3.2568 \(\int \frac{\sqrt{3+5 x}}{(1-2 x)^{5/2} (2+3 x)} \, dx\)

Optimal. Leaf size=79 \[ \frac{4 (5 x+3)^{3/2}}{231 (1-2 x)^{3/2}}+\frac{6 \sqrt{5 x+3}}{49 \sqrt{1-2 x}}+\frac{6 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]

[Out]

(6*Sqrt[3 + 5*x])/(49*Sqrt[1 - 2*x]) + (4*(3 + 5*x)^(3/2))/(231*(1 - 2*x)^(3/2))
+ (6*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(49*Sqrt[7])

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Rubi [A]  time = 0.123303, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{4 (5 x+3)^{3/2}}{231 (1-2 x)^{3/2}}+\frac{6 \sqrt{5 x+3}}{49 \sqrt{1-2 x}}+\frac{6 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(6*Sqrt[3 + 5*x])/(49*Sqrt[1 - 2*x]) + (4*(3 + 5*x)^(3/2))/(231*(1 - 2*x)^(3/2))
+ (6*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(49*Sqrt[7])

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Rubi in Sympy [A]  time = 10.4804, size = 73, normalized size = 0.92 \[ \frac{6 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{343} + \frac{6 \sqrt{5 x + 3}}{49 \sqrt{- 2 x + 1}} + \frac{4 \left (5 x + 3\right )^{\frac{3}{2}}}{231 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/343 + 6*sqrt(5*x + 3)/(
49*sqrt(-2*x + 1)) + 4*(5*x + 3)**(3/2)/(231*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.150137, size = 65, normalized size = 0.82 \[ \frac{2 \sqrt{5 x+3} (141-128 x)}{1617 (1-2 x)^{3/2}}+\frac{3 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(141 - 128*x)*Sqrt[3 + 5*x])/(1617*(1 - 2*x)^(3/2)) + (3*ArcTan[(-20 - 37*x)/
(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/(49*Sqrt[7])

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Maple [B]  time = 0.019, size = 154, normalized size = 2. \[ -{\frac{1}{11319\, \left ( -1+2\,x \right ) ^{2}} \left ( 396\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-396\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+99\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +1792\,x\sqrt{-10\,{x}^{2}-x+3}-1974\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/11319*(396*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-396
*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+99*7^(1/2)*arctan(
1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+1792*x*(-10*x^2-x+3)^(1/2)-1974*(-10
*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(-1+2*x)^2/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.50745, size = 117, normalized size = 1.48 \[ -\frac{3}{343} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{640 \, x}{1617 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{1}{1617 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{55 \, x}{21 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{11}{7 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/343*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 640/1617*x/sq
rt(-10*x^2 - x + 3) - 1/1617/sqrt(-10*x^2 - x + 3) + 55/21*x/(-10*x^2 - x + 3)^(
3/2) + 11/7/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 0.23042, size = 107, normalized size = 1.35 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (128 \, x - 141\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 99 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{11319 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/11319*sqrt(7)*(2*sqrt(7)*(128*x - 141)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 99*(4*x
^2 - 4*x + 1)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(
4*x^2 - 4*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)**(1/2)/(1-2*x)**(5/2)/(2+3*x),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.25412, size = 153, normalized size = 1.94 \[ -\frac{3}{3430} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{2 \,{\left (128 \, \sqrt{5}{\left (5 \, x + 3\right )} - 1089 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{40425 \,{\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/3430*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
)))) - 2/40425*(128*sqrt(5)*(5*x + 3) - 1089*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x +
5)/(2*x - 1)^2