3.2416 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^5 \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=151 \[ \frac{3 \sqrt{5 x+3} (1-2 x)^{7/2}}{28 (3 x+2)^4}+\frac{247 \sqrt{5 x+3} (1-2 x)^{5/2}}{168 (3 x+2)^3}+\frac{13585 \sqrt{5 x+3} (1-2 x)^{3/2}}{672 (3 x+2)^2}+\frac{149435 \sqrt{5 x+3} \sqrt{1-2 x}}{448 (3 x+2)}-\frac{1643785 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{448 \sqrt{7}} \]

[Out]

(3*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/(28*(2 + 3*x)^4) + (247*(1 - 2*x)^(5/2)*Sqrt[3
+ 5*x])/(168*(2 + 3*x)^3) + (13585*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(672*(2 + 3*x
)^2) + (149435*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(448*(2 + 3*x)) - (1643785*ArcTan[Sq
rt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(448*Sqrt[7])

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Rubi [A]  time = 0.218227, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{3 \sqrt{5 x+3} (1-2 x)^{7/2}}{28 (3 x+2)^4}+\frac{247 \sqrt{5 x+3} (1-2 x)^{5/2}}{168 (3 x+2)^3}+\frac{13585 \sqrt{5 x+3} (1-2 x)^{3/2}}{672 (3 x+2)^2}+\frac{149435 \sqrt{5 x+3} \sqrt{1-2 x}}{448 (3 x+2)}-\frac{1643785 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{448 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/(28*(2 + 3*x)^4) + (247*(1 - 2*x)^(5/2)*Sqrt[3
+ 5*x])/(168*(2 + 3*x)^3) + (13585*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(672*(2 + 3*x
)^2) + (149435*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(448*(2 + 3*x)) - (1643785*ArcTan[Sq
rt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(448*Sqrt[7])

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Rubi in Sympy [A]  time = 17.1594, size = 138, normalized size = 0.91 \[ \frac{3 \left (- 2 x + 1\right )^{\frac{7}{2}} \sqrt{5 x + 3}}{28 \left (3 x + 2\right )^{4}} + \frac{247 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{168 \left (3 x + 2\right )^{3}} + \frac{13585 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{672 \left (3 x + 2\right )^{2}} + \frac{149435 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{448 \left (3 x + 2\right )} - \frac{1643785 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{3136} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*(-2*x + 1)**(7/2)*sqrt(5*x + 3)/(28*(3*x + 2)**4) + 247*(-2*x + 1)**(5/2)*sqrt
(5*x + 3)/(168*(3*x + 2)**3) + 13585*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(672*(3*x +
2)**2) + 149435*sqrt(-2*x + 1)*sqrt(5*x + 3)/(448*(3*x + 2)) - 1643785*sqrt(7)*
atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/3136

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Mathematica [A]  time = 0.116955, size = 82, normalized size = 0.54 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (11637735 x^3+23794744 x^2+16236916 x+3699216\right )}{(3 x+2)^4}-4931355 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{18816} \]

Antiderivative was successfully verified.

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

[Out]

((14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(3699216 + 16236916*x + 23794744*x^2 + 11637735
*x^3))/(2 + 3*x)^4 - 4931355*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[
3 + 5*x])])/18816

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Maple [B]  time = 0.024, size = 250, normalized size = 1.7 \[{\frac{1}{18816\, \left ( 2+3\,x \right ) ^{4}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 399439755\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+333126416\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18816*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(399439755*7^(1/2)*arctan(1/14*(37*x+20)*7^(
1/2)/(-10*x^2-x+3)^(1/2))*x^4+1065172680*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(
-10*x^2-x+3)^(1/2))*x^3+1065172680*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^
2-x+3)^(1/2))*x^2+162928290*x^3*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/1
4*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+333126416*x^2*(-10*x^2-x+3)^(1/2)+789
01680*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*x*(-1
0*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^4

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Maxima [A]  time = 1.50179, size = 193, normalized size = 1.28 \[ \frac{1643785}{6272} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{49 \, \sqrt{-10 \, x^{2} - x + 3}}{36 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{1477 \, \sqrt{-10 \, x^{2} - x + 3}}{216 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{37091 \, \sqrt{-10 \, x^{2} - x + 3}}{864 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{3879245 \, \sqrt{-10 \, x^{2} - x + 3}}{12096 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1643785/6272*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 49/36*s
qrt(-10*x^2 - x + 3)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 1477/216*sqrt(-1
0*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 37091/864*sqrt(-10*x^2 - x + 3)/(9
*x^2 + 12*x + 4) + 3879245/12096*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.234013, size = 147, normalized size = 0.97 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (11637735 \, x^{3} + 23794744 \, x^{2} + 16236916 \, x + 3699216\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 4931355 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{18816 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18816*sqrt(7)*(2*sqrt(7)*(11637735*x^3 + 23794744*x^2 + 16236916*x + 3699216)*
sqrt(5*x + 3)*sqrt(-2*x + 1) + 4931355*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*
arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(81*x^4 + 216*x
^3 + 216*x^2 + 96*x + 16)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.414472, size = 512, normalized size = 3.39 \[ \frac{328757}{12544} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{6655 \,{\left (1947 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 1009736 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 213012800 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 16266432000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{672 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

328757/12544*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sq
rt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sq
rt(22)))) + 6655/672*(1947*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)))^7 + 1009736*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 + 213012800*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 + 16266432000*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)^4