Optimal. Leaf size=106 \[ \frac{3 d e^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2}}-\frac{e \sqrt{a+c x^2} \left (2 \left (c d^2-a e^2\right )+c d e x\right )}{a c^2}-\frac{(d+e x)^2 (a e-c d x)}{a c \sqrt{a+c x^2}} \]
[Out]
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Rubi [A] time = 0.180301, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{3 d e^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2}}-\frac{e \sqrt{a+c x^2} \left (2 \left (c d^2-a e^2\right )+c d e x\right )}{a c^2}-\frac{(d+e x)^2 (a e-c d x)}{a c \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^3/(a + c*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 22.925, size = 99, normalized size = 0.93 \[ \frac{3 d e^{2} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{c^{\frac{3}{2}}} - \frac{\left (d + e x\right )^{2} \left (a e - c d x\right )}{a c \sqrt{a + c x^{2}}} + \frac{e \sqrt{a + c x^{2}} \left (4 a e^{2} - 4 c d^{2} - 2 c d e x\right )}{2 a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**3/(c*x**2+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.159658, size = 91, normalized size = 0.86 \[ \frac{2 a^2 e^3+a c e \left (-3 d^2-3 d e x+e^2 x^2\right )+c^2 d^3 x}{a c^2 \sqrt{a+c x^2}}+\frac{3 d e^2 \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{c^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^3/(a + c*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.01, size = 118, normalized size = 1.1 \[{\frac{{d}^{3}x}{a}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{{e}^{3}{x}^{2}}{c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+2\,{\frac{a{e}^{3}}{{c}^{2}\sqrt{c{x}^{2}+a}}}-3\,{\frac{d{e}^{2}x}{c\sqrt{c{x}^{2}+a}}}+3\,{\frac{d{e}^{2}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ) }{{c}^{3/2}}}-3\,{\frac{{d}^{2}e}{c\sqrt{c{x}^{2}+a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^3/(c*x^2+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((e*x + d)^3/(c*x^2 + a)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.235336, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (a c e^{3} x^{2} - 3 \, a c d^{2} e + 2 \, a^{2} e^{3} +{\left (c^{2} d^{3} - 3 \, a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a} \sqrt{c} + 3 \,{\left (a c^{2} d e^{2} x^{2} + a^{2} c d e^{2}\right )} \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right )}{2 \,{\left (a c^{3} x^{2} + a^{2} c^{2}\right )} \sqrt{c}}, \frac{{\left (a c e^{3} x^{2} - 3 \, a c d^{2} e + 2 \, a^{2} e^{3} +{\left (c^{2} d^{3} - 3 \, a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a} \sqrt{-c} + 3 \,{\left (a c^{2} d e^{2} x^{2} + a^{2} c d e^{2}\right )} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right )}{{\left (a c^{3} x^{2} + a^{2} c^{2}\right )} \sqrt{-c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/(c*x^2 + a)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{3}}{\left (a + c x^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**3/(c*x**2+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.221405, size = 135, normalized size = 1.27 \[ -\frac{3 \, d e^{2}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{c^{\frac{3}{2}}} + \frac{x{\left (\frac{x e^{3}}{c} + \frac{c^{3} d^{3} - 3 \, a c^{2} d e^{2}}{a c^{3}}\right )} - \frac{3 \, a c^{2} d^{2} e - 2 \, a^{2} c e^{3}}{a c^{3}}}{\sqrt{c x^{2} + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/(c*x^2 + a)^(3/2),x, algorithm="giac")
[Out]