3.2527 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)} \, dx\)

Optimal. Leaf size=108 \[ \frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x}}+\frac{505}{84} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{475}{36} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{63 \sqrt{7}} \]

[Out]

(505*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/84 + (11*(3 + 5*x)^(3/2))/(7*Sqrt[1 - 2*x]) -
(475*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/36 + (2*ArcTan[Sqrt[1 - 2*x]/(S
qrt[7]*Sqrt[3 + 5*x])])/(63*Sqrt[7])

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Rubi [A]  time = 0.235279, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x}}+\frac{505}{84} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{475}{36} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{63 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(505*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/84 + (11*(3 + 5*x)^(3/2))/(7*Sqrt[1 - 2*x]) -
(475*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/36 + (2*ArcTan[Sqrt[1 - 2*x]/(S
qrt[7]*Sqrt[3 + 5*x])])/(63*Sqrt[7])

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Rubi in Sympy [A]  time = 23.7972, size = 99, normalized size = 0.92 \[ \frac{505 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{84} - \frac{475 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{72} + \frac{2 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{441} + \frac{11 \left (5 x + 3\right )^{\frac{3}{2}}}{7 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

505*sqrt(-2*x + 1)*sqrt(5*x + 3)/84 - 475*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/1
1)/72 + 2*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/441 + 11*(5*x +
3)**(3/2)/(7*sqrt(-2*x + 1))

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Mathematica [A]  time = 0.343426, size = 104, normalized size = 0.96 \[ \frac{\sqrt{5 x+3} (901-350 x)}{84 \sqrt{1-2 x}}+\frac{\tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{63 \sqrt{7}}-\frac{475}{72} \sqrt{\frac{5}{2}} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((901 - 350*x)*Sqrt[3 + 5*x])/(84*Sqrt[1 - 2*x]) + ArcTan[(-20 - 37*x)/(2*Sqrt[7
- 14*x]*Sqrt[3 + 5*x])]/(63*Sqrt[7]) - (475*Sqrt[5/2]*ArcTan[(1 + 20*x)/(2*Sqrt
[1 - 2*x]*Sqrt[30 + 50*x])])/72

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Maple [A]  time = 0.019, size = 146, normalized size = 1.4 \[ -{\frac{1}{-7056+14112\,x} \left ( 32\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+46550\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-16\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -23275\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -29400\,x\sqrt{-10\,{x}^{2}-x+3}+75684\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/7056*(32*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+46550*1
0^(1/2)*arcsin(20/11*x+1/11)*x-16*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2
-x+3)^(1/2))-23275*10^(1/2)*arcsin(20/11*x+1/11)-29400*x*(-10*x^2-x+3)^(1/2)+756
84*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(-1+2*x)/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.50059, size = 116, normalized size = 1.07 \[ -\frac{125 \, x^{2}}{6 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{475}{144} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{1}{441} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{3455 \, x}{84 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{901}{28 \, \sqrt{-10 \, x^{2} - x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-125/6*x^2/sqrt(-10*x^2 - x + 3) - 475/144*sqrt(10)*arcsin(20/11*x + 1/11) - 1/4
41*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 3455/84*x/sqrt(-1
0*x^2 - x + 3) + 901/28/sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.237054, size = 161, normalized size = 1.49 \[ \frac{\sqrt{7} \sqrt{2}{\left (6 \, \sqrt{7} \sqrt{2}{\left (350 \, x - 901\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 3325 \, \sqrt{7} \sqrt{5}{\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) - 8 \, \sqrt{2}{\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{7056 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7056*sqrt(7)*sqrt(2)*(6*sqrt(7)*sqrt(2)*(350*x - 901)*sqrt(5*x + 3)*sqrt(-2*x
+ 1) - 3325*sqrt(7)*sqrt(5)*(2*x - 1)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)/(sq
rt(5*x + 3)*sqrt(-2*x + 1))) - 8*sqrt(2)*(2*x - 1)*arctan(1/14*sqrt(7)*(37*x + 2
0)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(2*x - 1)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.272082, size = 243, normalized size = 2.25 \[ -\frac{1}{4410} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{475}{144} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{{\left (70 \, \sqrt{5}{\left (5 \, x + 3\right )} - 1111 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{420 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4410*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
)))) - 475/144*sqrt(10)*(pi + 2*arctan(-1/4*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)))) + 1/420*
(70*sqrt(5)*(5*x + 3) - 1111*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)