3.2253 \(\int \frac{\sqrt{1-2 x} \sqrt{3+5 x}}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=122 \[ \frac{37 \sqrt{1-2 x} (5 x+3)^{3/2}}{28 (3 x+2)^2}+\frac{(1-2 x)^{3/2} (5 x+3)^{3/2}}{7 (3 x+2)^3}-\frac{407 \sqrt{1-2 x} \sqrt{5 x+3}}{392 (3 x+2)}-\frac{4477 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{392 \sqrt{7}} \]

[Out]

(-407*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(392*(2 + 3*x)) + ((1 - 2*x)^(3/2)*(3 + 5*x)^
(3/2))/(7*(2 + 3*x)^3) + (37*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(28*(2 + 3*x)^2) - (
4477*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(392*Sqrt[7])

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Rubi [A]  time = 0.166629, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{37 \sqrt{1-2 x} (5 x+3)^{3/2}}{28 (3 x+2)^2}+\frac{(1-2 x)^{3/2} (5 x+3)^{3/2}}{7 (3 x+2)^3}-\frac{407 \sqrt{1-2 x} \sqrt{5 x+3}}{392 (3 x+2)}-\frac{4477 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{392 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2 + 3*x)^4,x]

[Out]

(-407*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(392*(2 + 3*x)) + ((1 - 2*x)^(3/2)*(3 + 5*x)^
(3/2))/(7*(2 + 3*x)^3) + (37*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(28*(2 + 3*x)^2) - (
4477*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(392*Sqrt[7])

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Rubi in Sympy [A]  time = 12.9723, size = 109, normalized size = 0.89 \[ - \frac{37 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{196 \left (3 x + 2\right )^{2}} + \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{7 \left (3 x + 2\right )^{3}} + \frac{407 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{392 \left (3 x + 2\right )} - \frac{4477 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{2744} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(1/2)*(3+5*x)**(1/2)/(2+3*x)**4,x)

[Out]

-37*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(196*(3*x + 2)**2) + (-2*x + 1)**(3/2)*(5*x
+ 3)**(3/2)/(7*(3*x + 2)**3) + 407*sqrt(-2*x + 1)*sqrt(5*x + 3)/(392*(3*x + 2))
- 4477*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/2744

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Mathematica [A]  time = 0.0792384, size = 77, normalized size = 0.63 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (3547 x^2+4902 x+1648\right )}{(3 x+2)^3}-4477 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{5488} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2 + 3*x)^4,x]

[Out]

((14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(1648 + 4902*x + 3547*x^2))/(2 + 3*x)^3 - 4477*
Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/5488

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Maple [B]  time = 0.019, size = 202, normalized size = 1.7 \[{\frac{1}{5488\, \left ( 2+3\,x \right ) ^{3}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 120879\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+241758\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+161172\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+49658\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+35816\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +68628\,x\sqrt{-10\,{x}^{2}-x+3}+23072\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4,x)

[Out]

1/5488*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(120879*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)
/(-10*x^2-x+3)^(1/2))*x^3+241758*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-
x+3)^(1/2))*x^2+161172*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2)
)*x+49658*x^2*(-10*x^2-x+3)^(1/2)+35816*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-
10*x^2-x+3)^(1/2))+68628*x*(-10*x^2-x+3)^(1/2)+23072*(-10*x^2-x+3)^(1/2))/(-10*x
^2-x+3)^(1/2)/(2+3*x)^3

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Maxima [A]  time = 1.50239, size = 163, normalized size = 1.34 \[ \frac{4477}{5488} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{185}{294} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{7 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{111 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{196 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{1369 \, \sqrt{-10 \, x^{2} - x + 3}}{1176 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*sqrt(-2*x + 1)/(3*x + 2)^4,x, algorithm="maxima")

[Out]

4477/5488*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 185/294*sq
rt(-10*x^2 - x + 3) + 1/7*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x + 8) +
111/196*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 1369/1176*sqrt(-10*x^2 - x
+ 3)/(3*x + 2)

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Fricas [A]  time = 0.225871, size = 127, normalized size = 1.04 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (3547 \, x^{2} + 4902 \, x + 1648\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 4477 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{5488 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*sqrt(-2*x + 1)/(3*x + 2)^4,x, algorithm="fricas")

[Out]

1/5488*sqrt(7)*(2*sqrt(7)*(3547*x^2 + 4902*x + 1648)*sqrt(5*x + 3)*sqrt(-2*x + 1
) + 4477*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x
+ 3)*sqrt(-2*x + 1))))/(27*x^3 + 54*x^2 + 36*x + 8)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3}}{\left (3 x + 2\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-2*x)**(1/2)*(3+5*x)**(1/2)/(2+3*x)**4,x)

[Out]

Integral(sqrt(-2*x + 1)*sqrt(5*x + 3)/(3*x + 2)**4, x)

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GIAC/XCAS [A]  time = 0.336566, size = 425, normalized size = 3.48 \[ \frac{121}{54880} \, \sqrt{5}{\left (37 \, \sqrt{70} \sqrt{2}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{280 \, \sqrt{2}{\left (37 \,{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 24640 \,{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - \frac{2900800 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{\sqrt{5 \, x + 3}} + \frac{11603200 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*sqrt(-2*x + 1)/(3*x + 2)^4,x, algorithm="giac")

[Out]

121/54880*sqrt(5)*(37*sqrt(70)*sqrt(2)*(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)))) - 280*sqrt(2)*(37*((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 - 24640*((sqrt
(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22)))^3 - 2900800*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x
+ 3) + 11603200*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*s
qrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x +
5) - sqrt(22)))^2 + 280)^3)