Optimal. Leaf size=63 \[ \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)} \]
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Rubi [A] time = 0.06, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {860} \[ \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)} \]
Antiderivative was successfully verified.
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Rule 860
Rubi steps
\begin {align*} \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{5/2}} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 52, normalized size = 0.83 \[ \frac {2 ((d+e x) (a e+c d x))^{3/2}}{3 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.95, size = 169, normalized size = 2.68 \[ \frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d x + a e\right )} \sqrt {e x + d} \sqrt {g x + f}}{3 \, {\left (c d^{2} f^{3} - a d e f^{2} g + {\left (c d e f g^{2} - a e^{2} g^{3}\right )} x^{3} + {\left (2 \, c d e f^{2} g - a d e g^{3} + {\left (c d^{2} - 2 \, a e^{2}\right )} f g^{2}\right )} x^{2} + {\left (c d e f^{3} - 2 \, a d e f g^{2} + {\left (2 \, c d^{2} - a e^{2}\right )} f^{2} g\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 0.01, size = 63, normalized size = 1.00 \[ -\frac {2 \left (c d x +a e \right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{3 \left (g x +f \right )^{\frac {3}{2}} \left (a e g -c d f \right ) \sqrt {e x +d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{\sqrt {e x + d} {\left (g x + f\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.92, size = 136, normalized size = 2.16 \[ -\frac {\left (\frac {2\,a\,e}{3\,a\,e\,g^2-3\,c\,d\,f\,g}+\frac {2\,c\,d\,x}{3\,a\,e\,g^2-3\,c\,d\,f\,g}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}-\frac {\sqrt {f+g\,x}\,\left (3\,c\,d\,f^2-3\,a\,e\,f\,g\right )\,\sqrt {d+e\,x}}{3\,a\,e\,g^2-3\,c\,d\,f\,g}} \]
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 \[ \int \frac {\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}{\sqrt {d + e x} \left (f + g x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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