Optimal. Leaf size=91 \[ -\frac{(b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{b^{5/2}}+\frac{\sqrt{c+d x^2} (b c-a d)}{b^2}+\frac{\left (c+d x^2\right )^{3/2}}{3 b} \]
[Out]
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Rubi [A] time = 0.19507, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{(b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{b^{5/2}}+\frac{\sqrt{c+d x^2} (b c-a d)}{b^2}+\frac{\left (c+d x^2\right )^{3/2}}{3 b} \]
Antiderivative was successfully verified.
[In] Int[(x*(c + d*x^2)^(3/2))/(a + b*x^2),x]
[Out]
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Rubi in Sympy [A] time = 26.2274, size = 75, normalized size = 0.82 \[ \frac{\left (c + d x^{2}\right )^{\frac{3}{2}}}{3 b} - \frac{\sqrt{c + d x^{2}} \left (a d - b c\right )}{b^{2}} + \frac{\left (a d - b c\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{b^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(d*x**2+c)**(3/2)/(b*x**2+a),x)
[Out]
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Mathematica [A] time = 0.150504, size = 83, normalized size = 0.91 \[ \frac{\sqrt{c+d x^2} \left (-3 a d+4 b c+b d x^2\right )}{3 b^2}-\frac{(b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{b^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(c + d*x^2)^(3/2))/(a + b*x^2),x]
[Out]
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Maple [B] time = 0.017, size = 1856, normalized size = 20.4 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(d*x^2+c)^(3/2)/(b*x^2+a),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((d*x^2 + c)^(3/2)*x/(b*x^2 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.271029, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (b c - a d\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} + 4 \,{\left (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt{d x^{2} + c} \sqrt{\frac{b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \,{\left (b d x^{2} + 4 \, b c - 3 \, a d\right )} \sqrt{d x^{2} + c}}{12 \, b^{2}}, -\frac{3 \,{\left (b c - a d\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{b d x^{2} + 2 \, b c - a d}{2 \, \sqrt{d x^{2} + c} b \sqrt{-\frac{b c - a d}{b}}}\right ) - 2 \,{\left (b d x^{2} + 4 \, b c - 3 \, a d\right )} \sqrt{d x^{2} + c}}{6 \, b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^(3/2)*x/(b*x^2 + a),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x \left (c + d x^{2}\right )^{\frac{3}{2}}}{a + b x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(d*x**2+c)**(3/2)/(b*x**2+a),x)
[Out]
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GIAC/XCAS [A] time = 0.23479, size = 151, normalized size = 1.66 \[ \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b^{2}} + \frac{{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{2} + 3 \, \sqrt{d x^{2} + c} b^{2} c - 3 \, \sqrt{d x^{2} + c} a b d}{3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^(3/2)*x/(b*x^2 + a),x, algorithm="giac")
[Out]