3.2542 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x)^4 \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=144 \[ -\frac{32735 \sqrt{5 x+3}}{15092 \sqrt{1-2 x}}+\frac{2865 \sqrt{5 x+3}}{392 \sqrt{1-2 x} (3 x+2)}+\frac{27 \sqrt{5 x+3}}{28 \sqrt{1-2 x} (3 x+2)^2}+\frac{\sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^3}-\frac{102345 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]

[Out]

(-32735*Sqrt[3 + 5*x])/(15092*Sqrt[1 - 2*x]) + Sqrt[3 + 5*x]/(7*Sqrt[1 - 2*x]*(2
+ 3*x)^3) + (27*Sqrt[3 + 5*x])/(28*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (2865*Sqrt[3 +
5*x])/(392*Sqrt[1 - 2*x]*(2 + 3*x)) - (102345*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt
[3 + 5*x])])/(2744*Sqrt[7])

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Rubi [A]  time = 0.326419, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{32735 \sqrt{5 x+3}}{15092 \sqrt{1-2 x}}+\frac{2865 \sqrt{5 x+3}}{392 \sqrt{1-2 x} (3 x+2)}+\frac{27 \sqrt{5 x+3}}{28 \sqrt{1-2 x} (3 x+2)^2}+\frac{\sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^3}-\frac{102345 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-32735*Sqrt[3 + 5*x])/(15092*Sqrt[1 - 2*x]) + Sqrt[3 + 5*x]/(7*Sqrt[1 - 2*x]*(2
+ 3*x)^3) + (27*Sqrt[3 + 5*x])/(28*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (2865*Sqrt[3 +
5*x])/(392*Sqrt[1 - 2*x]*(2 + 3*x)) - (102345*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt
[3 + 5*x])])/(2744*Sqrt[7])

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Rubi in Sympy [A]  time = 29.7378, size = 131, normalized size = 0.91 \[ - \frac{102345 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{19208} - \frac{32735 \sqrt{5 x + 3}}{15092 \sqrt{- 2 x + 1}} + \frac{2865 \sqrt{5 x + 3}}{392 \sqrt{- 2 x + 1} \left (3 x + 2\right )} + \frac{27 \sqrt{5 x + 3}}{28 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}} + \frac{\sqrt{5 x + 3}}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-102345*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/19208 - 32735*sqr
t(5*x + 3)/(15092*sqrt(-2*x + 1)) + 2865*sqrt(5*x + 3)/(392*sqrt(-2*x + 1)*(3*x
+ 2)) + 27*sqrt(5*x + 3)/(28*sqrt(-2*x + 1)*(3*x + 2)**2) + sqrt(5*x + 3)/(7*sqr
t(-2*x + 1)*(3*x + 2)**3)

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Mathematica [A]  time = 0.118313, size = 82, normalized size = 0.57 \[ \frac{\frac{14 \sqrt{5 x+3} \left (-1767690 x^3-1549935 x^2+377658 x+421184\right )}{\sqrt{1-2 x} (3 x+2)^3}-1125795 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{422576} \]

Antiderivative was successfully verified.

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

[Out]

((14*Sqrt[3 + 5*x]*(421184 + 377658*x - 1549935*x^2 - 1767690*x^3))/(Sqrt[1 - 2*
x]*(2 + 3*x)^3) - 1125795*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 +
5*x])])/422576

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Maple [B]  time = 0.024, size = 257, normalized size = 1.8 \[{\frac{1}{422576\, \left ( 2+3\,x \right ) ^{3} \left ( -1+2\,x \right ) } \left ( 60792930\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-22515900\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+21699090\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-9006360\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -5287212\,x\sqrt{-10\,{x}^{2}-x+3}-5896576\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/422576*(60792930*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^
4+91189395*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+202643
10*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+24747660*x^3*(
-10*x^2-x+3)^(1/2)-22515900*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^
(1/2))*x+21699090*x^2*(-10*x^2-x+3)^(1/2)-9006360*7^(1/2)*arctan(1/14*(37*x+20)*
7^(1/2)/(-10*x^2-x+3)^(1/2))-5287212*x*(-10*x^2-x+3)^(1/2)-5896576*(-10*x^2-x+3)
^(1/2))*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(2+3*x)^3/(-1+2*x)/(-10*x^2-x+3)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.229879, size = 147, normalized size = 1.02 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (1767690 \, x^{3} + 1549935 \, x^{2} - 377658 \, x - 421184\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 1125795 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{422576 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/422576*sqrt(7)*(2*sqrt(7)*(1767690*x^3 + 1549935*x^2 - 377658*x - 421184)*sqrt
(5*x + 3)*sqrt(-2*x + 1) + 1125795*(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(1/(1-2*x)**(3/2)/(2+3*x)**4/(3+5*x)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.430915, size = 464, normalized size = 3.22 \[ \frac{20469}{76832} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{32 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{132055 \,{\left (2 \, x - 1\right )}} + \frac{297 \,{\left (4937 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 1785280 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 188708800 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{9604 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

20469/76832*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)))) - 32/132055*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) + 297/9604*
(4937*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 + 1785280*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 + 188708800*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 + 280)^3