Optimal. Leaf size=140 \[ \frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^4}+\frac {85754 x+57069}{4116 \sqrt {1-2 x} (3 x+2)^4}-\frac {177635 \sqrt {1-2 x}}{403368 (3 x+2)}-\frac {177635 \sqrt {1-2 x}}{172872 (3 x+2)^2}-\frac {35527 \sqrt {1-2 x}}{12348 (3 x+2)^3}-\frac {177635 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{201684 \sqrt {21}} \]
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Rubi [A] time = 0.04, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {98, 144, 51, 63, 206} \[ \frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^4}+\frac {85754 x+57069}{4116 \sqrt {1-2 x} (3 x+2)^4}-\frac {177635 \sqrt {1-2 x}}{403368 (3 x+2)}-\frac {177635 \sqrt {1-2 x}}{172872 (3 x+2)^2}-\frac {35527 \sqrt {1-2 x}}{12348 (3 x+2)^3}-\frac {177635 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{201684 \sqrt {21}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 98
Rule 144
Rule 206
Rubi steps
\begin {align*} \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^5} \, dx &=\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^4}-\frac {1}{21} \int \frac {(-167-315 x) (3+5 x)}{(1-2 x)^{3/2} (2+3 x)^5} \, dx\\ &=\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^4}+\frac {57069+85754 x}{4116 \sqrt {1-2 x} (2+3 x)^4}+\frac {35527}{588} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=-\frac {35527 \sqrt {1-2 x}}{12348 (2+3 x)^3}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^4}+\frac {57069+85754 x}{4116 \sqrt {1-2 x} (2+3 x)^4}+\frac {177635 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx}{12348}\\ &=-\frac {35527 \sqrt {1-2 x}}{12348 (2+3 x)^3}-\frac {177635 \sqrt {1-2 x}}{172872 (2+3 x)^2}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^4}+\frac {57069+85754 x}{4116 \sqrt {1-2 x} (2+3 x)^4}+\frac {177635 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{57624}\\ &=-\frac {35527 \sqrt {1-2 x}}{12348 (2+3 x)^3}-\frac {177635 \sqrt {1-2 x}}{172872 (2+3 x)^2}-\frac {177635 \sqrt {1-2 x}}{403368 (2+3 x)}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^4}+\frac {57069+85754 x}{4116 \sqrt {1-2 x} (2+3 x)^4}+\frac {177635 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{403368}\\ &=-\frac {35527 \sqrt {1-2 x}}{12348 (2+3 x)^3}-\frac {177635 \sqrt {1-2 x}}{172872 (2+3 x)^2}-\frac {177635 \sqrt {1-2 x}}{403368 (2+3 x)}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^4}+\frac {57069+85754 x}{4116 \sqrt {1-2 x} (2+3 x)^4}-\frac {177635 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{403368}\\ &=-\frac {35527 \sqrt {1-2 x}}{12348 (2+3 x)^3}-\frac {177635 \sqrt {1-2 x}}{172872 (2+3 x)^2}-\frac {177635 \sqrt {1-2 x}}{403368 (2+3 x)}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^4}+\frac {57069+85754 x}{4116 \sqrt {1-2 x} (2+3 x)^4}-\frac {177635 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{201684 \sqrt {21}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 61, normalized size = 0.44 \[ \frac {875 x^2-3779 x+4000}{147 (1-2 x)^{3/2} (3 x+2)^4}-\frac {1136864 \sqrt {1-2 x} \, _2F_1\left (\frac {1}{2},5;\frac {3}{2};\frac {3}{7}-\frac {6 x}{7}\right )}{2470629} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 129, normalized size = 0.92 \[ \frac {177635 \, \sqrt {21} {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (19184580 \, x^{5} + 34105920 \, x^{4} + 10906789 \, x^{3} - 12952519 \, x^{2} - 10307138 \, x - 2094250\right )} \sqrt {-2 \, x + 1}}{8470728 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.31, size = 121, normalized size = 0.86 \[ \frac {177635}{8470728} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {484 \, {\left (360 \, x - 257\right )}}{352947 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {5346135 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 35859747 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 80180905 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 59762605 \, \sqrt {-2 \, x + 1}}{22588608 \, {\left (3 \, x + 2\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 84, normalized size = 0.60 \[ -\frac {177635 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{4235364}+\frac {5324}{50421 \left (-2 x +1\right )^{\frac {3}{2}}}+\frac {29040}{117649 \sqrt {-2 x +1}}+\frac {\frac {1782045 \left (-2 x +1\right )^{\frac {7}{2}}}{470596}-\frac {1707607 \left (-2 x +1\right )^{\frac {5}{2}}}{67228}+\frac {1636345 \left (-2 x +1\right )^{\frac {3}{2}}}{28812}-\frac {174235 \sqrt {-2 x +1}}{4116}}{\left (-6 x -4\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 128, normalized size = 0.91 \[ \frac {177635}{8470728} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {4796145 \, {\left (2 \, x - 1\right )}^{5} + 41033685 \, {\left (2 \, x - 1\right )}^{4} + 127080079 \, {\left (2 \, x - 1\right )}^{3} + 157094539 \, {\left (2 \, x - 1\right )}^{2} + 63748608 \, x - 83006000}{201684 \, {\left (81 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 756 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 2646 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 4116 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 2401 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.24, size = 108, normalized size = 0.77 \[ -\frac {177635\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{4235364}-\frac {\frac {15488\,x}{3969}+\frac {3206011\,{\left (2\,x-1\right )}^2}{333396}+\frac {2593471\,{\left (2\,x-1\right )}^3}{333396}+\frac {1953985\,{\left (2\,x-1\right )}^4}{777924}+\frac {177635\,{\left (2\,x-1\right )}^5}{605052}-\frac {60500}{11907}}{\frac {2401\,{\left (1-2\,x\right )}^{3/2}}{81}-\frac {1372\,{\left (1-2\,x\right )}^{5/2}}{27}+\frac {98\,{\left (1-2\,x\right )}^{7/2}}{3}-\frac {28\,{\left (1-2\,x\right )}^{9/2}}{3}+{\left (1-2\,x\right )}^{11/2}} \]
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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