3.2862 \(\int \frac{\sqrt{2+3 x}}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=125 \[ -\frac{74 \sqrt{1-2 x} \sqrt{3 x+2}}{363 \sqrt{5 x+3}}-\frac{2 \sqrt{1-2 x} \sqrt{3 x+2}}{33 (5 x+3)^{3/2}}-\frac{4 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{55 \sqrt{33}}+\frac{74 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{55 \sqrt{33}} \]

[Out]

(-2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(33*(3 + 5*x)^(3/2)) - (74*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x])/(363*Sqrt[3 + 5*x]) + (74*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35
/33])/(55*Sqrt[33]) - (4*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(55*
Sqrt[33])

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Rubi [A]  time = 0.269414, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{74 \sqrt{1-2 x} \sqrt{3 x+2}}{363 \sqrt{5 x+3}}-\frac{2 \sqrt{1-2 x} \sqrt{3 x+2}}{33 (5 x+3)^{3/2}}-\frac{4 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{55 \sqrt{33}}+\frac{74 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{55 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(33*(3 + 5*x)^(3/2)) - (74*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x])/(363*Sqrt[3 + 5*x]) + (74*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35
/33])/(55*Sqrt[33]) - (4*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(55*
Sqrt[33])

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Rubi in Sympy [A]  time = 26.3493, size = 114, normalized size = 0.91 \[ - \frac{74 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{363 \sqrt{5 x + 3}} - \frac{2 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{33 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{74 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1815} - \frac{4 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{1925} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-74*sqrt(-2*x + 1)*sqrt(3*x + 2)/(363*sqrt(5*x + 3)) - 2*sqrt(-2*x + 1)*sqrt(3*x
+ 2)/(33*(5*x + 3)**(3/2)) + 74*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1
)/7), 35/33)/1815 - 4*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/3
5)/1925

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Mathematica [A]  time = 0.399205, size = 97, normalized size = 0.78 \[ \frac{2 \left (-\frac{5 \sqrt{1-2 x} \sqrt{3 x+2} (185 x+122)}{(5 x+3)^{3/2}}+70 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-37 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{1815} \]

Antiderivative was successfully verified.

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

[Out]

(2*((-5*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(122 + 185*x))/(3 + 5*x)^(3/2) - 37*Sqrt[2]*
EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 70*Sqrt[2]*EllipticF[ArcSin
[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/1815

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Maple [C]  time = 0.031, size = 267, normalized size = 2.1 \[ -{\frac{2}{10890\,{x}^{2}+1815\,x-3630} \left ( 350\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-185\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+210\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -111\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +5550\,{x}^{3}+4585\,{x}^{2}-1240\,x-1220 \right ) \sqrt{1-2\,x}\sqrt{2+3\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/1815*(350*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)-185*2^(1/2)*Ellip
ticE(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)+210*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))-111*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))+5550*x^3+4585*x^2-1
240*x-1220)*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(6*x^2+x-2)/(3+5*x)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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