3.2770 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^{3/2}}{\sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=191 \[ \frac{2}{45} (3 x+2)^{3/2} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{106 (3 x+2)^{3/2} \sqrt{5 x+3} (1-2 x)^{3/2}}{1575}+\frac{8878 (3 x+2)^{3/2} \sqrt{5 x+3} \sqrt{1-2 x}}{118125}+\frac{21547 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{1771875}-\frac{509189 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8859375}-\frac{8024546 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8859375} \]

[Out]

(21547*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/1771875 + (8878*Sqrt[1 - 2*x]*
(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/118125 + (106*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*Sqr
t[3 + 5*x])/1575 + (2*(1 - 2*x)^(5/2)*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/45 - (80245
46*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/8859375 - (5091
89*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/8859375

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Rubi [A]  time = 0.399975, 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 \[ \frac{2}{45} (3 x+2)^{3/2} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{106 (3 x+2)^{3/2} \sqrt{5 x+3} (1-2 x)^{3/2}}{1575}+\frac{8878 (3 x+2)^{3/2} \sqrt{5 x+3} \sqrt{1-2 x}}{118125}+\frac{21547 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{1771875}-\frac{509189 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8859375}-\frac{8024546 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8859375} \]

Antiderivative was successfully verified.

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

[Out]

(21547*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/1771875 + (8878*Sqrt[1 - 2*x]*
(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/118125 + (106*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*Sqr
t[3 + 5*x])/1575 + (2*(1 - 2*x)^(5/2)*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/45 - (80245
46*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/8859375 - (5091
89*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/8859375

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Rubi in Sympy [A]  time = 42.3839, size = 172, normalized size = 0.9 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{45} + \frac{106 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{1575} + \frac{8878 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{118125} + \frac{21547 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{1771875} - \frac{8024546 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{26578125} - \frac{509189 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{26578125} \]

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)**(3/2)*sqrt(5*x + 3)/45 + 106*(-2*x + 1)**(3/2)*(3
*x + 2)**(3/2)*sqrt(5*x + 3)/1575 + 8878*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*
x + 3)/118125 + 21547*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/1771875 - 80245
46*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/26578125 - 509189
*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/26578125

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Mathematica [A]  time = 0.33754, size = 102, normalized size = 0.53 \[ \frac{15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (945000 x^3-1030500 x^2-113490 x+683887\right )+754145 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{26578125 \sqrt{2}} \]

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]*(683887 - 113490*x - 1030500*x^2 +
945000*x^3) + 16049092*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 754
145*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(26578125*Sqrt[2])

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Maple [C]  time = 0.017, size = 179, normalized size = 0.9 \[ -{\frac{1}{1594687500\,{x}^{3}[phone]\,{x}^{2}-372093750\,x-318937500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -850500000\,{x}^{6}+754145\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}-583495200\,{x}^{3}-681204930\,{x}^{2}[phone]\,x+123099660 \right ) } \]

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]

-1/53156250*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(-850500000*x^6+754145*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))[phone]*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))[phone]*x^5+1011636000*x^4-583495200*x^3-681204930*x^2+1
23188070*x+123099660)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((12*x^3 - 4*x^2 - 5*x + 2)*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)**(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. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{\sqrt{5 \, x + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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