Optimal. Leaf size=96 \[ \frac {2 \sqrt {a} \sqrt {\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {1-e} \sqrt {a+b x}}{\sqrt {a}}\right ),-\frac {a d}{(1-e) (b c-a d)}\right )}{b \sqrt {1-e} \sqrt {c+d x}} \]
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Rubi [A] time = 0.07, 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 = {121, 119} \[ \frac {2 \sqrt {a} \sqrt {\frac {b (c+d x)}{b c-a d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-e} \sqrt {a+b x}}{\sqrt {a}}\right )|-\frac {a d}{(b c-a d) (1-e)}\right )}{b \sqrt {1-e} \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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Rule 119
Rule 121
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx &=\frac {\sqrt {\frac {b (c+d x)}{b c-a d}} \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx}{\sqrt {c+d x}}\\ &=\frac {2 \sqrt {a} \sqrt {\frac {b (c+d x)}{b c-a d}} F\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 {c+d x}}\\ \end {align*}
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Mathematica [A] time = 0.53, size = 126, normalized size = 1.31 \[ -\frac {2 \sqrt {c+d x} \sqrt {\frac {\frac {a}{a+b x}+e-1}{e-1}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {-\frac {a}{e-1}}}{\sqrt {a+b x}}\right ),\frac {(e-1) (b c-a d)}{a d}\right )}{d \sqrt {-\frac {a}{e-1}} \sqrt {\frac {b (e-1) x}{a}+e} \sqrt {\frac {b (c+d x)}{d (a+b x)}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.11, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x + a} \sqrt {d x + c} a \sqrt {\frac {a e + {\left (b e - b\right )} x}{a}}}{a^{2} c e + {\left (b^{2} d e - b^{2} d\right )} x^{3} - {\left (b^{2} c + a b d - {\left (b^{2} c + 2 \, a b d\right )} e\right )} x^{2} - {\left (a b c - {\left (2 \, a b c + a^{2} d\right )} e\right )} x}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b {\left (e - 1\right )} x}{a} + e}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 207, normalized size = 2.16 \[ \frac {2 \sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {\frac {\left (b e x +a e -b x \right ) d}{a d e -b c e +b c}}\, \sqrt {-\frac {\left (b x +a \right ) \left (e -1\right )}{a}}\, \sqrt {-\frac {\left (d x +c \right ) \left (e -1\right ) b}{a d e -b c e +b c}}\, \left (a d e -b c e +b c \right ) \EllipticF \left (\sqrt {\frac {\left (b e x +a e -b x \right ) d}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{a d}}\right )}{\sqrt {\frac {b e x +a e -b x}{a}}\, \left (b d \,x^{2}+a d x +b c x +a c \right ) \left (e -1\right ) b d} \]
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 \[ \int \frac {1}{\sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b {\left (e - 1\right )} x}{a} + e}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {e+\frac {b\,x\,\left (e-1\right )}{a}}\,\sqrt {a+b\,x}\,\sqrt {c+d\,x}} \,d x \]
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 \[ \int \frac {1}{\sqrt {a + b x} \sqrt {c + d x} \sqrt {e + \frac {b e x}{a} - \frac {b x}{a}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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