3.2931 \(\int \frac{(2+3 x)^{3/2} \sqrt{3+5 x}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=123 \[ \frac{\sqrt{5 x+3} (3 x+2)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{67 \sqrt{5 x+3} \sqrt{3 x+2}}{33 \sqrt{1-2 x}}-\frac{2 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{\sqrt{33}}-\frac{133 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2 \sqrt{33}} \]

[Out]

(-67*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/(33*Sqrt[1 - 2*x]) + ((2 + 3*x)^(3/2)*Sqrt[3 +
5*x])/(3*(1 - 2*x)^(3/2)) - (133*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/
33])/(2*Sqrt[33]) - (2*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/Sqrt[3
3]

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Rubi [A]  time = 0.257972, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{\sqrt{5 x+3} (3 x+2)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{67 \sqrt{5 x+3} \sqrt{3 x+2}}{33 \sqrt{1-2 x}}-\frac{2 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{\sqrt{33}}-\frac{133 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-67*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/(33*Sqrt[1 - 2*x]) + ((2 + 3*x)^(3/2)*Sqrt[3 +
5*x])/(3*(1 - 2*x)^(3/2)) - (133*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/
33])/(2*Sqrt[33]) - (2*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/Sqrt[3
3]

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Rubi in Sympy [A]  time = 24.1358, size = 112, normalized size = 0.91 \[ - \frac{133 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{66} - \frac{2 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{33} - \frac{67 \sqrt{3 x + 2} \sqrt{5 x + 3}}{33 \sqrt{- 2 x + 1}} + \frac{\left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{3 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-133*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/66 - 2*sqrt(33)
*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/33 - 67*sqrt(3*x + 2)*sqrt(5
*x + 3)/(33*sqrt(-2*x + 1)) + (3*x + 2)**(3/2)*sqrt(5*x + 3)/(3*(-2*x + 1)**(3/2
))

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Mathematica [A]  time = 0.238522, size = 115, normalized size = 0.93 \[ -\frac{2 \sqrt{3 x+2} \sqrt{5 x+3} (45-167 x)-67 \sqrt{2-4 x} (2 x-1) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+133 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{66 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(2*(45 - 167*x)*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] + 133*Sqrt[2 - 4*x]*(-1 + 2*x)*Elli
pticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 67*Sqrt[2 - 4*x]*(-1 + 2*x)*Ell
ipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(66*(1 - 2*x)^(3/2))

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Maple [C]  time = 0.028, size = 276, normalized size = 2.2 \[{\frac{1}{66\, \left ( -1+2\,x \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 134\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-266\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-67\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +133\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +5010\,{x}^{3}+4996\,{x}^{2}+294\,x-540 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/66*(134*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)-266*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)-67*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))+133*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))+5010*x^3+4996*x^2+294*x
-540)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)/(-1+2*x)^2/(15*x^2+19*x+6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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