Optimal. Leaf size=191 \[ \frac{7 (3 x+2)^{7/2}}{11 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{37 \sqrt{1-2 x} (3 x+2)^{5/2}}{605 \sqrt{5 x+3}}+\frac{10851 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{15125}+\frac{502941 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{151250}+\frac{175111 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{68750}+\frac{2911577 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{34375} \]
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Rubi [A] time = 0.407807, antiderivative size = 191, 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 \[ \frac{7 (3 x+2)^{7/2}}{11 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{37 \sqrt{1-2 x} (3 x+2)^{5/2}}{605 \sqrt{5 x+3}}+\frac{10851 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{15125}+\frac{502941 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{151250}+\frac{175111 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{68750}+\frac{2911577 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{34375} \]
Antiderivative was successfully verified.
[In] Int[(2 + 3*x)^(9/2)/((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)),x]
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Rubi in Sympy [A] time = 40.977, size = 172, normalized size = 0.9 \[ - \frac{37 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}}}{605 \sqrt{5 x + 3}} + \frac{10851 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{15125} + \frac{502941 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{151250} + \frac{2911577 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{378125} + \frac{525333 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{2406250} + \frac{7 \left (3 x + 2\right )^{\frac{7}{2}}}{11 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2+3*x)**(9/2)/(1-2*x)**(3/2)/(3+5*x)**(3/2),x)
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Mathematica [A] time = 0.372633, size = 132, normalized size = 0.69 \[ \frac{10 \sqrt{3 x+2} \sqrt{5 x+3} \left (-490050 x^3-2188890 x^2+3684629 x+2892883\right )+5867645 \sqrt{2-4 x} (5 x+3) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-11646308 \sqrt{2-4 x} (5 x+3) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{1512500 \sqrt{1-2 x} (5 x+3)} \]
Antiderivative was successfully verified.
[In] Integrate[(2 + 3*x)^(9/2)/((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)),x]
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Maple [C] time = 0.035, size = 169, normalized size = 0.9 \[ -{\frac{1}{45375000\,{x}^{3}+34787500\,{x}^{2}-10587500\,x-9075000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 5867645\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -11646308\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -14701500\,{x}^{4}-75467700\,{x}^{3}+66761070\,{x}^{2}+160479070\,x+57857660 \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+3*x)^(9/2)/(1-2*x)^(3/2)/(3+5*x)^(3/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^(9/2)/((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt{3 \, x + 2}}{{\left (10 \, x^{2} + x - 3\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^(9/2)/((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)),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((2+3*x)**(9/2)/(1-2*x)**(3/2)/(3+5*x)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^(9/2)/((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)),x, algorithm="giac")
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