Optimal. Leaf size=88 \[ -\frac{50 \sqrt{1-2 x}}{1029 (3 x+2)}-\frac{50 \sqrt{1-2 x}}{441 (3 x+2)^2}+\frac{\sqrt{1-2 x}}{63 (3 x+2)^3}-\frac{100 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0224067, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {78, 51, 63, 206} \[ -\frac{50 \sqrt{1-2 x}}{1029 (3 x+2)}-\frac{50 \sqrt{1-2 x}}{441 (3 x+2)^2}+\frac{\sqrt{1-2 x}}{63 (3 x+2)^3}-\frac{100 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 78
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{3+5 x}{\sqrt{1-2 x} (2+3 x)^4} \, dx &=\frac{\sqrt{1-2 x}}{63 (2+3 x)^3}+\frac{100}{63} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=\frac{\sqrt{1-2 x}}{63 (2+3 x)^3}-\frac{50 \sqrt{1-2 x}}{441 (2+3 x)^2}+\frac{50}{147} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=\frac{\sqrt{1-2 x}}{63 (2+3 x)^3}-\frac{50 \sqrt{1-2 x}}{441 (2+3 x)^2}-\frac{50 \sqrt{1-2 x}}{1029 (2+3 x)}+\frac{50 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{1029}\\ &=\frac{\sqrt{1-2 x}}{63 (2+3 x)^3}-\frac{50 \sqrt{1-2 x}}{441 (2+3 x)^2}-\frac{50 \sqrt{1-2 x}}{1029 (2+3 x)}-\frac{50 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{1029}\\ &=\frac{\sqrt{1-2 x}}{63 (2+3 x)^3}-\frac{50 \sqrt{1-2 x}}{441 (2+3 x)^2}-\frac{50 \sqrt{1-2 x}}{1029 (2+3 x)}-\frac{100 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}}\\ \end{align*}
Mathematica [C] time = 0.0128606, size = 42, normalized size = 0.48 \[ \frac{\sqrt{1-2 x} \left (\frac{343}{(3 x+2)^3}-800 \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{21609} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.009, size = 57, normalized size = 0.7 \begin{align*} 216\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{3}} \left ({\frac{25\, \left ( 1-2\,x \right ) ^{5/2}}{6174}}-{\frac{100\, \left ( 1-2\,x \right ) ^{3/2}}{3969}}+{\frac{41\,\sqrt{1-2\,x}}{1134}} \right ) }-{\frac{100\,\sqrt{21}}{21609}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.90456, size = 124, normalized size = 1.41 \begin{align*} \frac{50}{21609} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{4 \,{\left (225 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 1400 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 2009 \, \sqrt{-2 \, x + 1}\right )}}{1029 \,{\left (27 \,{\left (2 \, x - 1\right )}^{3} + 189 \,{\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.3359, size = 238, normalized size = 2.7 \begin{align*} \frac{50 \, \sqrt{21}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \,{\left (450 \, x^{2} + 950 \, x + 417\right )} \sqrt{-2 \, x + 1}}{21609 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 2.29784, size = 113, normalized size = 1.28 \begin{align*} \frac{50}{21609} \, \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{225 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 1400 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 2009 \, \sqrt{-2 \, x + 1}}{2058 \,{\left (3 \, x + 2\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]