Optimal. Leaf size=52 \[ \frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{(d+e x) \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.02, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {650} \[ \frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{(d+e x) \left (c d^2-a e^2\right )} \]
Antiderivative was successfully verified.
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Rule 650
Rubi steps
\begin {align*} \int \frac {1}{(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\left (c d^2-a e^2\right ) (d+e x)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 42, normalized size = 0.81 \[ \frac {2 (a e+c d x)}{\left (c d^2-a e^2\right ) \sqrt {(d+e x) (a e+c d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.24, size = 59, normalized size = 1.13 \[ \frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{c d^{3} - a d e^{2} + {\left (c d^{2} e - a e^{3}\right )} x} \]
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 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 51, normalized size = 0.98 \[ -\frac {2 \left (c d x +a e \right )}{\left (a \,e^{2}-c \,d^{2}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.64, size = 50, normalized size = 0.96 \[ -\frac {2\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{\left (a\,e^2-c\,d^2\right )\,\left (d+e\,x\right )} \]
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 \[ \int \frac {1}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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