3.2450 \(\int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^5} \, dx\)

Optimal. Leaf size=151 \[ \frac{32735 \sqrt{1-2 x} \sqrt{5 x+3}}{21952 (3 x+2)}+\frac{305 \sqrt{1-2 x} \sqrt{5 x+3}}{1568 (3 x+2)^2}+\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{56 (3 x+2)^3}-\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{28 (3 x+2)^4}-\frac{375265 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{21952 \sqrt{7}} \]

[Out]

-(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(28*(2 + 3*x)^4) + (Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/
(56*(2 + 3*x)^3) + (305*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1568*(2 + 3*x)^2) + (32735
*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21952*(2 + 3*x)) - (375265*ArcTan[Sqrt[1 - 2*x]/(
Sqrt[7]*Sqrt[3 + 5*x])])/(21952*Sqrt[7])

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Rubi [A]  time = 0.29664, 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 \[ \frac{32735 \sqrt{1-2 x} \sqrt{5 x+3}}{21952 (3 x+2)}+\frac{305 \sqrt{1-2 x} \sqrt{5 x+3}}{1568 (3 x+2)^2}+\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{56 (3 x+2)^3}-\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{28 (3 x+2)^4}-\frac{375265 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{21952 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(28*(2 + 3*x)^4) + (Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/
(56*(2 + 3*x)^3) + (305*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1568*(2 + 3*x)^2) + (32735
*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21952*(2 + 3*x)) - (375265*ArcTan[Sqrt[1 - 2*x]/(
Sqrt[7]*Sqrt[3 + 5*x])])/(21952*Sqrt[7])

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Rubi in Sympy [A]  time = 28.9191, size = 134, normalized size = 0.89 \[ \frac{32735 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{21952 \left (3 x + 2\right )} + \frac{305 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1568 \left (3 x + 2\right )^{2}} + \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3}}{56 \left (3 x + 2\right )^{3}} - \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3}}{28 \left (3 x + 2\right )^{4}} - \frac{375265 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{153664} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

32735*sqrt(-2*x + 1)*sqrt(5*x + 3)/(21952*(3*x + 2)) + 305*sqrt(-2*x + 1)*sqrt(5
*x + 3)/(1568*(3*x + 2)**2) + sqrt(-2*x + 1)*sqrt(5*x + 3)/(56*(3*x + 2)**3) - s
qrt(-2*x + 1)*sqrt(5*x + 3)/(28*(3*x + 2)**4) - 375265*sqrt(7)*atan(sqrt(7)*sqrt
(-2*x + 1)/(7*sqrt(5*x + 3)))/153664

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Mathematica [A]  time = 0.120711, size = 82, normalized size = 0.54 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (883845 x^3+1806120 x^2+1230876 x+278960\right )}{(3 x+2)^4}-375265 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{307328} \]

Antiderivative was successfully verified.

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

[Out]

((14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(278960 + 1230876*x + 1806120*x^2 + 883845*x^3)
)/(2 + 3*x)^4 - 375265*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*
x])])/307328

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Maple [B]  time = 0.022, size = 250, normalized size = 1.7 \[{\frac{1}{307328\, \left ( 2+3\,x \right ) ^{4}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 30396465\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+25285680\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+6004240\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}+3905440\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/307328*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(30396465*7^(1/2)*arctan(1/14*(37*x+20)*7^(
1/2)/(-10*x^2-x+3)^(1/2))*x^4+81057240*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-1
0*x^2-x+3)^(1/2))*x^3+81057240*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+
3)^(1/2))*x^2+12373830*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+25285680*x^2*(-10*x^2-x+3)^(1/2)+6004240*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)+3905440*(-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 = 1.56804, size = 193, normalized size = 1.28 \[ \frac{375265}{307328} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{\sqrt{-10 \, x^{2} - x + 3}}{28 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{\sqrt{-10 \, x^{2} - x + 3}}{56 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{305 \, \sqrt{-10 \, x^{2} - x + 3}}{1568 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{32735 \, \sqrt{-10 \, x^{2} - x + 3}}{21952 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

375265/307328*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 1/28*s
qrt(-10*x^2 - x + 3)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 1/56*sqrt(-10*x^
2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 305/1568*sqrt(-10*x^2 - x + 3)/(9*x^2
+ 12*x + 4) + 32735/21952*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.224484, size = 147, normalized size = 0.97 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (883845 \, x^{3} + 1806120 \, x^{2} + 1230876 \, x + 278960\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 375265 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{307328 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/307328*sqrt(7)*(2*sqrt(7)*(883845*x^3 + 1806120*x^2 + 1230876*x + 278960)*sqrt
(5*x + 3)*sqrt(-2*x + 1) + 375265*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arcta
n(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(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.430446, size = 512, normalized size = 3.39 \[ \frac{75053}{614656} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{55 \,{\left (6823 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} - 7629720 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 1915892160 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 149136243200 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{10976 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

75053/614656*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)))) - 55/10976*(6823*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 - 7629720*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 - 1915892160*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(2
2)))^3 - 149136243200*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) - sq
rt(22)))^2 + 280)^4