Optimal. Leaf size=96 \[ \frac {2 \sqrt {a} \sqrt {c+d x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-e} \sqrt {a+b x}}{\sqrt {a}}\right )|-\frac {a d}{(b c-a d) (1-e)}\right )}{b \sqrt {1-e} \sqrt {\frac {b (c+d x)}{b c-a d}}} \]
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Rubi [A] time = 0.06, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {114, 113} \[ \frac {2 \sqrt {a} \sqrt {c+d x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-e} \sqrt {a+b x}}{\sqrt {a}}\right )|-\frac {a d}{(b c-a d) (1-e)}\right )}{b \sqrt {1-e} \sqrt {\frac {b (c+d x)}{b c-a d}}} \]
Antiderivative was successfully verified.
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Rule 113
Rule 114
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x}}{\sqrt {a+b x} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx &=\frac {\left (\sqrt {c+d x} \sqrt {\frac {b \left (e+\frac {b (-1+e) x}{a}\right )}{-b (-1+e)+b e}}\right ) \int \frac {\sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}}{\sqrt {a+b x} \sqrt {\frac {b e}{-b (-1+e)+b e}+\frac {b^2 (-1+e) x}{a (-b (-1+e)+b e)}}} \, dx}{\sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+\frac {b (-1+e) x}{a}}}\\ &=\frac {2 \sqrt {a} \sqrt {c+d x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-e} \sqrt {a+b x}}{\sqrt {a}}\right )|-\frac {a d}{(b c-a d) (1-e)}\right )}{b \sqrt {1-e} \sqrt {\frac {b (c+d x)}{b c-a d}}}\\ \end {align*}
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Mathematica [B] time = 1.38, size = 200, normalized size = 2.08 \[ \frac {2 \sqrt {\frac {\frac {a}{a+b x}+e-1}{e-1}} \left (b \sqrt {a+b x} (c+d x) \sqrt {a-\frac {b c}{d}} \sqrt {\frac {a e+b (e-1) x}{(e-1) (a+b x)}}-(a+b x) (b c-a d) \sqrt {\frac {b (c+d x)}{d (a+b x)}} E\left (\sin ^{-1}\left (\frac {\sqrt {a-\frac {b c}{d}}}{\sqrt {a+b x}}\right )|\frac {a d}{(b c-a d) (e-1)}\right )\right )}{b^2 \sqrt {c+d x} \sqrt {a-\frac {b c}{d}} \sqrt {\frac {b (e-1) x}{a}+e}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x + a} \sqrt {d x + c} a \sqrt {\frac {a e + {\left (b e - b\right )} x}{a}}}{a^{2} e + {\left (b^{2} e - b^{2}\right )} x^{2} + {\left (2 \, a b e - a b\right )} x}, x\right ) \]
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 \[ \int \frac {\sqrt {d x + c}}{\sqrt {b x + a} \sqrt {\frac {b {\left (e - 1\right )} x}{a} + e}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 822, normalized size = 8.56 \[ -\frac {2 \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {\frac {\left (b e x +a e -b x \right ) d}{a d e -b c e +b c}}\, \sqrt {-\frac {\left (b x +a \right ) \left (e -1\right )}{a}}\, \sqrt {-\frac {\left (d x +c \right ) \left (e -1\right ) b}{a d e -b c e +b c}}\, \left (a^{2} d^{2} e^{2} \EllipticF \left (\sqrt {\frac {\left (b e x +a e -b x \right ) d}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{a d}}\right )-2 a b c d \,e^{2} \EllipticF \left (\sqrt {\frac {\left (b e x +a e -b x \right ) d}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{a d}}\right )+b^{2} c^{2} e^{2} \EllipticF \left (\sqrt {\frac {\left (b e x +a e -b x \right ) d}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{a d}}\right )+a^{2} d^{2} e \EllipticE \left (\sqrt {\frac {\left (b e x +a e -b x \right ) d}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{a d}}\right )-a^{2} d^{2} e \EllipticF \left (\sqrt {\frac {\left (b e x +a e -b x \right ) d}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{a d}}\right )-a b c d e \EllipticE \left (\sqrt {\frac {\left (b e x +a e -b x \right ) d}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{a d}}\right )+3 a b c d e \EllipticF \left (\sqrt {\frac {\left (b e x +a e -b x \right ) d}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{a d}}\right )-2 b^{2} c^{2} e \EllipticF \left (\sqrt {\frac {\left (b e x +a e -b x \right ) d}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{a d}}\right )+a b c d \EllipticE \left (\sqrt {\frac {\left (b e x +a e -b x \right ) d}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{a d}}\right )-a b c d \EllipticF \left (\sqrt {\frac {\left (b e x +a e -b x \right ) d}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{a d}}\right )+b^{2} c^{2} \EllipticF \left (\sqrt {\frac {\left (b e x +a e -b x \right ) d}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{a d}}\right )\right )}{\sqrt {\frac {b e x +a e -b x}{a}}\, \left (b d \,x^{2}+a d x +b c x +a c \right ) \left (e -1\right )^{2} b^{2} d} \]
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 \[ \int \frac {\sqrt {d x + c}}{\sqrt {b x + a} \sqrt {\frac {b {\left (e - 1\right )} x}{a} + e}}\,{d x} \]
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 \[ \int \frac {\sqrt {c+d\,x}}{\sqrt {e+\frac {b\,x\,\left (e-1\right )}{a}}\,\sqrt {a+b\,x}} \,d x \]
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 \[ \int \frac {\sqrt {c + d x}}{\sqrt {a + b x} \sqrt {e + \frac {b e x}{a} - \frac {b x}{a}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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