Optimal. Leaf size=185 \[ \frac{5440 \sqrt{1-2 x} \sqrt{3 x+2}}{3 \sqrt{5 x+3}}-\frac{300 \sqrt{1-2 x} \sqrt{3 x+2}}{(5 x+3)^{3/2}}+\frac{404 \sqrt{1-2 x}}{9 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{14 \sqrt{1-2 x}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}-120 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-1088 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.433668, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{5440 \sqrt{1-2 x} \sqrt{3 x+2}}{3 \sqrt{5 x+3}}-\frac{300 \sqrt{1-2 x} \sqrt{3 x+2}}{(5 x+3)^{3/2}}+\frac{404 \sqrt{1-2 x}}{9 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{14 \sqrt{1-2 x}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}-120 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-1088 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
[In] Int[(1 - 2*x)^(3/2)/((2 + 3*x)^(5/2)*(3 + 5*x)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 40.3476, size = 170, normalized size = 0.92 \[ \frac{5440 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{3 \sqrt{5 x + 3}} - \frac{300 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{\left (5 x + 3\right )^{\frac{3}{2}}} + \frac{404 \sqrt{- 2 x + 1}}{9 \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{14 \sqrt{- 2 x + 1}}{9 \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{1088 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{3} - \frac{120 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{11} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(3/2)/(2+3*x)**(5/2)/(3+5*x)**(5/2),x)
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Mathematica [A] time = 0.306055, size = 104, normalized size = 0.56 \[ \frac{2}{3} \left (\frac{\sqrt{1-2 x} \left (122400 x^3+232590 x^2+147122 x+30977\right )}{(3 x+2)^{3/2} (5 x+3)^{3/2}}+2 \sqrt{2} \left (272 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-137 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(1 - 2*x)^(3/2)/((2 + 3*x)^(5/2)*(3 + 5*x)^(5/2)),x]
[Out]
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Maple [C] time = 0.035, size = 383, normalized size = 2.1 \[ -{\frac{2}{-3+6\,x}\sqrt{1-2\,x} \left ( 8160\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-4110\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+10336\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-5206\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+3264\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -1644\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -244800\,{x}^{4}-342780\,{x}^{3}-61654\,{x}^{2}+85168\,x+30977 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(3/2)/(2+3*x)^(5/2)/(3+5*x)^(5/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-2*x + 1)^(3/2)/((5*x + 3)^(5/2)*(3*x + 2)^(5/2)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-2*x + 1)^(3/2)/((5*x + 3)^(5/2)*(3*x + 2)^(5/2)),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((1-2*x)**(3/2)/(2+3*x)**(5/2)/(3+5*x)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-2*x + 1)^(3/2)/((5*x + 3)^(5/2)*(3*x + 2)^(5/2)),x, algorithm="giac")
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