Optimal. Leaf size=180 \[ \frac {33 \sqrt {1-2 x} (5 x+3)^{7/2}}{40 (3 x+2)^4}+\frac {(1-2 x)^{3/2} (5 x+3)^{7/2}}{5 (3 x+2)^5}-\frac {121 \sqrt {1-2 x} (5 x+3)^{5/2}}{560 (3 x+2)^3}-\frac {1331 \sqrt {1-2 x} (5 x+3)^{3/2}}{3136 (3 x+2)^2}-\frac {43923 \sqrt {1-2 x} \sqrt {5 x+3}}{43904 (3 x+2)}-\frac {483153 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{43904 \sqrt {7}} \]
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Rubi [A] time = 0.05, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {94, 93, 204} \[ \frac {33 \sqrt {1-2 x} (5 x+3)^{7/2}}{40 (3 x+2)^4}+\frac {(1-2 x)^{3/2} (5 x+3)^{7/2}}{5 (3 x+2)^5}-\frac {121 \sqrt {1-2 x} (5 x+3)^{5/2}}{560 (3 x+2)^3}-\frac {1331 \sqrt {1-2 x} (5 x+3)^{3/2}}{3136 (3 x+2)^2}-\frac {43923 \sqrt {1-2 x} \sqrt {5 x+3}}{43904 (3 x+2)}-\frac {483153 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{43904 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 204
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx &=\frac {(1-2 x)^{3/2} (3+5 x)^{7/2}}{5 (2+3 x)^5}+\frac {33}{10} \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx\\ &=\frac {(1-2 x)^{3/2} (3+5 x)^{7/2}}{5 (2+3 x)^5}+\frac {33 \sqrt {1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}+\frac {363}{80} \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=-\frac {121 \sqrt {1-2 x} (3+5 x)^{5/2}}{560 (2+3 x)^3}+\frac {(1-2 x)^{3/2} (3+5 x)^{7/2}}{5 (2+3 x)^5}+\frac {33 \sqrt {1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}+\frac {1331}{224} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=-\frac {1331 \sqrt {1-2 x} (3+5 x)^{3/2}}{3136 (2+3 x)^2}-\frac {121 \sqrt {1-2 x} (3+5 x)^{5/2}}{560 (2+3 x)^3}+\frac {(1-2 x)^{3/2} (3+5 x)^{7/2}}{5 (2+3 x)^5}+\frac {33 \sqrt {1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}+\frac {43923 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{6272}\\ &=-\frac {43923 \sqrt {1-2 x} \sqrt {3+5 x}}{43904 (2+3 x)}-\frac {1331 \sqrt {1-2 x} (3+5 x)^{3/2}}{3136 (2+3 x)^2}-\frac {121 \sqrt {1-2 x} (3+5 x)^{5/2}}{560 (2+3 x)^3}+\frac {(1-2 x)^{3/2} (3+5 x)^{7/2}}{5 (2+3 x)^5}+\frac {33 \sqrt {1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}+\frac {483153 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{87808}\\ &=-\frac {43923 \sqrt {1-2 x} \sqrt {3+5 x}}{43904 (2+3 x)}-\frac {1331 \sqrt {1-2 x} (3+5 x)^{3/2}}{3136 (2+3 x)^2}-\frac {121 \sqrt {1-2 x} (3+5 x)^{5/2}}{560 (2+3 x)^3}+\frac {(1-2 x)^{3/2} (3+5 x)^{7/2}}{5 (2+3 x)^5}+\frac {33 \sqrt {1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}+\frac {483153 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{43904}\\ &=-\frac {43923 \sqrt {1-2 x} \sqrt {3+5 x}}{43904 (2+3 x)}-\frac {1331 \sqrt {1-2 x} (3+5 x)^{3/2}}{3136 (2+3 x)^2}-\frac {121 \sqrt {1-2 x} (3+5 x)^{5/2}}{560 (2+3 x)^3}+\frac {(1-2 x)^{3/2} (3+5 x)^{7/2}}{5 (2+3 x)^5}+\frac {33 \sqrt {1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}-\frac {483153 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{43904 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 84, normalized size = 0.47 \[ \frac {\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} \left (15899035 x^4+46076650 x^3+47906548 x^2+21437032 x+3507552\right )}{(3 x+2)^5}-2415765 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1536640} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 131, normalized size = 0.73 \[ -\frac {2415765 \, \sqrt {7} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (15899035 \, x^{4} + 46076650 \, x^{3} + 47906548 \, x^{2} + 21437032 \, x + 3507552\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{3073280 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.59, size = 426, normalized size = 2.37 \[ \frac {483153}{6146560} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {161051 \, \sqrt {10} {\left (3 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} + 3920 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 2007040 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 307328000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {18439680000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {73758720000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{21952 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 298, normalized size = 1.66 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (587030895 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1956769650 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+222586490 \sqrt {-10 x^{2}-x +3}\, x^{4}+2609026200 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+645073100 \sqrt {-10 x^{2}-x +3}\, x^{3}+1739350800 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+670691672 \sqrt {-10 x^{2}-x +3}\, x^{2}+579783600 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+300118448 \sqrt {-10 x^{2}-x +3}\, x +77304480 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+49105728 \sqrt {-10 x^{2}-x +3}\right )}{3073280 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.51, size = 227, normalized size = 1.26 \[ \frac {90695}{230496} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{35 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {33 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{392 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {1221 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{5488 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {54417 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{153664 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {738705}{153664} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {483153}{614656} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {650859}{307328} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {215303 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{921984 \, {\left (3 \, x + 2\right )}} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^6} \,d x \]
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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