3.2414 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^3 \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=122 \[ \frac{7 \sqrt{5 x+3} (1-2 x)^{3/2}}{6 (3 x+2)^2}+\frac{637 \sqrt{5 x+3} \sqrt{1-2 x}}{36 (3 x+2)}-\frac{8}{27} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{3035}{108} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(7*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(6*(2 + 3*x)^2) + (637*Sqrt[1 - 2*x]*Sqrt[3 +
5*x])/(36*(2 + 3*x)) - (8*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/27 - (3035
*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/108

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Rubi [A]  time = 0.237294, antiderivative size = 122, 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 \[ \frac{7 \sqrt{5 x+3} (1-2 x)^{3/2}}{6 (3 x+2)^2}+\frac{637 \sqrt{5 x+3} \sqrt{1-2 x}}{36 (3 x+2)}-\frac{8}{27} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{3035}{108} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(7*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(6*(2 + 3*x)^2) + (637*Sqrt[1 - 2*x]*Sqrt[3 +
5*x])/(36*(2 + 3*x)) - (8*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/27 - (3035
*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/108

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Rubi in Sympy [A]  time = 22.3624, size = 109, normalized size = 0.89 \[ \frac{7 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{6 \left (3 x + 2\right )^{2}} + \frac{637 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{36 \left (3 x + 2\right )} - \frac{8 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{135} - \frac{3035 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{108} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(6*(3*x + 2)**2) + 637*sqrt(-2*x + 1)*sqrt(5*x
+ 3)/(36*(3*x + 2)) - 8*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/135 - 3035*sqr
t(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/108

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Mathematica [A]  time = 0.15431, size = 111, normalized size = 0.91 \[ \frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (261 x+188)}{36 (3 x+2)^2}-\frac{3035}{216} \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-\frac{4}{27} \sqrt{\frac{2}{5}} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(188 + 261*x))/(36*(2 + 3*x)^2) - (3035*Sqrt[7]*A
rcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/216 - (4*Sqrt[2/5]*ArcTan[
(1 + 20*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/27

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Maple [B]  time = 0.019, size = 191, normalized size = 1.6 \[{\frac{1}{1080\, \left ( 2+3\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 136575\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-288\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+182100\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-384\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+60700\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -128\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +54810\,x\sqrt{-10\,{x}^{2}-x+3}+39480\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1080*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(136575*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)
/(-10*x^2-x+3)^(1/2))*x^2-288*10^(1/2)*arcsin(20/11*x+1/11)*x^2+182100*7^(1/2)*a
rctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-384*10^(1/2)*arcsin(20/11*x+
1/11)*x+60700*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-128*10^
(1/2)*arcsin(20/11*x+1/11)+54810*x*(-10*x^2-x+3)^(1/2)+39480*(-10*x^2-x+3)^(1/2)
)/(-10*x^2-x+3)^(1/2)/(2+3*x)^2

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Maxima [A]  time = 1.50914, size = 117, normalized size = 0.96 \[ -\frac{4}{135} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{3035}{216} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{49 \, \sqrt{-10 \, x^{2} - x + 3}}{18 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{203 \, \sqrt{-10 \, x^{2} - x + 3}}{12 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/135*sqrt(10)*arcsin(20/11*x + 1/11) + 3035/216*sqrt(7)*arcsin(37/11*x/abs(3*x
+ 2) + 20/11/abs(3*x + 2)) + 49/18*sqrt(-10*x^2 - x + 3)/(9*x^2 + 12*x + 4) + 2
03/12*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.23226, size = 173, normalized size = 1.42 \[ \frac{\sqrt{5}{\left (3035 \, \sqrt{7} \sqrt{5}{\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 42 \, \sqrt{5}{\left (261 \, x + 188\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 32 \, \sqrt{2}{\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{1080 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1080*sqrt(5)*(3035*sqrt(7)*sqrt(5)*(9*x^2 + 12*x + 4)*arctan(1/14*sqrt(7)*(37*
x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) + 42*sqrt(5)*(261*x + 188)*sqrt(5*x + 3)
*sqrt(-2*x + 1) - 32*sqrt(2)*(9*x^2 + 12*x + 4)*arctan(1/20*sqrt(5)*sqrt(2)*(20*
x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(9*x^2 + 12*x + 4)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.350211, size = 437, normalized size = 3.58 \[ \frac{607}{432} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{4}{135} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{77 \,{\left (157 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 25480 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{18 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

607/432*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
)))) - 4/135*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)))) + 77/18*(1
57*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 + 25480*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))))/(((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)))^2 + 280)^2