3.2352 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^5 \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=151 \[ \frac{4148797 \sqrt{1-2 x} \sqrt{5 x+3}}{28224 (3 x+2)}+\frac{39667 \sqrt{1-2 x} \sqrt{5 x+3}}{2016 (3 x+2)^2}+\frac{227 \sqrt{1-2 x} \sqrt{5 x+3}}{72 (3 x+2)^3}+\frac{7 \sqrt{1-2 x} \sqrt{5 x+3}}{12 (3 x+2)^4}-\frac{5274027 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{3136 \sqrt{7}} \]

[Out]

(7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12*(2 + 3*x)^4) + (227*Sqrt[1 - 2*x]*Sqrt[3 + 5
*x])/(72*(2 + 3*x)^3) + (39667*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2016*(2 + 3*x)^2) +
(4148797*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(28224*(2 + 3*x)) - (5274027*ArcTan[Sqrt[
1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(3136*Sqrt[7])

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Rubi [A]  time = 0.303302, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{4148797 \sqrt{1-2 x} \sqrt{5 x+3}}{28224 (3 x+2)}+\frac{39667 \sqrt{1-2 x} \sqrt{5 x+3}}{2016 (3 x+2)^2}+\frac{227 \sqrt{1-2 x} \sqrt{5 x+3}}{72 (3 x+2)^3}+\frac{7 \sqrt{1-2 x} \sqrt{5 x+3}}{12 (3 x+2)^4}-\frac{5274027 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{3136 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12*(2 + 3*x)^4) + (227*Sqrt[1 - 2*x]*Sqrt[3 + 5
*x])/(72*(2 + 3*x)^3) + (39667*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2016*(2 + 3*x)^2) +
(4148797*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(28224*(2 + 3*x)) - (5274027*ArcTan[Sqrt[
1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(3136*Sqrt[7])

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Rubi in Sympy [A]  time = 28.1768, size = 138, normalized size = 0.91 \[ \frac{4148797 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{28224 \left (3 x + 2\right )} + \frac{39667 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2016 \left (3 x + 2\right )^{2}} + \frac{227 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{72 \left (3 x + 2\right )^{3}} + \frac{7 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{12 \left (3 x + 2\right )^{4}} - \frac{5274027 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{21952} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4148797*sqrt(-2*x + 1)*sqrt(5*x + 3)/(28224*(3*x + 2)) + 39667*sqrt(-2*x + 1)*sq
rt(5*x + 3)/(2016*(3*x + 2)**2) + 227*sqrt(-2*x + 1)*sqrt(5*x + 3)/(72*(3*x + 2)
**3) + 7*sqrt(-2*x + 1)*sqrt(5*x + 3)/(12*(3*x + 2)**4) - 5274027*sqrt(7)*atan(s
qrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/21952

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Mathematica [A]  time = 0.143134, size = 82, normalized size = 0.54 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (12446391 x^3+25448120 x^2+17365300 x+3956240\right )}{(3 x+2)^4}-5274027 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{43904} \]

Antiderivative was successfully verified.

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

[Out]

((14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(3956240 + 17365300*x + 25448120*x^2 + 12446391
*x^3))/(2 + 3*x)^4 - 5274027*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[
3 + 5*x])])/43904

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Maple [B]  time = 0.022, size = 250, normalized size = 1.7 \[{\frac{1}{43904\, \left ( 2+3\,x \right ) ^{4}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 427196187\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+356273680\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/43904*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(427196187*7^(1/2)*arctan(1/14*(37*x+20)*7^(
1/2)/(-10*x^2-x+3)^(1/2))*x^4+1139189832*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(
-10*x^2-x+3)^(1/2))*x^3+1139189832*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^
2-x+3)^(1/2))*x^2+174249474*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+356273680*x^2*(-10*x^2-x+3)^(1/2)+843
84432*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 = 1.51175, size = 193, normalized size = 1.28 \[ \frac{5274027}{43904} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{7 \, \sqrt{-10 \, x^{2} - x + 3}}{12 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{227 \, \sqrt{-10 \, x^{2} - x + 3}}{72 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{39667 \, \sqrt{-10 \, x^{2} - x + 3}}{2016 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{4148797 \, \sqrt{-10 \, x^{2} - x + 3}}{28224 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5274027/43904*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 7/12*s
qrt(-10*x^2 - x + 3)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 227/72*sqrt(-10*
x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 39667/2016*sqrt(-10*x^2 - x + 3)/(9*
x^2 + 12*x + 4) + 4148797/28224*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.21998, size = 147, normalized size = 0.97 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (12446391 \, x^{3} + 25448120 \, x^{2} + 17365300 \, x + 3956240\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 5274027 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{43904 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/43904*sqrt(7)*(2*sqrt(7)*(12446391*x^3 + 25448120*x^2 + 17365300*x + 3956240)*
sqrt(5*x + 3)*sqrt(-2*x + 1) + 5274027*(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)**(3/2)/(2+3*x)**5/(3+5*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.380449, size = 512, normalized size = 3.39 \[ \frac{5274027}{439040} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{121 \,{\left (113213 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 59365880 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 12529809600 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 956821824000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{1568 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5274027/439040*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)))) + 121/1568*(113213*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sq
rt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^7 + 59365880
*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 + 12529809600*sqrt(10)*((sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) -
sqrt(22)))^3 + 956821824000*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)*sqr
t(-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