3.2934 \(\int \frac{\sqrt{3+5 x}}{(1-2 x)^{5/2} (2+3 x)^{3/2}} \, dx\)

Optimal. Leaf size=156 \[ -\frac{458 \sqrt{1-2 x} \sqrt{5 x+3}}{3773 \sqrt{3 x+2}}+\frac{194 \sqrt{5 x+3}}{1617 \sqrt{1-2 x} \sqrt{3 x+2}}+\frac{2 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} \sqrt{3 x+2}}-\frac{178 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{343 \sqrt{33}}+\frac{458 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{343 \sqrt{33}} \]

[Out]

(2*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]) + (194*Sqrt[3 + 5*x])/(1617
*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]) - (458*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3773*Sqrt[2 +
3*x]) + (458*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(343*Sqrt[33])
- (178*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(343*Sqrt[33])

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Rubi [A]  time = 0.345364, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{458 \sqrt{1-2 x} \sqrt{5 x+3}}{3773 \sqrt{3 x+2}}+\frac{194 \sqrt{5 x+3}}{1617 \sqrt{1-2 x} \sqrt{3 x+2}}+\frac{2 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} \sqrt{3 x+2}}-\frac{178 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{343 \sqrt{33}}+\frac{458 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{343 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]) + (194*Sqrt[3 + 5*x])/(1617
*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]) - (458*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3773*Sqrt[2 +
3*x]) + (458*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(343*Sqrt[33])
- (178*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(343*Sqrt[33])

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Rubi in Sympy [A]  time = 30.1843, size = 143, normalized size = 0.92 \[ \frac{458 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{11319} - \frac{178 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{11319} + \frac{916 \sqrt{3 x + 2} \sqrt{5 x + 3}}{11319 \sqrt{- 2 x + 1}} - \frac{8 \sqrt{5 x + 3}}{49 \sqrt{- 2 x + 1} \sqrt{3 x + 2}} + \frac{2 \sqrt{5 x + 3}}{21 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

458*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/11319 - 178*sqrt
(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/11319 + 916*sqrt(3*x + 2
)*sqrt(5*x + 3)/(11319*sqrt(-2*x + 1)) - 8*sqrt(5*x + 3)/(49*sqrt(-2*x + 1)*sqrt
(3*x + 2)) + 2*sqrt(5*x + 3)/(21*(-2*x + 1)**(3/2)*sqrt(3*x + 2))

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Mathematica [A]  time = 0.196073, size = 99, normalized size = 0.63 \[ \frac{\sqrt{2} \left (3395 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-458 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )-\frac{2 \sqrt{5 x+3} \left (2748 x^2-1390 x-531\right )}{(1-2 x)^{3/2} \sqrt{3 x+2}}}{11319} \]

Antiderivative was successfully verified.

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

[Out]

((-2*Sqrt[3 + 5*x]*(-531 - 1390*x + 2748*x^2))/((1 - 2*x)^(3/2)*Sqrt[2 + 3*x]) +
Sqrt[2]*(-458*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 3395*Ellipti
cF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/11319

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Maple [C]  time = 0.033, size = 276, normalized size = 1.8 \[ -{\frac{1}{ \left ( 169785\,{x}^{2}+215061\,x+67914 \right ) \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 6790\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-916\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-3395\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +458\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +27480\,{x}^{3}+2588\,{x}^{2}-13650\,x-3186 \right ) } \]

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)^(3/2),x)

[Out]

-1/11319*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(6790*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)-916*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)
-3395*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))+458*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))+27480*x^3+2588*x^2-13650*x-3186)/(15*x^2+19*x+6)/(-1+2
*x)^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(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)**(1/2)/(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. \[ \int \frac{\sqrt{5 \, x + 3}}{{\left (3 \, x + 2\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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