Optimal. Leaf size=243 \[ \frac{1}{9} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^3+\frac{1679}{756} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^2+\frac{26291}{540} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)+\frac{46134551 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{38880}-\frac{2161804579 \sqrt{\frac{11}{6}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{54432 \sqrt{2 x-5}}+\frac{2629157597 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{163296 \sqrt{5-2 x}} \]
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Rubi [A] time = 0.729443, 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 \[ \frac{1}{9} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^3+\frac{1679}{756} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^2+\frac{26291}{540} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)+\frac{46134551 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{38880}-\frac{2161804579 \sqrt{\frac{11}{6}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{54432 \sqrt{2 x-5}}+\frac{2629157597 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{163296 \sqrt{5-2 x}} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*(7 + 5*x)^3)/Sqrt[-5 + 2*x],x]
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Rubi in Sympy [A] time = 164.158, size = 286, normalized size = 1.18 \[ - \frac{125 \left (- 3 x + 2\right )^{\frac{3}{2}} \sqrt{2 x - 5} \left (4 x + 1\right )^{\frac{5}{2}}}{432} - \frac{6025 \left (- 3 x + 2\right )^{\frac{3}{2}} \sqrt{2 x - 5} \left (4 x + 1\right )^{\frac{3}{2}}}{756} + \frac{575 \sqrt{- 3 x + 2} \left (2 x - 5\right )^{\frac{3}{2}} \left (4 x + 1\right )^{\frac{3}{2}}}{224} + \frac{61601 \sqrt{- 3 x + 2} \sqrt{2 x - 5} \left (4 x + 1\right )^{\frac{3}{2}}}{432} + \frac{21243319 \sqrt{- 3 x + 2} \sqrt{2 x - 5} \sqrt{4 x + 1}}{13608} + \frac{2629157597 \sqrt{11} \sqrt{\frac{12 x}{11} + \frac{3}{11}} \sqrt{2 x - 5} E\left (\operatorname{asin}{\left (\frac{2 \sqrt{11} \sqrt{- 3 x + 2}}{11} \right )}\middle | - \frac{1}{2}\right )}{163296 \sqrt{- \frac{6 x}{11} + \frac{15}{11}} \sqrt{4 x + 1}} - \frac{23779850369 \sqrt{33} \sqrt{- \frac{12 x}{11} + \frac{8}{11}} \sqrt{- \frac{4 x}{11} + \frac{10}{11}} F\left (\operatorname{asin}{\left (\frac{\sqrt{33} \sqrt{4 x + 1}}{11} \right )}\middle | \frac{1}{3}\right )}{653184 \sqrt{- 3 x + 2} \sqrt{2 x - 5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((7+5*x)**3*(2-3*x)**(1/2)*(1+4*x)**(1/2)/(-5+2*x)**(1/2),x)
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Mathematica [A] time = 0.395231, size = 130, normalized size = 0.53 \[ \frac{6 \sqrt{2-3 x} \sqrt{4 x+1} \left (1512000 x^4+8614800 x^3+21329208 x^2+51484034 x-455686385\right )-2161804579 \sqrt{66} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )+2629157597 \sqrt{66} \sqrt{5-2 x} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{326592 \sqrt{2 x-5}} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*(7 + 5*x)^3)/Sqrt[-5 + 2*x],x]
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Maple [A] time = 0.035, size = 161, normalized size = 0.7 \[ -{\frac{1}{7838208\,{x}^{3}-22861440\,{x}^{2}+6858432\,x+3265920}\sqrt{2-3\,x}\sqrt{-5+2\,x}\sqrt{1+4\,x} \left ( -108864000\,{x}^{6}+6485413737\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticF} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ) -5258315194\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticE} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ) -574905600\,{x}^{5}-1259114976\,{x}^{4}-2963596608\,{x}^{3}+34609891236\,{x}^{2}-13052783142\,x-5468236620 \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((7+5*x)^3*(2-3*x)^(1/2)*(1+4*x)^(1/2)/(-5+2*x)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 7\right )}^{3} \sqrt{4 \, x + 1} \sqrt{-3 \, x + 2}}{\sqrt{2 \, x - 5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 7)^3*sqrt(4*x + 1)*sqrt(-3*x + 2)/sqrt(2*x - 5),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (125 \, x^{3} + 525 \, x^{2} + 735 \, x + 343\right )} \sqrt{4 \, x + 1} \sqrt{-3 \, x + 2}}{\sqrt{2 \, x - 5}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 7)^3*sqrt(4*x + 1)*sqrt(-3*x + 2)/sqrt(2*x - 5),x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((7+5*x)**3*(2-3*x)**(1/2)*(1+4*x)**(1/2)/(-5+2*x)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 7\right )}^{3} \sqrt{4 \, x + 1} \sqrt{-3 \, x + 2}}{\sqrt{2 \, x - 5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 7)^3*sqrt(4*x + 1)*sqrt(-3*x + 2)/sqrt(2*x - 5),x, algorithm="giac")
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