Optimal. Leaf size=130 \[ -\frac {1840225 \sqrt {1-2 x}}{1369599 \sqrt {5 x+3}}-\frac {3830}{124509 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {3}{7 (1-2 x)^{3/2} (3 x+2) \sqrt {5 x+3}}-\frac {190}{1617 (1-2 x)^{3/2} \sqrt {5 x+3}}+\frac {3105 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{343 \sqrt {7}} \]
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Rubi [A] time = 0.05, antiderivative size = 130, 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 = {103, 152, 12, 93, 204} \begin {gather*} -\frac {1840225 \sqrt {1-2 x}}{1369599 \sqrt {5 x+3}}-\frac {3830}{124509 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {3}{7 (1-2 x)^{3/2} (3 x+2) \sqrt {5 x+3}}-\frac {190}{1617 (1-2 x)^{3/2} \sqrt {5 x+3}}+\frac {3105 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{343 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 103
Rule 152
Rule 204
Rubi steps
\begin {align*} \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}} \, dx &=\frac {3}{7 (1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}}+\frac {1}{7} \int \frac {-\frac {5}{2}-90 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac {190}{1617 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}}-\frac {2 \int \frac {-\frac {3785}{4}+2850 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}} \, dx}{1617}\\ &=-\frac {190}{1617 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {3830}{124509 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}}+\frac {4 \int \frac {\frac {299105}{8}-\frac {28725 x}{2}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{124509}\\ &=-\frac {190}{1617 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {3830}{124509 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1840225 \sqrt {1-2 x}}{1369599 \sqrt {3+5 x}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}}-\frac {8 \int \frac {12398265}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{1369599}\\ &=-\frac {190}{1617 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {3830}{124509 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1840225 \sqrt {1-2 x}}{1369599 \sqrt {3+5 x}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}}-\frac {3105}{686} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {190}{1617 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {3830}{124509 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1840225 \sqrt {1-2 x}}{1369599 \sqrt {3+5 x}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}}-\frac {3105}{343} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {190}{1617 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {3830}{124509 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1840225 \sqrt {1-2 x}}{1369599 \sqrt {3+5 x}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}}+\frac {3105 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{343 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 100, normalized size = 0.77 \begin {gather*} -\frac {12398265 \sqrt {7-14 x} \sqrt {5 x+3} \left (6 x^2+x-2\right ) \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )+7 \left (22082700 x^3-7613680 x^2-8760465 x+3499599\right )}{9587193 (1-2 x)^{3/2} (3 x+2) \sqrt {5 x+3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 122, normalized size = 0.94 \begin {gather*} \frac {\left (-\frac {1286250 (1-2 x)^3}{(5 x+3)^3}-\frac {12554985 (1-2 x)^2}{(5 x+3)^2}+\frac {45920 (1-2 x)}{5 x+3}+1568\right ) (5 x+3)^{3/2}}{1369599 (1-2 x)^{3/2} \left (\frac {1-2 x}{5 x+3}+7\right )}+\frac {3105 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{343 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 116, normalized size = 0.89 \begin {gather*} \frac {12398265 \, \sqrt {7} {\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\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 (22082700 \, x^{3} - 7613680 \, x^{2} - 8760465 \, x + 3499599\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{19174386 \, {\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.20, size = 291, normalized size = 2.24 \begin {gather*} -\frac {621}{9604} \, \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 {125}{2662} \, \sqrt {10} {\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 )} - \frac {1782 \, \sqrt {10} {\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 )}}{343 \, {\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 )}} - \frac {32 \, {\left (373 \, \sqrt {5} {\left (5 \, x + 3\right )} - 2244 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{34239975 \, {\left (2 \, x - 1\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 257, normalized size = 1.98 \begin {gather*} -\frac {\sqrt {-2 x +1}\, \left (743895900 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+198372240 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+309157800 \sqrt {-10 x^{2}-x +3}\, x^{3}-458735805 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-106591520 \sqrt {-10 x^{2}-x +3}\, x^{2}-61991325 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-122646510 \sqrt {-10 x^{2}-x +3}\, x +74389590 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+48994386 \sqrt {-10 x^{2}-x +3}\right )}{19174386 \left (3 x +2\right ) \left (2 x -1\right )^{2} \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{2} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \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 {1}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^2\,{\left (5\,x+3\right )}^{3/2}} \,d x \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 {1}{\left (1 - 2 x\right )^{\frac {5}{2}} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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