Optimal. Leaf size=171 \[ \frac {181 \sqrt {1-2 x} (5 x+3)^{5/2}}{108 (3 x+2)^2}-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}+\frac {331 \sqrt {1-2 x} (5 x+3)^{3/2}}{168 (3 x+2)}-\frac {39745 \sqrt {1-2 x} \sqrt {5 x+3}}{4536}-\frac {575}{243} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {326717 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{13608 \sqrt {7}} \]
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Rubi [A] time = 0.07, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ \frac {181 \sqrt {1-2 x} (5 x+3)^{5/2}}{108 (3 x+2)^2}-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}+\frac {331 \sqrt {1-2 x} (5 x+3)^{3/2}}{168 (3 x+2)}-\frac {39745 \sqrt {1-2 x} \sqrt {5 x+3}}{4536}-\frac {575}{243} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {326717 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{13608 \sqrt {7}} \]
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)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx &=-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {1}{9} \int \frac {\left (\frac {7}{2}-40 x\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^3} \, dx\\ &=-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {1}{54} \int \frac {\left (\frac {139}{4}-1065 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=\frac {331 \sqrt {1-2 x} (3+5 x)^{3/2}}{168 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {\int \frac {\left (\frac {8757}{8}-\frac {119235 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx}{1134}\\ &=-\frac {39745 \sqrt {1-2 x} \sqrt {3+5 x}}{4536}+\frac {331 \sqrt {1-2 x} (3+5 x)^{3/2}}{168 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{108 (2+3 x)^2}+\frac {\int \frac {-\frac {317283}{4}-241500 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{6804}\\ &=-\frac {39745 \sqrt {1-2 x} \sqrt {3+5 x}}{4536}+\frac {331 \sqrt {1-2 x} (3+5 x)^{3/2}}{168 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {2875}{243} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx+\frac {326717 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{27216}\\ &=-\frac {39745 \sqrt {1-2 x} \sqrt {3+5 x}}{4536}+\frac {331 \sqrt {1-2 x} (3+5 x)^{3/2}}{168 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{108 (2+3 x)^2}+\frac {326717 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{13608}-\frac {1}{243} \left (1150 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=-\frac {39745 \sqrt {1-2 x} \sqrt {3+5 x}}{4536}+\frac {331 \sqrt {1-2 x} (3+5 x)^{3/2}}{168 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {575}{243} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {326717 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{13608 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 139, normalized size = 0.81 \[ \frac {-21 \sqrt {-(1-2 x)^2} \sqrt {5 x+3} \left (75600 x^3+286791 x^2+275022 x+78416\right )-326717 \sqrt {14 x-7} (3 x+2)^3 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )+225400 \sqrt {10-20 x} (3 x+2)^3 \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{95256 \sqrt {2 x-1} (3 x+2)^3} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 1.17, size = 161, normalized size = 0.94 \[ -\frac {326717 \, \sqrt {7} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 225400 \, \sqrt {10} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 42 \, {\left (75600 \, x^{3} + 286791 \, x^{2} + 275022 \, x + 78416\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{190512 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.32, size = 396, normalized size = 2.32 \[ \frac {326717}{1905120} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {575}{486} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {10}{81} \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {11 \, \sqrt {10} {\left (2463 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 1767360 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {377652800 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {1510611200 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{756 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 270, normalized size = 1.58 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-6085800 \sqrt {10}\, x^{3} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+8821359 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-3175200 \sqrt {-10 x^{2}-x +3}\, x^{3}-12171600 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+17642718 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-12045222 \sqrt {-10 x^{2}-x +3}\, x^{2}-8114400 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+11761812 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-11550924 \sqrt {-10 x^{2}-x +3}\, x -1803200 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+2613736 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-3293472 \sqrt {-10 x^{2}-x +3}\right )}{190512 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.17, size = 161, normalized size = 0.94 \[ \frac {865}{2646} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{21 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {173 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{588 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {34805}{5292} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {575}{486} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {326717}{190512} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {152917}{31752} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {2507 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{3528 \, {\left (3 \, x + 2\right )}} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^4} \,d x \]
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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