Optimal. Leaf size=160 \[ -\frac {529}{378} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {1}{7} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}-\frac {20}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-\frac {2645}{378} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}-\frac {17587}{378} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {101, 154, 158, 113, 119} \[ -\frac {1}{7} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}-\frac {20}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-\frac {2645}{378} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}-\frac {529}{378} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {17587}{378} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 101
Rule 113
Rule 119
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \frac {\sqrt {2+3 x} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx &=-\frac {1}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {1}{7} \int \frac {(3+5 x)^{3/2} \left (\frac {131}{2}+100 x\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\frac {20}{21} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {1}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {1}{105} \int \frac {\left (-\frac {8595}{2}-\frac {13225 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\frac {2645}{378} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {20}{21} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {1}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {1}{945} \int \frac {\frac {556705}{4}+\frac {439675 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2645}{378} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {20}{21} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {1}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {5819}{756} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {17587}{378} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\frac {2645}{378} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {20}{21} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {1}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {17587}{378} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {529}{378} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\\ \end {align*}
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Mathematica [A] time = 0.25, size = 97, normalized size = 0.61 \[ \frac {-17717 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )-3 \sqrt {2-4 x} \sqrt {3 x+2} \sqrt {5 x+3} \left (1350 x^2+3420 x+4211\right )+35174 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{1134 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.24, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2 \, x - 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {3 \, x + 2}}{\sqrt {-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 150, normalized size = 0.94 \[ \frac {\sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \left (-243000 x^{5}-801900 x^{4}-1173240 x^{3}-388878 x^{2}+299982 x -35174 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+17717 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+151596\right )}{68040 x^{3}+52164 x^{2}-15876 x -13608} \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {3 \, x + 2}}{\sqrt {-2 \, x + 1}}\,{d x} \]
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 \[ \int \frac {\sqrt {3\,x+2}\,{\left (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}} \,d x \]
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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