Optimal. Leaf size=46 \[ -\frac {2 \sqrt {d+e x}}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
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Rubi [A] time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {648} \begin {gather*} -\frac {2 \sqrt {d+e x}}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rubi steps
\begin {align*} \int \frac {(d+e x)^{3/2}}{\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}}{c d \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 = 0.01, size = 35, normalized size = 0.76 \begin {gather*} -\frac {2 \sqrt {d+e x}}{c d \sqrt {(d+e x) (a e+c d x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 43, normalized size = 0.93 \begin {gather*} -\frac {2 (d+e x)^{3/2} (a e+c d x)}{c d ((d+e x) (a e+c d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 74, normalized size = 1.61 \begin {gather*} -\frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{c^{2} d^{2} e x^{2} + a c d^{2} e + {\left (c^{2} d^{3} + a c d e^{2}\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 50, normalized size = 1.09 \begin {gather*} -\frac {2 \left (c d x +a e \right ) \left (e x +d \right )^{\frac {3}{2}}}{\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}} c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 18, normalized size = 0.39 \begin {gather*} -\frac {2}{\sqrt {c d x + a e} c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.27, size = 82, normalized size = 1.78 \begin {gather*} -\frac {2\,\sqrt {d+e\,x}\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{c^2\,d^2\,e\,\left (\frac {a}{c}+x^2+\frac {x\,\left (c^2\,d^3+a\,c\,d\,e^2\right )}{c^2\,d^2\,e}\right )} \end {gather*}
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 \begin {gather*} \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}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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