Optimal. Leaf size=162 \[ -\frac {4543 \sqrt {\frac {11}{6}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{36 \sqrt {2 x-5}}+\frac {1}{4} \sqrt {2-3 x} \sqrt {2 x-5} (4 x+1)^{3/2}+\frac {95}{18} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}+\frac {1397 \sqrt {11} \sqrt {2 x-5} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{27 \sqrt {5-2 x}} \]
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Rubi [A] time = 0.06, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {154, 158, 114, 113, 121, 119} \[ \frac {1}{4} \sqrt {2-3 x} \sqrt {2 x-5} (4 x+1)^{3/2}+\frac {95}{18} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}-\frac {4543 \sqrt {\frac {11}{6}} \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{36 \sqrt {2 x-5}}+\frac {1397 \sqrt {11} \sqrt {2 x-5} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{27 \sqrt {5-2 x}} \]
Antiderivative was successfully verified.
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Rule 113
Rule 114
Rule 119
Rule 121
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)}{\sqrt {-5+2 x}} \, dx &=\frac {1}{4} \sqrt {2-3 x} \sqrt {-5+2 x} (1+4 x)^{3/2}+\frac {1}{20} \int \frac {\left (\frac {1065}{2}-950 x\right ) \sqrt {1+4 x}}{\sqrt {2-3 x} \sqrt {-5+2 x}} \, dx\\ &=\frac {95}{18} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}+\frac {1}{4} \sqrt {2-3 x} \sqrt {-5+2 x} (1+4 x)^{3/2}-\frac {1}{180} \int \frac {-\frac {29535}{2}+55880 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx\\ &=\frac {95}{18} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}+\frac {1}{4} \sqrt {2-3 x} \sqrt {-5+2 x} (1+4 x)^{3/2}-\frac {1397}{9} \int \frac {\sqrt {-5+2 x}}{\sqrt {2-3 x} \sqrt {1+4 x}} \, dx-\frac {49973}{72} \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx\\ &=\frac {95}{18} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}+\frac {1}{4} \sqrt {2-3 x} \sqrt {-5+2 x} (1+4 x)^{3/2}-\frac {\left (4543 \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}{36 \sqrt {-5+2 x}}-\frac {\left (1397 \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}{9 \sqrt {5-2 x}}\\ &=\frac {95}{18} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}+\frac {1}{4} \sqrt {2-3 x} \sqrt {-5+2 x} (1+4 x)^{3/2}+\frac {1397 \sqrt {11} \sqrt {-5+2 x} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{27 \sqrt {5-2 x}}-\frac {4543 \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 )}{36 \sqrt {-5+2 x}}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 120, normalized size = 0.74 \[ \frac {-4543 \sqrt {66} \sqrt {5-2 x} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )+6 \sqrt {2-3 x} \sqrt {4 x+1} \left (72 x^2+218 x-995\right )+5588 \sqrt {66} \sqrt {5-2 x} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{216 \sqrt {2 x-5}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (5 \, x + 7\right )} \sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{\sqrt {2 \, x - 5}}, x\right ) \]
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 \[ \int \frac {{\left (5 \, x + 7\right )} \sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{\sqrt {2 \, x - 5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 145, normalized size = 0.90 \[ -\frac {\sqrt {-3 x +2}\, \sqrt {4 x +1}\, \sqrt {2 x -5}\, \left (-5184 x^{4}-13536 x^{3}+79044 x^{2}-27234 x -11176 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, \EllipticE \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )+13629 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, \EllipticF \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )-11940\right )}{216 \left (24 x^{3}-70 x^{2}+21 x +10\right )} \]
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 \[ \int \frac {{\left (5 \, x + 7\right )} \sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{\sqrt {2 \, x - 5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\left (5\,x+7\right )}{\sqrt {2\,x-5}} \,d x \]
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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