Optimal. Leaf size=178 \[ \frac{3 c^2 d^{3/2} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/4}}+\frac{3 c^2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/4}}-\frac{c d \sqrt{b d+2 c d x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{d \sqrt{b d+2 c d x}}{2 \left (a+b x+c x^2\right )^2} \]
[Out]
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Rubi [A] time = 0.345454, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{3 c^2 d^{3/2} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/4}}+\frac{3 c^2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/4}}-\frac{c d \sqrt{b d+2 c d x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{d \sqrt{b d+2 c d x}}{2 \left (a+b x+c x^2\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(b*d + 2*c*d*x)^(3/2)/(a + b*x + c*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 80.0182, size = 173, normalized size = 0.97 \[ \frac{3 c^{2} d^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{7}{4}}} + \frac{3 c^{2} d^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{7}{4}}} - \frac{c d \sqrt{b d + 2 c d x}}{2 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )} - \frac{d \sqrt{b d + 2 c d x}}{2 \left (a + b x + c x^{2}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*d*x+b*d)**(3/2)/(c*x**2+b*x+a)**3,x)
[Out]
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Mathematica [A] time = 0.670905, size = 164, normalized size = 0.92 \[ (d (b+2 c x))^{3/2} \left (\frac{3 c^2 \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/4} (b+2 c x)^{3/2}}+\frac{3 c^2 \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/4} (b+2 c x)^{3/2}}+\frac{\frac{c (a+x (b+c x))}{4 a c-b^2}-1}{2 (b+2 c x) (a+x (b+c x))^2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(b*d + 2*c*d*x)^(3/2)/(a + b*x + c*x^2)^3,x]
[Out]
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Maple [B] time = 0.021, size = 431, normalized size = 2.4 \[ 2\,{\frac{{c}^{2}{d}^{3} \left ( 2\,cdx+bd \right ) ^{5/2}}{ \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) ^{2} \left ( 4\,ac-{b}^{2} \right ) }}-6\,{\frac{{c}^{2}{d}^{5}\sqrt{2\,cdx+bd}}{ \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) ^{2}}}+{\frac{3\,{c}^{2}{d}^{3}\sqrt{2}}{16\,ac-4\,{b}^{2}}\ln \left ({1 \left ( 2\,cdx+bd+\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) \left ( 2\,cdx+bd-\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) ^{-1}} \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{3}{4}}}}+{\frac{3\,{c}^{2}{d}^{3}\sqrt{2}}{8\,ac-2\,{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{3}{4}}}}-{\frac{3\,{c}^{2}{d}^{3}\sqrt{2}}{8\,ac-2\,{b}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{3}{4}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*d*x+b*d)^(3/2)/(c*x^2+b*x+a)^3,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)^(3/2)/(c*x^2 + b*x + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.243704, size = 1593, normalized size = 8.95 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^(3/2)/(c*x^2 + b*x + a)^3,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((2*c*d*x+b*d)**(3/2)/(c*x**2+b*x+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.24789, size = 765, normalized size = 4.3 \[ \frac{3 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2} d \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} + 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{\sqrt{2} b^{4} - 8 \, \sqrt{2} a b^{2} c + 16 \, \sqrt{2} a^{2} c^{2}} + \frac{3 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2} d \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} - 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{\sqrt{2} b^{4} - 8 \, \sqrt{2} a b^{2} c + 16 \, \sqrt{2} a^{2} c^{2}} + \frac{3 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2} d{\rm ln}\left (2 \, c d x + b d + \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{2 \,{\left (\sqrt{2} b^{4} - 8 \, \sqrt{2} a b^{2} c + 16 \, \sqrt{2} a^{2} c^{2}\right )}} - \frac{3 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2} d{\rm ln}\left (2 \, c d x + b d - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{2 \,{\left (\sqrt{2} b^{4} - 8 \, \sqrt{2} a b^{2} c + 16 \, \sqrt{2} a^{2} c^{2}\right )}} - \frac{2 \,{\left (3 \, \sqrt{2 \, c d x + b d} b^{2} c^{2} d^{5} - 12 \, \sqrt{2 \, c d x + b d} a c^{3} d^{5} +{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} c^{2} d^{3}\right )}}{{\left (b^{2} d^{2} - 4 \, a c d^{2} -{\left (2 \, c d x + b d\right )}^{2}\right )}^{2}{\left (b^{2} - 4 \, a c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^(3/2)/(c*x^2 + b*x + a)^3,x, algorithm="giac")
[Out]