Optimal. Leaf size=187 \[ -\frac {9124 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{26411 \sqrt {33}}+\frac {184636 \sqrt {1-2 x} \sqrt {5 x+3}}{290521 \sqrt {3 x+2}}+\frac {974 \sqrt {1-2 x} \sqrt {5 x+3}}{41503 (3 x+2)^{3/2}}+\frac {1072 \sqrt {5 x+3}}{17787 \sqrt {1-2 x} (3 x+2)^{3/2}}+\frac {4 \sqrt {5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{3/2}}-\frac {184636 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{26411 \sqrt {33}} \]
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Rubi [A] time = 0.07, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {104, 152, 158, 113, 119} \[ \frac {184636 \sqrt {1-2 x} \sqrt {5 x+3}}{290521 \sqrt {3 x+2}}+\frac {974 \sqrt {1-2 x} \sqrt {5 x+3}}{41503 (3 x+2)^{3/2}}+\frac {1072 \sqrt {5 x+3}}{17787 \sqrt {1-2 x} (3 x+2)^{3/2}}+\frac {4 \sqrt {5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{3/2}}-\frac {9124 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{26411 \sqrt {33}}-\frac {184636 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{26411 \sqrt {33}} \]
Antiderivative was successfully verified.
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Rule 104
Rule 113
Rule 119
Rule 152
Rule 158
Rubi steps
\begin {align*} \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx &=\frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{3/2}}-\frac {2}{231} \int \frac {-\frac {193}{2}-75 x}{(1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx\\ &=\frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {1072 \sqrt {3+5 x}}{17787 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {4 \int \frac {\frac {17541}{4}+6030 x}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{17787}\\ &=\frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {1072 \sqrt {3+5 x}}{17787 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {974 \sqrt {1-2 x} \sqrt {3+5 x}}{41503 (2+3 x)^{3/2}}+\frac {8 \int \frac {\frac {123867}{4}-\frac {21915 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{373527}\\ &=\frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {1072 \sqrt {3+5 x}}{17787 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {974 \sqrt {1-2 x} \sqrt {3+5 x}}{41503 (2+3 x)^{3/2}}+\frac {184636 \sqrt {1-2 x} \sqrt {3+5 x}}{290521 \sqrt {2+3 x}}+\frac {16 \int \frac {\frac {2718405}{8}+\frac {2077155 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2614689}\\ &=\frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {1072 \sqrt {3+5 x}}{17787 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {974 \sqrt {1-2 x} \sqrt {3+5 x}}{41503 (2+3 x)^{3/2}}+\frac {184636 \sqrt {1-2 x} \sqrt {3+5 x}}{290521 \sqrt {2+3 x}}+\frac {4562 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{26411}+\frac {184636 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{290521}\\ &=\frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {1072 \sqrt {3+5 x}}{17787 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {974 \sqrt {1-2 x} \sqrt {3+5 x}}{41503 (2+3 x)^{3/2}}+\frac {184636 \sqrt {1-2 x} \sqrt {3+5 x}}{290521 \sqrt {2+3 x}}-\frac {184636 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{26411 \sqrt {33}}-\frac {9124 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{26411 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 103, normalized size = 0.55 \[ \frac {2 \left (\sqrt {2} \left (92318 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-17045 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )+\frac {\sqrt {5 x+3} \left (3323448 x^3-1066908 x^2-1478206 x+597945\right )}{(1-2 x)^{3/2} (3 x+2)^{3/2}}\right )}{871563} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{1080 \, x^{7} + 1188 \, x^{6} - 666 \, x^{5} - 949 \, x^{4} + 117 \, x^{3} + 258 \, x^{2} - 4 \, x - 24}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 311, normalized size = 1.66 \[ \frac {2 \left (16617240 x^{4}+4635804 x^{3}-553908 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+102270 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-10591754 x^{2}-92318 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+17045 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-1444893 x +184636 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-34090 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+1793835\right ) \sqrt {-2 x +1}}{871563 \left (3 x +2\right )^{\frac {3}{2}} \left (2 x -1\right )^{2} \sqrt {5 x +3}} \]
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 \[ \int \frac {1}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{5/2}\,\sqrt {5\,x+3}} \,d x \]
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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