Optimal. Leaf size=84 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{\sqrt {e} \sqrt {c d^2-a e^2}} \]
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Rubi [A] time = 0.04, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {660, 205} \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{\sqrt {e} \sqrt {c d^2-a e^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 660
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=(2 e) \operatorname {Subst}\left (\int \frac {1}{2 c d^2 e-e \left (c d^2+a e^2\right )+e^2 x^2} \, dx,x,\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}}\right )\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{\sqrt {e} \sqrt {c d^2-a e^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 97, normalized size = 1.15 \begin {gather*} \frac {2 \sqrt {d+e x} \sqrt {a e+c d x} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{\sqrt {e} \sqrt {c d^2-a e^2} \sqrt {(d+e x) (a e+c d x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.33, size = 115, normalized size = 1.37 \begin {gather*} -\frac {4 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {d+e x} \sqrt {c d^2-a e^2}}{e \sqrt {a e (d+e x)-\frac {c d^2 (d+e x)}{e}+\frac {c d (d+e x)^2}{e}}-\sqrt {c d e} (d+e x)}\right )}{\sqrt {e} \sqrt {c d^2-a e^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 235, normalized size = 2.80 \begin {gather*} \left [-\frac {\sqrt {-c d^{2} e + a e^{3}} \log \left (-\frac {c d e^{2} x^{2} + 2 \, a e^{3} x - c d^{3} + 2 \, a d e^{2} - 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {-c d^{2} e + a e^{3}} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{c d^{2} e - a e^{3}}, -\frac {2 \, \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {c d^{2} e - a e^{3}} \sqrt {e x + d}}{c d e^{2} x^{2} + a d e^{2} + {\left (c d^{2} e + a e^{3}\right )} x}\right )}{\sqrt {c d^{2} e - a e^{3}}}\right ] \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 91, normalized size = 1.08 \begin {gather*} -\frac {2 \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, \arctanh \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right )}{\sqrt {e x +d}\, \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {d+e\,x}\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \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 {1}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \sqrt {d + e x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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