3.2893 \(\int \frac{(2+3 x)^{3/2} (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=188 \[ \frac{(3 x+2)^{3/2} (5 x+3)^{5/2}}{\sqrt{1-2 x}}+\frac{12}{7} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{167}{14} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}+\frac{3683}{42} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}+\frac{3683}{210} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{244879}{420} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(3683*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/42 + (167*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x]*(3 + 5*x)^(3/2))/14 + (12*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/7
+ ((2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/Sqrt[1 - 2*x] + (244879*Sqrt[11/3]*EllipticE
[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/420 + (3683*Sqrt[11/3]*EllipticF[ArcSi
n[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/210

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Rubi [A]  time = 0.390352, antiderivative size = 188, 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 \[ \frac{(3 x+2)^{3/2} (5 x+3)^{5/2}}{\sqrt{1-2 x}}+\frac{12}{7} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{167}{14} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}+\frac{3683}{42} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}+\frac{3683}{210} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{244879}{420} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(3683*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/42 + (167*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x]*(3 + 5*x)^(3/2))/14 + (12*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/7
+ ((2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/Sqrt[1 - 2*x] + (244879*Sqrt[11/3]*EllipticE
[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/420 + (3683*Sqrt[11/3]*EllipticF[ArcSi
n[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/210

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Rubi in Sympy [A]  time = 38.8181, size = 168, normalized size = 0.89 \[ \frac{12 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{5}{2}}}{7} + \frac{167 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{14} + \frac{3683 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{42} + \frac{244879 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1260} + \frac{3683 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{630} + \frac{\left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{\sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

12*sqrt(-2*x + 1)*sqrt(3*x + 2)*(5*x + 3)**(5/2)/7 + 167*sqrt(-2*x + 1)*sqrt(3*x
+ 2)*(5*x + 3)**(3/2)/14 + 3683*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/42 +
244879*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/1260 + 3683*
sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/630 + (3*x + 2)**(3/
2)*(5*x + 3)**(5/2)/sqrt(-2*x + 1)

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Mathematica [A]  time = 0.278442, size = 115, normalized size = 0.61 \[ \frac{-30 \sqrt{3 x+2} \sqrt{5 x+3} \left (450 x^3+1650 x^2+3349 x-6590\right )+123340 \sqrt{2-4 x} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-244879 \sqrt{2-4 x} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{1260 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(-30*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-6590 + 3349*x + 1650*x^2 + 450*x^3) - 244879*
Sqrt[2 - 4*x]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 123340*Sqrt[2
- 4*x]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(1260*Sqrt[1 - 2*x])

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Maple [C]  time = 0.026, size = 174, normalized size = 0.9 \[ -{\frac{1}{37800\,{x}^{3}+28980\,{x}^{2}-8820\,x-7560}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 123340\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -244879\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -202500\,{x}^{5}-999000\,{x}^{4}-2528550\,{x}^{3}+759570\,{x}^{2}+3153480\,x+1186200 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1260*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(123340*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))-244879*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))
-202500*x^5-999000*x^4-2528550*x^3+759570*x^2+3153480*x+1186200)/(30*x^3+23*x^2-
7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}{{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(75*x^3 + 140*x^2 + 87*x + 18)*sqrt(5*x + 3)*sqrt(3*x + 2)/((2*x - 1)*
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((2+3*x)**(3/2)*(3+5*x)**(5/2)/(1-2*x)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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