3.2738 \(\int (1-2 x)^{5/2} (2+3 x)^{3/2} \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=218 \[ \frac{2}{55} (3 x+2)^{3/2} (5 x+3)^{3/2} (1-2 x)^{5/2}+\frac{106 (3 x+2)^{3/2} (5 x+3)^{3/2} (1-2 x)^{3/2}}{2475}+\frac{2866 (3 x+2)^{3/2} (5 x+3)^{3/2} \sqrt{1-2 x}}{86625}+\frac{38729 \sqrt{3 x+2} (5 x+3)^{3/2} \sqrt{1-2 x}}{2165625}-\frac{4738087 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{19490625}-\frac{4738087 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8859375 \sqrt{33}}-\frac{326256461 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{17718750 \sqrt{33}} \]

[Out]

(-4738087*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/19490625 + (38729*Sqrt[1 -
2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/2165625 + (2866*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2
)*(3 + 5*x)^(3/2))/86625 + (106*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))
/2475 + (2*(1 - 2*x)^(5/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))/55 - (326256461*Elli
pticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(17718750*Sqrt[33]) - (4738087*El
lipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(8859375*Sqrt[33])

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Rubi [A]  time = 0.460922, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{2}{55} (3 x+2)^{3/2} (5 x+3)^{3/2} (1-2 x)^{5/2}+\frac{106 (3 x+2)^{3/2} (5 x+3)^{3/2} (1-2 x)^{3/2}}{2475}+\frac{2866 (3 x+2)^{3/2} (5 x+3)^{3/2} \sqrt{1-2 x}}{86625}+\frac{38729 \sqrt{3 x+2} (5 x+3)^{3/2} \sqrt{1-2 x}}{2165625}-\frac{4738087 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{19490625}-\frac{4738087 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8859375 \sqrt{33}}-\frac{326256461 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{17718750 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-4738087*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/19490625 + (38729*Sqrt[1 -
2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/2165625 + (2866*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2
)*(3 + 5*x)^(3/2))/86625 + (106*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))
/2475 + (2*(1 - 2*x)^(5/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))/55 - (326256461*Elli
pticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(17718750*Sqrt[33]) - (4738087*El
lipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(8859375*Sqrt[33])

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Rubi in Sympy [A]  time = 47.0519, size = 201, normalized size = 0.92 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{33} - \frac{59 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{495} - \frac{1324 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{3 x + 2} \sqrt{5 x + 3}}{5775} + \frac{10457 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \sqrt{5 x + 3}}{48125} + \frac{9592361 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{19490625} - \frac{326256461 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{584718750} - \frac{4738087 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{292359375} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*x + 1)**(5/2)*(3*x + 2)**(5/2)*sqrt(5*x + 3)/33 - 59*(-2*x + 1)**(5/2)*(3*
x + 2)**(3/2)*sqrt(5*x + 3)/495 - 1324*(-2*x + 1)**(5/2)*sqrt(3*x + 2)*sqrt(5*x
+ 3)/5775 + 10457*(-2*x + 1)**(3/2)*sqrt(3*x + 2)*sqrt(5*x + 3)/48125 + 9592361*
sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/19490625 - 326256461*sqrt(33)*ellipti
c_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/584718750 - 4738087*sqrt(33)*ellipti
c_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/292359375

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Mathematica [A]  time = 0.365066, size = 107, normalized size = 0.49 \[ \frac{15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (42525000 x^4-13702500 x^3-35750250 x^2+16294455 x+9437696\right )-169899590 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{292359375 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(9437696 + 16294455*x - 35750250*x
^2 - 13702500*x^3 + 42525000*x^4) + 326256461*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3
+ 5*x]], -33/2] - 169899590*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])
/(292359375*Sqrt[2])

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Maple [C]  time = 0.016, size = 184, normalized size = 0.8 \[{\frac{1}{17541562500\,{x}^{3}[phone]\,{x}^{2}-4093031250\,x-3508312500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 38272500000\,{x}^{7}[phone]\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -326256461\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -50560200000\,{x}^{5}-14779638000\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-4914918060\,x-1698785280 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/584718750*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(38272500000*x^7+170100000
00*x^6+169899590*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/1
1*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))-326256461*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))-50560200000*x^5-14779638000*x^4+29711
102850*x^3+9525219690*x^2-4914918060*x-1698785280)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((12*x^3 - 4*x^2 - 5*x + 2)*sqrt(5*x + 3)*sqrt(3*x + 2)*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((1-2*x)**(5/2)*(2+3*x)**(3/2)*(3+5*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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