Optimal. Leaf size=142 \[ \frac {7 (3 x+2)^4}{11 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {37 \sqrt {1-2 x} (3 x+2)^3}{605 \sqrt {5 x+3}}+\frac {8463 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2}{12100}+\frac {21 \sqrt {1-2 x} \sqrt {5 x+3} (841380 x+2027201)}{1936000}-\frac {2911419 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{16000 \sqrt {10}} \]
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Rubi [A] time = 0.04, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {98, 150, 153, 147, 54, 216} \[ \frac {7 (3 x+2)^4}{11 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {37 \sqrt {1-2 x} (3 x+2)^3}{605 \sqrt {5 x+3}}+\frac {8463 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2}{12100}+\frac {21 \sqrt {1-2 x} \sqrt {5 x+3} (841380 x+2027201)}{1936000}-\frac {2911419 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{16000 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 54
Rule 98
Rule 147
Rule 150
Rule 153
Rule 216
Rubi steps
\begin {align*} \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx &=\frac {7 (2+3 x)^4}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1}{11} \int \frac {(2+3 x)^3 \left (215+\frac {729 x}{2}\right )}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac {37 \sqrt {1-2 x} (2+3 x)^3}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^4}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {2}{605} \int \frac {(2+3 x)^2 \left (3843+\frac {25389 x}{4}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {37 \sqrt {1-2 x} (2+3 x)^3}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^4}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {8463 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}{12100}+\frac {\int \frac {\left (-\frac {1353933}{4}-\frac {4417245 x}{8}\right ) (2+3 x)}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{9075}\\ &=-\frac {37 \sqrt {1-2 x} (2+3 x)^3}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^4}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {8463 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}{12100}+\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} (2027201+841380 x)}{1936000}-\frac {2911419 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{32000}\\ &=-\frac {37 \sqrt {1-2 x} (2+3 x)^3}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^4}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {8463 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}{12100}+\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} (2027201+841380 x)}{1936000}-\frac {2911419 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{16000 \sqrt {5}}\\ &=-\frac {37 \sqrt {1-2 x} (2+3 x)^3}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^4}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {8463 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}{12100}+\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} (2027201+841380 x)}{1936000}-\frac {2911419 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{16000 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 98, normalized size = 0.69 \[ \frac {-352281699 \sqrt {10} \left (10 x^2+x-3\right ) \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \sqrt {2 x-1} \sqrt {5 x+3} \left (15681600 x^4+75663720 x^3+208989990 x^2-169670279 x-162727423\right )}{19360000 \sqrt {-(1-2 x)^2} (5 x+3)} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 1.08, size = 97, normalized size = 0.68 \[ \frac {352281699 \, \sqrt {10} {\left (10 \, x^{2} + x - 3\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (15681600 \, x^{4} + 75663720 \, x^{3} + 208989990 \, x^{2} - 169670279 \, x - 162727423\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{38720000 \, {\left (10 \, x^{2} + x - 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.38, size = 144, normalized size = 1.01 \[ -\frac {2911419}{160000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (6534 \, {\left (12 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} + 97 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 16325 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 1761451247 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{242000000 \, {\left (2 \, x - 1\right )}} - \frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{756250 \, \sqrt {5 \, x + 3}} + \frac {2 \, \sqrt {10} \sqrt {5 \, x + 3}}{378125 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 154, normalized size = 1.08 \[ -\frac {\sqrt {-2 x +1}\, \left (-313632000 \sqrt {-10 x^{2}-x +3}\, x^{4}-1513274400 \sqrt {-10 x^{2}-x +3}\, x^{3}+3522816990 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-4179799800 \sqrt {-10 x^{2}-x +3}\, x^{2}+352281699 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+3393405580 \sqrt {-10 x^{2}-x +3}\, x -1056845097 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+3254548460 \sqrt {-10 x^{2}-x +3}\right )}{38720000 \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.19, size = 92, normalized size = 0.65 \[ -\frac {81 \, x^{4}}{10 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {15633 \, x^{3}}{400 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {172719 \, x^{2}}{1600 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {2911419}{320000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {169670279 \, x}{1936000 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {162727423}{1936000 \, \sqrt {-10 \, x^{2} - x + 3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (3\,x+2\right )}^5}{{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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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