Optimal. Leaf size=42 \[ \frac {2 \sin ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {b (1-c)}{d}+b x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {d}} \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {63, 216} \begin {gather*} \frac {2 \sin ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {b (1-c)}{d}+b x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 216
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\frac {b-b c}{d}+b x} \sqrt {c-d x}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\frac {b-b c}{b}-\frac {d x^2}{b}}} \, dx,x,\sqrt {\frac {b-b c}{d}+b x}\right )}{b}\\ &=\frac {2 \sin ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {b (1-c)}{d}+b x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {d}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 67, normalized size = 1.60 \begin {gather*} \frac {2 \sqrt {-d} \sqrt {-c+d x+1} \sinh ^{-1}\left (\frac {\sqrt {-d} \sqrt {c-d x}}{\sqrt {d}}\right )}{d^{3/2} \sqrt {\frac {b (-c+d x+1)}{d}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.12, size = 61, normalized size = 1.45 \begin {gather*} -\frac {2 \sqrt {-\frac {b}{d}} \log \left (\sqrt {\frac {b}{d}-\frac {b (c-d x)}{d}}-\sqrt {-\frac {b}{d}} \sqrt {c-d x}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.85, size = 176, normalized size = 4.19 \begin {gather*} \left [-\frac {\sqrt {-b d} \log \left (8 \, b d^{2} x^{2} + 8 \, b c^{2} - 8 \, {\left (2 \, b c - b\right )} d x - 4 \, \sqrt {-b d} {\left (2 \, d x - 2 \, c + 1\right )} \sqrt {-d x + c} \sqrt {\frac {b d x - b c + b}{d}} - 8 \, b c + b\right )}{2 \, b d}, -\frac {\sqrt {b d} \arctan \left (\frac {\sqrt {b d} {\left (2 \, d x - 2 \, c + 1\right )} \sqrt {-d x + c} \sqrt {\frac {b d x - b c + b}{d}}}{2 \, {\left (b d^{2} x^{2} + b c^{2} - {\left (2 \, b c - b\right )} d x - b c\right )}}\right )}{b d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.08, size = 58, normalized size = 1.38 \begin {gather*} -\frac {2 \, b \log \left (-\sqrt {-b d} \sqrt {\frac {b d x - b c + b}{d}} + \sqrt {-{\left (b d x - b c + b\right )} b + b^{2}}\right )}{\sqrt {-b d} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.04, size = 118, normalized size = 2.81 \begin {gather*} \frac {\sqrt {\left (b x +\frac {\left (-c +1\right ) b}{d}\right ) \left (-d x +c \right )}\, \arctan \left (\frac {\sqrt {b d}\, \left (x -\frac {b c -\left (-c +1\right ) b}{2 b d}\right )}{\sqrt {-b d \,x^{2}+\frac {\left (-c +1\right ) b c}{d}+\left (b c -\left (-c +1\right ) b \right ) x}}\right )}{\sqrt {b x +\frac {\left (-c +1\right ) b}{d}}\, \sqrt {-d x +c}\, \sqrt {b d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.51, size = 63, normalized size = 1.50 \begin {gather*} -\frac {4\,\mathrm {atan}\left (-\frac {d\,\left (\sqrt {\frac {b-b\,c}{d}+b\,x}-\sqrt {\frac {b-b\,c}{d}}\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.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {b \left (- \frac {c}{d} + x + \frac {1}{d}\right )} \sqrt {c - d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________