Optimal. Leaf size=148 \[ \frac{59 (1-2 x)^{5/2}}{1890 (3 x+2)^5}-\frac{(1-2 x)^{5/2}}{378 (3 x+2)^6}-\frac{991 (1-2 x)^{3/2}}{4536 (3 x+2)^4}-\frac{991 \sqrt{1-2 x}}{444528 (3 x+2)}-\frac{991 \sqrt{1-2 x}}{190512 (3 x+2)^2}+\frac{991 \sqrt{1-2 x}}{13608 (3 x+2)^3}-\frac{991 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{222264 \sqrt{21}} \]
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Rubi [A] time = 0.0475829, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {89, 78, 47, 51, 63, 206} \[ \frac{59 (1-2 x)^{5/2}}{1890 (3 x+2)^5}-\frac{(1-2 x)^{5/2}}{378 (3 x+2)^6}-\frac{991 (1-2 x)^{3/2}}{4536 (3 x+2)^4}-\frac{991 \sqrt{1-2 x}}{444528 (3 x+2)}-\frac{991 \sqrt{1-2 x}}{190512 (3 x+2)^2}+\frac{991 \sqrt{1-2 x}}{13608 (3 x+2)^3}-\frac{991 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{222264 \sqrt{21}} \]
Antiderivative was successfully verified.
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Rule 89
Rule 78
Rule 47
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^7} \, dx &=-\frac{(1-2 x)^{5/2}}{378 (2+3 x)^6}+\frac{1}{378} \int \frac{(1-2 x)^{3/2} (1687+3150 x)}{(2+3 x)^6} \, dx\\ &=-\frac{(1-2 x)^{5/2}}{378 (2+3 x)^6}+\frac{59 (1-2 x)^{5/2}}{1890 (2+3 x)^5}+\frac{991}{378} \int \frac{(1-2 x)^{3/2}}{(2+3 x)^5} \, dx\\ &=-\frac{(1-2 x)^{5/2}}{378 (2+3 x)^6}+\frac{59 (1-2 x)^{5/2}}{1890 (2+3 x)^5}-\frac{991 (1-2 x)^{3/2}}{4536 (2+3 x)^4}-\frac{991 \int \frac{\sqrt{1-2 x}}{(2+3 x)^4} \, dx}{1512}\\ &=-\frac{(1-2 x)^{5/2}}{378 (2+3 x)^6}+\frac{59 (1-2 x)^{5/2}}{1890 (2+3 x)^5}-\frac{991 (1-2 x)^{3/2}}{4536 (2+3 x)^4}+\frac{991 \sqrt{1-2 x}}{13608 (2+3 x)^3}+\frac{991 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^3} \, dx}{13608}\\ &=-\frac{(1-2 x)^{5/2}}{378 (2+3 x)^6}+\frac{59 (1-2 x)^{5/2}}{1890 (2+3 x)^5}-\frac{991 (1-2 x)^{3/2}}{4536 (2+3 x)^4}+\frac{991 \sqrt{1-2 x}}{13608 (2+3 x)^3}-\frac{991 \sqrt{1-2 x}}{190512 (2+3 x)^2}+\frac{991 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{63504}\\ &=-\frac{(1-2 x)^{5/2}}{378 (2+3 x)^6}+\frac{59 (1-2 x)^{5/2}}{1890 (2+3 x)^5}-\frac{991 (1-2 x)^{3/2}}{4536 (2+3 x)^4}+\frac{991 \sqrt{1-2 x}}{13608 (2+3 x)^3}-\frac{991 \sqrt{1-2 x}}{190512 (2+3 x)^2}-\frac{991 \sqrt{1-2 x}}{444528 (2+3 x)}+\frac{991 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{444528}\\ &=-\frac{(1-2 x)^{5/2}}{378 (2+3 x)^6}+\frac{59 (1-2 x)^{5/2}}{1890 (2+3 x)^5}-\frac{991 (1-2 x)^{3/2}}{4536 (2+3 x)^4}+\frac{991 \sqrt{1-2 x}}{13608 (2+3 x)^3}-\frac{991 \sqrt{1-2 x}}{190512 (2+3 x)^2}-\frac{991 \sqrt{1-2 x}}{444528 (2+3 x)}-\frac{991 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{444528}\\ &=-\frac{(1-2 x)^{5/2}}{378 (2+3 x)^6}+\frac{59 (1-2 x)^{5/2}}{1890 (2+3 x)^5}-\frac{991 (1-2 x)^{3/2}}{4536 (2+3 x)^4}+\frac{991 \sqrt{1-2 x}}{13608 (2+3 x)^3}-\frac{991 \sqrt{1-2 x}}{190512 (2+3 x)^2}-\frac{991 \sqrt{1-2 x}}{444528 (2+3 x)}-\frac{991 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{222264 \sqrt{21}}\\ \end{align*}
Mathematica [C] time = 0.0266343, size = 47, normalized size = 0.32 \[ \frac{(1-2 x)^{5/2} \left (\frac{16807 (177 x+113)}{(3 x+2)^6}-31712 \, _2F_1\left (\frac{5}{2},5;\frac{7}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{31765230} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 84, normalized size = 0.6 \begin{align*} 23328\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{6}} \left ({\frac{991\, \left ( 1-2\,x \right ) ^{11/2}}{21337344}}-{\frac{16847\, \left ( 1-2\,x \right ) ^{9/2}}{27433728}}+{\frac{10303\, \left ( 1-2\,x \right ) ^{7/2}}{9797760}}+{\frac{29843\, \left ( 1-2\,x \right ) ^{5/2}}{9797760}}-{\frac{117929\, \left ( 1-2\,x \right ) ^{3/2}}{15116544}}+{\frac{48559\,\sqrt{1-2\,x}}{15116544}} \right ) }-{\frac{991\,\sqrt{21}}{4667544}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.77132, size = 197, normalized size = 1.33 \begin{align*} \frac{991}{9335088} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{1204065 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - 15920415 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + 27261738 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 78964578 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 202248235 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 83278685 \, \sqrt{-2 \, x + 1}}{1111320 \,{\left (729 \,{\left (2 \, x - 1\right )}^{6} + 10206 \,{\left (2 \, x - 1\right )}^{5} + 59535 \,{\left (2 \, x - 1\right )}^{4} + 185220 \,{\left (2 \, x - 1\right )}^{3} + 324135 \,{\left (2 \, x - 1\right )}^{2} + 605052 \, x - 184877\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33565, size = 419, normalized size = 2.83 \begin{align*} \frac{4955 \, \sqrt{21}{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \,{\left (1204065 \, x^{5} + 4950045 \, x^{4} - 6094818 \, x^{3} - 9658494 \, x^{2} - 1262200 \, x + 858112\right )} \sqrt{-2 \, x + 1}}{46675440 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.25493, size = 178, normalized size = 1.2 \begin{align*} \frac{991}{9335088} \, \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{1204065 \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + 15920415 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 27261738 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - 78964578 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + 202248235 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 83278685 \, \sqrt{-2 \, x + 1}}{71124480 \,{\left (3 \, x + 2\right )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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