3.2671 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=158 \[ -\frac{2780 \sqrt{1-2 x} \sqrt{3 x+2}}{21 \sqrt{5 x+3}}+\frac{92 \sqrt{1-2 x}}{7 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{2 \sqrt{1-2 x}}{3 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{184 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7 \sqrt{33}}+\frac{556}{7} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(2*Sqrt[1 - 2*x])/(3*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (92*Sqrt[1 - 2*x])/(7*Sqrt
[2 + 3*x]*Sqrt[3 + 5*x]) - (2780*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(21*Sqrt[3 + 5*x])
+ (556*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/7 + (184*E
llipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(7*Sqrt[33])

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Rubi [A]  time = 0.343884, antiderivative size = 158, 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 \[ -\frac{2780 \sqrt{1-2 x} \sqrt{3 x+2}}{21 \sqrt{5 x+3}}+\frac{92 \sqrt{1-2 x}}{7 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{2 \sqrt{1-2 x}}{3 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{184 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7 \sqrt{33}}+\frac{556}{7} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 - 2*x])/(3*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (92*Sqrt[1 - 2*x])/(7*Sqrt
[2 + 3*x]*Sqrt[3 + 5*x]) - (2780*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(21*Sqrt[3 + 5*x])
+ (556*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/7 + (184*E
llipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(7*Sqrt[33])

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Rubi in Sympy [A]  time = 31.0435, size = 143, normalized size = 0.91 \[ - \frac{2780 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{21 \sqrt{5 x + 3}} + \frac{92 \sqrt{- 2 x + 1}}{7 \sqrt{3 x + 2} \sqrt{5 x + 3}} + \frac{2 \sqrt{- 2 x + 1}}{3 \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}} + \frac{556 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{21} + \frac{184 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{245} \]

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)**(3/2),x)

[Out]

-2780*sqrt(-2*x + 1)*sqrt(3*x + 2)/(21*sqrt(5*x + 3)) + 92*sqrt(-2*x + 1)/(7*sqr
t(3*x + 2)*sqrt(5*x + 3)) + 2*sqrt(-2*x + 1)/(3*(3*x + 2)**(3/2)*sqrt(5*x + 3))
+ 556*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/21 + 184*sqrt(
35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/245

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Mathematica [A]  time = 0.31101, size = 100, normalized size = 0.63 \[ -\frac{2 \sqrt{1-2 x} \left (4170 x^2+5422 x+1759\right )}{7 (3 x+2)^{3/2} \sqrt{5 x+3}}-\frac{4}{21} \sqrt{2} \left (139 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-70 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x]*(1759 + 5422*x + 4170*x^2))/(7*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])
- (4*Sqrt[2]*(139*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 70*Ellipt
icF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/21

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Maple [C]  time = 0.033, size = 267, normalized size = 1.7 \[ -{\frac{2}{210\,{x}^{2}+21\,x-63}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 420\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-834\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+280\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -556\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +25020\,{x}^{3}+20022\,{x}^{2}-5712\,x-5277 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}} \]

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)^(3/2),x)

[Out]

-2/21*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(420*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)-834*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)+280*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))-556*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))+25020*x^3+20022*x^2-5712*x-5277)/(2+3*x)^(3/2)/(10*x^2+x-3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{-2 \, x + 1}}{{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(-2*x + 1)/((45*x^3 + 87*x^2 + 56*x + 12)*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)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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