Optimal. Leaf size=187 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^{7/2}}{33 (1-2 x)^{3/2}}-\frac{910 \sqrt{5 x+3} (3 x+2)^{5/2}}{363 \sqrt{1-2 x}}-\frac{27271 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{6050}-\frac{317384 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{15125}-\frac{663409 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{13750 \sqrt{33}}-\frac{44109377 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{27500 \sqrt{33}} \]
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Rubi [A] time = 0.413924, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^{7/2}}{33 (1-2 x)^{3/2}}-\frac{910 \sqrt{5 x+3} (3 x+2)^{5/2}}{363 \sqrt{1-2 x}}-\frac{27271 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{6050}-\frac{317384 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{15125}-\frac{663409 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{13750 \sqrt{33}}-\frac{44109377 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{27500 \sqrt{33}} \]
Antiderivative was successfully verified.
[In] Int[(2 + 3*x)^(9/2)/((1 - 2*x)^(5/2)*Sqrt[3 + 5*x]),x]
[Out]
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Rubi in Sympy [A] time = 38.6434, size = 172, normalized size = 0.92 \[ - \frac{27271 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{6050} - \frac{317384 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{15125} - \frac{44109377 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{907500} - \frac{663409 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{481250} - \frac{910 \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{363 \sqrt{- 2 x + 1}} + \frac{7 \left (3 x + 2\right )^{\frac{7}{2}} \sqrt{5 x + 3}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2+3*x)**(9/2)/(1-2*x)**(5/2)/(3+5*x)**(1/2),x)
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Mathematica [A] time = 0.358094, size = 125, normalized size = 0.67 \[ -\frac{10 \sqrt{3 x+2} \sqrt{5 x+3} \left (294030 x^3+1528956 x^2-9445541 x+3478434\right )-22216880 \sqrt{2-4 x} (2 x-1) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+44109377 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{907500 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(2 + 3*x)^(9/2)/((1 - 2*x)^(5/2)*Sqrt[3 + 5*x]),x]
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Maple [C] time = 0.03, size = 286, normalized size = 1.5 \[{\frac{1}{907500\, \left ( -1+2\,x \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 44433760\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-88218754\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-22216880\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +44109377\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -44104500\,{x}^{5}-285209100\,{x}^{4}+1108687710\,{x}^{3}+1181150330\,{x}^{2}-94170000\,x-208706040 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}\sqrt{2+3\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+3*x)^(9/2)/(1-2*x)^(5/2)/(3+5*x)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}{\sqrt{5 \, x + 3}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^(9/2)/(sqrt(5*x + 3)*(-2*x + 1)^(5/2)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt{3 \, x + 2}}{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^(9/2)/(sqrt(5*x + 3)*(-2*x + 1)^(5/2)),x, algorithm="fricas")
[Out]
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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((2+3*x)**(9/2)/(1-2*x)**(5/2)/(3+5*x)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}{\sqrt{5 \, x + 3}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^(9/2)/(sqrt(5*x + 3)*(-2*x + 1)^(5/2)),x, algorithm="giac")
[Out]