Optimal. Leaf size=61 \[ -\frac {2 \sqrt {d+e x} \sqrt {f+g x}}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)} \]
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Rubi [A] time = 0.07, antiderivative size = 61, 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} \[ -\frac {2 \sqrt {d+e x} \sqrt {f+g x}}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)} \]
Antiderivative was successfully verified.
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Rule 860
Rubi steps
\begin {align*} \int \frac {(d+e x)^{3/2}}{\sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=-\frac {2 \sqrt {d+e x} \sqrt {f+g x}}{(c d f-a e g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 50, normalized size = 0.82 \[ -\frac {2 \sqrt {d+e x} \sqrt {f+g x}}{\sqrt {(d+e x) (a e+c d x)} (c d f-a e g)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.98, size = 125, normalized size = 2.05 \[ -\frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f}}{a c d^{2} e f - a^{2} d e^{2} g + {\left (c^{2} d^{2} e f - a c d e^{2} g\right )} x^{2} + {\left ({\left (c^{2} d^{3} + a c d e^{2}\right )} f - {\left (a c d^{2} e + a^{2} e^{3}\right )} g\right )} x} \]
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 = 1.03 \[ \frac {2 \sqrt {g x +f}\, \left (c d x +a e \right ) \left (e x +d \right )^{\frac {3}{2}}}{\left (a e g -c d f \right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}} \]
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 \[ \int \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} \sqrt {g x + f}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.68, size = 147, normalized size = 2.41 \[ \frac {\left (\frac {2\,f\,\sqrt {d+e\,x}}{c\,d\,e\,\left (a\,e\,g-c\,d\,f\right )}+\frac {2\,g\,x\,\sqrt {d+e\,x}}{c\,d\,e\,\left (a\,e\,g-c\,d\,f\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x^2\,\sqrt {f+g\,x}+\frac {a\,\sqrt {f+g\,x}}{c}+\frac {x\,\sqrt {f+g\,x}\,\left (c\,d^2+a\,e^2\right )}{c\,d\,e}} \]
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 \[ \int \frac {\left (d + e x\right )^{\frac {3}{2}}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \sqrt {f + g x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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