Optimal. Leaf size=80 \[ -\frac {\sqrt {1-2 x} (2+3 x)^2}{110 (3+5 x)^2}-\frac {9 \sqrt {1-2 x} (432+715 x)}{6050 (3+5 x)}-\frac {1347 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3025 \sqrt {55}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {100, 151, 65,
212} \begin {gather*} -\frac {\sqrt {1-2 x} (3 x+2)^2}{110 (5 x+3)^2}-\frac {9 \sqrt {1-2 x} (715 x+432)}{6050 (5 x+3)}-\frac {1347 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3025 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 100
Rule 151
Rule 212
Rubi steps
\begin {align*} \int \frac {(2+3 x)^3}{\sqrt {1-2 x} (3+5 x)^3} \, dx &=-\frac {\sqrt {1-2 x} (2+3 x)^2}{110 (3+5 x)^2}-\frac {1}{110} \int \frac {(-144-195 x) (2+3 x)}{\sqrt {1-2 x} (3+5 x)^2} \, dx\\ &=-\frac {\sqrt {1-2 x} (2+3 x)^2}{110 (3+5 x)^2}-\frac {9 \sqrt {1-2 x} (432+715 x)}{6050 (3+5 x)}+\frac {1347 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{6050}\\ &=-\frac {\sqrt {1-2 x} (2+3 x)^2}{110 (3+5 x)^2}-\frac {9 \sqrt {1-2 x} (432+715 x)}{6050 (3+5 x)}-\frac {1347 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{6050}\\ &=-\frac {\sqrt {1-2 x} (2+3 x)^2}{110 (3+5 x)^2}-\frac {9 \sqrt {1-2 x} (432+715 x)}{6050 (3+5 x)}-\frac {1347 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3025 \sqrt {55}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.15, size = 58, normalized size = 0.72 \begin {gather*} \frac {-\frac {55 \sqrt {1-2 x} \left (11884+39405 x+32670 x^2\right )}{(3+5 x)^2}-2694 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{332750} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.12, size = 57, normalized size = 0.71
method | result | size |
risch | \(\frac {65340 x^{3}+46140 x^{2}-15637 x -11884}{6050 \left (3+5 x \right )^{2} \sqrt {1-2 x}}-\frac {1347 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{166375}\) | \(51\) |
derivativedivides | \(-\frac {27 \sqrt {1-2 x}}{125}+\frac {\frac {201 \left (1-2 x \right )^{\frac {3}{2}}}{3025}-\frac {203 \sqrt {1-2 x}}{1375}}{\left (-6-10 x \right )^{2}}-\frac {1347 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{166375}\) | \(57\) |
default | \(-\frac {27 \sqrt {1-2 x}}{125}+\frac {\frac {201 \left (1-2 x \right )^{\frac {3}{2}}}{3025}-\frac {203 \sqrt {1-2 x}}{1375}}{\left (-6-10 x \right )^{2}}-\frac {1347 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{166375}\) | \(57\) |
trager | \(-\frac {\left (32670 x^{2}+39405 x +11884\right ) \sqrt {1-2 x}}{6050 \left (3+5 x \right )^{2}}+\frac {1347 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{332750}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.49, size = 83, normalized size = 1.04 \begin {gather*} \frac {1347}{332750} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {27}{125} \, \sqrt {-2 \, x + 1} + \frac {1005 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2233 \, \sqrt {-2 \, x + 1}}{15125 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.12, size = 74, normalized size = 0.92 \begin {gather*} \frac {1347 \, \sqrt {55} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (32670 \, x^{2} + 39405 \, x + 11884\right )} \sqrt {-2 \, x + 1}}{332750 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
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]
________________________________________________________________________________________
Giac [A]
time = 1.55, size = 77, normalized size = 0.96 \begin {gather*} \frac {1347}{332750} \, \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 {27}{125} \, \sqrt {-2 \, x + 1} + \frac {1005 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2233 \, \sqrt {-2 \, x + 1}}{60500 \, {\left (5 \, x + 3\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.24, size = 63, normalized size = 0.79 \begin {gather*} -\frac {1347\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{166375}-\frac {27\,\sqrt {1-2\,x}}{125}-\frac {\frac {203\,\sqrt {1-2\,x}}{34375}-\frac {201\,{\left (1-2\,x\right )}^{3/2}}{75625}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________