Optimal. Leaf size=149 \[ -\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{9 (3 x+2)^3}+\frac {5 \sqrt {5 x+3} (1-2 x)^{3/2}}{12 (3 x+2)^2}+\frac {925 \sqrt {5 x+3} \sqrt {1-2 x}}{216 (3 x+2)}-\frac {8}{81} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {32765 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{648 \sqrt {7}} \]
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Rubi [A] time = 0.05, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {97, 149, 157, 54, 216, 93, 204} \[ -\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{9 (3 x+2)^3}+\frac {5 \sqrt {5 x+3} (1-2 x)^{3/2}}{12 (3 x+2)^2}+\frac {925 \sqrt {5 x+3} \sqrt {1-2 x}}{216 (3 x+2)}-\frac {8}{81} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {32765 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{648 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 54
Rule 93
Rule 97
Rule 149
Rule 157
Rule 204
Rule 216
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^4} \, dx &=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{9 (2+3 x)^3}+\frac {1}{9} \int \frac {\left (-\frac {25}{2}-30 x\right ) (1-2 x)^{3/2}}{(2+3 x)^3 \sqrt {3+5 x}} \, dx\\ &=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{9 (2+3 x)^3}+\frac {5 (1-2 x)^{3/2} \sqrt {3+5 x}}{12 (2+3 x)^2}-\frac {1}{54} \int \frac {\left (-\frac {1245}{4}-120 x\right ) \sqrt {1-2 x}}{(2+3 x)^2 \sqrt {3+5 x}} \, dx\\ &=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{9 (2+3 x)^3}+\frac {5 (1-2 x)^{3/2} \sqrt {3+5 x}}{12 (2+3 x)^2}+\frac {925 \sqrt {1-2 x} \sqrt {3+5 x}}{216 (2+3 x)}+\frac {1}{162} \int \frac {\frac {31485}{8}-240 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{9 (2+3 x)^3}+\frac {5 (1-2 x)^{3/2} \sqrt {3+5 x}}{12 (2+3 x)^2}+\frac {925 \sqrt {1-2 x} \sqrt {3+5 x}}{216 (2+3 x)}-\frac {40}{81} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx+\frac {32765 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{1296}\\ &=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{9 (2+3 x)^3}+\frac {5 (1-2 x)^{3/2} \sqrt {3+5 x}}{12 (2+3 x)^2}+\frac {925 \sqrt {1-2 x} \sqrt {3+5 x}}{216 (2+3 x)}+\frac {32765}{648} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )-\frac {1}{81} \left (16 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{9 (2+3 x)^3}+\frac {5 (1-2 x)^{3/2} \sqrt {3+5 x}}{12 (2+3 x)^2}+\frac {925 \sqrt {1-2 x} \sqrt {3+5 x}}{216 (2+3 x)}-\frac {8}{81} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {32765 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{648 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 134, normalized size = 0.90 \[ \frac {21 \sqrt {-(1-2 x)^2} \sqrt {5 x+3} \left (7689 x^2+11106 x+3856\right )-32765 \sqrt {14 x-7} (3 x+2)^3 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )+448 \sqrt {10-20 x} (3 x+2)^3 \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{4536 \sqrt {2 x-1} (3 x+2)^3} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.97, size = 156, normalized size = 1.05 \[ -\frac {32765 \, \sqrt {7} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 448 \, \sqrt {10} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 42 \, {\left (7689 \, x^{2} + 11106 \, x + 3856\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{9072 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.82, size = 377, normalized size = 2.53 \[ \frac {6553}{18144} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {4}{81} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {11 \, \sqrt {10} {\left (989 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 795200 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {72520000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {290080000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{108 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 253, normalized size = 1.70 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-12096 \sqrt {10}\, x^{3} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+884655 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-24192 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1769310 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+322938 \sqrt {-10 x^{2}-x +3}\, x^{2}-16128 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1179540 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+466452 \sqrt {-10 x^{2}-x +3}\, x -3584 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+262120 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+161952 \sqrt {-10 x^{2}-x +3}\right )}{9072 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.17, size = 132, normalized size = 0.89 \[ -\frac {4}{81} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {32765}{9072} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {145}{54} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {7 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{9 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {29 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{12 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {1105 \, \sqrt {-10 \, x^{2} - x + 3}}{216 \, {\left (3 \, x + 2\right )}} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^4} \,d x \]
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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