^{3} c^{5} d^{2} + 84 \, a b c^{6} d^{2}}{c^{5}}\right )} x + \frac{b^{4} c^{4} d^{2} + 144 \, a b^{2} c^{5} d^{2} + 48 \, a^{2} c^{6} d^{2}}{c^{5}}\right )} x - \frac{3 \, b^{5} c^{3} d^{2} - 32 \, a b^{3} c^{4} d^{2} - 48 \, a^{2} b c^{5} d^{2}}{c^{5}}\right )} - \frac{{\left (b^{6} d^{2} - 12 \, a b^{4} c d^{2} + 48 \, a^{2} b^{2} c^{2} d^{2} - 64 \, a^{3} c^{3} d^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{256 \, c^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^2*(c*x^2 + b*x + a)^(3/2),x, algorithm="giac")
[Out]