3.2703 \(\int (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=218 \[ \frac{2}{55} (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{7/2}+\frac{194 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}}{7425}-\frac{2377 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{155925}-\frac{22576 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{155925}-\frac{2930159 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{2806650}-\frac{2930159 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1275750 \sqrt{33}}-\frac{97540001 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1275750 \sqrt{33}} \]

[Out]

(-2930159*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/2806650 - (22576*Sqrt[1 - 2
*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/155925 - (2377*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3
+ 5*x)^(5/2))/155925 + (194*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(7/2))/7425 +
(2*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(7/2))/55 - (97540001*EllipticE[ArcS
in[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1275750*Sqrt[33]) - (2930159*EllipticF[Arc
Sin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1275750*Sqrt[33])

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Rubi [A]  time = 0.486633, 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 \[ \frac{2}{55} (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{7/2}+\frac{194 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}}{7425}-\frac{2377 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{155925}-\frac{22576 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{155925}-\frac{2930159 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{2806650}-\frac{2930159 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1275750 \sqrt{33}}-\frac{97540001 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1275750 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-2930159*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/2806650 - (22576*Sqrt[1 - 2
*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/155925 - (2377*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3
+ 5*x)^(5/2))/155925 + (194*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(7/2))/7425 +
(2*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(7/2))/55 - (97540001*EllipticE[ArcS
in[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1275750*Sqrt[33]) - (2930159*EllipticF[Arc
Sin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1275750*Sqrt[33])

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Rubi in Sympy [A]  time = 46.3733, size = 201, normalized size = 0.92 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{33} - \frac{115 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{891} + \frac{1228 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{6237} - \frac{19861 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{155925} - \frac{2930159 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{2806650} - \frac{97540001 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{42099750} - \frac{2930159 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{42099750} \]

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)**(1/2),x)

[Out]

2*(-2*x + 1)**(3/2)*(3*x + 2)**(3/2)*(5*x + 3)**(5/2)/33 - 115*(-2*x + 1)**(3/2)
*(3*x + 2)**(3/2)*(5*x + 3)**(3/2)/891 + 1228*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*(5
*x + 3)**(3/2)/6237 - 19861*sqrt(-2*x + 1)*sqrt(3*x + 2)*(5*x + 3)**(3/2)/155925
- 2930159*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/2806650 - 97540001*sqrt(33
)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/42099750 - 2930159*sqrt(33)
*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/42099750

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Mathematica [A]  time = 0.368606, size = 107, normalized size = 0.49 \[ \frac{-15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (25515000 x^4+24003000 x^3-10837350 x^2-14851260 x-201247\right )-98384755 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{42099750 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(-15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-201247 - 14851260*x - 10837350*
x^2 + 24003000*x^3 + 25515000*x^4) + 195080002*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[
3 + 5*x]], -33/2] - 98384755*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])
/(42099750*Sqrt[2])

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Maple [C]  time = 0.016, size = 184, normalized size = 0.8 \[{\frac{1}{2525985000\,{x}^{3}[phone]\,{x}^{2}-589396500\,x-505197000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -22963500000\,{x}^{7}-39208050000\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -195080002\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -1450305000\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}-4930627170\,{x}^{2}-2715488670\,x-36224460 \right ) } \]

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)^(1/2),x)

[Out]

1/84199500*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(-22963500000*x^7-392080500
00*x^6+98384755*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))-195080002*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))-1450305000*x^5+30477235500*x^4+1247318
8200*x^3-4930627170*x^2-2715488670*x-36224460)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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