3.2704 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{\sqrt{2+3 x}} \, dx\)

Optimal. Leaf size=191 \[ \frac{2}{27} (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{46}{567} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{499 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{2835}-\frac{11908 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{25515}-\frac{11908 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{127575}-\frac{886499 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{255150} \]

[Out]

(-11908*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/25515 - (499*Sqrt[1 - 2*x]*Sq
rt[2 + 3*x]*(3 + 5*x)^(3/2))/2835 + (46*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5
/2))/567 + (2*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/27 - (886499*Sqrt[1
1/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/255150 - (11908*Sqrt[11/
3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/127575

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Rubi [A]  time = 0.409541, 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 \[ \frac{2}{27} (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{46}{567} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{499 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{2835}-\frac{11908 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{25515}-\frac{11908 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{127575}-\frac{886499 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{255150} \]

Antiderivative was successfully verified.

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

[Out]

(-11908*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/25515 - (499*Sqrt[1 - 2*x]*Sq
rt[2 + 3*x]*(3 + 5*x)^(3/2))/2835 + (46*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5
/2))/567 + (2*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/27 - (886499*Sqrt[1
1/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/255150 - (11908*Sqrt[11/
3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/127575

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Rubi in Sympy [A]  time = 39.365, size = 172, normalized size = 0.9 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{5}{2}}}{27} - \frac{115 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{567} + \frac{766 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{2835} - \frac{11908 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{25515} - \frac{886499 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{765450} - \frac{11908 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{382725} \]

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)*sqrt(3*x + 2)*(5*x + 3)**(5/2)/27 - 115*(-2*x + 1)**(3/2)*sq
rt(3*x + 2)*(5*x + 3)**(3/2)/567 + 766*sqrt(-2*x + 1)*sqrt(3*x + 2)*(5*x + 3)**(
3/2)/2835 - 11908*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/25515 - 886499*sqrt
(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/765450 - 11908*sqrt(33)*
elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/382725

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Mathematica [A]  time = 0.357475, size = 105, normalized size = 0.55 \[ \frac{886499 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (94500 x^3+14400 x^2-62325 x-10259\right )+98707 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{382725 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(886499*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]*(-10259 - 62325*x + 14400*x^2 + 94500*x^3) + 98707*E
llipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/(382725*Sqrt[2])

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Maple [C]  time = 0.017, size = 179, normalized size = 0.9 \[{\frac{1}{22963500\,{x}^{3}[phone]\,{x}^{2}-5358150\,x-4592700}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -85050000\,{x}^{6}+493535\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -886499\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -78165000\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}-3417540\,{x}^{2}-13372890\,x-1846620 \right ) } \]

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]

1/765450*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(-85050000*x^6+493535*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))-886499*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))-78165000*x^5+66001500*x^4+72271350*x^3-3417540*x^2-13372890*x-18
46620)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(50*x^3 + 35*x^2 - 12*x - 9)*sqrt(5*x + 3)*sqrt(-2*x + 1)/sqrt(3*x + 2
), 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. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{\sqrt{3 \, x + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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