3.2935 \(\int \frac{\sqrt{3+5 x}}{(1-2 x)^{5/2} (2+3 x)^{5/2}} \, dx\)

Optimal. Leaf size=187 \[ \frac{338 \sqrt{1-2 x} \sqrt{5 x+3}}{26411 \sqrt{3 x+2}}-\frac{458 \sqrt{1-2 x} \sqrt{5 x+3}}{3773 (3 x+2)^{3/2}}+\frac{326 \sqrt{5 x+3}}{1617 \sqrt{1-2 x} (3 x+2)^{3/2}}+\frac{2 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^{3/2}}-\frac{992 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2401 \sqrt{33}}-\frac{338 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2401 \sqrt{33}} \]

[Out]

(2*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)) + (326*Sqrt[3 + 5*x])/(16
17*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)) - (458*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3773*(2 +
3*x)^(3/2)) + (338*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(26411*Sqrt[2 + 3*x]) - (338*El
lipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(2401*Sqrt[33]) - (992*Elliptic
F[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(2401*Sqrt[33])

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Rubi [A]  time = 0.424335, 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 \[ \frac{338 \sqrt{1-2 x} \sqrt{5 x+3}}{26411 \sqrt{3 x+2}}-\frac{458 \sqrt{1-2 x} \sqrt{5 x+3}}{3773 (3 x+2)^{3/2}}+\frac{326 \sqrt{5 x+3}}{1617 \sqrt{1-2 x} (3 x+2)^{3/2}}+\frac{2 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^{3/2}}-\frac{992 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2401 \sqrt{33}}-\frac{338 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2401 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)) + (326*Sqrt[3 + 5*x])/(16
17*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)) - (458*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3773*(2 +
3*x)^(3/2)) + (338*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(26411*Sqrt[2 + 3*x]) - (338*El
lipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(2401*Sqrt[33]) - (992*Elliptic
F[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(2401*Sqrt[33])

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Rubi in Sympy [A]  time = 37.3234, size = 172, normalized size = 0.92 \[ \frac{338 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{26411 \sqrt{3 x + 2}} - \frac{338 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{79233} - \frac{992 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{79233} + \frac{916 \sqrt{5 x + 3}}{11319 \sqrt{- 2 x + 1} \sqrt{3 x + 2}} - \frac{4 \sqrt{5 x + 3}}{49 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{2 \sqrt{5 x + 3}}{21 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

338*sqrt(-2*x + 1)*sqrt(5*x + 3)/(26411*sqrt(3*x + 2)) - 338*sqrt(33)*elliptic_e
(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/79233 - 992*sqrt(33)*elliptic_f(asin(sq
rt(21)*sqrt(-2*x + 1)/7), 35/33)/79233 + 916*sqrt(5*x + 3)/(11319*sqrt(-2*x + 1)
*sqrt(3*x + 2)) - 4*sqrt(5*x + 3)/(49*sqrt(-2*x + 1)*(3*x + 2)**(3/2)) + 2*sqrt(
5*x + 3)/(21*(-2*x + 1)**(3/2)*(3*x + 2)**(3/2))

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Mathematica [A]  time = 0.27618, size = 103, normalized size = 0.55 \[ \frac{2 \left (\frac{\sqrt{5 x+3} \left (6084 x^3-21264 x^2+727 x+7965\right )}{(1-2 x)^{3/2} (3 x+2)^{3/2}}+\sqrt{2} \left (8015 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+169 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{79233} \]

Antiderivative was successfully verified.

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

[Out]

(2*((Sqrt[3 + 5*x]*(7965 + 727*x - 21264*x^2 + 6084*x^3))/((1 - 2*x)^(3/2)*(2 +
3*x)^(3/2)) + Sqrt[2]*(169*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] +
8015*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/79233

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Maple [C]  time = 0.034, size = 383, normalized size = 2.1 \[ -{\frac{2}{79233\, \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 48090\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+1014\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+8015\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+169\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-16030\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -338\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -30420\,{x}^{4}+88068\,{x}^{3}+60157\,{x}^{2}-42006\,x-23895 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/79233*(1-2*x)^(1/2)*(48090*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
)+1014*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)+8015*2^(1/2)*Elliptic
F(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)+169*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)-16030*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/11*1
1^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))-338*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))-30420*x^4+88068*x^3+60157*x^2-42006*x-23895)/(
2+3*x)^(3/2)/(-1+2*x)^2/(3+5*x)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{5 \, x + 3}}{{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(5*x + 3)/((36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*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)**(1/2)/(1-2*x)**(5/2)/(2+3*x)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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