3.2785 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=160 \[ \frac{14 (1-2 x)^{3/2}}{9 (3 x+2)^{3/2} \sqrt{5 x+3}}-\frac{17804 \sqrt{3 x+2} \sqrt{1-2 x}}{27 \sqrt{5 x+3}}+\frac{1792 \sqrt{1-2 x}}{27 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{536}{45} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{17804}{45} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(14*(1 - 2*x)^(3/2))/(9*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (1792*Sqrt[1 - 2*x])/(2
7*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) - (17804*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(27*Sqrt[3
+ 5*x]) + (17804*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/4
5 + (536*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/45

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Rubi [A]  time = 0.346422, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{14 (1-2 x)^{3/2}}{9 (3 x+2)^{3/2} \sqrt{5 x+3}}-\frac{17804 \sqrt{3 x+2} \sqrt{1-2 x}}{27 \sqrt{5 x+3}}+\frac{1792 \sqrt{1-2 x}}{27 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{536}{45} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{17804}{45} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 - 2*x)^(5/2)/((2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)),x]

[Out]

(14*(1 - 2*x)^(3/2))/(9*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (1792*Sqrt[1 - 2*x])/(2
7*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) - (17804*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(27*Sqrt[3
+ 5*x]) + (17804*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/4
5 + (536*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/45

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Rubi in Sympy [A]  time = 32.9839, size = 143, normalized size = 0.89 \[ \frac{14 \left (- 2 x + 1\right )^{\frac{3}{2}}}{9 \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}} - \frac{17804 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{27 \sqrt{5 x + 3}} + \frac{1792 \sqrt{- 2 x + 1}}{27 \sqrt{3 x + 2} \sqrt{5 x + 3}} + \frac{17804 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{135} + \frac{5896 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{1575} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)/(2+3*x)**(5/2)/(3+5*x)**(3/2),x)

[Out]

14*(-2*x + 1)**(3/2)/(9*(3*x + 2)**(3/2)*sqrt(5*x + 3)) - 17804*sqrt(-2*x + 1)*s
qrt(3*x + 2)/(27*sqrt(5*x + 3)) + 1792*sqrt(-2*x + 1)/(27*sqrt(3*x + 2)*sqrt(5*x
+ 3)) + 17804*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/135 +
5896*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/1575

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Mathematica [A]  time = 0.223392, size = 100, normalized size = 0.62 \[ -\frac{2 \sqrt{1-2 x} \left (26706 x^2+34726 x+11265\right )}{9 (3 x+2)^{3/2} \sqrt{5 x+3}}-\frac{4}{135} \sqrt{2} \left (4451 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-2240 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)),x]

[Out]

(-2*Sqrt[1 - 2*x]*(11265 + 34726*x + 26706*x^2))/(9*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x
]) - (4*Sqrt[2]*(4451*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 2240*
EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/135

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Maple [C]  time = 0.034, size = 267, normalized size = 1.7 \[ -{\frac{2}{1350\,{x}^{2}+135\,x-405}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 13440\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-26706\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+8960\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -17804\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +801180\,{x}^{3}+641190\,{x}^{2}-182940\,x-168975 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)/(2+3*x)^(5/2)/(3+5*x)^(3/2),x)

[Out]

-2/135*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(13440*2^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2
)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1
-2*x)^(1/2)-26706*2^(1/2)*EllipticE(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11
^(1/2)*3^(1/2)*2^(1/2))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)+8960*2^(1/2)
*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*
x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))-17804*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/
2)*(1-2*x)^(1/2)*EllipticE(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^
(1/2)*2^(1/2))+801180*x^3+641190*x^2-182940*x-168975)/(2+3*x)^(3/2)/(10*x^2+x-3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(5/2)/((5*x + 3)^(3/2)*(3*x + 2)^(5/2)),x, algorithm="maxima")

[Out]

integrate((-2*x + 1)^(5/2)/((5*x + 3)^(3/2)*(3*x + 2)^(5/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{-2 \, x + 1}}{{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(5/2)/((5*x + 3)^(3/2)*(3*x + 2)^(5/2)),x, algorithm="fricas")

[Out]

integral((4*x^2 - 4*x + 1)*sqrt(-2*x + 1)/((45*x^3 + 87*x^2 + 56*x + 12)*sqrt(5*
x + 3)*sqrt(3*x + 2)), x)

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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)**(5/2)/(2+3*x)**(5/2)/(3+5*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(5/2)/((5*x + 3)^(3/2)*(3*x + 2)^(5/2)),x, algorithm="giac")

[Out]

integrate((-2*x + 1)^(5/2)/((5*x + 3)^(3/2)*(3*x + 2)^(5/2)), x)