3.2680 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=187 \[ \frac{184840 \sqrt{1-2 x} \sqrt{3 x+2}}{231 \sqrt{5 x+3}}-\frac{2780 \sqrt{1-2 x} \sqrt{3 x+2}}{21 (5 x+3)^{3/2}}+\frac{416 \sqrt{1-2 x}}{21 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{2 \sqrt{1-2 x}}{3 (3 x+2)^{3/2} (5 x+3)^{3/2}}-\frac{1112 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7 \sqrt{33}}-\frac{36968 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7 \sqrt{33}} \]

[Out]

(2*Sqrt[1 - 2*x])/(3*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (416*Sqrt[1 - 2*x])/(21*
Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (2780*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(21*(3 + 5*x
)^(3/2)) + (184840*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(231*Sqrt[3 + 5*x]) - (36968*Ell
ipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(7*Sqrt[33]) - (1112*EllipticF[A
rcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(7*Sqrt[33])

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Rubi [A]  time = 0.425303, 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 \[ \frac{184840 \sqrt{1-2 x} \sqrt{3 x+2}}{231 \sqrt{5 x+3}}-\frac{2780 \sqrt{1-2 x} \sqrt{3 x+2}}{21 (5 x+3)^{3/2}}+\frac{416 \sqrt{1-2 x}}{21 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{2 \sqrt{1-2 x}}{3 (3 x+2)^{3/2} (5 x+3)^{3/2}}-\frac{1112 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7 \sqrt{33}}-\frac{36968 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 - 2*x])/(3*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (416*Sqrt[1 - 2*x])/(21*
Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (2780*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(21*(3 + 5*x
)^(3/2)) + (184840*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(231*Sqrt[3 + 5*x]) - (36968*Ell
ipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(7*Sqrt[33]) - (1112*EllipticF[A
rcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(7*Sqrt[33])

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Rubi in Sympy [A]  time = 38.1557, size = 172, normalized size = 0.92 \[ \frac{184840 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{231 \sqrt{5 x + 3}} - \frac{2780 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{21 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{416 \sqrt{- 2 x + 1}}{21 \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{2 \sqrt{- 2 x + 1}}{3 \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{36968 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{231} - \frac{1112 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{231} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

184840*sqrt(-2*x + 1)*sqrt(3*x + 2)/(231*sqrt(5*x + 3)) - 2780*sqrt(-2*x + 1)*sq
rt(3*x + 2)/(21*(5*x + 3)**(3/2)) + 416*sqrt(-2*x + 1)/(21*sqrt(3*x + 2)*(5*x +
3)**(3/2)) + 2*sqrt(-2*x + 1)/(3*(3*x + 2)**(3/2)*(5*x + 3)**(3/2)) - 36968*sqrt
(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/231 - 1112*sqrt(33)*elli
ptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/231

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Mathematica [A]  time = 0.324012, size = 104, normalized size = 0.56 \[ \frac{2}{231} \left (\frac{\sqrt{1-2 x} \left (4158900 x^3+7902930 x^2+4998904 x+1052533\right )}{(3 x+2)^{3/2} (5 x+3)^{3/2}}+2 \sqrt{2} \left (9242 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-4655 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*((Sqrt[1 - 2*x]*(1052533 + 4998904*x + 7902930*x^2 + 4158900*x^3))/((2 + 3*x)
^(3/2)*(3 + 5*x)^(3/2)) + 2*Sqrt[2]*(9242*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5
*x]], -33/2] - 4655*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/231

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Maple [C]  time = 0.033, size = 383, normalized size = 2.1 \[ -{\frac{2}{-231+462\,x}\sqrt{1-2\,x} \left ( 277260\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-139650\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+351196\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-176890\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+110904\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -55860\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -8317800\,{x}^{4}-11646960\,{x}^{3}-2094878\,{x}^{2}+2893838\,x+1052533 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/231*(1-2*x)^(1/2)*(277260*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)
-139650*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)+351196*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)-176890*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)+110904*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))-55860*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))-8317800*x^4-11646960*x^3-2094878*x^2+
2893838*x+1052533)/(2+3*x)^(3/2)/(3+5*x)^(3/2)/(-1+2*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{-2 \, x + 1}}{{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(-2*x + 1)/((225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*sqrt(5*x + 3
)*sqrt(3*x + 2)), 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((1-2*x)**(1/2)/(2+3*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. \[ \int \frac{\sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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