3.2951 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{5/2}} \, dx\)

Optimal. Leaf size=191 \[ \frac{11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{3/2}}-\frac{169 \sqrt{1-2 x} \sqrt{5 x+3}}{7203 \sqrt{3 x+2}}+\frac{229 \sqrt{1-2 x} \sqrt{5 x+3}}{1029 (3 x+2)^{3/2}}-\frac{22 \sqrt{5 x+3}}{49 \sqrt{1-2 x} (3 x+2)^{3/2}}+\frac{496 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7203}+\frac{169 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7203} \]

[Out]

(-22*Sqrt[3 + 5*x])/(49*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)) + (229*Sqrt[1 - 2*x]*Sqrt
[3 + 5*x])/(1029*(2 + 3*x)^(3/2)) - (169*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7203*Sqrt
[2 + 3*x]) + (11*(3 + 5*x)^(3/2))/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)) + (169*Sq
rt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/7203 + (496*Sqrt[11/
3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/7203

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Rubi [A]  time = 0.421442, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{3/2}}-\frac{169 \sqrt{1-2 x} \sqrt{5 x+3}}{7203 \sqrt{3 x+2}}+\frac{229 \sqrt{1-2 x} \sqrt{5 x+3}}{1029 (3 x+2)^{3/2}}-\frac{22 \sqrt{5 x+3}}{49 \sqrt{1-2 x} (3 x+2)^{3/2}}+\frac{496 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7203}+\frac{169 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7203} \]

Antiderivative was successfully verified.

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

[Out]

(-22*Sqrt[3 + 5*x])/(49*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)) + (229*Sqrt[1 - 2*x]*Sqrt
[3 + 5*x])/(1029*(2 + 3*x)^(3/2)) - (169*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7203*Sqrt
[2 + 3*x]) + (11*(3 + 5*x)^(3/2))/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)) + (169*Sq
rt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/7203 + (496*Sqrt[11/
3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/7203

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Rubi in Sympy [A]  time = 37.5768, size = 172, normalized size = 0.9 \[ \frac{169 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{21609} + \frac{5456 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{252105} + \frac{338 \sqrt{3 x + 2} \sqrt{5 x + 3}}{21609 \sqrt{- 2 x + 1}} - \frac{209 \sqrt{5 x + 3}}{1029 \sqrt{- 2 x + 1} \sqrt{3 x + 2}} + \frac{31 \sqrt{5 x + 3}}{441 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{11 \left (5 x + 3\right )^{\frac{3}{2}}}{21 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

169*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/21609 + 5456*sqr
t(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/252105 + 338*sqrt(3*x
+ 2)*sqrt(5*x + 3)/(21609*sqrt(-2*x + 1)) - 209*sqrt(5*x + 3)/(1029*sqrt(-2*x +
1)*sqrt(3*x + 2)) + 31*sqrt(5*x + 3)/(441*sqrt(-2*x + 1)*(3*x + 2)**(3/2)) + 11*
(5*x + 3)**(3/2)/(21*(-2*x + 1)**(3/2)*(3*x + 2)**(3/2))

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Mathematica [A]  time = 0.279892, size = 105, normalized size = 0.55 \[ \frac{-\frac{6 \sqrt{5 x+3} \left (1014 x^3-3544 x^2-9883 x-4675\right )}{(1-2 x)^{3/2} (3 x+2)^{3/2}}-\sqrt{2} \left (8015 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+169 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{21609} \]

Antiderivative was successfully verified.

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

[Out]

((-6*Sqrt[3 + 5*x]*(-4675 - 9883*x - 3544*x^2 + 1014*x^3))/((1 - 2*x)^(3/2)*(2 +
3*x)^(3/2)) - Sqrt[2]*(169*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] +
8015*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/21609

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Maple [C]  time = 0.034, size = 383, normalized size = 2. \[{\frac{1}{21609\, \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 1014\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+48090\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+169\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+8015\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-338\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -16030\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -30420\,{x}^{4}+88068\,{x}^{3}+360282\,{x}^{2}+318144\,x+84150 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/21609*(1-2*x)^(1/2)*(1014*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^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)+
48090*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^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)+169*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)+8015*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)-338*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))-16030*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))-30420*x^4+88068*x^3+360282*x^2+318144*x+84150)/
(2+3*x)^(3/2)/(-1+2*x)^2/(3+5*x)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3}}{{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((25*x^2 + 30*x + 9)*sqrt(5*x + 3)/((36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)
*sqrt(3*x + 2)*sqrt(-2*x + 1)), 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((3+5*x)**(5/2)/(1-2*x)**(5/2)/(2+3*x)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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