Optimal. Leaf size=113 \[ \frac {7 (3 x+2)^3}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {203 (3 x+2)^2}{242 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {\sqrt {1-2 x} (991010 x+627287)}{2196150 (5 x+3)^{3/2}}+\frac {81 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{50 \sqrt {10}} \]
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Rubi [A] time = 0.03, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {98, 150, 145, 54, 216} \begin {gather*} \frac {7 (3 x+2)^3}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {203 (3 x+2)^2}{242 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {\sqrt {1-2 x} (991010 x+627287)}{2196150 (5 x+3)^{3/2}}+\frac {81 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{50 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 54
Rule 98
Rule 145
Rule 150
Rule 216
Rubi steps
\begin {align*} \int \frac {(2+3 x)^4}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx &=\frac {7 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1}{33} \int \frac {(2+3 x)^2 \left (78+\frac {297 x}{2}\right )}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=-\frac {203 (2+3 x)^2}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1}{363} \int \frac {\left (-\frac {2049}{2}-\frac {9801 x}{4}\right ) (2+3 x)}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx\\ &=-\frac {203 (2+3 x)^2}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {\sqrt {1-2 x} (627287+991010 x)}{2196150 (3+5 x)^{3/2}}+\frac {81}{100} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {203 (2+3 x)^2}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {\sqrt {1-2 x} (627287+991010 x)}{2196150 (3+5 x)^{3/2}}+\frac {81 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{50 \sqrt {5}}\\ &=-\frac {203 (2+3 x)^2}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {\sqrt {1-2 x} (627287+991010 x)}{2196150 (3+5 x)^{3/2}}+\frac {81 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{50 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 15.23, size = 65, normalized size = 0.58 \begin {gather*} \frac {49702040 x^3+51334383 x^2+7883562 x-3014813}{2196150 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {81 \sin ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{50 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 107, normalized size = 0.95 \begin {gather*} \frac {(5 x+3)^{3/2} \left (-\frac {20 (1-2 x)^3}{(5 x+3)^3}-\frac {1656 (1-2 x)^2}{(5 x+3)^2}-\frac {694575 (1-2 x)}{5 x+3}+120050\right )}{2196150 (1-2 x)^{3/2}}-\frac {81 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{50 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.36, size = 116, normalized size = 1.03 \begin {gather*} -\frac {3557763 \, \sqrt {10} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\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 ) - 20 \, {\left (49702040 \, x^{3} + 51334383 \, x^{2} + 7883562 \, x - 3014813\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{43923000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.55, size = 178, normalized size = 1.58 \begin {gather*} -\frac {1}{87846000} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {3300 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} + \frac {81}{500} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {343 \, {\left (232 \, \sqrt {5} {\left (5 \, x + 3\right )} - 891 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{2196150 \, {\left (2 \, x - 1\right )}^{2}} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {825 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{5490375 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 165, normalized size = 1.46 \begin {gather*} \frac {\sqrt {-2 x +1}\, \left (355776300 \sqrt {10}\, x^{4} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+71155260 \sqrt {10}\, x^{3} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+994040800 \sqrt {-10 x^{2}-x +3}\, x^{3}-209908017 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1026687660 \sqrt {-10 x^{2}-x +3}\, x^{2}-21346578 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+157671240 \sqrt {-10 x^{2}-x +3}\, x +32019867 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-60296260 \sqrt {-10 x^{2}-x +3}\right )}{43923000 \left (2 x -1\right )^{2} \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.21, size = 180, normalized size = 1.59 \begin {gather*} \frac {27}{1464100} \, x {\left (\frac {7220 \, x}{\sqrt {-10 \, x^{2} - x + 3}} + \frac {439230 \, x^{2}}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {361}{\sqrt {-10 \, x^{2} - x + 3}} + \frac {21901 \, x}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {87483}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}\right )} - \frac {81}{1000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {9747}{732050} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {1588351 \, x}{1098075 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {108 \, x^{2}}{5 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {34823}{1098075 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {86854 \, x}{9075 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {12682}{9075 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \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 (3\,x+2\right )}^4}{{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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