Optimal. Leaf size=67 \[ \frac {343}{44 \sqrt {1-2 x}}+\frac {162}{25} \sqrt {1-2 x}-\frac {9}{20} (1-2 x)^{3/2}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{275 \sqrt {55}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {89, 45, 65, 212}
\begin {gather*} -\frac {9}{20} (1-2 x)^{3/2}+\frac {162}{25} \sqrt {1-2 x}+\frac {343}{44 \sqrt {1-2 x}}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{275 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 65
Rule 89
Rule 212
Rubi steps
\begin {align*} \int \frac {(2+3 x)^3}{(1-2 x)^{3/2} (3+5 x)} \, dx &=\int \left (\frac {343}{44 (1-2 x)^{3/2}}-\frac {513}{100 \sqrt {1-2 x}}-\frac {27 x}{10 \sqrt {1-2 x}}+\frac {1}{275 \sqrt {1-2 x} (3+5 x)}\right ) \, dx\\ &=\frac {343}{44 \sqrt {1-2 x}}+\frac {513}{100} \sqrt {1-2 x}+\frac {1}{275} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx-\frac {27}{10} \int \frac {x}{\sqrt {1-2 x}} \, dx\\ &=\frac {343}{44 \sqrt {1-2 x}}+\frac {513}{100} \sqrt {1-2 x}-\frac {1}{275} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {27}{10} \int \left (\frac {1}{2 \sqrt {1-2 x}}-\frac {1}{2} \sqrt {1-2 x}\right ) \, dx\\ &=\frac {343}{44 \sqrt {1-2 x}}+\frac {162}{25} \sqrt {1-2 x}-\frac {9}{20} (1-2 x)^{3/2}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{275 \sqrt {55}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.09, size = 51, normalized size = 0.76 \begin {gather*} \frac {3802-3069 x-495 x^2}{275 \sqrt {1-2 x}}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{275 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.13, size = 47, normalized size = 0.70
method | result | size |
risch | \(-\frac {495 x^{2}+3069 x -3802}{275 \sqrt {1-2 x}}-\frac {2 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15125}\) | \(39\) |
derivativedivides | \(-\frac {9 \left (1-2 x \right )^{\frac {3}{2}}}{20}-\frac {2 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15125}+\frac {343}{44 \sqrt {1-2 x}}+\frac {162 \sqrt {1-2 x}}{25}\) | \(47\) |
default | \(-\frac {9 \left (1-2 x \right )^{\frac {3}{2}}}{20}-\frac {2 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15125}+\frac {343}{44 \sqrt {1-2 x}}+\frac {162 \sqrt {1-2 x}}{25}\) | \(47\) |
trager | \(\frac {\left (495 x^{2}+3069 x -3802\right ) \sqrt {1-2 x}}{-275+550 x}-\frac {\RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{15125}\) | \(73\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.52, size = 64, normalized size = 0.96 \begin {gather*} -\frac {9}{20} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{15125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {162}{25} \, \sqrt {-2 \, x + 1} + \frac {343}{44 \, \sqrt {-2 \, x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.90, size = 63, normalized size = 0.94 \begin {gather*} \frac {\sqrt {55} {\left (2 \, x - 1\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \, {\left (495 \, x^{2} + 3069 \, x - 3802\right )} \sqrt {-2 \, x + 1}}{15125 \, {\left (2 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 27.36, size = 95, normalized size = 1.42 \begin {gather*} - \frac {9 \left (1 - 2 x\right )^{\frac {3}{2}}}{20} + \frac {162 \sqrt {1 - 2 x}}{25} + \frac {2 \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 )}{275} + \frac {343}{44 \sqrt {1 - 2 x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.72, size = 67, normalized size = 1.00 \begin {gather*} -\frac {9}{20} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{15125} \, \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 {162}{25} \, \sqrt {-2 \, x + 1} + \frac {343}{44 \, \sqrt {-2 \, x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.06, size = 48, normalized size = 0.72 \begin {gather*} \frac {343}{44\,\sqrt {1-2\,x}}+\frac {162\,\sqrt {1-2\,x}}{25}-\frac {9\,{\left (1-2\,x\right )}^{3/2}}{20}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,2{}\mathrm {i}}{15125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________