Optimal. Leaf size=200 \[ \frac {1165 \sqrt {1-2 x} (5 x+3)^{5/2}}{2592 (3 x+2)^2}+\frac {185 (1-2 x)^{3/2} (5 x+3)^{5/2}}{216 (3 x+2)^3}-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}-\frac {3485 \sqrt {1-2 x} (5 x+3)^{3/2}}{4032 (3 x+2)}+\frac {249575 \sqrt {1-2 x} \sqrt {5 x+3}}{108864}+\frac {1850}{729} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {3304795 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{326592 \sqrt {7}} \]
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Rubi [A] time = 0.08, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {97, 149, 154, 157, 54, 216, 93, 204} \[ \frac {1165 \sqrt {1-2 x} (5 x+3)^{5/2}}{2592 (3 x+2)^2}+\frac {185 (1-2 x)^{3/2} (5 x+3)^{5/2}}{216 (3 x+2)^3}-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}-\frac {3485 \sqrt {1-2 x} (5 x+3)^{3/2}}{4032 (3 x+2)}+\frac {249575 \sqrt {1-2 x} \sqrt {5 x+3}}{108864}+\frac {1850}{729} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {3304795 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{326592 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 54
Rule 93
Rule 97
Rule 149
Rule 154
Rule 157
Rule 204
Rule 216
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx &=-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac {1}{12} \int \frac {\left (-\frac {5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{216 (2+3 x)^3}-\frac {1}{108} \int \frac {\left (-\frac {3655}{4}-1225 x\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^3} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{216 (2+3 x)^3}+\frac {1165 \sqrt {1-2 x} (3+5 x)^{5/2}}{2592 (2+3 x)^2}+\frac {1}{648} \int \frac {\left (\frac {20765}{8}-\frac {3975 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {3485 \sqrt {1-2 x} (3+5 x)^{3/2}}{4032 (2+3 x)}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{216 (2+3 x)^3}+\frac {1165 \sqrt {1-2 x} (3+5 x)^{5/2}}{2592 (2+3 x)^2}+\frac {\int \frac {\left (\frac {1308195}{16}-\frac {748725 x}{4}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx}{13608}\\ &=\frac {249575 \sqrt {1-2 x} \sqrt {3+5 x}}{108864}-\frac {3485 \sqrt {1-2 x} (3+5 x)^{3/2}}{4032 (2+3 x)}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{216 (2+3 x)^3}+\frac {1165 \sqrt {1-2 x} (3+5 x)^{5/2}}{2592 (2+3 x)^2}-\frac {\int \frac {-\frac {13271205}{8}-3108000 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{81648}\\ &=\frac {249575 \sqrt {1-2 x} \sqrt {3+5 x}}{108864}-\frac {3485 \sqrt {1-2 x} (3+5 x)^{3/2}}{4032 (2+3 x)}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{216 (2+3 x)^3}+\frac {1165 \sqrt {1-2 x} (3+5 x)^{5/2}}{2592 (2+3 x)^2}-\frac {3304795 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{653184}+\frac {9250}{729} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {249575 \sqrt {1-2 x} \sqrt {3+5 x}}{108864}-\frac {3485 \sqrt {1-2 x} (3+5 x)^{3/2}}{4032 (2+3 x)}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{216 (2+3 x)^3}+\frac {1165 \sqrt {1-2 x} (3+5 x)^{5/2}}{2592 (2+3 x)^2}-\frac {3304795 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{326592}+\frac {1}{729} \left (3700 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=\frac {249575 \sqrt {1-2 x} \sqrt {3+5 x}}{108864}-\frac {3485 \sqrt {1-2 x} (3+5 x)^{3/2}}{4032 (2+3 x)}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{216 (2+3 x)^3}+\frac {1165 \sqrt {1-2 x} (3+5 x)^{5/2}}{2592 (2+3 x)^2}+\frac {1850}{729} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {3304795 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{326592 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 144, normalized size = 0.72 \[ \frac {21 \sqrt {-(1-2 x)^2} \sqrt {5 x+3} \left (3628800 x^4+29475315 x^3+45563928 x^2+25998852 x+5093072\right )+3304795 \sqrt {14 x-7} (3 x+2)^4 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-5801600 \sqrt {10-20 x} (3 x+2)^4 \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{2286144 \sqrt {2 x-1} (3 x+2)^4} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 1.05, size = 181, normalized size = 0.90 \[ \frac {3304795 \, \sqrt {7} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 5801600 \, \sqrt {10} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 42 \, {\left (3628800 \, x^{4} + 29475315 \, x^{3} + 45563928 \, x^{2} + 25998852 \, x + 5093072\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{4572288 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 6.00, size = 454, normalized size = 2.27 \[ -\frac {660959}{9144576} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {925}{729} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {20}{243} \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {55 \, \sqrt {10} {\left (8191 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 7386792 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 2164545600 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {2731201984000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {10924807936000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{54432 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 332, normalized size = 1.66 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (469929600 \sqrt {10}\, x^{4} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-267688395 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+152409600 \sqrt {-10 x^{2}-x +3}\, x^{4}+1253145600 \sqrt {10}\, x^{3} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-713835720 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1237963230 \sqrt {-10 x^{2}-x +3}\, x^{3}+1253145600 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-713835720 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1913684976 \sqrt {-10 x^{2}-x +3}\, x^{2}+556953600 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-317260320 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1091951784 \sqrt {-10 x^{2}-x +3}\, x +92825600 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-52876720 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+213909024 \sqrt {-10 x^{2}-x +3}\right )}{4572288 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.17, size = 226, normalized size = 1.13 \[ \frac {5755}{49392} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{28 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {37 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{392 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {1151 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{10976 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {182225}{98784} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {1488395}{1778112} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {44881 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{197568 \, {\left (3 \, x + 2\right )}} - \frac {28675}{127008} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {925}{729} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {3304795}{4572288} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {1643795}{762048} \, \sqrt {-10 \, x^{2} - x + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^5} \,d x \]
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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