3.2642 \(\int \frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^{7/2}} \, dx\)

Optimal. Leaf size=160 \[ -\frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^{5/2}}+\frac{8314 \sqrt{1-2 x} \sqrt{5 x+3}}{6615 \sqrt{3 x+2}}-\frac{214 \sqrt{1-2 x} \sqrt{5 x+3}}{945 (3 x+2)^{3/2}}+\frac{824 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6615}-\frac{8314 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6615} \]

[Out]

(-214*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(945*(2 + 3*x)^(3/2)) + (8314*Sqrt[1 - 2*x]*S
qrt[3 + 5*x])/(6615*Sqrt[2 + 3*x]) - (2*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(15*(2 +
3*x)^(5/2)) - (8314*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33]
)/6615 + (824*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/6615

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Rubi [A]  time = 0.333205, 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 \[ -\frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^{5/2}}+\frac{8314 \sqrt{1-2 x} \sqrt{5 x+3}}{6615 \sqrt{3 x+2}}-\frac{214 \sqrt{1-2 x} \sqrt{5 x+3}}{945 (3 x+2)^{3/2}}+\frac{824 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6615}-\frac{8314 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6615} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(2 + 3*x)^(7/2),x]

[Out]

(-214*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(945*(2 + 3*x)^(3/2)) + (8314*Sqrt[1 - 2*x]*S
qrt[3 + 5*x])/(6615*Sqrt[2 + 3*x]) - (2*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(15*(2 +
3*x)^(5/2)) - (8314*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33]
)/6615 + (824*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/6615

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Rubi in Sympy [A]  time = 31.0573, size = 143, normalized size = 0.89 \[ \frac{8314 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{6615 \sqrt{3 x + 2}} - \frac{214 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{945 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{2 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{15 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{8314 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{19845} + \frac{9064 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{231525} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3+5*x)**(3/2)*(1-2*x)**(1/2)/(2+3*x)**(7/2),x)

[Out]

8314*sqrt(-2*x + 1)*sqrt(5*x + 3)/(6615*sqrt(3*x + 2)) - 214*sqrt(-2*x + 1)*sqrt
(5*x + 3)/(945*(3*x + 2)**(3/2)) - 2*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/(15*(3*x +
2)**(5/2)) - 8314*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/19
845 + 9064*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/231525

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Mathematica [A]  time = 0.31041, size = 99, normalized size = 0.62 \[ \frac{2 \left (\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (37413 x^2+45432 x+13807\right )}{(3 x+2)^{5/2}}+\sqrt{2} \left (4157 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-10955 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{19845} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(2 + 3*x)^(7/2),x]

[Out]

(2*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(13807 + 45432*x + 37413*x^2))/(2 + 3*x)^(5/2
) + Sqrt[2]*(4157*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 10955*Ell
ipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/19845

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Maple [C]  time = 0.027, size = 386, normalized size = 2.4 \[{\frac{2}{198450\,{x}^{2}+19845\,x-59535} \left ( 98595\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-37413\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+131460\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-49884\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+43820\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -16628\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +1122390\,{x}^{4}+1475199\,{x}^{3}+213789\,{x}^{2}-367467\,x-124263 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^(7/2),x)

[Out]

2/19845*(98595*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)-37413*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)+131460*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)-49884*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)+43
820*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))-16628*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))+1122390*x^4+1475199*x^3+213789*x^2-367467*x-124263)*(1
-2*x)^(1/2)*(3+5*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*sqrt(-2*x + 1)/(3*x + 2)^(7/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(3/2)*sqrt(-2*x + 1)/(3*x + 2)^(7/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{-2 \, x + 1}}{{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*sqrt(-2*x + 1)/(3*x + 2)^(7/2),x, algorithm="fricas")

[Out]

integral((5*x + 3)^(3/2)*sqrt(-2*x + 1)/((27*x^3 + 54*x^2 + 36*x + 8)*sqrt(3*x +
2)), 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)**(3/2)*(1-2*x)**(1/2)/(2+3*x)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*sqrt(-2*x + 1)/(3*x + 2)^(7/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(3/2)*sqrt(-2*x + 1)/(3*x + 2)^(7/2), x)