Optimal. Leaf size=140 \[ \frac {7 (3 x+2)^5}{11 \sqrt {1-2 x} (5 x+3)}-\frac {36 \sqrt {1-2 x} (3 x+2)^4}{605 (5 x+3)}+\frac {14517 \sqrt {1-2 x} (3 x+2)^3}{21175}+\frac {217152 \sqrt {1-2 x} (3 x+2)^2}{75625}+\frac {9 \sqrt {1-2 x} (1688625 x+5065808)}{378125}-\frac {402 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{378125 \sqrt {55}} \]
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Rubi [A] time = 0.05, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {98, 149, 153, 147, 63, 206} \[ \frac {7 (3 x+2)^5}{11 \sqrt {1-2 x} (5 x+3)}-\frac {36 \sqrt {1-2 x} (3 x+2)^4}{605 (5 x+3)}+\frac {14517 \sqrt {1-2 x} (3 x+2)^3}{21175}+\frac {217152 \sqrt {1-2 x} (3 x+2)^2}{75625}+\frac {9 \sqrt {1-2 x} (1688625 x+5065808)}{378125}-\frac {402 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{378125 \sqrt {55}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 98
Rule 147
Rule 149
Rule 153
Rule 206
Rubi steps
\begin {align*} \int \frac {(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)^2} \, dx &=\frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)}-\frac {1}{11} \int \frac {(2+3 x)^4 (243+417 x)}{\sqrt {1-2 x} (3+5 x)^2} \, dx\\ &=-\frac {36 \sqrt {1-2 x} (2+3 x)^4}{605 (3+5 x)}+\frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)}-\frac {1}{605} \int \frac {(2+3 x)^3 (8670+14517 x)}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {14517 \sqrt {1-2 x} (2+3 x)^3}{21175}-\frac {36 \sqrt {1-2 x} (2+3 x)^4}{605 (3+5 x)}+\frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)}+\frac {\int \frac {(-911757-1520064 x) (2+3 x)^2}{\sqrt {1-2 x} (3+5 x)} \, dx}{21175}\\ &=\frac {217152 \sqrt {1-2 x} (2+3 x)^2}{75625}+\frac {14517 \sqrt {1-2 x} (2+3 x)^3}{21175}-\frac {36 \sqrt {1-2 x} (2+3 x)^4}{605 (3+5 x)}+\frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)}-\frac {\int \frac {(2+3 x) (63828618+106383375 x)}{\sqrt {1-2 x} (3+5 x)} \, dx}{529375}\\ &=\frac {217152 \sqrt {1-2 x} (2+3 x)^2}{75625}+\frac {14517 \sqrt {1-2 x} (2+3 x)^3}{21175}-\frac {36 \sqrt {1-2 x} (2+3 x)^4}{605 (3+5 x)}+\frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)}+\frac {9 \sqrt {1-2 x} (5065808+1688625 x)}{378125}+\frac {201 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{378125}\\ &=\frac {217152 \sqrt {1-2 x} (2+3 x)^2}{75625}+\frac {14517 \sqrt {1-2 x} (2+3 x)^3}{21175}-\frac {36 \sqrt {1-2 x} (2+3 x)^4}{605 (3+5 x)}+\frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)}+\frac {9 \sqrt {1-2 x} (5065808+1688625 x)}{378125}-\frac {201 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{378125}\\ &=\frac {217152 \sqrt {1-2 x} (2+3 x)^2}{75625}+\frac {14517 \sqrt {1-2 x} (2+3 x)^3}{21175}-\frac {36 \sqrt {1-2 x} (2+3 x)^4}{605 (3+5 x)}+\frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)}+\frac {9 \sqrt {1-2 x} (5065808+1688625 x)}{378125}-\frac {402 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{378125 \sqrt {55}}\\ \end {align*}
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Mathematica [C] time = 0.13, size = 101, normalized size = 0.72 \[ \frac {\frac {8820 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {5}{11} (1-2 x)\right )}{\sqrt {1-2 x}}-\frac {55 \left (5011875 x^5+26663175 x^4+72309105 x^3+199582515 x^2-74439831 x-103960660\right )}{\sqrt {1-2 x} (5 x+3)}+546 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{13234375} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 85, normalized size = 0.61 \[ \frac {1407 \, \sqrt {55} {\left (10 \, x^{2} + x - 3\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \, {\left (55130625 \, x^{5} + 293294925 \, x^{4} + 795400155 \, x^{3} + 2195407665 \, x^{2} - 818846961 \, x - 1143572552\right )} \sqrt {-2 \, x + 1}}{145578125 \, {\left (10 \, x^{2} + x - 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.32, size = 118, normalized size = 0.84 \[ \frac {729}{2800} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {2187}{625} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {105057}{5000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {201}{20796875} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {315684}{3125} \, \sqrt {-2 \, x + 1} - \frac {1838265657 \, x + 1102959359}{3025000 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 81, normalized size = 0.58 \[ -\frac {402 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{20796875}-\frac {729 \left (-2 x +1\right )^{\frac {7}{2}}}{2800}+\frac {2187 \left (-2 x +1\right )^{\frac {5}{2}}}{625}-\frac {105057 \left (-2 x +1\right )^{\frac {3}{2}}}{5000}+\frac {315684 \sqrt {-2 x +1}}{3125}+\frac {117649}{1936 \sqrt {-2 x +1}}+\frac {2 \sqrt {-2 x +1}}{1890625 \left (-2 x -\frac {6}{5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 101, normalized size = 0.72 \[ -\frac {729}{2800} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {2187}{625} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {105057}{5000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {201}{20796875} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {315684}{3125} \, \sqrt {-2 \, x + 1} - \frac {1838265657 \, x + 1102959359}{3025000 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 84, normalized size = 0.60 \[ \frac {315684\,\sqrt {1-2\,x}}{3125}-\frac {105057\,{\left (1-2\,x\right )}^{3/2}}{5000}+\frac {2187\,{\left (1-2\,x\right )}^{5/2}}{625}-\frac {729\,{\left (1-2\,x\right )}^{7/2}}{2800}+\frac {\frac {1838265657\,x}{15125000}+\frac {1102959359}{15125000}}{\frac {11\,\sqrt {1-2\,x}}{5}-{\left (1-2\,x\right )}^{3/2}}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,402{}\mathrm {i}}{20796875} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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