3.2741 \(\int \frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{(2+3 x)^{3/2}} \, dx\)

Optimal. Leaf size=160 \[ -\frac{2 \sqrt{5 x+3} (1-2 x)^{5/2}}{3 \sqrt{3 x+2}}-\frac{8}{15} \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{1076}{675} \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{12758 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3375}+\frac{31588 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3375} \]

[Out]

(-2*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(3*Sqrt[2 + 3*x]) - (1076*Sqrt[1 - 2*x]*Sqrt[
2 + 3*x]*Sqrt[3 + 5*x])/675 - (8*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/15
+ (31588*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3375 - (
12758*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3375

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Rubi [A]  time = 0.330428, 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 \[ -\frac{2 \sqrt{5 x+3} (1-2 x)^{5/2}}{3 \sqrt{3 x+2}}-\frac{8}{15} \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{1076}{675} \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{12758 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3375}+\frac{31588 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3375} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(3*Sqrt[2 + 3*x]) - (1076*Sqrt[1 - 2*x]*Sqrt[
2 + 3*x]*Sqrt[3 + 5*x])/675 - (8*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/15
+ (31588*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3375 - (
12758*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3375

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Rubi in Sympy [A]  time = 31.556, size = 143, normalized size = 0.89 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{3 \sqrt{3 x + 2}} - \frac{8 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \sqrt{5 x + 3}}{15} - \frac{1076 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{675} + \frac{31588 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{10125} - \frac{140338 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{118125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(3*sqrt(3*x + 2)) - 8*(-2*x + 1)**(3/2)*sqrt(
3*x + 2)*sqrt(5*x + 3)/15 - 1076*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/675
+ 31588*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/10125 - 1403
38*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/118125

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Mathematica [A]  time = 0.305633, size = 102, normalized size = 0.64 \[ \frac{\frac{30 \sqrt{1-2 x} \sqrt{5 x+3} \left (180 x^2-534 x-1661\right )}{\sqrt{3 x+2}}+242095 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-31588 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{10125} \]

Antiderivative was successfully verified.

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

[Out]

((30*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-1661 - 534*x + 180*x^2))/Sqrt[2 + 3*x] - 3158
8*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 242095*Sqrt[2]*El
lipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/10125

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Maple [C]  time = 0.023, size = 169, normalized size = 1.1 \[ -{\frac{1}{303750\,{x}^{3}+232875\,{x}^{2}-70875\,x-60750}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 242095\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -31588\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -54000\,{x}^{4}+154800\,{x}^{3}+530520\,{x}^{2}+1770\,x-149490 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10125*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(242095*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))-31588*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))
-54000*x^4+154800*x^3+530520*x^2+1770*x-149490)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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