3.2497 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=108 \[ \frac{84235 \sqrt{1-2 x}}{2541 \sqrt{5 x+3}}-\frac{845 \sqrt{1-2 x}}{231 (5 x+3)^{3/2}}+\frac{3 \sqrt{1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}-\frac{1593 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{7 \sqrt{7}} \]

[Out]

(-845*Sqrt[1 - 2*x])/(231*(3 + 5*x)^(3/2)) + (3*Sqrt[1 - 2*x])/(7*(2 + 3*x)*(3 +
5*x)^(3/2)) + (84235*Sqrt[1 - 2*x])/(2541*Sqrt[3 + 5*x]) - (1593*ArcTan[Sqrt[1
- 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7])

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Rubi [A]  time = 0.243496, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{84235 \sqrt{1-2 x}}{2541 \sqrt{5 x+3}}-\frac{845 \sqrt{1-2 x}}{231 (5 x+3)^{3/2}}+\frac{3 \sqrt{1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}-\frac{1593 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{7 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-845*Sqrt[1 - 2*x])/(231*(3 + 5*x)^(3/2)) + (3*Sqrt[1 - 2*x])/(7*(2 + 3*x)*(3 +
5*x)^(3/2)) + (84235*Sqrt[1 - 2*x])/(2541*Sqrt[3 + 5*x]) - (1593*ArcTan[Sqrt[1
- 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7])

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Rubi in Sympy [A]  time = 20.948, size = 99, normalized size = 0.92 \[ \frac{84235 \sqrt{- 2 x + 1}}{2541 \sqrt{5 x + 3}} - \frac{845 \sqrt{- 2 x + 1}}{231 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{3 \sqrt{- 2 x + 1}}{7 \left (3 x + 2\right ) \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{1593 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{49} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

84235*sqrt(-2*x + 1)/(2541*sqrt(5*x + 3)) - 845*sqrt(-2*x + 1)/(231*(5*x + 3)**(
3/2)) + 3*sqrt(-2*x + 1)/(7*(3*x + 2)*(5*x + 3)**(3/2)) - 1593*sqrt(7)*atan(sqrt
(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/49

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Mathematica [A]  time = 0.0879518, size = 77, normalized size = 0.71 \[ \frac{\sqrt{1-2 x} \left (1263525 x^2+1572580 x+487909\right )}{2541 (3 x+2) (5 x+3)^{3/2}}-\frac{1593 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{14 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(487909 + 1572580*x + 1263525*x^2))/(2541*(2 + 3*x)*(3 + 5*x)^(3/
2)) - (1593*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/(14*Sqrt[7])

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Maple [B]  time = 0.023, size = 202, normalized size = 1.9 \[{\frac{1}{71148+106722\,x} \left ( 43369425\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+17689350\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}+6830726\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/35574*(43369425*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3
[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+5030853
3*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+17689350*x^2*(-10
*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/
2))[phone]*x*(-10*x^2-x+3)^(1/2)+6830726*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(2
+3*x)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.230399, size = 127, normalized size = 1.18 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (1263525 \, x^{2} + 1572580 \, x + 487909\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 578259 \,{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{35574 \,{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/35574*sqrt(7)*(2*sqrt(7)*(1263525*x^2 + 1572580*x + 487909)*sqrt(5*x + 3)*sqrt
(-2*x + 1) + 578259*(75*x^3 + 140*x^2 + 87*x + 18)*arctan(1/14*sqrt(7)*(37*x + 2
0)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(75*x^3 + 140*x^2 + 87*x + 18)

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.286716, size = 423, normalized size = 3.92 \[ -\frac{5}{5808} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + \frac{1593}{980} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{160}{121} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} + \frac{594 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{7 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/5808*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)))^3 + 1593/980*sqrt(70)*sqrt(10)*(pi
+ 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2
/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 160/121*sqrt(10)*((sqrt
(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22))) + 594/7*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqr
t(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sq
rt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x +
5) - sqrt(22)))^2 + 280)