Optimal. Leaf size=96 \[ \frac{45}{343 \sqrt{1-2 x}}-\frac{3}{14 (1-2 x)^{3/2} (3 x+2)}+\frac{5}{49 (1-2 x)^{3/2}}+\frac{1}{42 (1-2 x)^{3/2} (3 x+2)^2}-\frac{45}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0273081, antiderivative size = 96, normalized size of antiderivative = 1., 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 = {78, 51, 63, 206} \[ \frac{45}{343 \sqrt{1-2 x}}-\frac{3}{14 (1-2 x)^{3/2} (3 x+2)}+\frac{5}{49 (1-2 x)^{3/2}}+\frac{1}{42 (1-2 x)^{3/2} (3 x+2)^2}-\frac{45}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 78
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{3+5 x}{(1-2 x)^{5/2} (2+3 x)^3} \, dx &=\frac{1}{42 (1-2 x)^{3/2} (2+3 x)^2}+\frac{3}{2} \int \frac{1}{(1-2 x)^{5/2} (2+3 x)^2} \, dx\\ &=\frac{1}{42 (1-2 x)^{3/2} (2+3 x)^2}-\frac{3}{14 (1-2 x)^{3/2} (2+3 x)}+\frac{15}{14} \int \frac{1}{(1-2 x)^{5/2} (2+3 x)} \, dx\\ &=\frac{5}{49 (1-2 x)^{3/2}}+\frac{1}{42 (1-2 x)^{3/2} (2+3 x)^2}-\frac{3}{14 (1-2 x)^{3/2} (2+3 x)}+\frac{45}{98} \int \frac{1}{(1-2 x)^{3/2} (2+3 x)} \, dx\\ &=\frac{5}{49 (1-2 x)^{3/2}}+\frac{45}{343 \sqrt{1-2 x}}+\frac{1}{42 (1-2 x)^{3/2} (2+3 x)^2}-\frac{3}{14 (1-2 x)^{3/2} (2+3 x)}+\frac{135}{686} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{5}{49 (1-2 x)^{3/2}}+\frac{45}{343 \sqrt{1-2 x}}+\frac{1}{42 (1-2 x)^{3/2} (2+3 x)^2}-\frac{3}{14 (1-2 x)^{3/2} (2+3 x)}-\frac{135}{686} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{5}{49 (1-2 x)^{3/2}}+\frac{45}{343 \sqrt{1-2 x}}+\frac{1}{42 (1-2 x)^{3/2} (2+3 x)^2}-\frac{3}{14 (1-2 x)^{3/2} (2+3 x)}-\frac{45}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [C] time = 0.01764, size = 48, normalized size = 0.5 \[ -\frac{-12 (3 x+2)^2 \, _2F_1\left (-\frac{3}{2},2;-\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )-7}{294 (1-2 x)^{3/2} (3 x+2)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.012, size = 66, normalized size = 0.7 \begin{align*}{\frac{324}{2401\, \left ( -6\,x-4 \right ) ^{2}} \left ({\frac{59}{36} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{133}{36}\sqrt{1-2\,x}} \right ) }-{\frac{45\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{44}{1029} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{256}{2401}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.68887, size = 124, normalized size = 1.29 \begin{align*} \frac{45}{4802} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{1215 \,{\left (2 \, x - 1\right )}^{3} + 4725 \,{\left (2 \, x - 1\right )}^{2} + 7056 \, x - 5684}{1029 \,{\left (9 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 42 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 49 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.54602, size = 297, normalized size = 3.09 \begin{align*} \frac{135 \, \sqrt{7} \sqrt{3}{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 7 \,{\left (4860 \, x^{3} + 2160 \, x^{2} - 2277 \, x - 1087\right )} \sqrt{-2 \, x + 1}}{14406 \,{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\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 = 2.17242, size = 120, normalized size = 1.25 \begin{align*} \frac{45}{4802} \, \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{4 \,{\left (384 \, x - 269\right )}}{7203 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} + \frac{9 \,{\left (59 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 133 \, \sqrt{-2 \, x + 1}\right )}}{9604 \,{\left (3 \, x + 2\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]