3.2774 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^{5/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=129 \[ \frac{14 \sqrt{5 x+3} (1-2 x)^{3/2}}{9 (3 x+2)^{3/2}}+\frac{812 \sqrt{5 x+3} \sqrt{1-2 x}}{27 \sqrt{3 x+2}}-\frac{164}{135} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{3896}{135} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(14*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(9*(2 + 3*x)^(3/2)) + (812*Sqrt[1 - 2*x]*Sqrt
[3 + 5*x])/(27*Sqrt[2 + 3*x]) - (3896*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt
[1 - 2*x]], 35/33])/135 - (164*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*
x]], 35/33])/135

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Rubi [A]  time = 0.25988, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{14 \sqrt{5 x+3} (1-2 x)^{3/2}}{9 (3 x+2)^{3/2}}+\frac{812 \sqrt{5 x+3} \sqrt{1-2 x}}{27 \sqrt{3 x+2}}-\frac{164}{135} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{3896}{135} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(14*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(9*(2 + 3*x)^(3/2)) + (812*Sqrt[1 - 2*x]*Sqrt
[3 + 5*x])/(27*Sqrt[2 + 3*x]) - (3896*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt
[1 - 2*x]], 35/33])/135 - (164*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*
x]], 35/33])/135

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Rubi in Sympy [A]  time = 25.6196, size = 114, normalized size = 0.88 \[ \frac{14 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{9 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{812 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{27 \sqrt{3 x + 2}} - \frac{3896 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{405} - \frac{1804 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{4725} \]

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)**(1/2),x)

[Out]

14*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(9*(3*x + 2)**(3/2)) + 812*sqrt(-2*x + 1)*sqr
t(5*x + 3)/(27*sqrt(3*x + 2)) - 3896*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x
+ 1)/7), 35/33)/405 - 1804*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11)
, 33/35)/4725

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Mathematica [A]  time = 0.285075, size = 97, normalized size = 0.75 \[ \frac{2}{405} \left (\frac{735 \sqrt{1-2 x} \sqrt{5 x+3} (24 x+17)}{(3 x+2)^{3/2}}-595 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+1948 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*((735*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(17 + 24*x))/(2 + 3*x)^(3/2) + 1948*Sqrt[2]
*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 595*Sqrt[2]*EllipticF[ArcS
in[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/405

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Maple [C]  time = 0.028, size = 267, normalized size = 2.1 \[{\frac{2}{4050\,{x}^{2}+405\,x-1215} \left ( 1785\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-5844\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+1190\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -3896\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +176400\,{x}^{3}+142590\,{x}^{2}-40425\,x-37485 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}} \]

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)^(1/2),x)

[Out]

2/405*(1785*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)-5844*2^(1/2)*Ellip
ticE(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)+1190*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))-3896*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))+176400*x^3+142590
*x^2-40425*x-37485)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{-2 \, x + 1}}{{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((4*x^2 - 4*x + 1)*sqrt(-2*x + 1)/((9*x^2 + 12*x + 4)*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)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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