3.2729 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^{7/2}}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=220 \[ -\frac{6 \sqrt{1-2 x} (3 x+2)^{7/2}}{\sqrt{5 x+3}}-\frac{2 (1-2 x)^{3/2} (3 x+2)^{7/2}}{15 (5 x+3)^{3/2}}+\frac{622}{175} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}+\frac{3872 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{4375}+\frac{4801 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{21875}-\frac{24369 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{109375}-\frac{25643 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{109375} \]

[Out]

(-2*(1 - 2*x)^(3/2)*(2 + 3*x)^(7/2))/(15*(3 + 5*x)^(3/2)) - (6*Sqrt[1 - 2*x]*(2
+ 3*x)^(7/2))/Sqrt[3 + 5*x] + (4801*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/2
1875 + (3872*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/4375 + (622*Sqrt[1 - 2
*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/175 - (25643*Sqrt[11/3]*EllipticE[ArcSin[Sqrt
[3/7]*Sqrt[1 - 2*x]], 35/33])/109375 - (24369*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3
/7]*Sqrt[1 - 2*x]], 35/33])/109375

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Rubi [A]  time = 0.493695, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{6 \sqrt{1-2 x} (3 x+2)^{7/2}}{\sqrt{5 x+3}}-\frac{2 (1-2 x)^{3/2} (3 x+2)^{7/2}}{15 (5 x+3)^{3/2}}+\frac{622}{175} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}+\frac{3872 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{4375}+\frac{4801 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{21875}-\frac{24369 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{109375}-\frac{25643 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{109375} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(3/2)*(2 + 3*x)^(7/2))/(15*(3 + 5*x)^(3/2)) - (6*Sqrt[1 - 2*x]*(2
+ 3*x)^(7/2))/Sqrt[3 + 5*x] + (4801*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/2
1875 + (3872*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/4375 + (622*Sqrt[1 - 2
*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/175 - (25643*Sqrt[11/3]*EllipticE[ArcSin[Sqrt
[3/7]*Sqrt[1 - 2*x]], 35/33])/109375 - (24369*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3
/7]*Sqrt[1 - 2*x]], 35/33])/109375

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Rubi in Sympy [A]  time = 46.7345, size = 201, normalized size = 0.91 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{7}{2}}}{15 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{6 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{5}{2}}}{11 \sqrt{5 x + 3}} - \frac{508 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{1925} + \frac{3872 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{4375} + \frac{4801 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{21875} - \frac{25643 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{328125} - \frac{73107 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{3828125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(3/2)*(3*x + 2)**(7/2)/(15*(5*x + 3)**(3/2)) - 6*(-2*x + 1)**(3/2
)*(3*x + 2)**(5/2)/(11*sqrt(5*x + 3)) - 508*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*sqrt
(5*x + 3)/1925 + 3872*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/4375 + 4801*
sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/21875 - 25643*sqrt(33)*elliptic_e(asi
n(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/328125 - 73107*sqrt(35)*elliptic_f(asin(sqr
t(55)*sqrt(-2*x + 1)/11), 33/35)/3828125

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Mathematica [A]  time = 0.476057, size = 112, normalized size = 0.51 \[ \frac{\frac{10 \sqrt{1-2 x} \sqrt{3 x+2} \left (-202500 x^4-189000 x^3+174525 x^2+216050 x+52067\right )}{(5 x+3)^{3/2}}+168035 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+51286 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{656250} \]

Antiderivative was successfully verified.

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

[Out]

((10*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(52067 + 216050*x + 174525*x^2 - 189000*x^3 - 2
02500*x^4))/(3 + 5*x)^(3/2) + 51286*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]], -33/2] + 168035*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/
2])/656250

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Maple [C]  time = 0.029, size = 282, normalized size = 1.3 \[ -{\frac{1}{3937500\,{x}^{2}+656250\,x-1312500} \left ( 840175\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+256430\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,{x}^{6}+504105\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +153858\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}-12631500\,{x}^{4}-18488250\,{x}^{3}-1794020\,{x}^{2}+3800330\,x+1041340 \right ) \sqrt{2+3\,x}\sqrt{1-2\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/656250*(840175*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)+256430*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)[phone]*x^6+504105*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))+153858*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))[phone]*x^5-12631500*x^4-18488250*x^3-1794020*x^2+3800330*x+1041340)*(2
+3*x)^(1/2)*(1-2*x)^(1/2)/(6*x^2+x-2)/(3+5*x)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*sqrt(3*x + 2)*sqrt(-2*x + 1)/((2
5*x^2 + 30*x + 9)*sqrt(5*x + 3)), 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)*(2+3*x)**(7/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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