3.2705 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx\)

Optimal. Leaf size=191 \[ -\frac{32}{63} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{3 \sqrt{3 x+2}}+\frac{202}{63} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{1061}{567} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{1061 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2835}-\frac{2894 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2835} \]

[Out]

(-1061*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/567 + (202*Sqrt[1 - 2*x]*Sqrt[
2 + 3*x]*(3 + 5*x)^(3/2))/63 - (2*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(3*Sqrt[2 + 3
*x]) - (32*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/63 - (2894*Sqrt[11/3]*El
lipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2835 - (1061*Sqrt[11/3]*Ellipti
cF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2835

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Rubi [A]  time = 0.408329, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{32}{63} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{3 \sqrt{3 x+2}}+\frac{202}{63} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{1061}{567} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{1061 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2835}-\frac{2894 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2835} \]

Antiderivative was successfully verified.

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

[Out]

(-1061*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/567 + (202*Sqrt[1 - 2*x]*Sqrt[
2 + 3*x]*(3 + 5*x)^(3/2))/63 - (2*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(3*Sqrt[2 + 3
*x]) - (32*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/63 - (2894*Sqrt[11/3]*El
lipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2835 - (1061*Sqrt[11/3]*Ellipti
cF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2835

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Rubi in Sympy [A]  time = 42.5515, size = 172, normalized size = 0.9 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{3 \sqrt{3 x + 2}} - \frac{32 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{5}{2}}}{63} + \frac{202 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{63} - \frac{1061 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{567} - \frac{2894 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{8505} - \frac{1061 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{8505} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(3/2)*(5*x + 3)**(5/2)/(3*sqrt(3*x + 2)) - 32*sqrt(-2*x + 1)*sqrt
(3*x + 2)*(5*x + 3)**(5/2)/63 + 202*sqrt(-2*x + 1)*sqrt(3*x + 2)*(5*x + 3)**(3/2
)/63 - 1061*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/567 - 2894*sqrt(33)*ellip
tic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/8505 - 1061*sqrt(33)*elliptic_f(as
in(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/8505

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Mathematica [A]  time = 0.396741, size = 107, normalized size = 0.56 \[ \frac{\frac{30 \sqrt{1-2 x} \sqrt{5 x+3} \left (-2700 x^3+180 x^2+1767 x+200\right )}{\sqrt{3 x+2}}+29225 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+5788 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{17010} \]

Antiderivative was successfully verified.

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

[Out]

((30*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(200 + 1767*x + 180*x^2 - 2700*x^3))/Sqrt[2 + 3
*x] + 5788*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 29225*Sq
rt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/17010

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Maple [C]  time = 0.024, size = 174, normalized size = 0.9 \[ -{\frac{1}{510300\,{x}^{3}+391230\,{x}^{2}-119070\,x-102060}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 29225\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +5788\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +810000\,{x}^{5}+27000\,{x}^{4}-778500\,{x}^{3}-96810\,{x}^{2}+153030\,x+18000 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/17010*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(29225*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))+5788*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))+8
10000*x^5+27000*x^4-778500*x^3-96810*x^2+153030*x+18000)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(50*x^3 + 35*x^2 - 12*x - 9)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(3*x + 2)^(3
/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((1-2*x)**(3/2)*(3+5*x)**(5/2)/(2+3*x)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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