Optimal. Leaf size=113 \[ \frac {7 (3 x+2)^3}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}-\frac {1099 (3 x+2)^2}{726 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {\sqrt {1-2 x} (8200665 x+4898747)}{798600 \sqrt {5 x+3}}+\frac {4887 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{200 \sqrt {10}} \]
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Rubi [A] time = 0.03, antiderivative size = 113, 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 = {98, 150, 143, 54, 216} \[ \frac {7 (3 x+2)^3}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}-\frac {1099 (3 x+2)^2}{726 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {\sqrt {1-2 x} (8200665 x+4898747)}{798600 \sqrt {5 x+3}}+\frac {4887 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{200 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 54
Rule 98
Rule 143
Rule 150
Rule 216
Rubi steps
\begin {align*} \int \frac {(2+3 x)^4}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx &=\frac {7 (2+3 x)^3}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {1}{33} \int \frac {(2+3 x)^2 \left (148+\frac {507 x}{2}\right )}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=-\frac {1099 (2+3 x)^2}{726 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^3}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {1}{363} \int \frac {\left (-\frac {14369}{2}-\frac {49701 x}{4}\right ) (2+3 x)}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac {1099 (2+3 x)^2}{726 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^3}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {\sqrt {1-2 x} (4898747+8200665 x)}{798600 \sqrt {3+5 x}}+\frac {4887}{400} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {1099 (2+3 x)^2}{726 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^3}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {\sqrt {1-2 x} (4898747+8200665 x)}{798600 \sqrt {3+5 x}}+\frac {4887 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{200 \sqrt {5}}\\ &=-\frac {1099 (2+3 x)^2}{726 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^3}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {\sqrt {1-2 x} (4898747+8200665 x)}{798600 \sqrt {3+5 x}}+\frac {4887 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{200 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 94, normalized size = 0.83 \[ \frac {10 \sqrt {2 x-1} \left (6468660 x^3-40488772 x^2-12657123 x+8379147\right )+19513791 \sqrt {50 x+30} (1-2 x)^2 \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{7986000 \sqrt {1-2 x} (2 x-1)^{3/2} \sqrt {5 x+3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 106, normalized size = 0.94 \[ -\frac {19513791 \, \sqrt {10} {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (6468660 \, x^{3} - 40488772 \, x^{2} - 12657123 \, x + 8379147\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{15972000 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.49, size = 131, normalized size = 1.16 \[ \frac {4887}{2000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{332750 \, \sqrt {5 \, x + 3}} - \frac {{\left (4 \, {\left (323433 \, \sqrt {5} {\left (5 \, x + 3\right )} - 13033138 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 214579893 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{99825000 \, {\left (2 \, x - 1\right )}^{2}} + \frac {2 \, \sqrt {10} \sqrt {5 \, x + 3}}{166375 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 151, normalized size = 1.34 \[ \frac {\sqrt {-2 x +1}\, \left (390275820 \sqrt {10}\, x^{3} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-129373200 \sqrt {-10 x^{2}-x +3}\, x^{3}-156110328 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+809775440 \sqrt {-10 x^{2}-x +3}\, x^{2}-136596537 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+253142460 \sqrt {-10 x^{2}-x +3}\, x +58541373 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-167582940 \sqrt {-10 x^{2}-x +3}\right )}{15972000 \left (2 x -1\right )^{2} \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 95, normalized size = 0.84 \[ \frac {4887}{4000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {81 \, x^{2}}{20 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {18627221 \, x}{798600 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {3910543}{199650 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {2401}{264 \, {\left (2 \, \sqrt {-10 \, x^{2} - x + 3} x - \sqrt {-10 \, x^{2} - x + 3}\right )}} \]
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 \[ \int \frac {{\left (3\,x+2\right )}^4}{{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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