3.2782 \(\int \frac{(1-2 x)^{5/2} \sqrt{2+3 x}}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=160 \[ -\frac{2 \sqrt{3 x+2} (1-2 x)^{5/2}}{5 \sqrt{5 x+3}}-\frac{24}{125} \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{3028 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{5625}-\frac{28174 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{28125}+\frac{81164 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{28125} \]

[Out]

(-2*(1 - 2*x)^(5/2)*Sqrt[2 + 3*x])/(5*Sqrt[3 + 5*x]) - (3028*Sqrt[1 - 2*x]*Sqrt[
2 + 3*x]*Sqrt[3 + 5*x])/5625 - (24*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/
125 + (81164*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/28125
- (28174*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/28125

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Rubi [A]  time = 0.334653, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{2 \sqrt{3 x+2} (1-2 x)^{5/2}}{5 \sqrt{5 x+3}}-\frac{24}{125} \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{3028 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{5625}-\frac{28174 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{28125}+\frac{81164 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{28125} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(5/2)*Sqrt[2 + 3*x])/(5*Sqrt[3 + 5*x]) - (3028*Sqrt[1 - 2*x]*Sqrt[
2 + 3*x]*Sqrt[3 + 5*x])/5625 - (24*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/
125 + (81164*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/28125
- (28174*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/28125

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Rubi in Sympy [A]  time = 34.1973, size = 143, normalized size = 0.89 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{3 x + 2}}{5 \sqrt{5 x + 3}} - \frac{24 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \sqrt{5 x + 3}}{125} - \frac{3028 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{5625} + \frac{81164 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{84375} - \frac{309914 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{984375} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(5/2)*sqrt(3*x + 2)/(5*sqrt(5*x + 3)) - 24*(-2*x + 1)**(3/2)*sqrt
(3*x + 2)*sqrt(5*x + 3)/125 - 3028*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/56
25 + 81164*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/84375 - 3
09914*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/984375

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Mathematica [A]  time = 0.404501, size = 102, normalized size = 0.64 \[ \frac{\frac{30 \sqrt{1-2 x} \sqrt{3 x+2} \left (900 x^2-2530 x-7287\right )}{\sqrt{5 x+3}}+546035 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-81164 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{84375} \]

Antiderivative was successfully verified.

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

[Out]

((30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(-7287 - 2530*x + 900*x^2))/Sqrt[3 + 5*x] - 811
64*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 546035*Sqrt[2]*E
llipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/84375

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Maple [C]  time = 0.025, size = 169, normalized size = 1.1 \[ -{\frac{1}{2531250\,{x}^{3}+1940625\,{x}^{2}-590625\,x-506250}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 546035\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -81164\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -162000\,{x}^{4}+428400\,{x}^{3}+1441560\,{x}^{2}+66810\,x-437220 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/84375*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(546035*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))-81164*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))
-162000*x^4+428400*x^3+1441560*x^2+66810*x-437220)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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