Optimal. Leaf size=193 \[ \frac {72479 \sqrt {\frac {11}{6}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{756 \sqrt {2 x-5}}+\frac {5}{28} \sqrt {2-3 x} (2 x-5)^{3/2} (4 x+1)^{3/2}+\frac {136}{105} \sqrt {2-3 x} \sqrt {2 x-5} (4 x+1)^{3/2}-\frac {20911 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{3780}-\frac {954811 \sqrt {11} \sqrt {2 x-5} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{22680 \sqrt {5-2 x}} \]
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Rubi [A] time = 0.08, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ \frac {5}{28} \sqrt {2-3 x} (2 x-5)^{3/2} (4 x+1)^{3/2}+\frac {136}{105} \sqrt {2-3 x} \sqrt {2 x-5} (4 x+1)^{3/2}-\frac {20911 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{3780}+\frac {72479 \sqrt {\frac {11}{6}} \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{756 \sqrt {2 x-5}}-\frac {954811 \sqrt {11} \sqrt {2 x-5} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{22680 \sqrt {5-2 x}} \]
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 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x) \, dx &=\frac {5}{28} \sqrt {2-3 x} (-5+2 x)^{3/2} (1+4 x)^{3/2}+\frac {1}{28} \int \frac {\left (\frac {1249}{2}-1088 x\right ) \sqrt {-5+2 x} \sqrt {1+4 x}}{\sqrt {2-3 x}} \, dx\\ &=\frac {136}{105} \sqrt {2-3 x} \sqrt {-5+2 x} (1+4 x)^{3/2}+\frac {5}{28} \sqrt {2-3 x} (-5+2 x)^{3/2} (1+4 x)^{3/2}-\frac {1}{840} \int \frac {(38731-41822 x) \sqrt {1+4 x}}{\sqrt {2-3 x} \sqrt {-5+2 x}} \, dx\\ &=-\frac {20911 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{3780}+\frac {136}{105} \sqrt {2-3 x} \sqrt {-5+2 x} (1+4 x)^{3/2}+\frac {5}{28} \sqrt {2-3 x} (-5+2 x)^{3/2} (1+4 x)^{3/2}+\frac {\int \frac {-787710+1909622 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{7560}\\ &=-\frac {20911 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{3780}+\frac {136}{105} \sqrt {2-3 x} \sqrt {-5+2 x} (1+4 x)^{3/2}+\frac {5}{28} \sqrt {2-3 x} (-5+2 x)^{3/2} (1+4 x)^{3/2}+\frac {954811 \int \frac {\sqrt {-5+2 x}}{\sqrt {2-3 x} \sqrt {1+4 x}} \, dx}{7560}+\frac {797269 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{1512}\\ &=-\frac {20911 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{3780}+\frac {136}{105} \sqrt {2-3 x} \sqrt {-5+2 x} (1+4 x)^{3/2}+\frac {5}{28} \sqrt {2-3 x} (-5+2 x)^{3/2} (1+4 x)^{3/2}+\frac {\left (72479 \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 (954811 \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}{7560 \sqrt {5-2 x}}\\ &=-\frac {20911 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{3780}+\frac {136}{105} \sqrt {2-3 x} \sqrt {-5+2 x} (1+4 x)^{3/2}+\frac {5}{28} \sqrt {2-3 x} (-5+2 x)^{3/2} (1+4 x)^{3/2}-\frac {954811 \sqrt {11} \sqrt {-5+2 x} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{22680 \sqrt {5-2 x}}+\frac {72479 \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 = 0.25, size = 125, normalized size = 0.65 \[ \frac {724790 \sqrt {66} \sqrt {5-2 x} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )+24 \sqrt {2-3 x} \sqrt {4 x+1} \left (5400 x^3-6066 x^2-37975 x+48475\right )-954811 \sqrt {66} \sqrt {5-2 x} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{45360 \sqrt {2 x-5}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (5 \, x + 7\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}, x\right ) \]
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 \[ \int {\left (5 \, x + 7\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 150, normalized size = 0.78 \[ \frac {\sqrt {-3 x +2}\, \sqrt {2 x -5}\, \sqrt {4 x +1}\, \left (777600 x^{5}-1197504 x^{4}-5234040 x^{3}+9404484 x^{2}-1997100 x -954811 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, \EllipticE \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )+1087185 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, \EllipticF \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )-1163400\right )}{544320 x^{3}-1587600 x^{2}+476280 x +226800} \]
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 \[ \int {\left (5 \, x + 7\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}\,{d x} \]
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 \[ \int \sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}\,\left (5\,x+7\right ) \,d x \]
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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