Optimal. Leaf size=171 \[ \frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {511 (3 x+2)^4}{242 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {7591 \sqrt {1-2 x} (3 x+2)^3}{39930 (5 x+3)^{3/2}}+\frac {261331 \sqrt {1-2 x} (3 x+2)^2}{2196150 \sqrt {5 x+3}}-\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} (78981180 x+190406711)}{117128000}+\frac {753543 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{8000 \sqrt {10}} \]
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Rubi [A] time = 0.06, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {98, 150, 147, 54, 216} \begin {gather*} \frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {511 (3 x+2)^4}{242 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {7591 \sqrt {1-2 x} (3 x+2)^3}{39930 (5 x+3)^{3/2}}+\frac {261331 \sqrt {1-2 x} (3 x+2)^2}{2196150 \sqrt {5 x+3}}-\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} (78981180 x+190406711)}{117128000}+\frac {753543 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{8000 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 54
Rule 98
Rule 147
Rule 150
Rule 216
Rubi steps
\begin {align*} \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx &=\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1}{33} \int \frac {(2+3 x)^4 \left (204+\frac {717 x}{2}\right )}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=-\frac {511 (2+3 x)^4}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1}{363} \int \frac {\left (-\frac {32415}{2}-\frac {115641 x}{4}\right ) (2+3 x)^3}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {7591 \sqrt {1-2 x} (2+3 x)^3}{39930 (3+5 x)^{3/2}}-\frac {511 (2+3 x)^4}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {2 \int \frac {\left (-880026-\frac {11995011 x}{8}\right ) (2+3 x)^2}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx}{59895}\\ &=\frac {7591 \sqrt {1-2 x} (2+3 x)^3}{39930 (3+5 x)^{3/2}}-\frac {511 (2+3 x)^4}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {261331 \sqrt {1-2 x} (2+3 x)^2}{2196150 \sqrt {3+5 x}}-\frac {4 \int \frac {\left (-\frac {127241163}{8}-\frac {414651195 x}{16}\right ) (2+3 x)}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{3294225}\\ &=\frac {7591 \sqrt {1-2 x} (2+3 x)^3}{39930 (3+5 x)^{3/2}}-\frac {511 (2+3 x)^4}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {261331 \sqrt {1-2 x} (2+3 x)^2}{2196150 \sqrt {3+5 x}}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (190406711+78981180 x)}{117128000}+\frac {753543 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{16000}\\ &=\frac {7591 \sqrt {1-2 x} (2+3 x)^3}{39930 (3+5 x)^{3/2}}-\frac {511 (2+3 x)^4}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {261331 \sqrt {1-2 x} (2+3 x)^2}{2196150 \sqrt {3+5 x}}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (190406711+78981180 x)}{117128000}+\frac {753543 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{8000 \sqrt {5}}\\ &=\frac {7591 \sqrt {1-2 x} (2+3 x)^3}{39930 (3+5 x)^{3/2}}-\frac {511 (2+3 x)^4}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {261331 \sqrt {1-2 x} (2+3 x)^2}{2196150 \sqrt {3+5 x}}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (190406711+78981180 x)}{117128000}+\frac {753543 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{8000 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 15.20, size = 75, normalized size = 0.44 \begin {gather*} -\frac {12807946800 x^5+97980793020 x^4-252342435560 x^3-274128335769 x^2+19932058554 x+44437106459}{351384000 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {753543 \sin ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{8000 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.21, size = 157, normalized size = 0.92 \begin {gather*} \frac {(5 x+3)^{3/2} \left (-\frac {3200 (1-2 x)^5}{(5 x+3)^5}-\frac {394240 (1-2 x)^4}{(5 x+3)^4}-\frac {165483078425 (1-2 x)^3}{(5 x+3)^3}-\frac {110301771622 (1-2 x)^2}{(5 x+3)^2}-\frac {14050652000 (1-2 x)}{5 x+3}+941192000\right )}{351384000 (1-2 x)^{3/2} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^2}-\frac {753543 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{8000 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.36, size = 126, normalized size = 0.74 \begin {gather*} -\frac {33097869189 \, \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 (12807946800 \, x^{5} + 97980793020 \, x^{4} - 252342435560 \, x^{3} - 274128335769 \, x^{2} + 19932058554 \, x + 44437106459\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{7027680000 \, {\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 [A] time = 1.53, size = 204, normalized size = 1.19 \begin {gather*} -\frac {1}{2196150000} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {4884 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} + \frac {753543}{80000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (32019867 \, {\left (4 \, \sqrt {5} {\left (5 \, x + 3\right )} + 93 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 110347010662 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 1820310410259 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{219615000000 \, {\left (2 \, x - 1\right )}^{2}} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {1221 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{137259375 \, {\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 = 199, normalized size = 1.16 \begin {gather*} \frac {\sqrt {-2 x +1}\, \left (-256158936000 \sqrt {-10 x^{2}-x +3}\, x^{5}+3309786918900 \sqrt {10}\, x^{4} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-1959615860400 \sqrt {-10 x^{2}-x +3}\, x^{4}+661957383780 \sqrt {10}\, x^{3} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+5046848711200 \sqrt {-10 x^{2}-x +3}\, x^{3}-1952774282151 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+5482566715380 \sqrt {-10 x^{2}-x +3}\, x^{2}-198587215134 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-398641171080 \sqrt {-10 x^{2}-x +3}\, x +297880822701 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-888742129180 \sqrt {-10 x^{2}-x +3}\right )}{7027680000 \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 [A] time = 1.34, size = 214, normalized size = 1.25 \begin {gather*} -\frac {729 \, x^{5}}{20 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {111537 \, x^{4}}{400 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {251181}{234256000} \, 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 {753543}{160000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {90676341}{117128000} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {170985889 \, x}{7027680 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {766611 \, x^{2}}{1000 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {1005653687}{878460000 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {416356591 \, x}{3630000 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {496819753}{3630000 \, {\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 )}^6}{{\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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