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 = 0.09, antiderivative size = 53, normalized size of antiderivative = 1.00, 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*}
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Mathematica [C] time = 0.31, size = 244, normalized size = 4.60 \[ \frac {\sqrt {1-\frac {b^2 x^2}{a^2}} \left (a^2 d+b^2 c\right ) x^{-\frac {b^2 c}{a^2 d+b^2 c}} \left (b^2 d x^2 \, _2F_1\left (\frac {1}{2},\frac {2 d a^2+b^2 c}{2 d a^2+2 b^2 c};\frac {4 d a^2+3 b^2 c}{2 d a^2+2 b^2 c};\frac {b^2 x^2}{a^2}\right )-\left (2 a^2 d+b^2 c\right ) \, _2F_1\left (\frac {1}{2},-\frac {b^2 c}{2 \left (d a^2+b^2 c\right )};\frac {2 d a^2+b^2 c}{2 d a^2+2 b^2 c};\frac {b^2 x^2}{a^2}\right )\right )}{b^2 \sqrt {b x-a} \sqrt {a+b x} \left (2 a^2 d+b^2 c\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 65, normalized size = 1.23 \[ \frac {{\left (b^{2} c + a^{2} d\right )} \sqrt {b x + a} \sqrt {b x - a} x}{a^{2} b^{2} x^{\frac {2 \, b^{2} c + a^{2} d}{b^{2} c + a^{2} d}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {d x^{2} + c}{\sqrt {b x + a} \sqrt {b x - a} x^{\frac {2 \, b^{2} c + a^{2} d}{b^{2} c + a^{2} d}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 66, normalized size = 1.25 \[ \frac {\left (a^{2} d +b^{2} c \right ) \sqrt {b x +a}\, \sqrt {b x -a}\, x \,x^{-\frac {a^{2} d +2 b^{2} c}{a^{2} d +b^{2} c}}}{a^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.24, size = 79, normalized size = 1.49 \[ \frac {{\left (b^{2} c + a^{2} d\right )} \sqrt {b x + a} \sqrt {b x - a} x e^{\left (-\frac {2 \, b^{2} c \log \relax (x)}{b^{2} c + a^{2} d} - \frac {a^{2} d \log \relax (x)}{b^{2} c + a^{2} d}\right )}}{a^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.27, size = 96, normalized size = 1.81 \[ -\frac {\frac {x\,\left (d\,a^4+c\,a^2\,b^2\right )}{a^2\,b^2}-\frac {x^3\,\left (d\,a^2\,b^2+c\,b^4\right )}{a^2\,b^2}}{x^{\frac {d\,a^2+2\,c\,b^2}{d\,a^2+c\,b^2}}\,\sqrt {a+b\,x}\,\sqrt {b\,x-a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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