Optimal. Leaf size=88 \[ -\frac {294}{625} \sqrt {1-2 x}+\frac {21}{125} \sqrt {1-2 x} (2+3 x)^2-\frac {\sqrt {1-2 x} (2+3 x)^3}{5 (3+5 x)}-\frac {196 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{625 \sqrt {55}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {99, 158, 12, 81,
65, 212} \begin {gather*} -\frac {\sqrt {1-2 x} (3 x+2)^3}{5 (5 x+3)}+\frac {21}{125} \sqrt {1-2 x} (3 x+2)^2-\frac {294}{625} \sqrt {1-2 x}-\frac {196 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{625 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 65
Rule 81
Rule 99
Rule 158
Rule 212
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (2+3 x)^3}{(3+5 x)^2} \, dx &=-\frac {\sqrt {1-2 x} (2+3 x)^3}{5 (3+5 x)}+\frac {1}{5} \int \frac {(7-21 x) (2+3 x)^2}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {21}{125} \sqrt {1-2 x} (2+3 x)^2-\frac {\sqrt {1-2 x} (2+3 x)^3}{5 (3+5 x)}-\frac {1}{125} \int -\frac {98 (2+3 x)}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {21}{125} \sqrt {1-2 x} (2+3 x)^2-\frac {\sqrt {1-2 x} (2+3 x)^3}{5 (3+5 x)}+\frac {98}{125} \int \frac {2+3 x}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {294}{625} \sqrt {1-2 x}+\frac {21}{125} \sqrt {1-2 x} (2+3 x)^2-\frac {\sqrt {1-2 x} (2+3 x)^3}{5 (3+5 x)}+\frac {98}{625} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {294}{625} \sqrt {1-2 x}+\frac {21}{125} \sqrt {1-2 x} (2+3 x)^2-\frac {\sqrt {1-2 x} (2+3 x)^3}{5 (3+5 x)}-\frac {98}{625} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {294}{625} \sqrt {1-2 x}+\frac {21}{125} \sqrt {1-2 x} (2+3 x)^2-\frac {\sqrt {1-2 x} (2+3 x)^3}{5 (3+5 x)}-\frac {196 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{625 \sqrt {55}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.12, size = 63, normalized size = 0.72 \begin {gather*} \frac {\sqrt {1-2 x} \left (-622-90 x+2385 x^2+1350 x^3\right )}{625 (3+5 x)}-\frac {196 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{625 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.12, size = 63, normalized size = 0.72
method | result | size |
risch | \(-\frac {2700 x^{4}+3420 x^{3}-2565 x^{2}-1154 x +622}{625 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {196 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{34375}\) | \(56\) |
derivativedivides | \(\frac {27 \left (1-2 x \right )^{\frac {5}{2}}}{250}-\frac {117 \left (1-2 x \right )^{\frac {3}{2}}}{250}+\frac {18 \sqrt {1-2 x}}{625}+\frac {2 \sqrt {1-2 x}}{3125 \left (-\frac {6}{5}-2 x \right )}-\frac {196 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{34375}\) | \(63\) |
default | \(\frac {27 \left (1-2 x \right )^{\frac {5}{2}}}{250}-\frac {117 \left (1-2 x \right )^{\frac {3}{2}}}{250}+\frac {18 \sqrt {1-2 x}}{625}+\frac {2 \sqrt {1-2 x}}{3125 \left (-\frac {6}{5}-2 x \right )}-\frac {196 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{34375}\) | \(63\) |
trager | \(\frac {\left (1350 x^{3}+2385 x^{2}-90 x -622\right ) \sqrt {1-2 x}}{1875+3125 x}-\frac {98 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{34375}\) | \(78\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.52, size = 80, normalized size = 0.91 \begin {gather*} \frac {27}{250} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {117}{250} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {98}{34375} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {18}{625} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{625 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.94, size = 69, normalized size = 0.78 \begin {gather*} \frac {98 \, \sqrt {55} {\left (5 \, x + 3\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \, {\left (1350 \, x^{3} + 2385 \, x^{2} - 90 \, x - 622\right )} \sqrt {-2 \, x + 1}}{34375 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 59.48, size = 216, normalized size = 2.45 \begin {gather*} \frac {27 \left (1 - 2 x\right )^{\frac {5}{2}}}{250} - \frac {117 \left (1 - 2 x\right )^{\frac {3}{2}}}{250} + \frac {18 \sqrt {1 - 2 x}}{625} - \frac {44 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{625} + \frac {194 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: x < - \frac {3}{5} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: x > - \frac {3}{5} \end {cases}\right )}{625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.29, size = 90, normalized size = 1.02 \begin {gather*} \frac {27}{250} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {117}{250} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {98}{34375} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {18}{625} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{625 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.05, size = 64, normalized size = 0.73 \begin {gather*} \frac {18\,\sqrt {1-2\,x}}{625}-\frac {2\,\sqrt {1-2\,x}}{3125\,\left (2\,x+\frac {6}{5}\right )}-\frac {117\,{\left (1-2\,x\right )}^{3/2}}{250}+\frac {27\,{\left (1-2\,x\right )}^{5/2}}{250}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,196{}\mathrm {i}}{34375} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________