3.2950 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{3/2}} \, dx\)

Optimal. Leaf size=158 \[ \frac{11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} \sqrt{3 x+2}}+\frac{438 \sqrt{1-2 x} \sqrt{5 x+3}}{343 \sqrt{3 x+2}}-\frac{143 \sqrt{5 x+3}}{49 \sqrt{1-2 x} \sqrt{3 x+2}}-\frac{17}{343} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{146}{343} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(-143*Sqrt[3 + 5*x])/(49*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]) + (438*Sqrt[1 - 2*x]*Sqrt[
3 + 5*x])/(343*Sqrt[2 + 3*x]) + (11*(3 + 5*x)^(3/2))/(21*(1 - 2*x)^(3/2)*Sqrt[2
+ 3*x]) - (146*Sqrt[33]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/343 -
(17*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/343

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Rubi [A]  time = 0.34163, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} \sqrt{3 x+2}}+\frac{438 \sqrt{1-2 x} \sqrt{5 x+3}}{343 \sqrt{3 x+2}}-\frac{143 \sqrt{5 x+3}}{49 \sqrt{1-2 x} \sqrt{3 x+2}}-\frac{17}{343} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{146}{343} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-143*Sqrt[3 + 5*x])/(49*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]) + (438*Sqrt[1 - 2*x]*Sqrt[
3 + 5*x])/(343*Sqrt[2 + 3*x]) + (11*(3 + 5*x)^(3/2))/(21*(1 - 2*x)^(3/2)*Sqrt[2
+ 3*x]) - (146*Sqrt[33]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/343 -
(17*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/343

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Rubi in Sympy [A]  time = 30.0254, size = 143, normalized size = 0.91 \[ - \frac{146 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{343} - \frac{187 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{12005} - \frac{292 \sqrt{3 x + 2} \sqrt{5 x + 3}}{343 \sqrt{- 2 x + 1}} + \frac{3 \sqrt{5 x + 3}}{49 \sqrt{- 2 x + 1} \sqrt{3 x + 2}} + \frac{11 \left (5 x + 3\right )^{\frac{3}{2}}}{21 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-146*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/343 - 187*sqrt(
35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/12005 - 292*sqrt(3*x + 2
)*sqrt(5*x + 3)/(343*sqrt(-2*x + 1)) + 3*sqrt(5*x + 3)/(49*sqrt(-2*x + 1)*sqrt(3
*x + 2)) + 11*(5*x + 3)**(3/2)/(21*(-2*x + 1)**(3/2)*sqrt(3*x + 2))

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Mathematica [A]  time = 0.230418, size = 102, normalized size = 0.65 \[ \frac{\frac{2 \sqrt{5 x+3} \left (5256 x^2+3445 x-72\right )}{(1-2 x)^{3/2} \sqrt{3 x+2}}-315 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+876 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{2058} \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[3 + 5*x]*(-72 + 3445*x + 5256*x^2))/((1 - 2*x)^(3/2)*Sqrt[2 + 3*x]) + 8
76*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 315*Sqrt[2]*Elli
pticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/2058

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Maple [C]  time = 0.034, size = 276, normalized size = 1.8 \[{\frac{1}{ \left ( 30870\,{x}^{2}+39102\,x+12348 \right ) \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 630\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-1752\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-315\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +876\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +52560\,{x}^{3}+65986\,{x}^{2}+19950\,x-432 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2058*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(630*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)-1752*2^(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))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-3
15*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))+876*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))+52560*x^3+65986*x^2+19950*x-432)/(15*x^2+19*x+6)/(-1+2*x)
^2

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

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)^(5/2)),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3}}{{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((25*x^2 + 30*x + 9)*sqrt(5*x + 3)/((12*x^3 - 4*x^2 - 5*x + 2)*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((3+5*x)**(5/2)/(1-2*x)**(5/2)/(2+3*x)**(3/2),x)

[Out]

Timed out

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

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)^(5/2)),x, algorithm="giac")

[Out]

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