Optimal. Leaf size=151 \[ -\frac {3993 \sqrt {1-2 x} \sqrt {3+5 x}}{3136 (2+3 x)}-\frac {121 \sqrt {1-2 x} (3+5 x)^{3/2}}{224 (2+3 x)^2}+\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{4 (2+3 x)^4}+\frac {11 \sqrt {1-2 x} (3+5 x)^{5/2}}{8 (2+3 x)^3}-\frac {43923 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{3136 \sqrt {7}} \]
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Rubi [A]
time = 0.03, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {96, 95, 210}
\begin {gather*} -\frac {43923 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3136 \sqrt {7}}+\frac {11 \sqrt {1-2 x} (5 x+3)^{5/2}}{8 (3 x+2)^3}+\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{4 (3 x+2)^4}-\frac {121 \sqrt {1-2 x} (5 x+3)^{3/2}}{224 (3 x+2)^2}-\frac {3993 \sqrt {1-2 x} \sqrt {5 x+3}}{3136 (3 x+2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 96
Rule 210
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx &=\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{4 (2+3 x)^4}+\frac {33}{8} \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx\\ &=\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{4 (2+3 x)^4}+\frac {11 \sqrt {1-2 x} (3+5 x)^{5/2}}{8 (2+3 x)^3}+\frac {121}{16} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=-\frac {121 \sqrt {1-2 x} (3+5 x)^{3/2}}{224 (2+3 x)^2}+\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{4 (2+3 x)^4}+\frac {11 \sqrt {1-2 x} (3+5 x)^{5/2}}{8 (2+3 x)^3}+\frac {3993}{448} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {3993 \sqrt {1-2 x} \sqrt {3+5 x}}{3136 (2+3 x)}-\frac {121 \sqrt {1-2 x} (3+5 x)^{3/2}}{224 (2+3 x)^2}+\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{4 (2+3 x)^4}+\frac {11 \sqrt {1-2 x} (3+5 x)^{5/2}}{8 (2+3 x)^3}+\frac {43923 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{6272}\\ &=-\frac {3993 \sqrt {1-2 x} \sqrt {3+5 x}}{3136 (2+3 x)}-\frac {121 \sqrt {1-2 x} (3+5 x)^{3/2}}{224 (2+3 x)^2}+\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{4 (2+3 x)^4}+\frac {11 \sqrt {1-2 x} (3+5 x)^{5/2}}{8 (2+3 x)^3}+\frac {43923 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{3136}\\ &=-\frac {3993 \sqrt {1-2 x} \sqrt {3+5 x}}{3136 (2+3 x)}-\frac {121 \sqrt {1-2 x} (3+5 x)^{3/2}}{224 (2+3 x)^2}+\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{4 (2+3 x)^4}+\frac {11 \sqrt {1-2 x} (3+5 x)^{5/2}}{8 (2+3 x)^3}-\frac {43923 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{3136 \sqrt {7}}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 79, normalized size = 0.52 \begin {gather*} \frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (32400+145940 x+213240 x^2+100159 x^3\right )}{(2+3 x)^4}-43923 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{21952} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(249\) vs.
\(2(118)=236\).
time = 0.14, size = 250, normalized size = 1.66
method | result | size |
risch | \(-\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (100159 x^{3}+213240 x^{2}+145940 x +32400\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{3136 \left (2+3 x \right )^{4} \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {43923 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{43904 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(129\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (3557763 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+9487368 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+9487368 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+1402226 x^{3} \sqrt {-10 x^{2}-x +3}+4216608 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +2985360 x^{2} \sqrt {-10 x^{2}-x +3}+702768 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+2043160 x \sqrt {-10 x^{2}-x +3}+453600 \sqrt {-10 x^{2}-x +3}\right )}{43904 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{4}}\) | \(250\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.59, size = 186, normalized size = 1.23 \begin {gather*} \frac {8245}{16464} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{28 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {111 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{392 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {4947 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{10976 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {67155}{10976} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {43923}{43904} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {59169}{21952} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {19573 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{65856 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.66, size = 116, normalized size = 0.77 \begin {gather*} -\frac {43923 \, \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 ) - 14 \, {\left (100159 \, x^{3} + 213240 \, x^{2} + 145940 \, x + 32400\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{43904 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1 - 2 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{\left (3 x + 2\right )^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 368 vs.
\(2 (118) = 236\).
time = 1.55, size = 368, normalized size = 2.44 \begin {gather*} \frac {43923}{439040} \, \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 {14641 \, \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 )}^{7} + 3080 \, {\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} - 862400 \, {\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 {65856000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {263424000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{1568 \, {\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}} \end {gather*}
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 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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