Optimal. Leaf size=73 \[ \frac {\sqrt {1-2 x} (5 x+3)^2}{21 (3 x+2)}-\frac {10}{189} \sqrt {1-2 x} (95 x+214)-\frac {208 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{189 \sqrt {21}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 73, 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 = {98, 147, 63, 206} \[ \frac {\sqrt {1-2 x} (5 x+3)^2}{21 (3 x+2)}-\frac {10}{189} \sqrt {1-2 x} (95 x+214)-\frac {208 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{189 \sqrt {21}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 98
Rule 147
Rule 206
Rubi steps
\begin {align*} \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^2} \, dx &=\frac {\sqrt {1-2 x} (3+5 x)^2}{21 (2+3 x)}-\frac {1}{21} \int \frac {(-92-190 x) (3+5 x)}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {\sqrt {1-2 x} (3+5 x)^2}{21 (2+3 x)}-\frac {10}{189} \sqrt {1-2 x} (214+95 x)+\frac {104}{189} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {\sqrt {1-2 x} (3+5 x)^2}{21 (2+3 x)}-\frac {10}{189} \sqrt {1-2 x} (214+95 x)-\frac {104}{189} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {\sqrt {1-2 x} (3+5 x)^2}{21 (2+3 x)}-\frac {10}{189} \sqrt {1-2 x} (214+95 x)-\frac {208 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{189 \sqrt {21}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 58, normalized size = 0.79 \[ \frac {-\frac {21 \sqrt {1-2 x} \left (2625 x^2+8050 x+4199\right )}{3 x+2}-208 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3969} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.89, size = 64, normalized size = 0.88 \[ \frac {104 \, \sqrt {21} {\left (3 \, x + 2\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (2625 \, x^{2} + 8050 \, x + 4199\right )} \sqrt {-2 \, x + 1}}{3969 \, {\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.05, size = 74, normalized size = 1.01 \[ \frac {125}{54} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {104}{3969} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {725}{54} \, \sqrt {-2 \, x + 1} + \frac {\sqrt {-2 \, x + 1}}{189 \, {\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 54, normalized size = 0.74 \[ -\frac {208 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{3969}+\frac {125 \left (-2 x +1\right )^{\frac {3}{2}}}{54}-\frac {725 \sqrt {-2 x +1}}{54}-\frac {2 \sqrt {-2 x +1}}{567 \left (-2 x -\frac {4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.16, size = 71, normalized size = 0.97 \[ \frac {125}{54} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {104}{3969} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {725}{54} \, \sqrt {-2 \, x + 1} + \frac {\sqrt {-2 \, x + 1}}{189 \, {\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.18, size = 55, normalized size = 0.75 \[ \frac {2\,\sqrt {1-2\,x}}{567\,\left (2\,x+\frac {4}{3}\right )}-\frac {725\,\sqrt {1-2\,x}}{54}+\frac {125\,{\left (1-2\,x\right )}^{3/2}}{54}+\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,208{}\mathrm {i}}{3969} \]
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]
________________________________________________________________________________________