Optimal. Leaf size=67 \[ \frac{1}{2} d x \sqrt{a+c x^2}+\frac{a d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 \sqrt{c}}+\frac{e \left (a+c x^2\right )^{3/2}}{3 c} \]
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Rubi [A] time = 0.0616367, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{1}{2} d x \sqrt{a+c x^2}+\frac{a d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 \sqrt{c}}+\frac{e \left (a+c x^2\right )^{3/2}}{3 c} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)*Sqrt[a + c*x^2],x]
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Rubi in Sympy [A] time = 6.84611, size = 58, normalized size = 0.87 \[ \frac{a d \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{2 \sqrt{c}} + \frac{d x \sqrt{a + c x^{2}}}{2} + \frac{e \left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)*(c*x**2+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.052258, size = 67, normalized size = 1. \[ \frac{\sqrt{a+c x^2} (2 a e+c x (3 d+2 e x))+3 a \sqrt{c} d \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{6 c} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)*Sqrt[a + c*x^2],x]
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Maple [A] time = 0.007, size = 53, normalized size = 0.8 \[{\frac{dx}{2}\sqrt{c{x}^{2}+a}}+{\frac{ad}{2}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{e}{3\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)*(c*x^2+a)^(1/2),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(sqrt(c*x^2 + a)*(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.2561, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a c d \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right ) + 2 \,{\left (2 \, c e x^{2} + 3 \, c d x + 2 \, a e\right )} \sqrt{c x^{2} + a} \sqrt{c}}{12 \, c^{\frac{3}{2}}}, \frac{3 \, a c d \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) +{\left (2 \, c e x^{2} + 3 \, c d x + 2 \, a e\right )} \sqrt{c x^{2} + a} \sqrt{-c}}{6 \, \sqrt{-c} c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + a)*(e*x + d),x, algorithm="fricas")
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Sympy [A] time = 7.70669, size = 70, normalized size = 1.04 \[ \frac{\sqrt{a} d x \sqrt{1 + \frac{c x^{2}}{a}}}{2} + \frac{a d \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 \sqrt{c}} + e \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)*(c*x**2+a)**(1/2),x)
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GIAC/XCAS [A] time = 0.214083, size = 77, normalized size = 1.15 \[ -\frac{a d{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{2 \, \sqrt{c}} + \frac{1}{6} \, \sqrt{c x^{2} + a}{\left ({\left (2 \, x e + 3 \, d\right )} x + \frac{2 \, a e}{c}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + a)*(e*x + d),x, algorithm="giac")
[Out]