Optimal. Leaf size=125 \[ \frac {2 g \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d e}-\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (2 a e^2 g-c d (3 e f-d g)\right )}{3 c^2 d^2 e \sqrt {d+e x}} \]
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Rubi [A] time = 0.09, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {794, 648} \[ \frac {2 g \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d e}-\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (2 a e^2 g-c d (3 e f-d g)\right )}{3 c^2 d^2 e \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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Rule 648
Rule 794
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x} (f+g x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {2 g \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d e}+\frac {1}{3} \left (3 f-\frac {d g}{e}-\frac {2 a e g}{c d}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\\ &=-\frac {2 \left (2 a e^2 g-c d (3 e f-d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2 e \sqrt {d+e x}}+\frac {2 g \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d e}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 53, normalized size = 0.42 \[ \frac {2 \sqrt {(d+e x) (a e+c d x)} (c d (3 f+g x)-2 a e g)}{3 c^2 d^2 \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 71, normalized size = 0.57 \[ \frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d g x + 3 \, c d f - 2 \, a e g\right )} \sqrt {e x + d}}{3 \, {\left (c^{2} d^{2} e x + c^{2} d^{3}\right )}} \]
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 \[ \int \frac {\sqrt {e x + d} {\left (g x + f\right )}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 67, normalized size = 0.54 \[ -\frac {2 \left (c d x +a e \right ) \left (-c d g x +2 a e g -3 c d f \right ) \sqrt {e x +d}}{3 \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 65, normalized size = 0.52 \[ \frac {2 \, \sqrt {c d x + a e} f}{c d} + \frac {2 \, {\left (c^{2} d^{2} x^{2} - a c d e x - 2 \, a^{2} e^{2}\right )} g}{3 \, \sqrt {c d x + a e} c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.23, size = 88, normalized size = 0.70 \[ -\frac {\left (\frac {\left (4\,a\,e\,g-6\,c\,d\,f\right )\,\sqrt {d+e\,x}}{3\,c^2\,d^2\,e}-\frac {2\,g\,x\,\sqrt {d+e\,x}}{3\,c\,d\,e}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x+\frac {d}{e}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d + e x} \left (f + g x\right )}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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