3.2415 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^4 \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=122 \[ \frac{\sqrt{5 x+3} (1-2 x)^{5/2}}{3 (3 x+2)^3}+\frac{55 \sqrt{5 x+3} (1-2 x)^{3/2}}{12 (3 x+2)^2}+\frac{605 \sqrt{5 x+3} \sqrt{1-2 x}}{8 (3 x+2)}-\frac{6655 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{8 \sqrt{7}} \]

[Out]

((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(3*(2 + 3*x)^3) + (55*(1 - 2*x)^(3/2)*Sqrt[3 + 5
*x])/(12*(2 + 3*x)^2) + (605*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(8*(2 + 3*x)) - (6655*
ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(8*Sqrt[7])

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Rubi [A]  time = 0.166898, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{\sqrt{5 x+3} (1-2 x)^{5/2}}{3 (3 x+2)^3}+\frac{55 \sqrt{5 x+3} (1-2 x)^{3/2}}{12 (3 x+2)^2}+\frac{605 \sqrt{5 x+3} \sqrt{1-2 x}}{8 (3 x+2)}-\frac{6655 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{8 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - 2*x)^(5/2)/((2 + 3*x)^4*Sqrt[3 + 5*x]),x]

[Out]

((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(3*(2 + 3*x)^3) + (55*(1 - 2*x)^(3/2)*Sqrt[3 + 5
*x])/(12*(2 + 3*x)^2) + (605*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(8*(2 + 3*x)) - (6655*
ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(8*Sqrt[7])

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Rubi in Sympy [A]  time = 13.6088, size = 109, normalized size = 0.89 \[ \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{3 \left (3 x + 2\right )^{3}} + \frac{55 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{12 \left (3 x + 2\right )^{2}} + \frac{605 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{8 \left (3 x + 2\right )} - \frac{6655 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{56} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)/(2+3*x)**4/(3+5*x)**(1/2),x)

[Out]

(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(3*(3*x + 2)**3) + 55*(-2*x + 1)**(3/2)*sqrt(5*x
+ 3)/(12*(3*x + 2)**2) + 605*sqrt(-2*x + 1)*sqrt(5*x + 3)/(8*(3*x + 2)) - 6655*
sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/56

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Mathematica [A]  time = 0.10472, size = 77, normalized size = 0.63 \[ \frac{\sqrt{1-2 x} \sqrt{5 x+3} \left (15707 x^2+21638 x+7488\right )}{24 (3 x+2)^3}-\frac{6655 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{16 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^4*Sqrt[3 + 5*x]),x]

[Out]

(Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(7488 + 21638*x + 15707*x^2))/(24*(2 + 3*x)^3) - (6
655*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/(16*Sqrt[7])

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Maple [B]  time = 0.02, size = 202, normalized size = 1.7 \[{\frac{1}{336\, \left ( 2+3\,x \right ) ^{3}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 539055\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+1078110\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+718740\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+219898\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+159720\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +302932\,x\sqrt{-10\,{x}^{2}-x+3}+104832\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)/(2+3*x)^4/(3+5*x)^(1/2),x)

[Out]

1/336*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(539055*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/
(-10*x^2-x+3)^(1/2))*x^3+1078110*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-
x+3)^(1/2))*x^2+718740*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2)
)*x+219898*x^2*(-10*x^2-x+3)^(1/2)+159720*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/
(-10*x^2-x+3)^(1/2))+302932*x*(-10*x^2-x+3)^(1/2)+104832*(-10*x^2-x+3)^(1/2))/(-
10*x^2-x+3)^(1/2)/(2+3*x)^3

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Maxima [A]  time = 1.52475, size = 144, normalized size = 1.18 \[ \frac{6655}{112} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{49 \, \sqrt{-10 \, x^{2} - x + 3}}{27 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{1043 \, \sqrt{-10 \, x^{2} - x + 3}}{108 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{15707 \, \sqrt{-10 \, x^{2} - x + 3}}{216 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^4),x, algorithm="maxima")

[Out]

6655/112*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 49/27*sqrt(
-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 1043/108*sqrt(-10*x^2 - x + 3)/(
9*x^2 + 12*x + 4) + 15707/216*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.233959, size = 127, normalized size = 1.04 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (15707 \, x^{2} + 21638 \, x + 7488\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 19965 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{336 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^4),x, algorithm="fricas")

[Out]

1/336*sqrt(7)*(2*sqrt(7)*(15707*x^2 + 21638*x + 7488)*sqrt(5*x + 3)*sqrt(-2*x +
1) + 19965*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*
x + 3)*sqrt(-2*x + 1))))/(27*x^3 + 54*x^2 + 36*x + 8)

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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)**4/(3+5*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.367804, size = 429, normalized size = 3.52 \[ \frac{1331}{224} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{1331 \,{\left (33 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 11200 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 1176000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{12 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^4),x, algorithm="giac")

[Out]

1331/224*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(2
2)))) + 1331/12*(33*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)))^5 + 11200*sqrt(10)*((sq
rt(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 + 1176000*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))))/(((sqr
t(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)^3