Optimal. Leaf size=117 \[ \frac {3840 \sqrt {1-2 x}}{343 (3 x+2)}+\frac {55 \sqrt {1-2 x}}{49 (3 x+2)^2}+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3}+\frac {88310}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-250 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {103, 151, 156, 63, 206} \[ \frac {3840 \sqrt {1-2 x}}{343 (3 x+2)}+\frac {55 \sqrt {1-2 x}}{49 (3 x+2)^2}+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3}+\frac {88310}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-250 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 103
Rule 151
Rule 156
Rule 206
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)} \, dx &=\frac {\sqrt {1-2 x}}{7 (2+3 x)^3}+\frac {1}{21} \int \frac {60-75 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)} \, dx\\ &=\frac {\sqrt {1-2 x}}{7 (2+3 x)^3}+\frac {55 \sqrt {1-2 x}}{49 (2+3 x)^2}+\frac {1}{294} \int \frac {4380-4950 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)} \, dx\\ &=\frac {\sqrt {1-2 x}}{7 (2+3 x)^3}+\frac {55 \sqrt {1-2 x}}{49 (2+3 x)^2}+\frac {3840 \sqrt {1-2 x}}{343 (2+3 x)}+\frac {\int \frac {188130-115200 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{2058}\\ &=\frac {\sqrt {1-2 x}}{7 (2+3 x)^3}+\frac {55 \sqrt {1-2 x}}{49 (2+3 x)^2}+\frac {3840 \sqrt {1-2 x}}{343 (2+3 x)}-\frac {132465}{343} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+625 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {\sqrt {1-2 x}}{7 (2+3 x)^3}+\frac {55 \sqrt {1-2 x}}{49 (2+3 x)^2}+\frac {3840 \sqrt {1-2 x}}{343 (2+3 x)}+\frac {132465}{343} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-625 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {\sqrt {1-2 x}}{7 (2+3 x)^3}+\frac {55 \sqrt {1-2 x}}{49 (2+3 x)^2}+\frac {3840 \sqrt {1-2 x}}{343 (2+3 x)}+\frac {88310}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-250 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 87, normalized size = 0.74 \[ \frac {3 \sqrt {1-2 x} \left (11520 x^2+15745 x+5393\right )}{343 (3 x+2)^3}+\frac {88310}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-250 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.92, size = 142, normalized size = 1.21 \[ \frac {300125 \, \sqrt {11} \sqrt {5} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 485705 \, \sqrt {7} \sqrt {3} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 231 \, {\left (11520 \, x^{2} + 15745 \, x + 5393\right )} \sqrt {-2 \, x + 1}}{26411 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.23, size = 123, normalized size = 1.05 \[ \frac {125}{11} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {44155}{2401} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {3 \, {\left (5760 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 27265 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 32291 \, \sqrt {-2 \, x + 1}\right )}}{686 \, {\left (3 \, x + 2\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 75, normalized size = 0.64 \[ \frac {88310 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{2401}-\frac {250 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{11}-\frac {162 \left (\frac {1280 \left (-2 x +1\right )^{\frac {5}{2}}}{1029}-\frac {7790 \left (-2 x +1\right )^{\frac {3}{2}}}{1323}+\frac {1318 \sqrt {-2 x +1}}{189}\right )}{\left (-6 x -4\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.23, size = 128, normalized size = 1.09 \[ \frac {125}{11} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {44155}{2401} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {12 \, {\left (5760 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 27265 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 32291 \, \sqrt {-2 \, x + 1}\right )}}{343 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.10, size = 89, normalized size = 0.76 \[ \frac {88310\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401}-\frac {250\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{11}+\frac {\frac {2636\,\sqrt {1-2\,x}}{63}-\frac {15580\,{\left (1-2\,x\right )}^{3/2}}{441}+\frac {2560\,{\left (1-2\,x\right )}^{5/2}}{343}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________