Optimal. Leaf size=43 \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a-b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} \sqrt {d}} \]
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Rubi [A] time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {63, 217, 203} \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a-b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} \sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 217
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a-b x} \sqrt {c+d x}} \, dx &=-\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\frac {a d}{b}-\frac {d x^2}{b}}} \, dx,x,\sqrt {a-b x}\right )}{b}\\ &=-\frac {2 \operatorname {Subst}\left (\int \frac {1}{1+\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a-b x}}{\sqrt {c+d x}}\right )}{b}\\ &=-\frac {2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a-b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} \sqrt {d}}\\ \end {align*}
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Mathematica [B] time = 0.08, size = 103, normalized size = 2.40 \begin {gather*} \frac {2 \sqrt {-b} \sqrt {-a d-b c} \sqrt {\frac {b (c+d x)}{a d+b c}} \sin ^{-1}\left (\frac {\sqrt {-b} \sqrt {d} \sqrt {a-b x}}{\sqrt {b} \sqrt {-a d-b c}}\right )}{b^{3/2} \sqrt {d} \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 43, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a-b x}}\right )}{\sqrt {b} \sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.73, size = 185, normalized size = 4.30 \begin {gather*} \left [-\frac {\sqrt {-b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c - a d\right )} \sqrt {-b d} \sqrt {-b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d - a b d^{2}\right )} x\right )}{2 \, b d}, -\frac {\sqrt {b d} \arctan \left (\frac {{\left (2 \, b d x + b c - a d\right )} \sqrt {b d} \sqrt {-b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} - a b c d + {\left (b^{2} c d - a b d^{2}\right )} x\right )}}\right )}{b d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.16, size = 54, normalized size = 1.26 \begin {gather*} \frac {2 \, b \log \left ({\left | -\sqrt {-b d} \sqrt {-b x + a} + \sqrt {b^{2} c + {\left (b x - a\right )} b d + a b d} \right |}\right )}{\sqrt {-b d} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 84, normalized size = 1.95 \begin {gather*} \frac {\sqrt {\left (-b x +a \right ) \left (d x +c \right )}\, \arctan \left (\frac {\sqrt {b d}\, \left (x -\frac {a d -b c}{2 b d}\right )}{\sqrt {-b d \,x^{2}+a c +\left (a d -b c \right ) x}}\right )}{\sqrt {-b x +a}\, \sqrt {d x +c}\, \sqrt {b d}} \end {gather*}
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.34, size = 44, normalized size = 1.02 \begin {gather*} -\frac {4\,\mathrm {atan}\left (\frac {d\,\left (\sqrt {a-b\,x}-\sqrt {a}\right )}{\sqrt {b\,d}\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}\right )}{\sqrt {b\,d}} \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 {a - b x} \sqrt {c + d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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