Optimal. Leaf size=59 \[ \frac {(1-2 x)^{3/2}}{21 (3 x+2)}+\frac {8}{7} \sqrt {1-2 x}-\frac {8 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}} \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {78, 50, 63, 206} \begin {gather*} \frac {(1-2 x)^{3/2}}{21 (3 x+2)}+\frac {8}{7} \sqrt {1-2 x}-\frac {8 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 78
Rule 206
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (3+5 x)}{(2+3 x)^2} \, dx &=\frac {(1-2 x)^{3/2}}{21 (2+3 x)}+\frac {12}{7} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx\\ &=\frac {8}{7} \sqrt {1-2 x}+\frac {(1-2 x)^{3/2}}{21 (2+3 x)}+4 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {8}{7} \sqrt {1-2 x}+\frac {(1-2 x)^{3/2}}{21 (2+3 x)}-4 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {8}{7} \sqrt {1-2 x}+\frac {(1-2 x)^{3/2}}{21 (2+3 x)}-\frac {8 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 55, normalized size = 0.93 \begin {gather*} \frac {7 \sqrt {1-2 x} (10 x+7)-8 \sqrt {21} (3 x+2) \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{63 x+42} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.12, size = 59, normalized size = 1.00 \begin {gather*} \frac {2 (5 (1-2 x)-12) \sqrt {1-2 x}}{3 (3 (1-2 x)-7)}-\frac {8 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.20, size = 59, normalized size = 1.00 \begin {gather*} \frac {4 \, \sqrt {21} {\left (3 \, x + 2\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 7 \, {\left (10 \, x + 7\right )} \sqrt {-2 \, x + 1}}{21 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.20, size = 65, normalized size = 1.10 \begin {gather*} \frac {4}{21} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {10}{9} \, \sqrt {-2 \, x + 1} + \frac {\sqrt {-2 \, x + 1}}{9 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 45, normalized size = 0.76 \begin {gather*} -\frac {8 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{21}+\frac {10 \sqrt {-2 x +1}}{9}-\frac {2 \sqrt {-2 x +1}}{27 \left (-2 x -\frac {4}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.21, size = 62, normalized size = 1.05 \begin {gather*} \frac {4}{21} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {10}{9} \, \sqrt {-2 \, x + 1} + \frac {\sqrt {-2 \, x + 1}}{9 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.06, size = 44, normalized size = 0.75 \begin {gather*} \frac {2\,\sqrt {1-2\,x}}{27\,\left (2\,x+\frac {4}{3}\right )}-\frac {8\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{21}+\frac {10\,\sqrt {1-2\,x}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 112.41, size = 178, normalized size = 3.02 \begin {gather*} \frac {10 \sqrt {1 - 2 x}}{9} + \frac {28 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {2}{3} \end {cases}\right )}{9} + \frac {74 \left (\begin {cases} - \frac {\sqrt {21} \operatorname {acoth}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: 2 x - 1 < - \frac {7}{3} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: 2 x - 1 > - \frac {7}{3} \end {cases}\right )}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________