Optimal. Leaf size=133 \[ \frac{\tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{32 c^{3/2} d^5 \left (b^2-4 a c\right )^{3/2}}+\frac{\sqrt{a+b x+c x^2}}{16 c d^5 \left (b^2-4 a c\right ) (b+2 c x)^2}-\frac{\sqrt{a+b x+c x^2}}{8 c d^5 (b+2 c x)^4} \]
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Rubi [A] time = 0.224488, antiderivative size = 133, 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 \[ \frac{\tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{32 c^{3/2} d^5 \left (b^2-4 a c\right )^{3/2}}+\frac{\sqrt{a+b x+c x^2}}{16 c d^5 \left (b^2-4 a c\right ) (b+2 c x)^2}-\frac{\sqrt{a+b x+c x^2}}{8 c d^5 (b+2 c x)^4} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*x + c*x^2]/(b*d + 2*c*d*x)^5,x]
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Rubi in Sympy [A] time = 55.0965, size = 121, normalized size = 0.91 \[ \frac{\sqrt{a + b x + c x^{2}}}{16 c d^{5} \left (b + 2 c x\right )^{2} \left (- 4 a c + b^{2}\right )} - \frac{\sqrt{a + b x + c x^{2}}}{8 c d^{5} \left (b + 2 c x\right )^{4}} + \frac{\operatorname{atan}{\left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} \right )}}{32 c^{\frac{3}{2}} d^{5} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)**(1/2)/(2*c*d*x+b*d)**5,x)
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Mathematica [A] time = 0.842007, size = 162, normalized size = 1.22 \[ \frac{\frac{\log \left (2 c \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)}+4 a c^{3/2}+b^2 \left (-\sqrt{c}\right )\right )}{\left (4 a c-b^2\right )^{3/2}}+\frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (4 c \left (2 a+c x^2\right )-b^2+4 b c x\right )}{\left (b^2-4 a c\right ) (b+2 c x)^4}-\frac{\log (b+2 c x)}{\left (4 a c-b^2\right )^{3/2}}}{32 c^{3/2} d^5} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b*x + c*x^2]/(b*d + 2*c*d*x)^5,x]
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Maple [B] time = 0.021, size = 400, normalized size = 3. \[ -{\frac{1}{32\,{c}^{4}{d}^{5} \left ( 4\,ac-{b}^{2} \right ) } \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{b}{2\,c}} \right ) ^{-4}}+{\frac{1}{16\,{c}^{2}{d}^{5} \left ( 4\,ac-{b}^{2} \right ) ^{2}} \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{b}{2\,c}} \right ) ^{-2}}-{\frac{1}{32\,{d}^{5}c \left ( 4\,ac-{b}^{2} \right ) ^{2}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}}+{\frac{a}{8\,{d}^{5}c \left ( 4\,ac-{b}^{2} \right ) ^{2}}\ln \left ({1 \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}}-{\frac{{b}^{2}}{32\,{c}^{2}{d}^{5} \left ( 4\,ac-{b}^{2} \right ) ^{2}}\ln \left ({1 \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^(1/2)/(2*c*d*x+b*d)^5,x)
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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(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d)^5,x, algorithm="maxima")
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Fricas [A] time = 0.434168, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c\right )} \sqrt{-b^{2} c + 4 \, a c^{2}} \sqrt{c x^{2} + b x + a} -{\left (16 \, c^{4} x^{4} + 32 \, b c^{3} x^{3} + 24 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x + b^{4}\right )} \log \left (-\frac{{\left (4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c\right )} \sqrt{-b^{2} c + 4 \, a c^{2}} - 4 \,{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{c x^{2} + b x + a}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right )}{64 \,{\left (16 \,{\left (b^{2} c^{5} - 4 \, a c^{6}\right )} d^{5} x^{4} + 32 \,{\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} d^{5} x^{3} + 24 \,{\left (b^{4} c^{3} - 4 \, a b^{2} c^{4}\right )} d^{5} x^{2} + 8 \,{\left (b^{5} c^{2} - 4 \, a b^{3} c^{3}\right )} d^{5} x +{\left (b^{6} c - 4 \, a b^{4} c^{2}\right )} d^{5}\right )} \sqrt{-b^{2} c + 4 \, a c^{2}}}, \frac{2 \,{\left (4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c\right )} \sqrt{b^{2} c - 4 \, a c^{2}} \sqrt{c x^{2} + b x + a} -{\left (16 \, c^{4} x^{4} + 32 \, b c^{3} x^{3} + 24 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x + b^{4}\right )} \arctan \left (\frac{\sqrt{b^{2} c - 4 \, a c^{2}}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{32 \,{\left (16 \,{\left (b^{2} c^{5} - 4 \, a c^{6}\right )} d^{5} x^{4} + 32 \,{\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} d^{5} x^{3} + 24 \,{\left (b^{4} c^{3} - 4 \, a b^{2} c^{4}\right )} d^{5} x^{2} + 8 \,{\left (b^{5} c^{2} - 4 \, a b^{3} c^{3}\right )} d^{5} x +{\left (b^{6} c - 4 \, a b^{4} c^{2}\right )} d^{5}\right )} \sqrt{b^{2} c - 4 \, a c^{2}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d)^5,x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{\sqrt{a + b x + c x^{2}}}{b^{5} + 10 b^{4} c x + 40 b^{3} c^{2} x^{2} + 80 b^{2} c^{3} x^{3} + 80 b c^{4} x^{4} + 32 c^{5} x^{5}}\, dx}{d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)**(1/2)/(2*c*d*x+b*d)**5,x)
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GIAC/XCAS [A] time = 0.361057, size = 860, normalized size = 6.47 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d)^5,x, algorithm="giac")
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