Optimal. Leaf size=82 \[ -\frac {5}{21} (1-2 x)^{7/2}-\frac {2}{45} (1-2 x)^{5/2}-\frac {14}{81} (1-2 x)^{3/2}-\frac {98}{81} \sqrt {1-2 x}+\frac {98}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {80, 50, 63, 206} \begin {gather*} -\frac {5}{21} (1-2 x)^{7/2}-\frac {2}{45} (1-2 x)^{5/2}-\frac {14}{81} (1-2 x)^{3/2}-\frac {98}{81} \sqrt {1-2 x}+\frac {98}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 80
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)}{2+3 x} \, dx &=-\frac {5}{21} (1-2 x)^{7/2}-\frac {1}{3} \int \frac {(1-2 x)^{5/2}}{2+3 x} \, dx\\ &=-\frac {2}{45} (1-2 x)^{5/2}-\frac {5}{21} (1-2 x)^{7/2}-\frac {7}{9} \int \frac {(1-2 x)^{3/2}}{2+3 x} \, dx\\ &=-\frac {14}{81} (1-2 x)^{3/2}-\frac {2}{45} (1-2 x)^{5/2}-\frac {5}{21} (1-2 x)^{7/2}-\frac {49}{27} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx\\ &=-\frac {98}{81} \sqrt {1-2 x}-\frac {14}{81} (1-2 x)^{3/2}-\frac {2}{45} (1-2 x)^{5/2}-\frac {5}{21} (1-2 x)^{7/2}-\frac {343}{81} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {98}{81} \sqrt {1-2 x}-\frac {14}{81} (1-2 x)^{3/2}-\frac {2}{45} (1-2 x)^{5/2}-\frac {5}{21} (1-2 x)^{7/2}+\frac {343}{81} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {98}{81} \sqrt {1-2 x}-\frac {14}{81} (1-2 x)^{3/2}-\frac {2}{45} (1-2 x)^{5/2}-\frac {5}{21} (1-2 x)^{7/2}+\frac {98}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 56, normalized size = 0.68 \begin {gather*} \frac {3 \sqrt {1-2 x} \left (5400 x^3-8604 x^2+5534 x-4721\right )+3430 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{8505} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.06, size = 70, normalized size = 0.85 \begin {gather*} \frac {98}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {\left (675 (1-2 x)^3+126 (1-2 x)^2+490 (1-2 x)+3430\right ) \sqrt {1-2 x}}{2835} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.67, size = 62, normalized size = 0.76 \begin {gather*} \frac {49}{243} \, \sqrt {7} \sqrt {3} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + \frac {1}{2835} \, {\left (5400 \, x^{3} - 8604 \, x^{2} + 5534 \, x - 4721\right )} \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.06, size = 90, normalized size = 1.10 \begin {gather*} \frac {5}{21} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {2}{45} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {14}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {49}{243} \, \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 {98}{81} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 56, normalized size = 0.68 \begin {gather*} \frac {98 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{243}-\frac {14 \left (-2 x +1\right )^{\frac {3}{2}}}{81}-\frac {2 \left (-2 x +1\right )^{\frac {5}{2}}}{45}-\frac {5 \left (-2 x +1\right )^{\frac {7}{2}}}{21}-\frac {98 \sqrt {-2 x +1}}{81} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.28, size = 73, normalized size = 0.89 \begin {gather*} -\frac {5}{21} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {2}{45} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {14}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {49}{243} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {98}{81} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.18, size = 57, normalized size = 0.70 \begin {gather*} -\frac {98\,\sqrt {1-2\,x}}{81}-\frac {14\,{\left (1-2\,x\right )}^{3/2}}{81}-\frac {2\,{\left (1-2\,x\right )}^{5/2}}{45}-\frac {5\,{\left (1-2\,x\right )}^{7/2}}{21}-\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,98{}\mathrm {i}}{243} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 34.94, size = 116, normalized size = 1.41 \begin {gather*} - \frac {5 \left (1 - 2 x\right )^{\frac {7}{2}}}{21} - \frac {2 \left (1 - 2 x\right )^{\frac {5}{2}}}{45} - \frac {14 \left (1 - 2 x\right )^{\frac {3}{2}}}{81} - \frac {98 \sqrt {1 - 2 x}}{81} - \frac {686 \left (\begin {cases} - \frac {\sqrt {21} \operatorname {acoth}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: 2 x - 1 < - \frac {7}{3} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: 2 x - 1 > - \frac {7}{3} \end {cases}\right )}{81} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________