Optimal. Leaf size=141 \[ -\frac {(1-2 x)^{5/2} (3 x+2)^4}{5 (5 x+3)}+\frac {39}{275} (1-2 x)^{5/2} (3 x+2)^3-\frac {32 (1-2 x)^{5/2} (3 x+2)^2}{4125}+\frac {254 (1-2 x)^{3/2}}{46875}-\frac {(1-2 x)^{5/2} (1110975 x+1347116)}{3609375}+\frac {2794 \sqrt {1-2 x}}{78125}-\frac {2794 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125} \]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {97, 153, 147, 50, 63, 206} \begin {gather*} -\frac {(1-2 x)^{5/2} (3 x+2)^4}{5 (5 x+3)}+\frac {39}{275} (1-2 x)^{5/2} (3 x+2)^3-\frac {32 (1-2 x)^{5/2} (3 x+2)^2}{4125}+\frac {254 (1-2 x)^{3/2}}{46875}-\frac {(1-2 x)^{5/2} (1110975 x+1347116)}{3609375}+\frac {2794 \sqrt {1-2 x}}{78125}-\frac {2794 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 97
Rule 147
Rule 153
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^2} \, dx &=-\frac {(1-2 x)^{5/2} (2+3 x)^4}{5 (3+5 x)}+\frac {1}{5} \int \frac {(2-39 x) (1-2 x)^{3/2} (2+3 x)^3}{3+5 x} \, dx\\ &=\frac {39}{275} (1-2 x)^{5/2} (2+3 x)^3-\frac {(1-2 x)^{5/2} (2+3 x)^4}{5 (3+5 x)}-\frac {1}{275} \int \frac {(-337-96 x) (1-2 x)^{3/2} (2+3 x)^2}{3+5 x} \, dx\\ &=-\frac {32 (1-2 x)^{5/2} (2+3 x)^2}{4125}+\frac {39}{275} (1-2 x)^{5/2} (2+3 x)^3-\frac {(1-2 x)^{5/2} (2+3 x)^4}{5 (3+5 x)}+\frac {\int \frac {(1-2 x)^{3/2} (2+3 x) (29178+44439 x)}{3+5 x} \, dx}{12375}\\ &=-\frac {32 (1-2 x)^{5/2} (2+3 x)^2}{4125}+\frac {39}{275} (1-2 x)^{5/2} (2+3 x)^3-\frac {(1-2 x)^{5/2} (2+3 x)^4}{5 (3+5 x)}-\frac {(1-2 x)^{5/2} (1347116+1110975 x)}{3609375}+\frac {127 \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx}{3125}\\ &=\frac {254 (1-2 x)^{3/2}}{46875}-\frac {32 (1-2 x)^{5/2} (2+3 x)^2}{4125}+\frac {39}{275} (1-2 x)^{5/2} (2+3 x)^3-\frac {(1-2 x)^{5/2} (2+3 x)^4}{5 (3+5 x)}-\frac {(1-2 x)^{5/2} (1347116+1110975 x)}{3609375}+\frac {1397 \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx}{15625}\\ &=\frac {2794 \sqrt {1-2 x}}{78125}+\frac {254 (1-2 x)^{3/2}}{46875}-\frac {32 (1-2 x)^{5/2} (2+3 x)^2}{4125}+\frac {39}{275} (1-2 x)^{5/2} (2+3 x)^3-\frac {(1-2 x)^{5/2} (2+3 x)^4}{5 (3+5 x)}-\frac {(1-2 x)^{5/2} (1347116+1110975 x)}{3609375}+\frac {15367 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{78125}\\ &=\frac {2794 \sqrt {1-2 x}}{78125}+\frac {254 (1-2 x)^{3/2}}{46875}-\frac {32 (1-2 x)^{5/2} (2+3 x)^2}{4125}+\frac {39}{275} (1-2 x)^{5/2} (2+3 x)^3-\frac {(1-2 x)^{5/2} (2+3 x)^4}{5 (3+5 x)}-\frac {(1-2 x)^{5/2} (1347116+1110975 x)}{3609375}-\frac {15367 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{78125}\\ &=\frac {2794 \sqrt {1-2 x}}{78125}+\frac {254 (1-2 x)^{3/2}}{46875}-\frac {32 (1-2 x)^{5/2} (2+3 x)^2}{4125}+\frac {39}{275} (1-2 x)^{5/2} (2+3 x)^3-\frac {(1-2 x)^{5/2} (2+3 x)^4}{5 (3+5 x)}-\frac {(1-2 x)^{5/2} (1347116+1110975 x)}{3609375}-\frac {2794 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 78, normalized size = 0.55 \begin {gather*} \frac {\frac {5 \sqrt {1-2 x} \left (212625000 x^6+237037500 x^5-173598750 x^4-214071975 x^3+85482115 x^2+50081215 x-15982128\right )}{5 x+3}-645414 