3.2307 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^2 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=103 \[ \frac{2495 \sqrt{1-2 x}}{33 \sqrt{5 x+3}}-\frac{25 \sqrt{1-2 x}}{3 (5 x+3)^{3/2}}+\frac{\sqrt{1-2 x}}{(3 x+2) (5 x+3)^{3/2}}-\frac{519 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{\sqrt{7}} \]

[Out]

(-25*Sqrt[1 - 2*x])/(3*(3 + 5*x)^(3/2)) + Sqrt[1 - 2*x]/((2 + 3*x)*(3 + 5*x)^(3/
2)) + (2495*Sqrt[1 - 2*x])/(33*Sqrt[3 + 5*x]) - (519*ArcTan[Sqrt[1 - 2*x]/(Sqrt[
7]*Sqrt[3 + 5*x])])/Sqrt[7]

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Rubi [A]  time = 0.225076, antiderivative size = 103, 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 \[ \frac{2495 \sqrt{1-2 x}}{33 \sqrt{5 x+3}}-\frac{25 \sqrt{1-2 x}}{3 (5 x+3)^{3/2}}+\frac{\sqrt{1-2 x}}{(3 x+2) (5 x+3)^{3/2}}-\frac{519 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{\sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-25*Sqrt[1 - 2*x])/(3*(3 + 5*x)^(3/2)) + Sqrt[1 - 2*x]/((2 + 3*x)*(3 + 5*x)^(3/
2)) + (2495*Sqrt[1 - 2*x])/(33*Sqrt[3 + 5*x]) - (519*ArcTan[Sqrt[1 - 2*x]/(Sqrt[
7]*Sqrt[3 + 5*x])])/Sqrt[7]

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Rubi in Sympy [A]  time = 21.3321, size = 95, normalized size = 0.92 \[ \frac{2495 \sqrt{- 2 x + 1}}{33 \sqrt{5 x + 3}} - \frac{25 \sqrt{- 2 x + 1}}{3 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{\sqrt{- 2 x + 1}}{\left (3 x + 2\right ) \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{519 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2495*sqrt(-2*x + 1)/(33*sqrt(5*x + 3)) - 25*sqrt(-2*x + 1)/(3*(5*x + 3)**(3/2))
+ sqrt(-2*x + 1)/((3*x + 2)*(5*x + 3)**(3/2)) - 519*sqrt(7)*atan(sqrt(7)*sqrt(-2
*x + 1)/(7*sqrt(5*x + 3)))/7

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Mathematica [A]  time = 0.0918607, size = 77, normalized size = 0.75 \[ \frac{\sqrt{1-2 x} \left (37425 x^2+46580 x+14453\right )}{33 (3 x+2) (5 x+3)^{3/2}}-\frac{519 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{2 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(14453 + 46580*x + 37425*x^2))/(33*(2 + 3*x)*(3 + 5*x)^(3/2)) - (
519*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/(2*Sqrt[7])

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Maple [B]  time = 0.02, size = 202, normalized size = 2. \[{\frac{1}{924+1386\,x} \left ( 1284525\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+2397780\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+1490049\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+523950\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+308286\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +652120\,x\sqrt{-10\,{x}^{2}-x+3}+202342\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/462*(1284525*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+23
97780*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+1490049*7^(
1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+523950*x^2*(-10*x^2-x+
3)^(1/2)+308286*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+65212
0*x*(-10*x^2-x+3)^(1/2)+202342*(-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 [A]  time = 1.50547, size = 163, normalized size = 1.58 \[ \frac{519}{14} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{4990 \, x}{33 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{2605}{33 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{38 \, x}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{49}{9 \,{\left (3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 2 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} - \frac{185}{9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

519/14*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 4990/33*x/sqr
t(-10*x^2 - x + 3) + 2605/33/sqrt(-10*x^2 - x + 3) + 38*x/(-10*x^2 - x + 3)^(3/2
) + 49/9/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) - 185/9/(-10*
x^2 - x + 3)^(3/2)

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Fricas [A]  time = 0.220467, size = 127, normalized size = 1.23 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (37425 \, x^{2} + 46580 \, x + 14453\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 17127 \,{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{462 \,{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/462*sqrt(7)*(2*sqrt(7)*(37425*x^2 + 46580*x + 14453)*sqrt(5*x + 3)*sqrt(-2*x +
1) + 17127*(75*x^3 + 140*x^2 + 87*x + 18)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt
(5*x + 3)*sqrt(-2*x + 1))))/(75*x^3 + 140*x^2 + 87*x + 18)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.307327, size = 429, normalized size = 4.17 \[ -\frac{1}{18480} \, \sqrt{5}{\left (35 \, \sqrt{2}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 68508 \, \sqrt{70} \sqrt{2}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - 55440 \, \sqrt{2}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{3659040 \, \sqrt{2}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/18480*sqrt(5)*(35*sqrt(2)*((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 - 68508*sqrt(70)*sqrt
(2)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqr
t(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 55440*sqrt(2)*(
(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sq
rt(-10*x + 5) - sqrt(22))) - 3659040*sqrt(2)*((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)))/(((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)))^2 + 280))