Integrand size = 24, antiderivative size = 93 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} (3+5 x)^2} \, dx=-\frac {84 \sqrt {1-2 x} (2+3 x)^2}{1375}-\frac {\sqrt {1-2 x} (2+3 x)^3}{55 (3+5 x)}-\frac {21 \sqrt {1-2 x} (1144+375 x)}{6875}-\frac {266 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{6875 \sqrt {55}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {100, 158, 152, 65, 212} \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} (3+5 x)^2} \, dx=-\frac {266 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{6875 \sqrt {55}}-\frac {\sqrt {1-2 x} (3 x+2)^3}{55 (5 x+3)}-\frac {84 \sqrt {1-2 x} (3 x+2)^2}{1375}-\frac {21 \sqrt {1-2 x} (375 x+1144)}{6875} \]
[In]
[Out]
Rule 65
Rule 100
Rule 152
Rule 158
Rule 212
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1-2 x} (2+3 x)^3}{55 (3+5 x)}-\frac {1}{55} \int \frac {(-77-84 x) (2+3 x)^2}{\sqrt {1-2 x} (3+5 x)} \, dx \\ & = -\frac {84 \sqrt {1-2 x} (2+3 x)^2}{1375}-\frac {\sqrt {1-2 x} (2+3 x)^3}{55 (3+5 x)}+\frac {\int \frac {(2+3 x) (4858+7875 x)}{\sqrt {1-2 x} (3+5 x)} \, dx}{1375} \\ & = -\frac {84 \sqrt {1-2 x} (2+3 x)^2}{1375}-\frac {\sqrt {1-2 x} (2+3 x)^3}{55 (3+5 x)}-\frac {21 \sqrt {1-2 x} (1144+375 x)}{6875}+\frac {133 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{6875} \\ & = -\frac {84 \sqrt {1-2 x} (2+3 x)^2}{1375}-\frac {\sqrt {1-2 x} (2+3 x)^3}{55 (3+5 x)}-\frac {21 \sqrt {1-2 x} (1144+375 x)}{6875}-\frac {133 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{6875} \\ & = -\frac {84 \sqrt {1-2 x} (2+3 x)^2}{1375}-\frac {\sqrt {1-2 x} (2+3 x)^3}{55 (3+5 x)}-\frac {21 \sqrt {1-2 x} (1144+375 x)}{6875}-\frac {266 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{6875 \sqrt {55}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.68 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} (3+5 x)^2} \, dx=\frac {-\frac {55 \sqrt {1-2 x} \left (78112+171765 x+82665 x^2+22275 x^3\right )}{3+5 x}-266 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{378125} \]
[In]
[Out]
Time = 1.06 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.60
method | result | size |
risch | \(\frac {44550 x^{4}+143055 x^{3}+260865 x^{2}-15541 x -78112}{6875 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {266 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{378125}\) | \(56\) |
pseudoelliptic | \(\frac {-266 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right ) \sqrt {55}-55 \sqrt {1-2 x}\, \left (22275 x^{3}+82665 x^{2}+171765 x +78112\right )}{1134375+1890625 x}\) | \(57\) |
derivativedivides | \(-\frac {81 \left (1-2 x \right )^{\frac {5}{2}}}{500}+\frac {333 \left (1-2 x \right )^{\frac {3}{2}}}{250}-\frac {12393 \sqrt {1-2 x}}{2500}+\frac {2 \sqrt {1-2 x}}{34375 \left (-\frac {6}{5}-2 x \right )}-\frac {266 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{378125}\) | \(63\) |
default | \(-\frac {81 \left (1-2 x \right )^{\frac {5}{2}}}{500}+\frac {333 \left (1-2 x \right )^{\frac {3}{2}}}{250}-\frac {12393 \sqrt {1-2 x}}{2500}+\frac {2 \sqrt {1-2 x}}{34375 \left (-\frac {6}{5}-2 x \right )}-\frac {266 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{378125}\) | \(63\) |
trager | \(-\frac {\left (22275 x^{3}+82665 x^{2}+171765 x +78112\right ) \sqrt {1-2 x}}{6875 \left (3+5 x \right )}+\frac {133 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{378125}\) | \(77\) |
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.74 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} (3+5 x)^2} \, dx=\frac {133 \, \sqrt {55} {\left (5 \, x + 3\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (22275 \, x^{3} + 82665 \, x^{2} + 171765 \, x + 78112\right )} \sqrt {-2 \, x + 1}}{378125 \, {\left (5 \, x + 3\right )}} \]
[In]
[Out]
Time = 41.58 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.12 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} (3+5 x)^2} \, dx=- \frac {81 \left (1 - 2 x\right )^{\frac {5}{2}}}{500} + \frac {333 \left (1 - 2 x\right )^{\frac {3}{2}}}{250} - \frac {12393 \sqrt {1 - 2 x}}{2500} + \frac {12 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{34375} - \frac {4 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{625} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.86 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} (3+5 x)^2} \, dx=-\frac {81}{500} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {333}{250} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {133}{378125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {12393}{2500} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{6875 \, {\left (5 \, x + 3\right )}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.97 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} (3+5 x)^2} \, dx=-\frac {81}{500} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {333}{250} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {133}{378125} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {12393}{2500} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{6875 \, {\left (5 \, x + 3\right )}} \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.69 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} (3+5 x)^2} \, dx=\frac {333\,{\left (1-2\,x\right )}^{3/2}}{250}-\frac {12393\,\sqrt {1-2\,x}}{2500}-\frac {2\,\sqrt {1-2\,x}}{34375\,\left (2\,x+\frac {6}{5}\right )}-\frac {81\,{\left (1-2\,x\right )}^{5/2}}{500}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,266{}\mathrm {i}}{378125} \]
[In]
[Out]