3.2731 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^{3/2}}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=160 \[ -\frac{62 \sqrt{1-2 x} (3 x+2)^{3/2}}{25 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{3/2} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}+\frac{178}{125} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}-\frac{582}{625} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{496}{625} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(-2*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2))/(15*(3 + 5*x)^(3/2)) - (62*Sqrt[1 - 2*x]*(2
+ 3*x)^(3/2))/(25*Sqrt[3 + 5*x]) + (178*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*
x])/125 + (496*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/625
- (582*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/625

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Rubi [A]  time = 0.333466, 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 \[ -\frac{62 \sqrt{1-2 x} (3 x+2)^{3/2}}{25 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{3/2} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}+\frac{178}{125} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}-\frac{582}{625} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{496}{625} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2))/(15*(3 + 5*x)^(3/2)) - (62*Sqrt[1 - 2*x]*(2
+ 3*x)^(3/2))/(25*Sqrt[3 + 5*x]) + (178*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*
x])/125 + (496*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/625
- (582*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/625

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Rubi in Sympy [A]  time = 34.5952, size = 143, normalized size = 0.89 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{3}{2}}}{15 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{62 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2}}{275 \sqrt{5 x + 3}} - \frac{212 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{1375} + \frac{496 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1875} - \frac{1746 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{21875} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(3/2)*(3*x + 2)**(3/2)/(15*(5*x + 3)**(3/2)) - 62*(-2*x + 1)**(3/
2)*sqrt(3*x + 2)/(275*sqrt(5*x + 3)) - 212*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x
+ 3)/1375 + 496*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/187
5 - 1746*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/21875

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Mathematica [A]  time = 0.392152, size = 102, normalized size = 0.64 \[ \frac{-\frac{10 \sqrt{1-2 x} \sqrt{3 x+2} \left (150 x^2+800 x+437\right )}{(5 x+3)^{3/2}}+3115 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-496 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{1875} \]

Antiderivative was successfully verified.

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

[Out]

((-10*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(437 + 800*x + 150*x^2))/(3 + 5*x)^(3/2) - 496
*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 3115*Sqrt[2]*Ellip
ticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/1875

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Maple [C]  time = 0.026, size = 272, normalized size = 1.7 \[ -{\frac{1}{11250\,{x}^{2}+1875\,x-3750} \left ( 15575\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-2480\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+9345\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -1488\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +9000\,{x}^{4}+49500\,{x}^{3}+31220\,{x}^{2}-11630\,x-8740 \right ) \sqrt{2+3\,x}\sqrt{1-2\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1875*(15575*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)-2480*2^(1/2)*El
lipticE(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)+9345*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))-1488*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))+9000*x^4+49500
*x^3+31220*x^2-11630*x-8740)*(2+3*x)^(1/2)*(1-2*x)^(1/2)/(6*x^2+x-2)/(3+5*x)^(3/
2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (6 \, x^{2} + x - 2\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(6*x^2 + x - 2)*sqrt(3*x + 2)*sqrt(-2*x + 1)/((25*x^2 + 30*x + 9)*sqrt
(5*x + 3)), 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)**(3/2)*(2+3*x)**(3/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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