Optimal. Leaf size=113 \[ -\frac {1099 (2+3 x)^2}{726 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^3}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {\sqrt {1-2 x} (4898747+8200665 x)}{798600 \sqrt {3+5 x}}+\frac {4887 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{200 \sqrt {10}} \]
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Rubi [A]
time = 0.02, 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 = {100, 155, 148,
56, 222} \begin {gather*} \frac {4887 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{200 \sqrt {10}}+\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 56
Rule 100
Rule 148
Rule 155
Rule 222
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 \text {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 = 0.15, size = 88, normalized size = 0.78 \begin {gather*} -\frac {10 \sqrt {3+5 x} \left (8379147-12657123 x-40488772 x^2+6468660 x^3\right )-19513791 \sqrt {10-20 x} \left (-3+x+10 x^2\right ) \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{7986000 (1-2 x)^{3/2} (3+5 x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 151, normalized size = 1.34
method | result | size |
default | \(\frac {\sqrt {1-2 x}\, \left (390275820 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{3}-156110328 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-129373200 x^{3} \sqrt {-10 x^{2}-x +3}-136596537 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +809775440 x^{2} \sqrt {-10 x^{2}-x +3}+58541373 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+253142460 x \sqrt {-10 x^{2}-x +3}-167582940 \sqrt {-10 x^{2}-x +3}\right )}{15972000 \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\) | \(151\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 95, normalized size = 0.84 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 106, normalized size = 0.94 \begin {gather*} -\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 )}} \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 )^{4}}{\left (1 - 2 x\right )^{\frac {5}{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 [A]
time = 2.69, size = 131, normalized size = 1.16 \begin {gather*} \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 )}} \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 )}^4}{{\left (1-2\,x\right )}^{5/2}\,{\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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