3.2769 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^{5/2}}{\sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=218 \[ \frac{2}{55} (1-2 x)^{5/2} \sqrt{5 x+3} (3 x+2)^{5/2}+\frac{62 (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^{5/2}}{1485}+\frac{4258 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}}{155925}+\frac{181333 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{3898125}-\frac{2865161 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{19490625}-\frac{3963068 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8859375 \sqrt{33}}-\frac{231061879 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{17718750 \sqrt{33}} \]

[Out]

(-2865161*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/19490625 + (181333*Sqrt[1 -
2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/3898125 + (4258*Sqrt[1 - 2*x]*(2 + 3*x)^(5/
2)*Sqrt[3 + 5*x])/155925 + (62*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/14
85 + (2*(1 - 2*x)^(5/2)*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/55 - (231061879*EllipticE
[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(17718750*Sqrt[33]) - (3963068*Ellipti
cF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(8859375*Sqrt[33])

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Rubi [A]  time = 0.4873, 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 \[ \frac{2}{55} (1-2 x)^{5/2} \sqrt{5 x+3} (3 x+2)^{5/2}+\frac{62 (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^{5/2}}{1485}+\frac{4258 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}}{155925}+\frac{181333 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{3898125}-\frac{2865161 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{19490625}-\frac{3963068 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8859375 \sqrt{33}}-\frac{231061879 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{17718750 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-2865161*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/19490625 + (181333*Sqrt[1 -
2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/3898125 + (4258*Sqrt[1 - 2*x]*(2 + 3*x)^(5/
2)*Sqrt[3 + 5*x])/155925 + (62*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/14
85 + (2*(1 - 2*x)^(5/2)*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/55 - (231061879*EllipticE
[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(17718750*Sqrt[33]) - (3963068*Ellipti
cF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(8859375*Sqrt[33])

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Rubi in Sympy [A]  time = 50.4115, size = 201, normalized size = 0.92 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{55} + \frac{62 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{1485} + \frac{4258 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{155925} + \frac{181333 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{3898125} - \frac{2865161 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{19490625} - \frac{231061879 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{584718750} - \frac{3963068 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{292359375} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*x + 1)**(5/2)*(3*x + 2)**(5/2)*sqrt(5*x + 3)/55 + 62*(-2*x + 1)**(3/2)*(3*
x + 2)**(5/2)*sqrt(5*x + 3)/1485 + 4258*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*sqrt(5*x
+ 3)/155925 + 181333*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/3898125 - 28
65161*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/19490625 - 231061879*sqrt(33)*e
lliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/584718750 - 3963068*sqrt(33)*e
lliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/292359375

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Mathematica [A]  time = 0.510876, size = 107, normalized size = 0.49 \[ \frac{15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (25515000 x^4-6142500 x^3-23717250 x^2+9526995 x+7167169\right )-100280635 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{292359375 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(7167169 + 9526995*x - 23717250*x^
2 - 6142500*x^3 + 25515000*x^4) + 231061879*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]], -33/2] - 100280635*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(
292359375*Sqrt[2])

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Maple [C]  time = 0.017, size = 184, normalized size = 0.8 \[{\frac{1}{17541562500\,{x}^{3}[phone]\,{x}^{2}-4093031250\,x-3508312500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 22963500000\,{x}^{7}[phone]\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -231061879\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -30942000000\,{x}^{5}-11093382000\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-3219964590\,x-1290090420 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/584718750*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(22963500000*x^7+120771000
00*x^6+100280635*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))-231061879*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))-30942000000*x^5-11093382000*x^4+19110
351150*x^3+7213782660*x^2-3219964590*x-1290090420)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{\sqrt{5 \, x + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*sqrt(3*x + 2)*sqrt(-2*x + 1)/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)**(5/2)*(2+3*x)**(5/2)/(3+5*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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