Optimal. Leaf size=243 \[ -\frac {2161804579 \sqrt {\frac {11}{6}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{54432 \sqrt {2 x-5}}+\frac {1}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3+\frac {1679}{756} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2+\frac {26291}{540} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)+\frac {46134551 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{38880}+\frac {2629157597 \sqrt {11} \sqrt {2 x-5} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{163296 \sqrt {5-2 x}} \]
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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 = {162, 1600, 1615, 158, 114, 113, 121, 119} \[ \frac {1}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3+\frac {1679}{756} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2+\frac {26291}{540} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)+\frac {46134551 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{38880}-\frac {2161804579 \sqrt {\frac {11}{6}} \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{54432 \sqrt {2 x-5}}+\frac {2629157597 \sqrt {11} \sqrt {2 x-5} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{163296 \sqrt {5-2 x}} \]
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)^3}{\sqrt {-5+2 x}} \, dx &=\frac {1}{9} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3-\frac {1}{18} \int \frac {(7+5 x)^2 \left (-699-565 x+3358 x^2\right )}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx\\ &=\frac {1679}{756} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {1}{9} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3+\frac {\int \frac {(7+5 x) \left (1987250-276290 x-8833776 x^2\right )}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{3024}\\ &=\frac {26291}{540} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)+\frac {1679}{756} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {1}{9} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3-\frac {\int \frac {-3851232672+4914194640 x+15501209136 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{362880}\\ &=\frac {46134551 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{38880}+\frac {26291}{540} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)+\frac {1679}{756} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {1}{9} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3-\frac {\int \frac {-904221216360+3785986939680 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{39191040}\\ &=\frac {46134551 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{38880}+\frac {26291}{540} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)+\frac {1679}{756} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {1}{9} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3-\frac {2629157597 \int \frac {\sqrt {-5+2 x}}{\sqrt {2-3 x} \sqrt {1+4 x}} \, dx}{54432}-\frac {23779850369 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{108864}\\ &=\frac {46134551 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{38880}+\frac {26291}{540} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)+\frac {1679}{756} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {1}{9} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3-\frac {\left (2161804579 \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}{54432 \sqrt {-5+2 x}}-\frac {\left (2629157597 \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}{54432 \sqrt {5-2 x}}\\ &=\frac {46134551 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{38880}+\frac {26291}{540} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)+\frac {1679}{756} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {1}{9} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3+\frac {2629157597 \sqrt {11} \sqrt {-5+2 x} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{163296 \sqrt {5-2 x}}-\frac {2161804579 \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 )}{54432 \sqrt {-5+2 x}}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 130, normalized size = 0.53 \[ \frac {-2161804579 \sqrt {66} \sqrt {5-2 x} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )+6 \sqrt {2-3 x} \sqrt {4 x+1} \left (1512000 x^4+8614800 x^3+21329208 x^2+51484034 x-455686385\right )+2629157597 \sqrt {66} \sqrt {5-2 x} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{326592 \sqrt {2 x-5}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (125 \, x^{3} + 525 \, x^{2} + 735 \, x + 343\right )} \sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{\sqrt {2 \, x - 5}}, x\right ) \]
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 \[ \int \frac {{\left (5 \, x + 7\right )}^{3} \sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{\sqrt {2 \, x - 5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 155, normalized size = 0.64 \[ -\frac {\sqrt {-3 x +2}\, \sqrt {4 x +1}\, \sqrt {2 x -5}\, \left (-108864000 x^{6}-574905600 x^{5}-1259114976 x^{4}-2963596608 x^{3}+34609891236 x^{2}-13052783142 x -5258315194 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, \EllipticE \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )+6485413737 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, \EllipticF \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )-5468236620\right )}{326592 \left (24 x^{3}-70 x^{2}+21 x +10\right )} \]
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 \[ \int \frac {{\left (5 \, x + 7\right )}^{3} \sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{\sqrt {2 \, x - 5}}\,{d x} \]
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 \[ \int \frac {\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,{\left (5\,x+7\right )}^3}{\sqrt {2\,x-5}} \,d x \]
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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