Optimal. Leaf size=86 \[ -\frac {58 \sqrt {5 x+3}}{539 \sqrt {1-2 x}}+\frac {3 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}-\frac {123 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{49 \sqrt {7}} \]
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Rubi [A] time = 0.02, antiderivative size = 86, 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 = {103, 152, 12, 93, 204} \begin {gather*} -\frac {58 \sqrt {5 x+3}}{539 \sqrt {1-2 x}}+\frac {3 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}-\frac {123 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{49 \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)^{3/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx &=\frac {3 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)}+\frac {1}{7} \int \frac {\frac {1}{2}-30 x}{(1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {58 \sqrt {3+5 x}}{539 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)}-\frac {2}{539} \int -\frac {1353}{4 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {58 \sqrt {3+5 x}}{539 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)}+\frac {123}{98} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {58 \sqrt {3+5 x}}{539 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)}+\frac {123}{49} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {58 \sqrt {3+5 x}}{539 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)}-\frac {123 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{49 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 78, normalized size = 0.91 \begin {gather*} \frac {7 (115-174 x) \sqrt {5 x+3}-1353 \sqrt {7-14 x} (3 x+2) \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3773 \sqrt {1-2 x} (3 x+2)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.13, size = 90, normalized size = 1.05 \begin {gather*} \frac {\sqrt {5 x+3} \left (\frac {1097 (1-2 x)}{5 x+3}+56\right )}{539 \sqrt {1-2 x} \left (\frac {1-2 x}{5 x+3}+7\right )}-\frac {123 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{49 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 82, normalized size = 0.95 \begin {gather*} -\frac {1353 \, \sqrt {7} {\left (6 \, x^{2} + x - 2\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 (174 \, x - 115\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{7546 \, {\left (6 \, x^{2} + x - 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.57, size = 219, normalized size = 2.55 \begin {gather*} \frac {123}{6860} \, \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 {8 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{2695 \, {\left (2 \, x - 1\right )}} + \frac {198 \, \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 )}}{49 \, {\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 161, normalized size = 1.87 \begin {gather*} \frac {\left (8118 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1353 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+2436 \sqrt {-10 x^{2}-x +3}\, x -2706 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-1610 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{7546 \left (3 x +2\right ) \left (2 x -1\right ) \sqrt {-10 x^{2}-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}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{2} {\left (-2 \, x + 1\right )}^{\frac {3}{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 )}^{3/2}\,{\left (3\,x+2\right )}^2\,\sqrt {5\,x+3}} \,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 {3}{2}} \left (3 x + 2\right )^{2} \sqrt {5 x + 3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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