3.2520 \(\int \frac{(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx\)

Optimal. Leaf size=151 \[ -\frac{415 \sqrt{1-2 x} \sqrt{5 x+3}}{8232 (3 x+2)}-\frac{145 \sqrt{1-2 x} \sqrt{5 x+3}}{588 (3 x+2)^2}-\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{3 (3 x+2)^3}+\frac{11 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^3}-\frac{2805 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]

[Out]

(11*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x
])/(3*(2 + 3*x)^3) - (145*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(588*(2 + 3*x)^2) - (415*
Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(8232*(2 + 3*x)) - (2805*ArcTan[Sqrt[1 - 2*x]/(Sqrt
[7]*Sqrt[3 + 5*x])])/(2744*Sqrt[7])

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Rubi [A]  time = 0.301797, 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 \[ -\frac{415 \sqrt{1-2 x} \sqrt{5 x+3}}{8232 (3 x+2)}-\frac{145 \sqrt{1-2 x} \sqrt{5 x+3}}{588 (3 x+2)^2}-\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{3 (3 x+2)^3}+\frac{11 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^3}-\frac{2805 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(11*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x
])/(3*(2 + 3*x)^3) - (145*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(588*(2 + 3*x)^2) - (415*
Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(8232*(2 + 3*x)) - (2805*ArcTan[Sqrt[1 - 2*x]/(Sqrt
[7]*Sqrt[3 + 5*x])])/(2744*Sqrt[7])

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Rubi in Sympy [A]  time = 29.2741, size = 138, normalized size = 0.91 \[ - \frac{415 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{8232 \left (3 x + 2\right )} - \frac{145 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{588 \left (3 x + 2\right )^{2}} - \frac{2 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{3 \left (3 x + 2\right )^{3}} - \frac{2805 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{19208} + \frac{11 \sqrt{5 x + 3}}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-415*sqrt(-2*x + 1)*sqrt(5*x + 3)/(8232*(3*x + 2)) - 145*sqrt(-2*x + 1)*sqrt(5*x
+ 3)/(588*(3*x + 2)**2) - 2*sqrt(-2*x + 1)*sqrt(5*x + 3)/(3*(3*x + 2)**3) - 280
5*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/19208 + 11*sqrt(5*x + 3
)/(7*sqrt(-2*x + 1)*(3*x + 2)**3)

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Mathematica [A]  time = 0.115583, size = 82, normalized size = 0.54 \[ \frac{\frac{14 \sqrt{5 x+3} \left (2490 x^3+6135 x^2+3782 x+576\right )}{\sqrt{1-2 x} (3 x+2)^3}-2805 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{38416} \]

Antiderivative was successfully verified.

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

[Out]

((14*Sqrt[3 + 5*x]*(576 + 3782*x + 6135*x^2 + 2490*x^3))/(Sqrt[1 - 2*x]*(2 + 3*x
)^3) - 2805*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/38416

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Maple [B]  time = 0.023, size = 257, normalized size = 1.7 \[{\frac{1}{38416\, \left ( 2+3\,x \right ) ^{3} \left ( -1+2\,x \right ) } \left ( 151470\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+227205\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+50490\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-34860\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-56100\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-85890\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-22440\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -52948\,x\sqrt{-10\,{x}^{2}-x+3}-8064\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/38416*(151470*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+2
27205*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+50490*7^(1/
2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-34860*x^3*(-10*x^2-x+3
)^(1/2)-56100*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-85890
*x^2*(-10*x^2-x+3)^(1/2)-22440*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+
3)^(1/2))-52948*x*(-10*x^2-x+3)^(1/2)-8064*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3
+5*x)^(1/2)/(2+3*x)^3/(-1+2*x)/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.50499, size = 285, normalized size = 1.89 \[ \frac{2805}{38416} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{2075 \, x}{12348 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{4415}{24696 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{1}{189 \,{\left (27 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt{-10 \, x^{2} - x + 3} x + 8 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{53}{756 \,{\left (9 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt{-10 \, x^{2} - x + 3} x + 4 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} - \frac{275}{1176 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2805/38416*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 2075/1234
8*x/sqrt(-10*x^2 - x + 3) + 4415/24696/sqrt(-10*x^2 - x + 3) - 1/189/(27*sqrt(-1
0*x^2 - x + 3)*x^3 + 54*sqrt(-10*x^2 - x + 3)*x^2 + 36*sqrt(-10*x^2 - x + 3)*x +
8*sqrt(-10*x^2 - x + 3)) + 53/756/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^
2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3)) - 275/1176/(3*sqrt(-10*x^2 - x + 3)*x +
2*sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 0.237093, size = 147, normalized size = 0.97 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (2490 \, x^{3} + 6135 \, x^{2} + 3782 \, x + 576\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 2805 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{38416 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/38416*sqrt(7)*(2*sqrt(7)*(2490*x^3 + 6135*x^2 + 3782*x + 576)*sqrt(5*x + 3)*s
qrt(-2*x + 1) - 2805*(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*arctan(1/14*sqrt(7)*(
37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8
)

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.476243, size = 464, normalized size = 3.07 \[ \frac{561}{76832} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{88 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{12005 \,{\left (2 \, x - 1\right )}} - \frac{11 \,{\left (1849 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 1386560 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 15601600 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{9604 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

561/76832*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)))) - 88/12005*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) - 11/9604*(184
9*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 + 1386560*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) - sqr
t(22)))^3 + 15601600*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) - sqr
t(22)))^2 + 280)^3