Optimal. Leaf size=205 \[ -\frac {1679161 \sqrt {\frac {11}{6}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{756 \sqrt {2 x-5}}+\frac {1}{7} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2+\frac {173}{60} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)+\frac {73207 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{1080}+\frac {8198333 \sqrt {11} \sqrt {2 x-5} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{9072 \sqrt {5-2 x}} \]
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Rubi [A] time = 0.21, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {162, 1600, 1615, 158, 114, 113, 121, 119} \[ \frac {1}{7} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2+\frac {173}{60} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)+\frac {73207 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{1080}-\frac {1679161 \sqrt {\frac {11}{6}} \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{756 \sqrt {2 x-5}}+\frac {8198333 \sqrt {11} \sqrt {2 x-5} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{9072 \sqrt {5-2 x}} \]
Antiderivative was successfully verified.
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Rule 113
Rule 114
Rule 119
Rule 121
Rule 158
Rule 162
Rule 1600
Rule 1615
Rubi steps
\begin {align*} \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)^2}{\sqrt {-5+2 x}} \, dx &=\frac {1}{7} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2-\frac {1}{14} \int \frac {(7+5 x) \left (-543-175 x+2422 x^2\right )}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx\\ &=\frac {173}{60} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)+\frac {1}{7} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {\int \frac {1054354-1137830 x-4099592 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{1680}\\ &=\frac {73207 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1080}+\frac {173}{60} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)+\frac {1}{7} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {\int \frac {243007380-983799960 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{181440}\\ &=\frac {73207 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1080}+\frac {173}{60} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)+\frac {1}{7} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2-\frac {8198333 \int \frac {\sqrt {-5+2 x}}{\sqrt {2-3 x} \sqrt {1+4 x}} \, dx}{3024}-\frac {18470771 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{1512}\\ &=\frac {73207 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1080}+\frac {173}{60} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)+\frac {1}{7} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2-\frac {\left (1679161 \sqrt {\frac {11}{2}} \sqrt {5-2 x}\right ) \int \frac {1}{\sqrt {2-3 x} \sqrt {\frac {10}{11}-\frac {4 x}{11}} \sqrt {1+4 x}} \, dx}{756 \sqrt {-5+2 x}}-\frac {\left (8198333 \sqrt {-5+2 x}\right ) \int \frac {\sqrt {\frac {15}{11}-\frac {6 x}{11}}}{\sqrt {2-3 x} \sqrt {\frac {3}{11}+\frac {12 x}{11}}} \, dx}{3024 \sqrt {5-2 x}}\\ &=\frac {73207 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1080}+\frac {173}{60} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)+\frac {1}{7} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {8198333 \sqrt {11} \sqrt {-5+2 x} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{9072 \sqrt {5-2 x}}-\frac {1679161 \sqrt {\frac {11}{6}} \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )}{756 \sqrt {-5+2 x}}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 125, normalized size = 0.61 \[ \frac {-6716644 \sqrt {66} \sqrt {5-2 x} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )+12 \sqrt {2-3 x} \sqrt {4 x+1} \left (10800 x^3+46836 x^2+102592 x-717955\right )+8198333 \sqrt {66} \sqrt {5-2 x} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{18144 \sqrt {2 x-5}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (25 \, x^{2} + 70 \, x + 49\right )} \sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{\sqrt {2 \, x - 5}}, x\right ) \]
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 \[ \int \frac {{\left (5 \, x + 7\right )}^{2} \sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{\sqrt {2 \, x - 5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 150, normalized size = 0.73 \[ -\frac {\sqrt {-3 x +2}\, \sqrt {4 x +1}\, \sqrt {2 x -5}\, \left (-777600 x^{5}-3048192 x^{4}-5851944 x^{3}+55332552 x^{2}-20307546 x -8198333 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, \EllipticE \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )+10074966 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, \EllipticF \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )-8615460\right )}{9072 \left (24 x^{3}-70 x^{2}+21 x +10\right )} \]
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 \[ \int \frac {{\left (5 \, x + 7\right )}^{2} \sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{\sqrt {2 \, x - 5}}\,{d x} \]
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 \[ \int \frac {\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,{\left (5\,x+7\right )}^2}{\sqrt {2\,x-5}} \,d x \]
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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