Optimal. Leaf size=218 \[ -\frac {12996374 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{15946875 \sqrt {33}}+\frac {2}{55} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{5/2}+\frac {326 (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}}{7425}+\frac {30362 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{779625}-\frac {78797 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{3898125}-\frac {12996374 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{35083125}-\frac {829177897 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{31893750 \sqrt {33}} \]
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Rubi [A] time = 0.08, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ \frac {2}{55} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{5/2}+\frac {326 (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}}{7425}+\frac {30362 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{779625}-\frac {78797 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{3898125}-\frac {12996374 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{35083125}-\frac {12996374 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15946875 \sqrt {33}}-\frac {829177897 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{31893750 \sqrt {33}} \]
Antiderivative was successfully verified.
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Rule 101
Rule 113
Rule 119
Rule 154
Rule 158
Rubi steps
\begin {align*} \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2} \, dx &=\frac {2}{55} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {2}{55} \int \frac {\left (-\frac {111}{2}-\frac {163 x}{2}\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx\\ &=\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}}{7425}+\frac {2}{55} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {4 \int \frac {\left (-2809-\frac {15181 x}{4}\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx}{7425}\\ &=\frac {30362 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{779625}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}}{7425}+\frac {2}{55} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {8 \int \frac {\left (-\frac {466273}{8}-\frac {236391 x}{8}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{779625}\\ &=-\frac {78797 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{3898125}+\frac {30362 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{779625}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}}{7425}+\frac {2}{55} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {8 \int \frac {\sqrt {3+5 x} \left (\frac {48347127}{16}+\frac {19494561 x}{4}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{11694375}\\ &=-\frac {12996374 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{35083125}-\frac {78797 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{3898125}+\frac {30362 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{779625}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}}{7425}+\frac {2}{55} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {8 \int \frac {-\frac {1578296283}{16}-\frac {2487533691 x}{16}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{105249375}\\ &=-\frac {12996374 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{35083125}-\frac {78797 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{3898125}+\frac {30362 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{779625}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}}{7425}+\frac {2}{55} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {6498187 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{15946875}+\frac {829177897 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{350831250}\\ &=-\frac {12996374 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{35083125}-\frac {78797 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{3898125}+\frac {30362 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{779625}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}}{7425}+\frac {2}{55} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {829177897 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{31893750 \sqrt {33}}-\frac {12996374 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15946875 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 107, normalized size = 0.49 \[ \frac {-400297555 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+15 \sqrt {2-4 x} \sqrt {3 x+2} \sqrt {5 x+3} \left (127575000 x^4-51502500 x^3-95024250 x^2+48272535 x+22517617\right )+829177897 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{526246875 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.29, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}, x\right ) \]
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 \[ \int {\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 160, normalized size = 0.73 \[ -\frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \left (-114817500000 x^{7}-41674500000 x^{6}+147849300000 x^{5}+34269426000 x^{4}-82799446950 x^{3}-22504288380 x^{2}+13417755870 x +829177897 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-400297555 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+4053171060\right )}{1052493750 \left (30 x^{3}+23 x^{2}-7 x -6\right )} \]
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 \[ \int {\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (1-2\,x\right )}^{5/2}\,\sqrt {3\,x+2}\,{\left (5\,x+3\right )}^{3/2} \,d x \]
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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