3.2948 \(\int \frac{\sqrt{2+3 x} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=160 \[ \frac{\sqrt{3 x+2} (5 x+3)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{104 \sqrt{3 x+2} (5 x+3)^{3/2}}{21 \sqrt{1-2 x}}-\frac{695}{42} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{139}{42} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{4621}{42} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(-695*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/42 - (104*Sqrt[2 + 3*x]*(3 + 5*
x)^(3/2))/(21*Sqrt[1 - 2*x]) + (Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/(3*(1 - 2*x)^(3/2
)) - (4621*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/42 - (1
39*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/42

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Rubi [A]  time = 0.331876, antiderivative size = 160, 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 \[ \frac{\sqrt{3 x+2} (5 x+3)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{104 \sqrt{3 x+2} (5 x+3)^{3/2}}{21 \sqrt{1-2 x}}-\frac{695}{42} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{139}{42} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{4621}{42} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-695*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/42 - (104*Sqrt[2 + 3*x]*(3 + 5*
x)^(3/2))/(21*Sqrt[1 - 2*x]) + (Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/(3*(1 - 2*x)^(3/2
)) - (4621*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/42 - (1
39*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/42

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Rubi in Sympy [A]  time = 31.0433, size = 141, normalized size = 0.88 \[ - \frac{695 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{42} - \frac{4621 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{126} - \frac{139 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{126} - \frac{104 \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{21 \sqrt{- 2 x + 1}} + \frac{\sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{5}{2}}}{3 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-695*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/42 - 4621*sqrt(33)*elliptic_e(as
in(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/126 - 139*sqrt(33)*elliptic_f(asin(sqrt(21
)*sqrt(-2*x + 1)/7), 35/33)/126 - 104*sqrt(3*x + 2)*(5*x + 3)**(3/2)/(21*sqrt(-2
*x + 1)) + sqrt(3*x + 2)*(5*x + 3)**(5/2)/(3*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.208972, size = 120, normalized size = 0.75 \[ -\frac{6 \sqrt{3 x+2} \sqrt{5 x+3} \left (350 x^2-3408 x+1193\right )-4655 \sqrt{2-4 x} (2 x-1) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+9242 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{252 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(6*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(1193 - 3408*x + 350*x^2) + 9242*Sqrt[2 - 4*x]*(
-1 + 2*x)*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 4655*Sqrt[2 - 4*x
]*(-1 + 2*x)*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(252*(1 - 2*x)^
(3/2))

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Maple [C]  time = 0.028, size = 286, normalized size = 1.8 \[{\frac{1}{ \left ( 7560\,{x}^{3}+5796\,{x}^{2}-1764\,x-1512 \right ) \left ( -1+2\,x \right ) } \left ( 9310\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-18484\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-4655\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +9242\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -31500\,{x}^{4}+266820\,{x}^{3}+268542\,{x}^{2}-13314\,x-42948 \right ) \sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/252*(9310*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)-18484*2^(1/2)*Elli
pticE(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)-4655*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))+9242*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))-31500*x^4+266820
*x^3+268542*x^2-13314*x-42948)*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*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. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{3 \, x + 2}}{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((25*x^2 + 30*x + 9)*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((3+5*x)**(5/2)*(2+3*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. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{3 \, x + 2}}{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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