3.2496 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=77 \[ \frac{950 \sqrt{1-2 x}}{363 \sqrt{5 x+3}}-\frac{10 \sqrt{1-2 x}}{33 (5 x+3)^{3/2}}-\frac{18 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{\sqrt{7}} \]

[Out]

(-10*Sqrt[1 - 2*x])/(33*(3 + 5*x)^(3/2)) + (950*Sqrt[1 - 2*x])/(363*Sqrt[3 + 5*x
]) - (18*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/Sqrt[7]

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Rubi [A]  time = 0.16571, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{950 \sqrt{1-2 x}}{363 \sqrt{5 x+3}}-\frac{10 \sqrt{1-2 x}}{33 (5 x+3)^{3/2}}-\frac{18 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{\sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[1 - 2*x])/(33*(3 + 5*x)^(3/2)) + (950*Sqrt[1 - 2*x])/(363*Sqrt[3 + 5*x
]) - (18*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/Sqrt[7]

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Rubi in Sympy [A]  time = 14.5261, size = 73, normalized size = 0.95 \[ \frac{950 \sqrt{- 2 x + 1}}{363 \sqrt{5 x + 3}} - \frac{10 \sqrt{- 2 x + 1}}{33 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{18 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

950*sqrt(-2*x + 1)/(363*sqrt(5*x + 3)) - 10*sqrt(-2*x + 1)/(33*(5*x + 3)**(3/2))
- 18*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/7

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Mathematica [A]  time = 0.0941768, size = 63, normalized size = 0.82 \[ \frac{10 \sqrt{1-2 x} (475 x+274)}{363 (5 x+3)^{3/2}}-\frac{9 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{\sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*(274 + 475*x))/(363*(3 + 5*x)^(3/2)) - (9*ArcTan[(-20 - 37*x)/
(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/Sqrt[7]

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Maple [B]  time = 0.022, size = 147, normalized size = 1.9 \[{\frac{1}{2541} \left ( 81675\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+98010\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+29403\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +33250\,x\sqrt{-10\,{x}^{2}-x+3}+19180\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2541*(81675*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+980
10*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+29403*7^(1/2)*ar
ctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+33250*x*(-10*x^2-x+3)^(1/2)+191
80*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-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. \[ \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )} \sqrt{-2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.233107, size = 107, normalized size = 1.39 \[ \frac{\sqrt{7}{\left (10 \, \sqrt{7}{\left (475 \, x + 274\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 3267 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{2541 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2541*sqrt(7)*(10*sqrt(7)*(475*x + 274)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 3267*(25
*x^2 + 30*x + 9)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1)))
)/(25*x^2 + 30*x + 9)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.247934, size = 262, normalized size = 3.4 \[ -\frac{1}{5808} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + \frac{9}{70} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{31}{242} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5808*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 + 9/70*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)))) + 31/242*sqrt(10)*((sqrt(2)*s
qrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x +
5) - sqrt(22)))