3.2895 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{3/2} \sqrt{2+3 x}} \, dx\)

Optimal. Leaf size=129 \[ \frac{11 \sqrt{3 x+2} (5 x+3)^{3/2}}{7 \sqrt{1-2 x}}+\frac{335}{63} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}+\frac{67}{63} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{4451}{126} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(335*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/63 + (11*Sqrt[2 + 3*x]*(3 + 5*x)
^(3/2))/(7*Sqrt[1 - 2*x]) + (4451*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 -
2*x]], 35/33])/126 + (67*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]],
35/33])/63

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Rubi [A]  time = 0.258495, antiderivative size = 129, 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 \[ \frac{11 \sqrt{3 x+2} (5 x+3)^{3/2}}{7 \sqrt{1-2 x}}+\frac{335}{63} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}+\frac{67}{63} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{4451}{126} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(335*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/63 + (11*Sqrt[2 + 3*x]*(3 + 5*x)
^(3/2))/(7*Sqrt[1 - 2*x]) + (4451*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 -
2*x]], 35/33])/126 + (67*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]],
35/33])/63

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Rubi in Sympy [A]  time = 24.2255, size = 114, normalized size = 0.88 \[ \frac{335 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{63} + \frac{4451 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{378} + \frac{737 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{2205} + \frac{11 \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{7 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

335*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/63 + 4451*sqrt(33)*elliptic_e(asi
n(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/378 + 737*sqrt(35)*elliptic_f(asin(sqrt(55)
*sqrt(-2*x + 1)/11), 33/35)/2205 + 11*sqrt(3*x + 2)*(5*x + 3)**(3/2)/(7*sqrt(-2*
x + 1))

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Mathematica [A]  time = 0.143931, size = 105, normalized size = 0.81 \[ \frac{6 \sqrt{3 x+2} \sqrt{5 x+3} (632-175 x)+2240 \sqrt{2-4 x} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-4451 \sqrt{2-4 x} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{378 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(6*(632 - 175*x)*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] - 4451*Sqrt[2 - 4*x]*EllipticE[ArcS
in[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 2240*Sqrt[2 - 4*x]*EllipticF[ArcSin[Sqrt[
2/11]*Sqrt[3 + 5*x]], -33/2])/(378*Sqrt[1 - 2*x])

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Maple [C]  time = 0.027, size = 164, normalized size = 1.3 \[ -{\frac{1}{11340\,{x}^{3}+8694\,{x}^{2}-2646\,x-2268}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 2240\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -4451\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -15750\,{x}^{3}+36930\,{x}^{2}+65748\,x+22752 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/378*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(2240*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))-4451*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*El
lipticE(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))-1575
0*x^3+36930*x^2+65748*x+22752)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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