Optimal. Leaf size=84 \[ \frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {\sqrt {1-2 x} (24439+38770 x)}{99825 (3+5 x)^{3/2}}-\frac {27 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{25 \sqrt {10}} \]
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Rubi [A]
time = 0.01, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {100, 150, 56,
222} \begin {gather*} -\frac {27 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{25 \sqrt {10}}+\frac {7 (3 x+2)^2}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}-\frac {\sqrt {1-2 x} (38770 x+24439)}{99825 (5 x+3)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 56
Rule 100
Rule 150
Rule 222
Rubi steps
\begin {align*} \int \frac {(2+3 x)^3}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {1}{11} \int \frac {(2+3 x) \left (19+\frac {99 x}{2}\right )}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {\sqrt {1-2 x} (24439+38770 x)}{99825 (3+5 x)^{3/2}}-\frac {27}{50} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {\sqrt {1-2 x} (24439+38770 x)}{99825 (3+5 x)^{3/2}}-\frac {27 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{25 \sqrt {5}}\\ &=\frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {\sqrt {1-2 x} (24439+38770 x)}{99825 (3+5 x)^{3/2}}-\frac {27 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{25 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 64, normalized size = 0.76 \begin {gather*} \frac {229661+772408 x+649265 x^2}{99825 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {27 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{25 \sqrt {10}} \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. \(133\) vs.
\(2(63)=126\).
time = 0.09, size = 134, normalized size = 1.60
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \left (5390550 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{3}+3773385 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-1293732 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +12985300 x^{2} \sqrt {-10 x^{2}-x +3}-970299 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+15448160 x \sqrt {-10 x^{2}-x +3}+4593220 \sqrt {-10 x^{2}-x +3}\right )}{1996500 \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(134\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.57, size = 78, normalized size = 0.93 \begin {gather*} -\frac {27}{500} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {129853 \, x}{99825 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {382849}{499125 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {2}{4125 \, {\left (5 \, \sqrt {-10 \, x^{2} - x + 3} x + 3 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.67, size = 101, normalized size = 1.20 \begin {gather*} \frac {107811 \, \sqrt {10} {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, 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 (649265 \, x^{2} + 772408 \, x + 229661\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1996500 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\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 (3 x + 2\right )^{3}}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {5}{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. 165 vs.
\(2 (63) = 126\).
time = 1.28, size = 165, normalized size = 1.96 \begin {gather*} -\frac {1}{7986000} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {2460 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} - \frac {27}{250} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {343 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{6655 \, {\left (2 \, x - 1\right )}} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {615 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{499125 \, {\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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (3\,x+2\right )}^3}{{\left (1-2\,x\right )}^{3/2}\,{\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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