Optimal. Leaf size=53 \[ \sqrt{b x-a} \sqrt{a+b x} \left (\frac{c}{a^2}+\frac{d}{b^2}\right ) x^{-\frac{b^2 c}{a^2 d+b^2 c}} \]
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Rubi [A] time = 0.0907204, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 57, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.018, Rules used = {450} \[ \sqrt{b x-a} \sqrt{a+b x} \left (\frac{c}{a^2}+\frac{d}{b^2}\right ) x^{-\frac{b^2 c}{a^2 d+b^2 c}} \]
Antiderivative was successfully verified.
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Rule 450
Rubi steps
\begin{align*} \int \frac{x^{-\frac{2 b^2 c+a^2 d}{b^2 c+a^2 d}} \left (c+d x^2\right )}{\sqrt{-a+b x} \sqrt{a+b x}} \, dx &=\left (\frac{c}{a^2}+\frac{d}{b^2}\right ) x^{-\frac{b^2 c}{b^2 c+a^2 d}} \sqrt{-a+b x} \sqrt{a+b x}\\ \end{align*}
Mathematica [C] time = 0.258403, size = 244, normalized size = 4.6 \[ \frac{\sqrt{1-\frac{b^2 x^2}{a^2}} \left (a^2 d+b^2 c\right ) x^{-\frac{b^2 c}{a^2 d+b^2 c}} \left (b^2 d x^2 \, _2F_1\left (\frac{1}{2},\frac{2 d a^2+b^2 c}{2 d a^2+2 b^2 c};\frac{4 d a^2+3 b^2 c}{2 d a^2+2 b^2 c};\frac{b^2 x^2}{a^2}\right )-\left (2 a^2 d+b^2 c\right ) \, _2F_1\left (\frac{1}{2},-\frac{b^2 c}{2 \left (d a^2+b^2 c\right )};\frac{2 d a^2+b^2 c}{2 d a^2+2 b^2 c};\frac{b^2 x^2}{a^2}\right )\right )}{b^2 \sqrt{b x-a} \sqrt{a+b x} \left (2 a^2 d+b^2 c\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 66, normalized size = 1.3 \begin{align*}{\frac{x \left ({a}^{2}d+{b}^{2}c \right ) }{{b}^{2}{a}^{2}}\sqrt{bx+a}\sqrt{bx-a} \left ({x}^{{\frac{{a}^{2}d+2\,{b}^{2}c}{{a}^{2}d+{b}^{2}c}}} \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.61435, size = 107, normalized size = 2.02 \begin{align*} \frac{{\left (b^{2} c + a^{2} d\right )} \sqrt{b x + a} \sqrt{b x - a} x e^{\left (-\frac{2 \, b^{2} c \log \left (x\right )}{b^{2} c + a^{2} d} - \frac{a^{2} d \log \left (x\right )}{b^{2} c + a^{2} d}\right )}}{a^{2} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79243, size = 128, normalized size = 2.42 \begin{align*} \frac{{\left (b^{2} c + a^{2} d\right )} \sqrt{b x + a} \sqrt{b x - a} x}{a^{2} b^{2} x^{\frac{2 \, b^{2} c + a^{2} d}{b^{2} c + a^{2} d}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{d x^{2} + c}{\sqrt{b x + a} \sqrt{b x - a} x^{\frac{2 \, b^{2} c + a^{2} d}{b^{2} c + a^{2} d}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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