3.2499 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^4 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=166 \[ \frac{2184369575 \sqrt{1-2 x}}{996072 \sqrt{5 x+3}}-\frac{21891025 \sqrt{1-2 x}}{90552 (5 x+3)^{3/2}}+\frac{79335 \sqrt{1-2 x}}{2744 (3 x+2) (5 x+3)^{3/2}}+\frac{325 \sqrt{1-2 x}}{196 (3 x+2)^2 (5 x+3)^{3/2}}+\frac{\sqrt{1-2 x}}{7 (3 x+2)^3 (5 x+3)^{3/2}}-\frac{41307885 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]

[Out]

(-21891025*Sqrt[1 - 2*x])/(90552*(3 + 5*x)^(3/2)) + Sqrt[1 - 2*x]/(7*(2 + 3*x)^3
*(3 + 5*x)^(3/2)) + (325*Sqrt[1 - 2*x])/(196*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (793
35*Sqrt[1 - 2*x])/(2744*(2 + 3*x)*(3 + 5*x)^(3/2)) + (2184369575*Sqrt[1 - 2*x])/
(996072*Sqrt[3 + 5*x]) - (41307885*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]
)/(2744*Sqrt[7])

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Rubi [A]  time = 0.40114, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{2184369575 \sqrt{1-2 x}}{996072 \sqrt{5 x+3}}-\frac{21891025 \sqrt{1-2 x}}{90552 (5 x+3)^{3/2}}+\frac{79335 \sqrt{1-2 x}}{2744 (3 x+2) (5 x+3)^{3/2}}+\frac{325 \sqrt{1-2 x}}{196 (3 x+2)^2 (5 x+3)^{3/2}}+\frac{\sqrt{1-2 x}}{7 (3 x+2)^3 (5 x+3)^{3/2}}-\frac{41307885 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-21891025*Sqrt[1 - 2*x])/(90552*(3 + 5*x)^(3/2)) + Sqrt[1 - 2*x]/(7*(2 + 3*x)^3
*(3 + 5*x)^(3/2)) + (325*Sqrt[1 - 2*x])/(196*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (793
35*Sqrt[1 - 2*x])/(2744*(2 + 3*x)*(3 + 5*x)^(3/2)) + (2184369575*Sqrt[1 - 2*x])/
(996072*Sqrt[3 + 5*x]) - (41307885*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]
)/(2744*Sqrt[7])

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Rubi in Sympy [A]  time = 35.3827, size = 151, normalized size = 0.91 \[ \frac{2184369575 \sqrt{- 2 x + 1}}{996072 \sqrt{5 x + 3}} - \frac{21891025 \sqrt{- 2 x + 1}}{90552 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{79335 \sqrt{- 2 x + 1}}{2744 \left (3 x + 2\right ) \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{325 \sqrt{- 2 x + 1}}{196 \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{\sqrt{- 2 x + 1}}{7 \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{41307885 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{19208} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2184369575*sqrt(-2*x + 1)/(996072*sqrt(5*x + 3)) - 21891025*sqrt(-2*x + 1)/(9055
2*(5*x + 3)**(3/2)) + 79335*sqrt(-2*x + 1)/(2744*(3*x + 2)*(5*x + 3)**(3/2)) + 3
25*sqrt(-2*x + 1)/(196*(3*x + 2)**2*(5*x + 3)**(3/2)) + sqrt(-2*x + 1)/(7*(3*x +
2)**3*(5*x + 3)**(3/2)) - 41307885*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(
5*x + 3)))/19208

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Mathematica [A]  time = 0.13611, size = 87, normalized size = 0.52 \[ \frac{\sqrt{1-2 x} \left (294889892625 x^4+760212086400 x^3+734310313245 x^2+314968389410 x+50617099616\right )}{996072 (3 x+2)^3 (5 x+3)^{3/2}}-\frac{41307885 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{5488 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(50617099616 + 314968389410*x + 734310313245*x^2 + 760212086400*x
^3 + 294889892625*x^4))/(996072*(2 + 3*x)^3*(3 + 5*x)^(3/2)) - (41307885*ArcTan[
(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/(5488*Sqrt[7])

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Maple [B]  time = 0.024, size = 298, normalized size = 1.8 \[{\frac{1}{13945008\, \left ( 2+3\,x \right ) ^{3}} \left ( 10121464522125\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+10280344385430\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/13945008*(10121464522125*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(
1/2))*x^5+32388686470800*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/
2))*x^4+41430528110565*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2)
)*x^3+4128458496750*x^4*(-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^2+10642969209600*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+1028
0344385430*x^2*(-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))[phone]*x*(-10*x^2-x+3)^(1/2)[phone]*(-1
0*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(2+3*x)^3/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.234379, size = 167, normalized size = 1.01 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (294889892625 \, x^{4} + 760212086400 \, x^{3} + 734310313245 \, x^{2} + 314968389410 \, x + 50617099616\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 14994762255 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{13945008 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/13945008*sqrt(7)*(2*sqrt(7)*(294889892625*x^4 + 760212086400*x^3 + 73431031324
5*x^2 + 314968389410*x + 50617099616)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 14994762255
*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*arctan(1/14*sqrt(7)*(37
*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*
x^2 + 564*x + 72)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.408014, size = 591, normalized size = 3.56 \[ -\frac{125}{5808} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + \frac{8261577}{76832} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{8125}{121} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} + \frac{1485 \,{\left (13759 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 6614720 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 818950720 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{1372 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-125/5808*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 + 8261577/76832*sqrt(70)*sqrt(1
0)*(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)))) + 8125/121*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))) + 1485/1372*(13759*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 + 6614720*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 + 818950720*sqrt(10)*((
sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqr
t(-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