3.2939 \(\int \frac{(2+3 x)^{3/2} (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=158 \[ \frac{(3 x+2)^{3/2} (5 x+3)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{34 \sqrt{3 x+2} (5 x+3)^{3/2}}{11 \sqrt{1-2 x}}-\frac{225}{22} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{15}{2} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-68 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(-225*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/22 - (34*Sqrt[2 + 3*x]*(3 + 5*x
)^(3/2))/(11*Sqrt[1 - 2*x]) + ((2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))/(3*(1 - 2*x)^(3/
2)) - 68*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33] - (15*Sqrt
[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2

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Rubi [A]  time = 0.319269, antiderivative size = 158, 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 \[ \frac{(3 x+2)^{3/2} (5 x+3)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{34 \sqrt{3 x+2} (5 x+3)^{3/2}}{11 \sqrt{1-2 x}}-\frac{225}{22} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{15}{2} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-68 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-225*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/22 - (34*Sqrt[2 + 3*x]*(3 + 5*x
)^(3/2))/(11*Sqrt[1 - 2*x]) + ((2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))/(3*(1 - 2*x)^(3/
2)) - 68*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33] - (15*Sqrt
[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2

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Rubi in Sympy [A]  time = 30.8067, size = 141, normalized size = 0.89 \[ - \frac{137 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{14} - \frac{68 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{3} - \frac{9 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{14} - \frac{34 \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{7 \sqrt{- 2 x + 1}} + \frac{\left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-137*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/14 - 68*sqrt(33)*elliptic_e(asin
(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/3 - 9*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt
(-2*x + 1)/11), 33/35)/14 - 34*(3*x + 2)**(3/2)*sqrt(5*x + 3)/(7*sqrt(-2*x + 1))
+ (3*x + 2)**(3/2)*(5*x + 3)**(3/2)/(3*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.272116, size = 120, normalized size = 0.76 \[ -\frac{2 \sqrt{3 x+2} \sqrt{5 x+3} \left (30 x^2-302 x+105\right )-137 \sqrt{2-4 x} (2 x-1) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+272 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{12 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(2*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(105 - 302*x + 30*x^2) + 272*Sqrt[2 - 4*x]*(-1 +
2*x)*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 137*Sqrt[2 - 4*x]*(-1
+ 2*x)*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(12*(1 - 2*x)^(3/2))

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Maple [C]  time = 0.027, size = 286, normalized size = 1.8 \[{\frac{1}{ \left ( 360\,{x}^{3}+276\,{x}^{2}-84\,x-72 \right ) \left ( -1+2\,x \right ) } \left ( 274\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-544\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-137\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +272\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -900\,{x}^{4}+7920\,{x}^{3}+7966\,{x}^{2}-366\,x-1260 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(274*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)-544*2^(1/2)*Elliptic
E(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)-137*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))+272*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))-900*x^4+7920*x^3+7966*
x^2-366*x-1260)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)/(30*x^3+23*x^2-7*x-6)/
(-1+2*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (15 \, x^{2} + 19 \, x + 6\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((15*x^2 + 19*x + 6)*sqrt(5*x + 3)*sqrt(3*x + 2)/((4*x^2 - 4*x + 1)*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((2+3*x)**(3/2)*(3+5*x)**(3/2)/(1-2*x)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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