Optimal. Leaf size=243 \[ -\frac{2161804579 \sqrt{\frac{11}{6}} \sqrt{5-2 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right ),\frac{1}{3}\right )}{54432 \sqrt{2 x-5}}+\frac{1}{9} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^3+\frac{1679}{756} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^2+\frac{26291}{540} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)+\frac{46134551 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{38880}+\frac{2629157597 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{163296 \sqrt{5-2 x}} \]
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Rubi [A] time = 0.296547, antiderivative size = 243, normalized size of antiderivative = 1., 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 162
Rule 1600
Rule 1615
Rule 158
Rule 114
Rule 113
Rule 121
Rule 119
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*}
Mathematica [A] time = 0.367859, size = 130, normalized size = 0.53 \[ \frac{-2161804579 \sqrt{66} \sqrt{5-2 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right ),\frac{1}{3}\right )+6 \sqrt{2-3 x} \sqrt{4 x+1} \left (1512000 x^4+8614800 x^3+21329208 x^2+51484034 x-455686385\right )+2629157597 \sqrt{66} \sqrt{5-2 x} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{326592 \sqrt{2 x-5}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 155, normalized size = 0.6 \begin{align*} -{\frac{1}{7838208\,{x}^{3}-22861440\,{x}^{2}+6858432\,x+3265920}\sqrt{2-3\,x}\sqrt{2\,x-5}\sqrt{4\,x+1} \left ( -108864000\,{x}^{6}+6485413737\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{4\,x+1}{\it EllipticF} \left ( 2/11\,\sqrt{22-33\,x},i/2\sqrt{2} \right ) -5258315194\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{4\,x+1}{\it EllipticE} \left ( 2/11\,\sqrt{22-33\,x},i/2\sqrt{2} \right ) -574905600\,{x}^{5}-1259114976\,{x}^{4}-2963596608\,{x}^{3}+34609891236\,{x}^{2}-13052783142\,x-5468236620 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (5 \, x + 7\right )}^{3} \sqrt{4 \, x + 1} \sqrt{-3 \, x + 2}}{\sqrt{2 \, x - 5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (5 \, x + 7\right )}^{3} \sqrt{4 \, x + 1} \sqrt{-3 \, x + 2}}{\sqrt{2 \, x - 5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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