Optimal. Leaf size=130 \[ -\frac {190}{1617 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {3830}{124509 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1840225 \sqrt {1-2 x}}{1369599 \sqrt {3+5 x}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}}+\frac {3105 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{343 \sqrt {7}} \]
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Rubi [A]
time = 0.03, 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 = {105, 157, 12,
95, 210} \begin {gather*} \frac {3105 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{343 \sqrt {7}}-\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 105
Rule 157
Rule 210
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} \text {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 = 0.16, size = 79, normalized size = 0.61 \begin {gather*} \frac {-3499599+8760465 x+7613680 x^2-22082700 x^3}{1369599 (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 {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(256\) vs.
\(2(97)=194\).
time = 0.09, size = 257, normalized size = 1.98
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \left (743895900 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+198372240 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}-458735805 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+309157800 x^{3} \sqrt {-10 x^{2}-x +3}-61991325 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -106591520 x^{2} \sqrt {-10 x^{2}-x +3}+74389590 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-122646510 x \sqrt {-10 x^{2}-x +3}+48994386 \sqrt {-10 x^{2}-x +3}\right )}{19174386 \left (2+3 x \right ) \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\) | \(257\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, 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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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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 291 vs.
\(2 (97) = 194\).
time = 1.31, 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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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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