Integrand size = 24, antiderivative size = 120 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^2} \, dx=-\frac {7588 (2+3 x)^2}{6655 \sqrt {1-2 x}}-\frac {38 (2+3 x)^3}{1815 \sqrt {1-2 x} (3+5 x)}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {6 \sqrt {1-2 x} (114092+38025 x)}{33275}-\frac {68 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{33275 \sqrt {55}} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {100, 154, 155, 152, 65, 212} \[ \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^2} \, dx=-\frac {68 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{33275 \sqrt {55}}+\frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)}-\frac {38 (3 x+2)^3}{1815 \sqrt {1-2 x} (5 x+3)}-\frac {7588 (3 x+2)^2}{6655 \sqrt {1-2 x}}-\frac {6 \sqrt {1-2 x} (38025 x+114092)}{33275} \]
[In]
[Out]
Rule 65
Rule 100
Rule 152
Rule 154
Rule 155
Rule 212
Rubi steps \begin{align*} \text {integral}& = \frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {1}{33} \int \frac {(2+3 x)^3 (176+306 x)}{(1-2 x)^{3/2} (3+5 x)^2} \, dx \\ & = -\frac {38 (2+3 x)^3}{1815 \sqrt {1-2 x} (3+5 x)}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {\int \frac {(2+3 x)^2 (6162+10440 x)}{(1-2 x)^{3/2} (3+5 x)} \, dx}{1815} \\ & = -\frac {7588 (2+3 x)^2}{6655 \sqrt {1-2 x}}-\frac {38 (2+3 x)^3}{1815 \sqrt {1-2 x} (3+5 x)}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {\int \frac {(-410772-684450 x) (2+3 x)}{\sqrt {1-2 x} (3+5 x)} \, dx}{19965} \\ & = -\frac {7588 (2+3 x)^2}{6655 \sqrt {1-2 x}}-\frac {38 (2+3 x)^3}{1815 \sqrt {1-2 x} (3+5 x)}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {6 \sqrt {1-2 x} (114092+38025 x)}{33275}+\frac {34 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{33275} \\ & = -\frac {7588 (2+3 x)^2}{6655 \sqrt {1-2 x}}-\frac {38 (2+3 x)^3}{1815 \sqrt {1-2 x} (3+5 x)}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {6 \sqrt {1-2 x} (114092+38025 x)}{33275}-\frac {34 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{33275} \\ & = -\frac {7588 (2+3 x)^2}{6655 \sqrt {1-2 x}}-\frac {38 (2+3 x)^3}{1815 \sqrt {1-2 x} (3+5 x)}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {6 \sqrt {1-2 x} (114092+38025 x)}{33275}-\frac {68 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{33275 \sqrt {55}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.71 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^2} \, dx=-\frac {55 \left (7204728-10671002 x-28677318 x^2+16171650 x^3+1617165 x^4\right )-204 \sqrt {55} \sqrt {1-2 x} \left (-3+x+10 x^2\right ) \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{5490375 (1-2 x)^{3/2} (3+5 x)} \]
[In]
[Out]
Time = 3.55 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.52
method | result | size |
risch | \(\frac {1617165 x^{4}+16171650 x^{3}-28677318 x^{2}-10671002 x +7204728}{99825 \sqrt {1-2 x}\, \left (3+5 x \right ) \left (-1+2 x \right )}-\frac {68 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1830125}\) | \(63\) |
pseudoelliptic | \(\frac {204 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (10 x^{2}+x -3\right ) \sqrt {55}-88944075 x^{4}-889440750 x^{3}+1577252490 x^{2}+586905110 x -396260040}{\left (1-2 x \right )^{\frac {3}{2}} \left (16471125+27451875 x \right )}\) | \(70\) |
derivativedivides | \(\frac {81 \left (1-2 x \right )^{\frac {3}{2}}}{200}-\frac {8829 \sqrt {1-2 x}}{1000}+\frac {2 \sqrt {1-2 x}}{831875 \left (-\frac {6}{5}-2 x \right )}-\frac {68 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1830125}+\frac {16807}{2904 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {228095}{10648 \sqrt {1-2 x}}\) | \(72\) |
default | \(\frac {81 \left (1-2 x \right )^{\frac {3}{2}}}{200}-\frac {8829 \sqrt {1-2 x}}{1000}+\frac {2 \sqrt {1-2 x}}{831875 \left (-\frac {6}{5}-2 x \right )}-\frac {68 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1830125}+\frac {16807}{2904 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {228095}{10648 \sqrt {1-2 x}}\) | \(72\) |
trager | \(-\frac {\left (1617165 x^{4}+16171650 x^{3}-28677318 x^{2}-10671002 x +7204728\right ) \sqrt {1-2 x}}{99825 \left (-1+2 x \right )^{2} \left (3+5 x \right )}+\frac {34 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{1830125}\) | \(89\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.78 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^2} \, dx=\frac {102 \, \sqrt {55} {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (1617165 \, x^{4} + 16171650 \, x^{3} - 28677318 \, x^{2} - 10671002 \, x + 7204728\right )} \sqrt {-2 \, x + 1}}{5490375 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \]
[In]
[Out]
Time = 81.18 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.74 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^2} \, dx=\frac {81 \left (1 - 2 x\right )^{\frac {3}{2}}}{200} - \frac {8829 \sqrt {1 - 2 x}}{1000} + \frac {169 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{9150625} - \frac {4 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{15125} - \frac {228095}{10648 \sqrt {1 - 2 x}} + \frac {16807}{2904 \left (1 - 2 x\right )^{\frac {3}{2}}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.77 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^2} \, dx=\frac {81}{200} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {34}{1830125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {8829}{1000} \, \sqrt {-2 \, x + 1} - \frac {427678077 \, {\left (2 \, x - 1\right )}^{2} + 2112880000 \, x - 802234125}{3993000 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 11 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.79 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^2} \, dx=\frac {81}{200} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {34}{1830125} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {8829}{1000} \, \sqrt {-2 \, x + 1} - \frac {2401 \, {\left (285 \, x - 104\right )}}{15972 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {\sqrt {-2 \, x + 1}}{166375 \, {\left (5 \, x + 3\right )}} \]
[In]
[Out]
Time = 1.40 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.62 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^2} \, dx=\frac {\frac {38416\,x}{363}+\frac {142559359\,{\left (2\,x-1\right )}^2}{6655000}-\frac {194481}{4840}}{\frac {11\,{\left (1-2\,x\right )}^{3/2}}{5}-{\left (1-2\,x\right )}^{5/2}}-\frac {8829\,\sqrt {1-2\,x}}{1000}+\frac {81\,{\left (1-2\,x\right )}^{3/2}}{200}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,68{}\mathrm {i}}{1830125} \]
[In]
[Out]