3.2979 \(\int \frac{\sqrt{2+3 x}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=187 \[ -\frac{22090 \sqrt{1-2 x} \sqrt{3 x+2}}{307461 \sqrt{5 x+3}}-\frac{2470 \sqrt{1-2 x} \sqrt{3 x+2}}{27951 (5 x+3)^{3/2}}+\frac{118 \sqrt{3 x+2}}{847 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{2 \sqrt{3 x+2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{988 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{9317 \sqrt{33}}+\frac{4418 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{9317 \sqrt{33}} \]

[Out]

(2*Sqrt[2 + 3*x])/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + (118*Sqrt[2 + 3*x])/(84
7*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (2470*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(27951*(3
+ 5*x)^(3/2)) - (22090*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(307461*Sqrt[3 + 5*x]) + (44
18*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(9317*Sqrt[33]) - (988*Ell
ipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(9317*Sqrt[33])

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Rubi [A]  time = 0.429626, antiderivative size = 187, 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 \[ -\frac{22090 \sqrt{1-2 x} \sqrt{3 x+2}}{307461 \sqrt{5 x+3}}-\frac{2470 \sqrt{1-2 x} \sqrt{3 x+2}}{27951 (5 x+3)^{3/2}}+\frac{118 \sqrt{3 x+2}}{847 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{2 \sqrt{3 x+2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{988 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{9317 \sqrt{33}}+\frac{4418 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{9317 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[2 + 3*x])/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + (118*Sqrt[2 + 3*x])/(84
7*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (2470*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(27951*(3
+ 5*x)^(3/2)) - (22090*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(307461*Sqrt[3 + 5*x]) + (44
18*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(9317*Sqrt[33]) - (988*Ell
ipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(9317*Sqrt[33])

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Rubi in Sympy [A]  time = 38.5524, size = 172, normalized size = 0.92 \[ \frac{4418 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{307461} - \frac{988 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{307461} + \frac{8836 \sqrt{3 x + 2} \sqrt{5 x + 3}}{307461 \sqrt{- 2 x + 1}} - \frac{490 \sqrt{3 x + 2}}{3993 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} - \frac{20 \sqrt{3 x + 2}}{363 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{2 \sqrt{3 x + 2}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4418*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/307461 - 988*sq
rt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/307461 + 8836*sqrt(3*x
+ 2)*sqrt(5*x + 3)/(307461*sqrt(-2*x + 1)) - 490*sqrt(3*x + 2)/(3993*sqrt(-2*x
+ 1)*sqrt(5*x + 3)) - 20*sqrt(3*x + 2)/(363*sqrt(-2*x + 1)*(5*x + 3)**(3/2)) + 2
*sqrt(3*x + 2)/(33*(-2*x + 1)**(3/2)*(5*x + 3)**(3/2))

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Mathematica [A]  time = 0.28685, size = 103, normalized size = 0.55 \[ \frac{2 \left (\frac{\sqrt{3 x+2} \left (-220900 x^3+34020 x^2+88821 x-15986\right )}{(1-2 x)^{3/2} (5 x+3)^{3/2}}+\sqrt{2} \left (10360 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-2209 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{307461} \]

Antiderivative was successfully verified.

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

[Out]

(2*((Sqrt[2 + 3*x]*(-15986 + 88821*x + 34020*x^2 - 220900*x^3))/((1 - 2*x)^(3/2)
*(3 + 5*x)^(3/2)) + Sqrt[2]*(-2209*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -
33/2] + 10360*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/307461

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Maple [C]  time = 0.035, size = 383, normalized size = 2.1 \[ -{\frac{2}{307461\, \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 103600\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-22090\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+10360\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-2209\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-31080\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +6627\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +662700\,{x}^{4}+339740\,{x}^{3}-334503\,{x}^{2}-129684\,x+31972 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{2+3\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/307461*(1-2*x)^(1/2)*(103600*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^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1
/2)-22090*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^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)+10360*2^(1/2)*Elli
pticF(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)-2209*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)-31080*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))+6627*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))+662700*x^4+339740*x^3-334503*x^2-129684*
x+31972)/(3+5*x)^(3/2)/(-1+2*x)^2/(2+3*x)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{3 \, x + 2}}{{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(3*x + 2)/((100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*sqrt(5*x + 3)*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)**(1/2)/(1-2*x)**(5/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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