Optimal. Leaf size=87 \[ \frac{a (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{3/2}}+\frac{x \sqrt{a+b x^2} (4 b c-a d)}{8 b}+\frac{d x \left (a+b x^2\right )^{3/2}}{4 b} \]
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Rubi [A] time = 0.0789974, antiderivative size = 87, 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 \[ \frac{a (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{3/2}}+\frac{x \sqrt{a+b x^2} (4 b c-a d)}{8 b}+\frac{d x \left (a+b x^2\right )^{3/2}}{4 b} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*x^2]*(c + d*x^2),x]
[Out]
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Rubi in Sympy [A] time = 9.62465, size = 75, normalized size = 0.86 \[ - \frac{a \left (a d - 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{8 b^{\frac{3}{2}}} + \frac{d x \left (a + b x^{2}\right )^{\frac{3}{2}}}{4 b} - \frac{x \sqrt{a + b x^{2}} \left (a d - 4 b c\right )}{8 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**(1/2)*(d*x**2+c),x)
[Out]
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Mathematica [A] time = 0.0657133, size = 78, normalized size = 0.9 \[ \sqrt{a+b x^2} \left (\frac{x (a d+4 b c)}{8 b}+\frac{d x^3}{4}\right )-\frac{a (a d-4 b c) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{8 b^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b*x^2]*(c + d*x^2),x]
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Maple [A] time = 0.007, size = 96, normalized size = 1.1 \[{\frac{cx}{2}\sqrt{b{x}^{2}+a}}+{\frac{ac}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}}+{\frac{dx}{4\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{adx}{8\,b}\sqrt{b{x}^{2}+a}}-{\frac{{a}^{2}d}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^(1/2)*(d*x^2+c),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(b*x^2 + a)*(d*x^2 + c),x, algorithm="maxima")
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Fricas [A] time = 0.232479, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \, b d x^{3} +{\left (4 \, b c + a d\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} -{\left (4 \, a b c - a^{2} d\right )} \log \left (2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{16 \, b^{\frac{3}{2}}}, \frac{{\left (2 \, b d x^{3} +{\left (4 \, b c + a d\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} +{\left (4 \, a b c - a^{2} d\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{8 \, \sqrt{-b} b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^2 + a)*(d*x^2 + c),x, algorithm="fricas")
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Sympy [A] time = 17.1212, size = 144, normalized size = 1.66 \[ \frac{a^{\frac{3}{2}} d x}{8 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{\sqrt{a} c x \sqrt{1 + \frac{b x^{2}}{a}}}{2} + \frac{3 \sqrt{a} d x^{3}}{8 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{a^{2} d \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 b^{\frac{3}{2}}} + \frac{a c \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 \sqrt{b}} + \frac{b d x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**(1/2)*(d*x**2+c),x)
[Out]
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GIAC/XCAS [A] time = 0.232104, size = 95, normalized size = 1.09 \[ \frac{1}{8} \, \sqrt{b x^{2} + a}{\left (2 \, d x^{2} + \frac{4 \, b^{2} c + a b d}{b^{2}}\right )} x - \frac{{\left (4 \, a b c - a^{2} d\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{8 \, b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^2 + a)*(d*x^2 + c),x, algorithm="giac")
[Out]