Optimal. Leaf size=114 \[ \frac {2 \sqrt {-e^2+d f} \sqrt {a x} \sqrt {\frac {e (e+f x)}{e^2-d f}} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {d+e x}}{\sqrt {-e^2+d f}}\right )|1-\frac {e^2}{d f}\right )}{e \sqrt {f} \sqrt {-\frac {e x}{d}} \sqrt {e+f x}} \]
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Rubi [A]
time = 0.04, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {115, 114}
\begin {gather*} \frac {2 \sqrt {a x} \sqrt {d f-e^2} \sqrt {\frac {e (e+f x)}{e^2-d f}} E\left (\text {ArcSin}\left (\frac {\sqrt {f} \sqrt {d+e x}}{\sqrt {d f-e^2}}\right )|1-\frac {e^2}{d f}\right )}{e \sqrt {f} \sqrt {-\frac {e x}{d}} \sqrt {e+f x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 114
Rule 115
Rubi steps
\begin {align*} \int \frac {\sqrt {a x}}{\sqrt {d+e x} \sqrt {e+f x}} \, dx &=\frac {\left (\sqrt {a x} \sqrt {\frac {e (e+f x)}{e^2-d f}}\right ) \int \frac {\sqrt {-\frac {e x}{d}}}{\sqrt {d+e x} \sqrt {\frac {e^2}{e^2-d f}+\frac {e f x}{e^2-d f}}} \, dx}{\sqrt {-\frac {e x}{d}} \sqrt {e+f x}}\\ &=\frac {2 \sqrt {-e^2+d f} \sqrt {a x} \sqrt {\frac {e (e+f x)}{e^2-d f}} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {d+e x}}{\sqrt {-e^2+d f}}\right )|1-\frac {e^2}{d f}\right )}{e \sqrt {f} \sqrt {-\frac {e x}{d}} \sqrt {e+f x}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.91, size = 106, normalized size = 0.93 \begin {gather*} -\frac {2 i e \sqrt {a x} \sqrt {1+\frac {f x}{e}} \left (E\left (i \sinh ^{-1}\left (\sqrt {\frac {e x}{d}}\right )|\frac {d f}{e^2}\right )-F\left (i \sinh ^{-1}\left (\sqrt {\frac {e x}{d}}\right )|\frac {d f}{e^2}\right )\right )}{f \sqrt {\frac {e x}{d+e x}} \sqrt {d+e x} \sqrt {e+f x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 191, normalized size = 1.68
method | result | size |
default | \(-\frac {2 \left (d \EllipticF \left (\sqrt {\frac {f x +e}{e}}, \sqrt {-\frac {e^{2}}{d f -e^{2}}}\right ) f -\EllipticE \left (\sqrt {\frac {f x +e}{e}}, \sqrt {-\frac {e^{2}}{d f -e^{2}}}\right ) d f +\EllipticE \left (\sqrt {\frac {f x +e}{e}}, \sqrt {-\frac {e^{2}}{d f -e^{2}}}\right ) e^{2}\right ) \sqrt {-\frac {f x}{e}}\, \sqrt {\frac {\left (e x +d \right ) f}{d f -e^{2}}}\, \sqrt {\frac {f x +e}{e}}\, \sqrt {a x}\, \sqrt {e x +d}\, \sqrt {f x +e}}{f^{2} x \left (e f \,x^{2}+d f x +e^{2} x +d e \right )}\) | \(191\) |
elliptic | \(\frac {2 \sqrt {a x}\, \sqrt {\left (e x +d \right ) \left (f x +e \right ) a x}\, e \sqrt {\frac {\left (x +\frac {e}{f}\right ) f}{e}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {e}{f}+\frac {d}{e}}}\, \sqrt {-\frac {f x}{e}}\, \left (\left (-\frac {e}{f}+\frac {d}{e}\right ) \EllipticE \left (\sqrt {\frac {\left (x +\frac {e}{f}\right ) f}{e}}, \sqrt {-\frac {e}{f \left (-\frac {e}{f}+\frac {d}{e}\right )}}\right )-\frac {d \EllipticF \left (\sqrt {\frac {\left (x +\frac {e}{f}\right ) f}{e}}, \sqrt {-\frac {e}{f \left (-\frac {e}{f}+\frac {d}{e}\right )}}\right )}{e}\right )}{\sqrt {e x +d}\, \sqrt {f x +e}\, x f \sqrt {a e f \,x^{3}+a d f \,x^{2}+a \,e^{2} x^{2}+a d e x}}\) | \(215\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.13, size = 250, normalized size = 2.19 \begin {gather*} -\frac {2 \, {\left (\sqrt {a f} {\left (d f + e^{2}\right )} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} f^{2} - d f e^{2} + e^{4}\right )} e^{\left (-2\right )}}{3 \, f^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} - 3 \, d^{2} f^{2} e^{2} - 3 \, d f e^{4} + 2 \, e^{6}\right )} e^{\left (-3\right )}}{27 \, f^{3}}, \frac {{\left (3 \, f x e + d f + e^{2}\right )} e^{\left (-1\right )}}{3 \, f}\right ) + 3 \, \sqrt {a f} f e^{\frac {3}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (d^{2} f^{2} - d f e^{2} + e^{4}\right )} e^{\left (-2\right )}}{3 \, f^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} - 3 \, d^{2} f^{2} e^{2} - 3 \, d f e^{4} + 2 \, e^{6}\right )} e^{\left (-3\right )}}{27 \, f^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} f^{2} - d f e^{2} + e^{4}\right )} e^{\left (-2\right )}}{3 \, f^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} - 3 \, d^{2} f^{2} e^{2} - 3 \, d f e^{4} + 2 \, e^{6}\right )} e^{\left (-3\right )}}{27 \, f^{3}}, \frac {{\left (3 \, f x e + d f + e^{2}\right )} e^{\left (-1\right )}}{3 \, f}\right )\right )\right )} e^{\left (-2\right )}}{3 \, f^{2}} \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 {\sqrt {a x}}{\sqrt {d + e x} \sqrt {e + f x}}\, dx \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*} \text {could not integrate} \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 {\sqrt {a\,x}}{\sqrt {e+f\,x}\,\sqrt {d+e\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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