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{90234375} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.13, size = 108, normalized size = 0.77 \begin {gather*} -\frac {\sqrt {1-2 x} \left (26578125 (1-2 x)^6-218728125 (1-2 x)^5+608169375 (1-2 x)^4-562886775 (1-2 x)^3-782320 (1-2 x)^2-8605520 (1-2 x)+28398216\right )}{72187500 (5 (1-2 x)-11)}-\frac {2794 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.65, size = 90, normalized size = 0.64 \begin {gather*} \frac {322707 \, \sqrt {11} \sqrt {5} {\left (5 \, x + 3\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 5 \, {\left (212625000 \, x^{6} + 237037500 \, x^{5} - 173598750 \, x^{4} - 214071975 \, x^{3} + 85482115 \, x^{2} + 50081215 \, x - 15982128\right )} \sqrt {-2 \, x + 1}}{90234375 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.98, size = 138, normalized size = 0.98 \begin {gather*} \frac {81}{1100} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {111}{250} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {12393}{17500} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {24}{15625} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {52}{9375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1397}{390625} \, \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 {2816}{78125} \, \sqrt {-2 \, x + 1} - \frac {121 \, \sqrt {-2 \, x + 1}}{78125 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 90, normalized size = 0.64 \begin {gather*} -\frac {2794 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{390625}-\frac {81 \left (-2 x +1\right )^{\frac {11}{2}}}{1100}+\frac {111 \left (-2 x +1\right )^{\frac {9}{2}}}{250}-\frac {12393 \left (-2 x +1\right )^{\frac {7}{2}}}{17500}+\frac {24 \left (-2 x +1\right )^{\frac {5}{2}}}{15625}+\frac {52 \left (-2 x +1\right )^{\frac {3}{2}}}{9375}+\frac {2816 \sqrt {-2 x +1}}{78125}+\frac {242 \sqrt {-2 x +1}}{390625 \left (-2 x -\frac {6}{5}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.27, size = 107, normalized size = 0.76 \begin {gather*} -\frac {81}{1100} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {111}{250} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {12393}{17500} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {24}{15625} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {52}{9375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1397}{390625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2816}{78125} \, \sqrt {-2 \, x + 1} - \frac {121 \, \sqrt {-2 \, x + 1}}{78125 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.07, size = 91, normalized size = 0.65 \begin {gather*} \frac {2816\,\sqrt {1-2\,x}}{78125}-\frac {242\,\sqrt {1-2\,x}}{390625\,\left (2\,x+\frac {6}{5}\right )}+\frac {52\,{\left (1-2\,x\right )}^{3/2}}{9375}+\frac {24\,{\left (1-2\,x\right )}^{5/2}}{15625}-\frac {12393\,{\left (1-2\,x\right )}^{7/2}}{17500}+\frac {111\,{\left (1-2\,x\right )}^{9/2}}{250}-\frac {81\,{\left (1-2\,x\right )}^{11/2}}{1100}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,2794{}\mathrm {i}}{390625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________