3.2498 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=137 \[ \frac{20743985 \sqrt{1-2 x}}{71148 \sqrt{5 x+3}}-\frac{207895 \sqrt{1-2 x}}{6468 (5 x+3)^{3/2}}+\frac{753 \sqrt{1-2 x}}{196 (3 x+2) (5 x+3)^{3/2}}+\frac{3 \sqrt{1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}-\frac{392283 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{196 \sqrt{7}} \]

[Out]

(-207895*Sqrt[1 - 2*x])/(6468*(3 + 5*x)^(3/2)) + (3*Sqrt[1 - 2*x])/(14*(2 + 3*x)
^2*(3 + 5*x)^(3/2)) + (753*Sqrt[1 - 2*x])/(196*(2 + 3*x)*(3 + 5*x)^(3/2)) + (207
43985*Sqrt[1 - 2*x])/(71148*Sqrt[3 + 5*x]) - (392283*ArcTan[Sqrt[1 - 2*x]/(Sqrt[
7]*Sqrt[3 + 5*x])])/(196*Sqrt[7])

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Rubi [A]  time = 0.321724, antiderivative size = 137, 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 \[ \frac{20743985 \sqrt{1-2 x}}{71148 \sqrt{5 x+3}}-\frac{207895 \sqrt{1-2 x}}{6468 (5 x+3)^{3/2}}+\frac{753 \sqrt{1-2 x}}{196 (3 x+2) (5 x+3)^{3/2}}+\frac{3 \sqrt{1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}-\frac{392283 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{196 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-207895*Sqrt[1 - 2*x])/(6468*(3 + 5*x)^(3/2)) + (3*Sqrt[1 - 2*x])/(14*(2 + 3*x)
^2*(3 + 5*x)^(3/2)) + (753*Sqrt[1 - 2*x])/(196*(2 + 3*x)*(3 + 5*x)^(3/2)) + (207
43985*Sqrt[1 - 2*x])/(71148*Sqrt[3 + 5*x]) - (392283*ArcTan[Sqrt[1 - 2*x]/(Sqrt[
7]*Sqrt[3 + 5*x])])/(196*Sqrt[7])

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Rubi in Sympy [A]  time = 28.0494, size = 126, normalized size = 0.92 \[ \frac{20743985 \sqrt{- 2 x + 1}}{71148 \sqrt{5 x + 3}} - \frac{207895 \sqrt{- 2 x + 1}}{6468 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{753 \sqrt{- 2 x + 1}}{196 \left (3 x + 2\right ) \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{3 \sqrt{- 2 x + 1}}{14 \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{392283 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{1372} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

20743985*sqrt(-2*x + 1)/(71148*sqrt(5*x + 3)) - 207895*sqrt(-2*x + 1)/(6468*(5*x
+ 3)**(3/2)) + 753*sqrt(-2*x + 1)/(196*(3*x + 2)*(5*x + 3)**(3/2)) + 3*sqrt(-2*
x + 1)/(14*(3*x + 2)**2*(5*x + 3)**(3/2)) - 392283*sqrt(7)*atan(sqrt(7)*sqrt(-2*
x + 1)/(7*sqrt(5*x + 3)))/1372

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Mathematica [A]  time = 0.110367, size = 82, normalized size = 0.6 \[ \frac{\sqrt{1-2 x} \left (933479325 x^3+1784145090 x^2+1135041037 x+240342364\right )}{71148 (3 x+2)^2 (5 x+3)^{3/2}}-\frac{392283 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{392 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(240342364 + 1135041037*x + 1784145090*x^2 + 933479325*x^3))/(711
48*(2 + 3*x)^2*(3 + 5*x)^(3/2)) - (392283*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*
Sqrt[3 + 5*x])])/(392*Sqrt[7])

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Maple [B]  time = 0.024, size = 250, normalized size = 1.8 \[{\frac{1}{996072\, \left ( 2+3\,x \right ) ^{2}} \left ( 32039714025\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+24978031260\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/996072*(32039714025*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))
*x^4+81167275530*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+
77037712389*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+13068
710550*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+24978031260*x^2*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*a
rctan(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]*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(2+3*x)^2/(-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. \[ \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{3} \sqrt{-2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.231444, size = 147, normalized size = 1.07 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (933479325 \, x^{3} + 1784145090 \, x^{2} + 1135041037 \, x + 240342364\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 142398729 \,{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{996072 \,{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/996072*sqrt(7)*(2*sqrt(7)*(933479325*x^3 + 1784145090*x^2 + 1135041037*x + 240
342364)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 142398729*(225*x^4 + 570*x^3 + 541*x^2 +
228*x + 36)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(22
5*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.336747, size = 509, normalized size = 3.72 \[ -\frac{25}{5808} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + \frac{392283}{27440} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{2425}{242} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} + \frac{297 \,{\left (461 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 110600 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{98 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25/5808*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 + 392283/27440*sqrt(70)*sqrt(10)
*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(2
2))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 2425/242*sqrt(10)*
((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*s
qrt(-10*x + 5) - sqrt(22))) + 297/98*(461*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - s
qrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3
+ 110600*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) - sqr
t(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 +
280)^2