Optimal. Leaf size=86 \[ -\frac{2}{d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{4 \sqrt{c} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{d \left (b^2-4 a c\right )^{3/2}} \]
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Rubi [A] time = 0.151581, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{2}{d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{4 \sqrt{c} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{d \left (b^2-4 a c\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[1/((b*d + 2*c*d*x)*(a + b*x + c*x^2)^(3/2)),x]
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Rubi in Sympy [A] time = 37.8107, size = 80, normalized size = 0.93 \[ - \frac{4 \sqrt{c} \operatorname{atan}{\left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} \right )}}{d \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} - \frac{2}{d \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(2*c*d*x+b*d)/(c*x**2+b*x+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.426581, size = 130, normalized size = 1.51 \[ \frac{2 \left (-\frac{2 \sqrt{c} \log \left (2 c \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)}+4 a c^{3/2}+b^2 \left (-\sqrt{c}\right )\right )}{\left (4 a c-b^2\right )^{3/2}}-\frac{1}{\left (b^2-4 a c\right ) \sqrt{a+x (b+c x)}}+\frac{2 \sqrt{c} \log (b+2 c x)}{\left (4 a c-b^2\right )^{3/2}}\right )}{d} \]
Antiderivative was successfully verified.
[In] Integrate[1/((b*d + 2*c*d*x)*(a + b*x + c*x^2)^(3/2)),x]
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Maple [B] time = 0.014, size = 158, normalized size = 1.8 \[ 2\,{\frac{1}{d \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{ \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+1/4\,{\frac{4\,ac-{b}^{2}}{c}}}}}}-4\,{\frac{1}{d \left ( 4\,ac-{b}^{2} \right ) }\ln \left ({1 \left ( 1/2\,{\frac{4\,ac-{b}^{2}}{c}}+1/2\,\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}} \right ) \left ( x+1/2\,{\frac{b}{c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(2*c*d*x+b*d)/(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(1/((2*c*d*x + b*d)*(c*x^2 + b*x + a)^(3/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.285707, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left ({\left (c x^{2} + b x + a\right )} \sqrt{-\frac{c}{b^{2} - 4 \, a c}} \log \left (-\frac{4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c + 4 \, \sqrt{c x^{2} + b x + a}{\left (b^{2} - 4 \, a c\right )} \sqrt{-\frac{c}{b^{2} - 4 \, a c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) + \sqrt{c x^{2} + b x + a}\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} d x^{2} +{\left (b^{3} - 4 \, a b c\right )} d x +{\left (a b^{2} - 4 \, a^{2} c\right )} d}, \frac{2 \,{\left (2 \,{\left (c x^{2} + b x + a\right )} \sqrt{\frac{c}{b^{2} - 4 \, a c}} \arctan \left (\frac{1}{2 \, \sqrt{c x^{2} + b x + a} \sqrt{\frac{c}{b^{2} - 4 \, a c}}}\right ) - \sqrt{c x^{2} + b x + a}\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} d x^{2} +{\left (b^{3} - 4 \, a b c\right )} d x +{\left (a b^{2} - 4 \, a^{2} c\right )} d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((2*c*d*x + b*d)*(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. \[ \frac{\int \frac{1}{a b \sqrt{a + b x + c x^{2}} + 2 a c x \sqrt{a + b x + c x^{2}} + b^{2} x \sqrt{a + b x + c x^{2}} + 3 b c x^{2} \sqrt{a + b x + c x^{2}} + 2 c^{2} x^{3} \sqrt{a + b x + c x^{2}}}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(2*c*d*x+b*d)/(c*x**2+b*x+a)**(3/2),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(1/((2*c*d*x + b*d)*(c*x^2 + b*x + a)^(3/2)),x, algorithm="giac")
[Out]