3.2488 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=115 \[ -\frac{90415 \sqrt{1-2 x}}{2156 \sqrt{5 x+3}}+\frac{543 \sqrt{1-2 x}}{196 (3 x+2) \sqrt{5 x+3}}+\frac{3 \sqrt{1-2 x}}{14 (3 x+2)^2 \sqrt{5 x+3}}+\frac{56421 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{196 \sqrt{7}} \]

[Out]

(-90415*Sqrt[1 - 2*x])/(2156*Sqrt[3 + 5*x]) + (3*Sqrt[1 - 2*x])/(14*(2 + 3*x)^2*
Sqrt[3 + 5*x]) + (543*Sqrt[1 - 2*x])/(196*(2 + 3*x)*Sqrt[3 + 5*x]) + (56421*ArcT
an[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(196*Sqrt[7])

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Rubi [A]  time = 0.241999, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{90415 \sqrt{1-2 x}}{2156 \sqrt{5 x+3}}+\frac{543 \sqrt{1-2 x}}{196 (3 x+2) \sqrt{5 x+3}}+\frac{3 \sqrt{1-2 x}}{14 (3 x+2)^2 \sqrt{5 x+3}}+\frac{56421 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{196 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-90415*Sqrt[1 - 2*x])/(2156*Sqrt[3 + 5*x]) + (3*Sqrt[1 - 2*x])/(14*(2 + 3*x)^2*
Sqrt[3 + 5*x]) + (543*Sqrt[1 - 2*x])/(196*(2 + 3*x)*Sqrt[3 + 5*x]) + (56421*ArcT
an[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(196*Sqrt[7])

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Rubi in Sympy [A]  time = 21.2501, size = 105, normalized size = 0.91 \[ - \frac{90415 \sqrt{- 2 x + 1}}{2156 \sqrt{5 x + 3}} + \frac{543 \sqrt{- 2 x + 1}}{196 \left (3 x + 2\right ) \sqrt{5 x + 3}} + \frac{3 \sqrt{- 2 x + 1}}{14 \left (3 x + 2\right )^{2} \sqrt{5 x + 3}} + \frac{56421 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{1372} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-90415*sqrt(-2*x + 1)/(2156*sqrt(5*x + 3)) + 543*sqrt(-2*x + 1)/(196*(3*x + 2)*s
qrt(5*x + 3)) + 3*sqrt(-2*x + 1)/(14*(3*x + 2)**2*sqrt(5*x + 3)) + 56421*sqrt(7)
*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/1372

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Mathematica [A]  time = 0.0915852, size = 77, normalized size = 0.67 \[ \frac{56421 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{392 \sqrt{7}}-\frac{\sqrt{1-2 x} \left (813735 x^2+1067061 x+349252\right )}{2156 (3 x+2)^2 \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(349252 + 1067061*x + 813735*x^2))/(2156*(2 + 3*x)^2*Sqrt[3 + 5*
x]) + (56421*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/(392*Sqrt[7]
)

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Maple [B]  time = 0.023, size = 202, normalized size = 1.8 \[ -{\frac{1}{30184\, \left ( 2+3\,x \right ) ^{2}} \left ( 27928395\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+53994897\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+34755336\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+11392290\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+7447572\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +14938854\,x\sqrt{-10\,{x}^{2}-x+3}+4889528\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/30184*(27928395*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^
3+53994897*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+347553
36*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+11392290*x^2*(-1
0*x^2-x+3)^(1/2)+7447572*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/
2))+14938854*x*(-10*x^2-x+3)^(1/2)+4889528*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(2
+3*x)^2/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.233186, size = 127, normalized size = 1.1 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (813735 \, x^{2} + 1067061 \, x + 349252\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 620631 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{30184 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/30184*sqrt(7)*(2*sqrt(7)*(813735*x^2 + 1067061*x + 349252)*sqrt(5*x + 3)*sqrt
(-2*x + 1) + 620631*(45*x^3 + 87*x^2 + 56*x + 12)*arctan(1/14*sqrt(7)*(37*x + 20
)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(45*x^3 + 87*x^2 + 56*x + 12)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.292462, size = 427, normalized size = 3.71 \[ -\frac{56421}{27440} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{25}{22} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{297 \,{\left (107 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 23800 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{98 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-56421/27440*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sq
rt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sq
rt(22)))) - 25/22*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) -
 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) - 297/98*(107*sqrt(10)*((
sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqr
t(-10*x + 5) - sqrt(22)))^3 + 23800*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22
))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))/(((sqr
t(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-
10*x + 5) - sqrt(22)))^2 + 280)^2