3.2469 \(\int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x} (2+3 x)^5} \, dx\)

Optimal. Leaf size=151 \[ \frac{3 \sqrt{1-2 x} (5 x+3)^{7/2}}{28 (3 x+2)^4}-\frac{\sqrt{1-2 x} (5 x+3)^{5/2}}{24 (3 x+2)^3}-\frac{55 \sqrt{1-2 x} (5 x+3)^{3/2}}{672 (3 x+2)^2}-\frac{605 \sqrt{1-2 x} \sqrt{5 x+3}}{3136 (3 x+2)}-\frac{6655 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{3136 \sqrt{7}} \]

[Out]

(-605*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3136*(2 + 3*x)) - (55*Sqrt[1 - 2*x]*(3 + 5*x
)^(3/2))/(672*(2 + 3*x)^2) - (Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(24*(2 + 3*x)^3) +
(3*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))/(28*(2 + 3*x)^4) - (6655*ArcTan[Sqrt[1 - 2*x]/
(Sqrt[7]*Sqrt[3 + 5*x])])/(3136*Sqrt[7])

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Rubi [A]  time = 0.216009, 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 \[ \frac{3 \sqrt{1-2 x} (5 x+3)^{7/2}}{28 (3 x+2)^4}-\frac{\sqrt{1-2 x} (5 x+3)^{5/2}}{24 (3 x+2)^3}-\frac{55 \sqrt{1-2 x} (5 x+3)^{3/2}}{672 (3 x+2)^2}-\frac{605 \sqrt{1-2 x} \sqrt{5 x+3}}{3136 (3 x+2)}-\frac{6655 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{3136 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-605*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3136*(2 + 3*x)) - (55*Sqrt[1 - 2*x]*(3 + 5*x
)^(3/2))/(672*(2 + 3*x)^2) - (Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(24*(2 + 3*x)^3) +
(3*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))/(28*(2 + 3*x)^4) - (6655*ArcTan[Sqrt[1 - 2*x]/
(Sqrt[7]*Sqrt[3 + 5*x])])/(3136*Sqrt[7])

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Rubi in Sympy [A]  time = 16.5343, size = 136, normalized size = 0.9 \[ - \frac{605 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{3136 \left (3 x + 2\right )} - \frac{55 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{672 \left (3 x + 2\right )^{2}} - \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{24 \left (3 x + 2\right )^{3}} + \frac{3 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{7}{2}}}{28 \left (3 x + 2\right )^{4}} - \frac{6655 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{21952} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-605*sqrt(-2*x + 1)*sqrt(5*x + 3)/(3136*(3*x + 2)) - 55*sqrt(-2*x + 1)*(5*x + 3)
**(3/2)/(672*(3*x + 2)**2) - sqrt(-2*x + 1)*(5*x + 3)**(5/2)/(24*(3*x + 2)**3) +
3*sqrt(-2*x + 1)*(5*x + 3)**(7/2)/(28*(3*x + 2)**4) - 6655*sqrt(7)*atan(sqrt(7)
*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/21952

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Mathematica [A]  time = 0.0905075, size = 82, normalized size = 0.54 \[ \frac{\sqrt{1-2 x} \sqrt{5 x+3} \left (12945 x^3+6920 x^2-6484 x-3600\right )}{9408 (3 x+2)^4}-\frac{6655 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{6272 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-3600 - 6484*x + 6920*x^2 + 12945*x^3))/(9408*(2 +
3*x)^4) - (6655*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/(6272*Sq
rt[7])

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Maple [B]  time = 0.023, size = 250, normalized size = 1.7 \[{\frac{1}{131712\, \left ( 2+3\,x \right ) ^{4}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 1617165\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+4312440\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+4312440\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+181230\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+1916640\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+96880\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+319440\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -90776\,x\sqrt{-10\,{x}^{2}-x+3}-50400\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/131712*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(1617165*7^(1/2)*arctan(1/14*(37*x+20)*7^(1
/2)/(-10*x^2-x+3)^(1/2))*x^4+4312440*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*
x^2-x+3)^(1/2))*x^3+4312440*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^
(1/2))*x^2+181230*x^3*(-10*x^2-x+3)^(1/2)+1916640*7^(1/2)*arctan(1/14*(37*x+20)*
7^(1/2)/(-10*x^2-x+3)^(1/2))*x+96880*x^2*(-10*x^2-x+3)^(1/2)+319440*7^(1/2)*arct
an(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-90776*x*(-10*x^2-x+3)^(1/2)-50400
*(-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 = 1.51684, size = 193, normalized size = 1.28 \[ \frac{6655}{43904} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{\sqrt{-10 \, x^{2} - x + 3}}{252 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{83 \, \sqrt{-10 \, x^{2} - x + 3}}{1512 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} - \frac{1355 \, \sqrt{-10 \, x^{2} - x + 3}}{6048 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{4315 \, \sqrt{-10 \, x^{2} - x + 3}}{84672 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6655/43904*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 1/252*sqr
t(-10*x^2 - x + 3)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 83/1512*sqrt(-10*x
^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) - 1355/6048*sqrt(-10*x^2 - x + 3)/(9*x^
2 + 12*x + 4) + 4315/84672*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.221541, size = 147, normalized size = 0.97 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (12945 \, x^{3} + 6920 \, x^{2} - 6484 \, x - 3600\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 19965 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{131712 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/131712*sqrt(7)*(2*sqrt(7)*(12945*x^3 + 6920*x^2 - 6484*x - 3600)*sqrt(5*x + 3)
*sqrt(-2*x + 1) + 19965*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan(1/14*sqr
t(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((3+5*x)**(5/2)/(2+3*x)**5/(1-2*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.429224, size = 512, normalized size = 3.39 \[ \frac{1331}{87808} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{6655 \,{\left (3 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 3080 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 1144640 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 2956800 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{4704 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1331/87808*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)))) - 6655/4704*(3*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 + 3080*sqrt(10)*((
sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqr
t(-10*x + 5) - sqrt(22)))^5 + 1144640*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 + 2
956800*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(2
2))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 28
0)^4