Optimal. Leaf size=243 \[ \frac {5592499 \sqrt {\frac {11}{6}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{3888 \sqrt {2 x-5}}+\frac {2}{45} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3-\frac {61}{270} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2-\frac {8141 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}{2700}-\frac {5256763 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{97200}-\frac {17746949 \sqrt {11} \sqrt {2 x-5} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{29160 \sqrt {5-2 x}} \]
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Rubi [A] time = 0.30, antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {161, 1600, 1615, 158, 114, 113, 121, 119} \[ \frac {2}{45} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3-\frac {61}{270} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2-\frac {8141 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}{2700}-\frac {5256763 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{97200}+\frac {5592499 \sqrt {\frac {11}{6}} \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{3888 \sqrt {2 x-5}}-\frac {17746949 \sqrt {11} \sqrt {2 x-5} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{29160 \sqrt {5-2 x}} \]
Antiderivative was successfully verified.
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Rule 113
Rule 114
Rule 119
Rule 121
Rule 158
Rule 161
Rule 1600
Rule 1615
Rubi steps
\begin {align*} \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2 \, dx &=\frac {2}{45} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3+\frac {1}{45} \int \frac {(7+5 x)^2 \left (-3-1190 x+854 x^2\right )}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx\\ &=-\frac {61}{270} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {2}{45} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3-\frac {\int \frac {(7+5 x) \left (299866+1013390 x-1367688 x^2\right )}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{7560}\\ &=-\frac {8141 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)}{2700}-\frac {61}{270} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {2}{45} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3+\frac {\int \frac {-589706376-121654680 x+1766272368 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{907200}\\ &=-\frac {5256763 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{97200}-\frac {8141 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)}{2700}-\frac {61}{270} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {2}{45} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3+\frac {\int \frac {-119325868200+357778491840 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{97977600}\\ &=-\frac {5256763 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{97200}-\frac {8141 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)}{2700}-\frac {61}{270} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {2}{45} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3+\frac {17746949 \int \frac {\sqrt {-5+2 x}}{\sqrt {2-3 x} \sqrt {1+4 x}} \, dx}{9720}+\frac {61517489 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{7776}\\ &=-\frac {5256763 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{97200}-\frac {8141 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)}{2700}-\frac {61}{270} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {2}{45} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3+\frac {\left (5592499 \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}{3888 \sqrt {-5+2 x}}+\frac {\left (17746949 \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}{9720 \sqrt {5-2 x}}\\ &=-\frac {5256763 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{97200}-\frac {8141 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)}{2700}-\frac {61}{270} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {2}{45} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3-\frac {17746949 \sqrt {11} \sqrt {-5+2 x} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{29160 \sqrt {5-2 x}}+\frac {5592499 \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 )}{3888 \sqrt {-5+2 x}}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 130, normalized size = 0.53 \[ \frac {27962495 \sqrt {66} \sqrt {5-2 x} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )+6 \sqrt {2-3 x} \sqrt {4 x+1} \left (216000 x^4+147600 x^3-1649952 x^2-2933650 x+6902575\right )-35493898 \sqrt {66} \sqrt {5-2 x} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{116640 \sqrt {2 x-5}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.16, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (25 \, x^{2} + 70 \, x + 49\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}, x\right ) \]
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 \[ \int {\left (5 \, x + 7\right )}^{2} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 155, normalized size = 0.64 \[ \frac {\sqrt {-3 x +2}\, \sqrt {2 x -5}\, \sqrt {4 x +1}\, \left (15552000 x^{6}+4147200 x^{5}-125816544 x^{4}-163495440 x^{3}+604794324 x^{2}-171873450 x -70987796 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, \EllipticE \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )+83887485 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, \EllipticF \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )-82830900\right )}{2799360 x^{3}-8164800 x^{2}+2449440 x +1166400} \]
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 \[ \int {\left (5 \, x + 7\right )}^{2} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}\,{\left (5\,x+7\right )}^2 \,d x \]
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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