Optimal. Leaf size=158 \[ -\frac {6856 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1715 \sqrt {33}}+\frac {20644 \sqrt {1-2 x} \sqrt {5 x+3}}{1715 \sqrt {3 x+2}}+\frac {296 \sqrt {1-2 x} \sqrt {5 x+3}}{245 (3 x+2)^{3/2}}+\frac {6 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}-\frac {20644 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715} \]
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Rubi [A] time = 0.05, antiderivative size = 158, 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 = {104, 152, 158, 113, 119} \[ \frac {20644 \sqrt {1-2 x} \sqrt {5 x+3}}{1715 \sqrt {3 x+2}}+\frac {296 \sqrt {1-2 x} \sqrt {5 x+3}}{245 (3 x+2)^{3/2}}+\frac {6 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}-\frac {6856 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715 \sqrt {33}}-\frac {20644 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715} \]
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}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx &=\frac {6 \sqrt {1-2 x} \sqrt {3+5 x}}{35 (2+3 x)^{5/2}}+\frac {2}{35} \int \frac {44-45 x}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx\\ &=\frac {6 \sqrt {1-2 x} \sqrt {3+5 x}}{35 (2+3 x)^{5/2}}+\frac {296 \sqrt {1-2 x} \sqrt {3+5 x}}{245 (2+3 x)^{3/2}}+\frac {4}{735} \int \frac {\frac {3681}{2}-1110 x}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=\frac {6 \sqrt {1-2 x} \sqrt {3+5 x}}{35 (2+3 x)^{5/2}}+\frac {296 \sqrt {1-2 x} \sqrt {3+5 x}}{245 (2+3 x)^{3/2}}+\frac {20644 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 \sqrt {2+3 x}}+\frac {8 \int \frac {24510+\frac {77415 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{5145}\\ &=\frac {6 \sqrt {1-2 x} \sqrt {3+5 x}}{35 (2+3 x)^{5/2}}+\frac {296 \sqrt {1-2 x} \sqrt {3+5 x}}{245 (2+3 x)^{3/2}}+\frac {20644 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 \sqrt {2+3 x}}+\frac {3428 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1715}+\frac {20644 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1715}\\ &=\frac {6 \sqrt {1-2 x} \sqrt {3+5 x}}{35 (2+3 x)^{5/2}}+\frac {296 \sqrt {1-2 x} \sqrt {3+5 x}}{245 (2+3 x)^{3/2}}+\frac {20644 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 \sqrt {2+3 x}}-\frac {20644 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715}-\frac {6856 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 101, normalized size = 0.64 \[ \frac {4 \left (\sqrt {2} \left (5161 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-2590 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )+\frac {3 \sqrt {1-2 x} \sqrt {5 x+3} \left (92898 x^2+126972 x+43507\right )}{2 (3 x+2)^{5/2}}\right )}{5145} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{810 \, x^{6} + 2241 \, x^{5} + 2133 \, x^{4} + 528 \, x^{3} - 392 \, x^{2} - 272 \, x - 48}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {7}{2}} \sqrt {-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 314, normalized size = 1.99 \[ \frac {2 \left (2786940 x^{4}+4087854 x^{3}-92898 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+46620 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+850044 x^{2}-123864 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+62160 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-1012227 x -41288 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+20720 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-391563\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{5145 \left (10 x^{2}+x -3\right ) \left (3 x +2\right )^{\frac {5}{2}}} \]
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 \[ \int \frac {1}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {7}{2}} \sqrt {-2 \, x + 1}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{7/2}\,\sqrt {5\,x+3}} \,d x \]
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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