Optimal. Leaf size=67 \[ \frac{a}{c^2 x \sqrt{d x-c} \sqrt{c+d x}}-\frac{x \left (2 a d^2+b c^2\right )}{c^4 \sqrt{d x-c} \sqrt{c+d x}} \]
[Out]
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Rubi [A] time = 0.239412, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{a}{c^2 x \sqrt{d x-c} \sqrt{c+d x}}-\frac{x \left (2 a d^2+b c^2\right )}{c^4 \sqrt{d x-c} \sqrt{c+d x}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)/(x^2*(-c + d*x)^(3/2)*(c + d*x)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 11.3995, size = 56, normalized size = 0.84 \[ \frac{a}{c^{2} x \sqrt{- c + d x} \sqrt{c + d x}} - \frac{x \left (2 a d^{2} + b c^{2}\right )}{c^{4} \sqrt{- c + d x} \sqrt{c + d x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)/x**2/(d*x-c)**(3/2)/(d*x+c)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0849212, size = 51, normalized size = 0.76 \[ \frac{a \left (c^2-2 d^2 x^2\right )-b c^2 x^2}{c^4 x \sqrt{d x-c} \sqrt{c+d x}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)/(x^2*(-c + d*x)^(3/2)*(c + d*x)^(3/2)),x]
[Out]
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Maple [A] time = 0.007, size = 48, normalized size = 0.7 \[{\frac{-2\,a{d}^{2}{x}^{2}-b{c}^{2}{x}^{2}+a{c}^{2}}{x{c}^{4}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{dx-c}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)/x^2/(d*x-c)^(3/2)/(d*x+c)^(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((b*x^2 + a)/((d*x + c)^(3/2)*(d*x - c)^(3/2)*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.237032, size = 117, normalized size = 1.75 \[ \frac{b d x^{2} - \sqrt{d x + c} \sqrt{d x - c} b x + a d}{2 \, d^{4} x^{4} - 2 \, c^{2} d^{2} x^{2} -{\left (2 \, d^{3} x^{3} - c^{2} d x\right )} \sqrt{d x + c} \sqrt{d x - c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)/((d*x + c)^(3/2)*(d*x - c)^(3/2)*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 152.285, size = 165, normalized size = 2.46 \[ a \left (- \frac{d{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{7}{4}, \frac{9}{4}, 1 & \frac{3}{2}, \frac{5}{2}, 3 \\\frac{7}{4}, 2, \frac{9}{4}, \frac{5}{2}, 3 & 0 \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} c^{4}} + \frac{i d{G_{6, 6}^{2, 6}\left (\begin{matrix} \frac{1}{2}, 1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 1 & \\\frac{5}{4}, \frac{7}{4} & \frac{1}{2}, 1, 2, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} c^{4}}\right ) + b \left (- \frac{{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & \frac{1}{2}, \frac{3}{2}, 2 \\\frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2}, 2 & 0 \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} c^{2} d} + \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1 & \\\frac{1}{4}, \frac{3}{4} & - \frac{1}{2}, 0, 1, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} c^{2} d}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)/x**2/(d*x-c)**(3/2)/(d*x+c)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.277904, size = 296, normalized size = 4.42 \[ -\frac{{\left (b c^{2} + a d^{2}\right )} \sqrt{d x + c}}{2 \, \sqrt{d x - c} c^{4} d} - \frac{2 \,{\left (b c^{2}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + a d^{2}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 4 \, a c d^{2}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2} + 4 \, b c^{4} + 12 \, a c^{2} d^{2}\right )}}{{\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{6} + 2 \, c{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 4 \, c^{2}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2} + 8 \, c^{3}\right )} c^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)/((d*x + c)^(3/2)*(d*x - c)^(3/2)*x^2),x, algorithm="giac")
[Out]