Optimal. Leaf size=136 \[ \frac{1935}{67228 \sqrt{1-2 x}}-\frac{129}{2744 (1-2 x)^{3/2} (3 x+2)}+\frac{215}{9604 (1-2 x)^{3/2}}-\frac{129}{2744 (1-2 x)^{3/2} (3 x+2)^2}-\frac{43}{588 (1-2 x)^{3/2} (3 x+2)^3}+\frac{1}{84 (1-2 x)^{3/2} (3 x+2)^4}-\frac{1935 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{67228} \]
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Rubi [A] time = 0.0496317, antiderivative size = 150, normalized size of antiderivative = 1.1, number of steps used = 8, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {78, 51, 63, 206} \[ -\frac{5805 \sqrt{1-2 x}}{134456 (3 x+2)}-\frac{1935 \sqrt{1-2 x}}{19208 (3 x+2)^2}-\frac{387 \sqrt{1-2 x}}{1372 (3 x+2)^3}+\frac{387}{686 \sqrt{1-2 x} (3 x+2)^3}+\frac{43}{294 (1-2 x)^{3/2} (3 x+2)^3}+\frac{1}{84 (1-2 x)^{3/2} (3 x+2)^4}-\frac{1935 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{67228} \]
Antiderivative was successfully verified.
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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)^5} \, dx &=\frac{1}{84 (1-2 x)^{3/2} (2+3 x)^4}+\frac{43}{28} \int \frac{1}{(1-2 x)^{5/2} (2+3 x)^4} \, dx\\ &=\frac{1}{84 (1-2 x)^{3/2} (2+3 x)^4}+\frac{43}{294 (1-2 x)^{3/2} (2+3 x)^3}+\frac{387}{196} \int \frac{1}{(1-2 x)^{3/2} (2+3 x)^4} \, dx\\ &=\frac{1}{84 (1-2 x)^{3/2} (2+3 x)^4}+\frac{43}{294 (1-2 x)^{3/2} (2+3 x)^3}+\frac{387}{686 \sqrt{1-2 x} (2+3 x)^3}+\frac{1161}{196} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^4} \, dx\\ &=\frac{1}{84 (1-2 x)^{3/2} (2+3 x)^4}+\frac{43}{294 (1-2 x)^{3/2} (2+3 x)^3}+\frac{387}{686 \sqrt{1-2 x} (2+3 x)^3}-\frac{387 \sqrt{1-2 x}}{1372 (2+3 x)^3}+\frac{1935 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^3} \, dx}{1372}\\ &=\frac{1}{84 (1-2 x)^{3/2} (2+3 x)^4}+\frac{43}{294 (1-2 x)^{3/2} (2+3 x)^3}+\frac{387}{686 \sqrt{1-2 x} (2+3 x)^3}-\frac{387 \sqrt{1-2 x}}{1372 (2+3 x)^3}-\frac{1935 \sqrt{1-2 x}}{19208 (2+3 x)^2}+\frac{5805 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{19208}\\ &=\frac{1}{84 (1-2 x)^{3/2} (2+3 x)^4}+\frac{43}{294 (1-2 x)^{3/2} (2+3 x)^3}+\frac{387}{686 \sqrt{1-2 x} (2+3 x)^3}-\frac{387 \sqrt{1-2 x}}{1372 (2+3 x)^3}-\frac{1935 \sqrt{1-2 x}}{19208 (2+3 x)^2}-\frac{5805 \sqrt{1-2 x}}{134456 (2+3 x)}+\frac{5805 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{134456}\\ &=\frac{1}{84 (1-2 x)^{3/2} (2+3 x)^4}+\frac{43}{294 (1-2 x)^{3/2} (2+3 x)^3}+\frac{387}{686 \sqrt{1-2 x} (2+3 x)^3}-\frac{387 \sqrt{1-2 x}}{1372 (2+3 x)^3}-\frac{1935 \sqrt{1-2 x}}{19208 (2+3 x)^2}-\frac{5805 \sqrt{1-2 x}}{134456 (2+3 x)}-\frac{5805 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{134456}\\ &=\frac{1}{84 (1-2 x)^{3/2} (2+3 x)^4}+\frac{43}{294 (1-2 x)^{3/2} (2+3 x)^3}+\frac{387}{686 \sqrt{1-2 x} (2+3 x)^3}-\frac{387 \sqrt{1-2 x}}{1372 (2+3 x)^3}-\frac{1935 \sqrt{1-2 x}}{19208 (2+3 x)^2}-\frac{5805 \sqrt{1-2 x}}{134456 (2+3 x)}-\frac{1935 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{67228}\\ \end{align*}
Mathematica [C] time = 0.0175627, size = 42, normalized size = 0.31 \[ \frac{688 \, _2F_1\left (-\frac{3}{2},4;-\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )+\frac{2401}{(3 x+2)^4}}{201684 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 84, normalized size = 0.6 \begin{align*}{\frac{3888}{117649\, \left ( -6\,x-4 \right ) ^{4}} \left ({\frac{5225}{192} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{119623}{576} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{921935}{1728} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{2378705}{5184}\sqrt{1-2\,x}} \right ) }-{\frac{1935\,\sqrt{21}}{470596}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{176}{50421} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{2080}{117649}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.89816, size = 173, normalized size = 1.27 \begin{align*} \frac{1935}{941192} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{470205 \,{\left (2 \, x - 1\right )}^{5} + 4022865 \,{\left (2 \, x - 1\right )}^{4} + 12458691 \,{\left (2 \, x - 1\right )}^{3} + 15872031 \,{\left (2 \, x - 1\right )}^{2} + 11327232 \, x - 7353920}{201684 \,{\left (81 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - 756 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + 2646 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 4116 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 2401 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59007, size = 416, normalized size = 3.06 \begin{align*} \frac{5805 \, \sqrt{7} \sqrt{3}{\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )} \log \left (\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 7 \,{\left (1880820 \, x^{5} + 3343680 \, x^{4} + 1069281 \, x^{3} - 1034451 \, x^{2} - 611202 \, x - 48490\right )} \sqrt{-2 \, x + 1}}{2823576 \,{\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [A] time = 1.56851, size = 163, normalized size = 1.2 \begin{align*} \frac{1935}{941192} \, \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{16 \,{\left (780 \, x - 467\right )}}{352947 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{3 \,{\left (141075 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 1076607 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 2765805 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 2378705 \, \sqrt{-2 \, x + 1}\right )}}{7529536 \,{\left (3 \, x + 2\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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