3.2379 \(\int \frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=151 \[ \frac{(5 x+3)^{3/2} (1-2 x)^{5/2}}{4 (3 x+2)^4}+\frac{55 (5 x+3)^{3/2} (1-2 x)^{3/2}}{24 (3 x+2)^3}+\frac{605 (5 x+3)^{3/2} \sqrt{1-2 x}}{32 (3 x+2)^2}-\frac{6655 \sqrt{5 x+3} \sqrt{1-2 x}}{448 (3 x+2)}-\frac{73205 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{448 \sqrt{7}} \]

[Out]

(-6655*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(448*(2 + 3*x)) + ((1 - 2*x)^(5/2)*(3 + 5*x)
^(3/2))/(4*(2 + 3*x)^4) + (55*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(24*(2 + 3*x)^3)
+ (605*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(32*(2 + 3*x)^2) - (73205*ArcTan[Sqrt[1 -
2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(448*Sqrt[7])

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Rubi [A]  time = 0.215887, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{(5 x+3)^{3/2} (1-2 x)^{5/2}}{4 (3 x+2)^4}+\frac{55 (5 x+3)^{3/2} (1-2 x)^{3/2}}{24 (3 x+2)^3}+\frac{605 (5 x+3)^{3/2} \sqrt{1-2 x}}{32 (3 x+2)^2}-\frac{6655 \sqrt{5 x+3} \sqrt{1-2 x}}{448 (3 x+2)}-\frac{73205 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{448 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-6655*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(448*(2 + 3*x)) + ((1 - 2*x)^(5/2)*(3 + 5*x)
^(3/2))/(4*(2 + 3*x)^4) + (55*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(24*(2 + 3*x)^3)
+ (605*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(32*(2 + 3*x)^2) - (73205*ArcTan[Sqrt[1 -
2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(448*Sqrt[7])

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Rubi in Sympy [A]  time = 17.0809, size = 136, normalized size = 0.9 \[ - \frac{55 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{168 \left (3 x + 2\right )^{3}} + \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{4 \left (3 x + 2\right )^{4}} + \frac{605 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{672 \left (3 x + 2\right )^{2}} + \frac{6655 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{448 \left (3 x + 2\right )} - \frac{73205 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{3136} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-55*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(168*(3*x + 2)**3) + (-2*x + 1)**(5/2)*(5*x
+ 3)**(3/2)/(4*(3*x + 2)**4) + 605*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(672*(3*x + 2
)**2) + 6655*sqrt(-2*x + 1)*sqrt(5*x + 3)/(448*(3*x + 2)) - 73205*sqrt(7)*atan(s
qrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/3136

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Mathematica [A]  time = 0.0876901, size = 82, normalized size = 0.54 \[ \frac{\sqrt{1-2 x} \sqrt{5 x+3} \left (518715 x^3+1059032 x^2+723428 x+164688\right )}{1344 (3 x+2)^4}-\frac{73205 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{896 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(164688 + 723428*x + 1059032*x^2 + 518715*x^3))/(13
44*(2 + 3*x)^4) - (73205*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/
(896*Sqrt[7])

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Maple [B]  time = 0.017, size = 250, normalized size = 1.7 \[{\frac{1}{18816\, \left ( 2+3\,x \right ) ^{4}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 17788815\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+7262010\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+14826448\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+3513840\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}+2305632\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18816*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(17788815*7^(1/2)*arctan(1/14*(37*x+20)*7^(1
/2)/(-10*x^2-x+3)^(1/2))*x^4+47436840*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10
*x^2-x+3)^(1/2))*x^3+47436840*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3
)^(1/2))*x^2+7262010*x^3*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+
20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+14826448*x^2*(-10*x^2-x+3)^(1/2)+3513840*7^(1
/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*x*(-10*x^2-x+3)^
(1/2)+2305632*(-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 = 1.55112, size = 212, normalized size = 1.4 \[ \frac{73205}{6272} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{3025}{336} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{7 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{12 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{17 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{8 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{1815 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{224 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{22385 \, \sqrt{-10 \, x^{2} - x + 3}}{1344 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

73205/6272*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 3025/336*
sqrt(-10*x^2 - x + 3) + 7/12*(-10*x^2 - x + 3)^(3/2)/(81*x^4 + 216*x^3 + 216*x^2
+ 96*x + 16) + 17/8*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 1815
/224*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 22385/1344*sqrt(-10*x^2 - x +
3)/(3*x + 2)

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Fricas [A]  time = 0.22421, size = 147, normalized size = 0.97 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (518715 \, x^{3} + 1059032 \, x^{2} + 723428 \, x + 164688\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 219615 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{18816 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18816*sqrt(7)*(2*sqrt(7)*(518715*x^3 + 1059032*x^2 + 723428*x + 164688)*sqrt(5
*x + 3)*sqrt(-2*x + 1) + 219615*(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 + 21
6*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)*(3+5*x)**(1/2)/(2+3*x)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.415004, size = 512, normalized size = 3.39 \[ \frac{14641}{12544} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{73205 \,{\left (3 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} - 4088 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 862400 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 65856000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{672 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

14641/12544*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqr
t(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqr
t(22)))) - 73205/672*(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 - 4088*sqrt(10)*(
(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sq
rt(-10*x + 5) - sqrt(22)))^5 - 862400*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 - 6
5856000*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 + 2
80)^4