3.2938 \(\int \frac{(2+3 x)^{5/2} (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=191 \[ \frac{(5 x+3)^{3/2} (3 x+2)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{45 (5 x+3)^{3/2} (3 x+2)^{3/2}}{11 \sqrt{1-2 x}}-\frac{807}{110} \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}-\frac{6231}{110} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}-\frac{2077}{50} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{37663}{100} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(-6231*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/110 - (807*Sqrt[1 - 2*x]*Sqrt[
2 + 3*x]*(3 + 5*x)^(3/2))/110 - (45*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))/(11*Sqrt[1
- 2*x]) + ((2 + 3*x)^(5/2)*(3 + 5*x)^(3/2))/(3*(1 - 2*x)^(3/2)) - (37663*Sqrt[11
/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/100 - (2077*Sqrt[3/11]*El
lipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/50

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Rubi [A]  time = 0.395255, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{(5 x+3)^{3/2} (3 x+2)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{45 (5 x+3)^{3/2} (3 x+2)^{3/2}}{11 \sqrt{1-2 x}}-\frac{807}{110} \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}-\frac{6231}{110} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}-\frac{2077}{50} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{37663}{100} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-6231*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/110 - (807*Sqrt[1 - 2*x]*Sqrt[
2 + 3*x]*(3 + 5*x)^(3/2))/110 - (45*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))/(11*Sqrt[1
- 2*x]) + ((2 + 3*x)^(5/2)*(3 + 5*x)^(3/2))/(3*(1 - 2*x)^(3/2)) - (37663*Sqrt[11
/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/100 - (2077*Sqrt[3/11]*El
lipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/50

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Rubi in Sympy [A]  time = 38.1935, size = 170, normalized size = 0.89 \[ - \frac{163 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{14} - \frac{271 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{5} - \frac{37663 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{300} - \frac{2077 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{550} - \frac{45 \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{7 \sqrt{- 2 x + 1}} + \frac{\left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-163*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/14 - 271*sqrt(-2*x + 1)*sqrt(
3*x + 2)*sqrt(5*x + 3)/5 - 37663*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1
)/7), 35/33)/300 - 2077*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/
33)/550 - 45*(3*x + 2)**(5/2)*sqrt(5*x + 3)/(7*sqrt(-2*x + 1)) + (3*x + 2)**(5/2
)*(5*x + 3)**(3/2)/(3*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.317014, size = 125, normalized size = 0.65 \[ -\frac{10 \sqrt{3 x+2} \sqrt{5 x+3} \left (270 x^3+1344 x^2-8039 x+2976\right )-18970 \sqrt{2-4 x} (2 x-1) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+37663 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{300 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(10*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(2976 - 8039*x + 1344*x^2 + 270*x^3) + 37663*Sq
rt[2 - 4*x]*(-1 + 2*x)*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 1897
0*Sqrt[2 - 4*x]*(-1 + 2*x)*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(
300*(1 - 2*x)^(3/2))

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Maple [C]  time = 0.029, size = 286, normalized size = 1.5 \[{\frac{1}{300\, \left ( -1+2\,x \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 37940\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-75326\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-18970\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +37663\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -40500\,{x}^{5}-252900\,{x}^{4}+934290\,{x}^{3}+1000370\,{x}^{2}-83100\,x-178560 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/300*(37940*2^(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))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-75326*2^(1/2)*Ell
ipticE(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))*x*(3+
5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-18970*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))+37663*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))-40500*x^5-252
900*x^4+934290*x^3+1000370*x^2-83100*x-178560)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*
x)^(1/2)/(-1+2*x)^2/(15*x^2+19*x+6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((45*x^3 + 87*x^2 + 56*x + 12)*sqrt(5*x + 3)*sqrt(3*x + 2)/((4*x^2 - 4*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)**(5/2)*(3+5*x)**(3/2)/(1-2*x)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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