3.2693 \(\int (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{3/2} \, dx\)

Optimal. Leaf size=191 \[ \frac{2}{45} (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{194 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{4725}-\frac{839 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{23625}-\frac{76163 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{212625}-\frac{76163 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1063125}-\frac{4971289 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2126250} \]

[Out]

(-76163*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/212625 - (839*Sqrt[1 - 2*x]*S
qrt[2 + 3*x]*(3 + 5*x)^(3/2))/23625 + (194*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)
^(5/2))/4725 + (2*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/45 - (4971289*S
qrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2126250 - (76163*Sq
rt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1063125

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Rubi [A]  time = 0.404634, 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 \[ \frac{2}{45} (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{194 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{4725}-\frac{839 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{23625}-\frac{76163 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{212625}-\frac{76163 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1063125}-\frac{4971289 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2126250} \]

Antiderivative was successfully verified.

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

[Out]

(-76163*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/212625 - (839*Sqrt[1 - 2*x]*S
qrt[2 + 3*x]*(3 + 5*x)^(3/2))/23625 + (194*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)
^(5/2))/4725 + (2*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/45 - (4971289*S
qrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2126250 - (76163*Sq
rt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1063125

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Rubi in Sympy [A]  time = 38.3437, size = 172, normalized size = 0.9 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{27} - \frac{37 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{189} + \frac{4126 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{14175} - \frac{70226 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{212625} - \frac{4971289 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{6378750} - \frac{76163 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{3189375} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*x + 1)**(3/2)*(3*x + 2)**(3/2)*(5*x + 3)**(3/2)/27 - 37*(-2*x + 1)**(3/2)*
(3*x + 2)**(3/2)*sqrt(5*x + 3)/189 + 4126*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5
*x + 3)/14175 - 70226*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/212625 - 497128
9*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/6378750 - 76163*sq
rt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/3189375

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Mathematica [A]  time = 0.342226, size = 105, normalized size = 0.55 \[ \frac{4971289 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (472500 x^3+112500 x^2-337545 x-64804\right )+491582 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{3189375 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(4971289*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 5*(3*Sqrt[2 - 4*x]
*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-64804 - 337545*x + 112500*x^2 + 472500*x^3) + 491
582*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/(3189375*Sqrt[2])

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Maple [C]  time = 0.017, size = 179, normalized size = 0.9 \[{\frac{1}{191362500\,{x}^{3}[phone]\,{x}^{2}-44651250\,x-38272500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -425250000\,{x}^{6}+2457910\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -4971289\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -427275000\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}-5919690\,{x}^{2}-74366940\,x-11664720 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6378750*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(-425250000*x^6+2457910*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))-4971289*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))-427275000*x^5+325390500*x^4+399904650*x^3-5919690*x^2-743669
40*x-11664720)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-{\left (10 \, x^{2} + x - 3\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(10*x^2 + x - 3)*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)**(3/2)*(2+3*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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