Optimal. Leaf size=143 \[ -\frac{\sqrt{b d+2 c d x}}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{6 c \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt{d} \left (b^2-4 a c\right )^{7/4}}+\frac{6 c \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt{d} \left (b^2-4 a c\right )^{7/4}} \]
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Rubi [A] time = 0.264924, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{\sqrt{b d+2 c d x}}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{6 c \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt{d} \left (b^2-4 a c\right )^{7/4}}+\frac{6 c \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt{d} \left (b^2-4 a c\right )^{7/4}} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[b*d + 2*c*d*x]*(a + b*x + c*x^2)^2),x]
[Out]
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Rubi in Sympy [A] time = 62.3926, size = 138, normalized size = 0.97 \[ \frac{6 c \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{\sqrt{d} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}}} + \frac{6 c \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{\sqrt{d} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}}} - \frac{\sqrt{b d + 2 c d x}}{d \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(2*c*d*x+b*d)**(1/2)/(c*x**2+b*x+a)**2,x)
[Out]
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Mathematica [A] time = 0.252383, size = 149, normalized size = 1.04 \[ \frac{-\left (b^2-4 a c\right )^{3/4} (b+2 c x)+6 c \sqrt{b+2 c x} (a+x (b+c x)) \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )+6 c \sqrt{b+2 c x} (a+x (b+c x)) \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/4} (a+x (b+c x)) \sqrt{d (b+2 c x)}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[b*d + 2*c*d*x]*(a + b*x + c*x^2)^2),x]
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Maple [B] time = 0.013, size = 345, normalized size = 2.4 \[ 4\,{\frac{c{d}^{3}\sqrt{2\,cdx+bd}}{ \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) }}+{\frac{3\,c{d}^{3}\sqrt{2}}{2}\ln \left ({1 \left ( 2\,cdx+bd+\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) \left ( 2\,cdx+bd-\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) ^{-1}} \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{7}{4}}}}+3\,{\frac{c{d}^{3}\sqrt{2}}{ \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{7/4}}\arctan \left ({\frac{\sqrt{2}\sqrt{2\,cdx+bd}}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}+1 \right ) }-3\,{\frac{c{d}^{3}\sqrt{2}}{ \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{7/4}}\arctan \left ( -{\frac{\sqrt{2}\sqrt{2\,cdx+bd}}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}+1 \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(2*c*d*x+b*d)^(1/2)/(c*x^2+b*x+a)^2,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(1/(sqrt(2*c*d*x + b*d)*(c*x^2 + b*x + a)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.246349, size = 1307, normalized size = 9.14 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(2*c*d*x + b*d)*(c*x^2 + b*x + a)^2),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(2*c*d*x+b*d)**(1/2)/(c*x**2+b*x+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.230603, size = 684, normalized size = 4.78 \[ \frac{3 \, \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} + 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{b^{4} d - 8 \, a b^{2} c d + 16 \, a^{2} c^{2} d} + \frac{3 \, \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} - 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{b^{4} d - 8 \, a b^{2} c d + 16 \, a^{2} c^{2} d} + \frac{3 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c{\rm ln}\left (2 \, c d x + b d + \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{\sqrt{2} b^{4} d - 8 \, \sqrt{2} a b^{2} c d + 16 \, \sqrt{2} a^{2} c^{2} d} - \frac{3 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c{\rm ln}\left (2 \, c d x + b d - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{\sqrt{2} b^{4} d - 8 \, \sqrt{2} a b^{2} c d + 16 \, \sqrt{2} a^{2} c^{2} d} + \frac{4 \, \sqrt{2 \, c d x + b d} c d}{{\left (b^{2} d^{2} - 4 \, a c d^{2} -{\left (2 \, c d x + b d\right )}^{2}\right )}{\left (b^{2} - 4 \, a c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(2*c*d*x + b*d)*(c*x^2 + b*x + a)^2),x, algorithm="giac")
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