3.2467 \(\int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x} (2+3 x)^3} \, dx\)

Optimal. Leaf size=120 \[ \frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{42 (3 x+2)^2}+\frac{239 \sqrt{1-2 x} \sqrt{5 x+3}}{1764 (3 x+2)}+\frac{25}{27} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{17687 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{5292 \sqrt{7}} \]

[Out]

(239*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1764*(2 + 3*x)) + (Sqrt[1 - 2*x]*(3 + 5*x)^(3
/2))/(42*(2 + 3*x)^2) + (25*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/27 + (176
87*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(5292*Sqrt[7])

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Rubi [A]  time = 0.232804, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{42 (3 x+2)^2}+\frac{239 \sqrt{1-2 x} \sqrt{5 x+3}}{1764 (3 x+2)}+\frac{25}{27} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{17687 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{5292 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(239*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1764*(2 + 3*x)) + (Sqrt[1 - 2*x]*(3 + 5*x)^(3
/2))/(42*(2 + 3*x)^2) + (25*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/27 + (176
87*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(5292*Sqrt[7])

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Rubi in Sympy [A]  time = 22.8996, size = 107, normalized size = 0.89 \[ \frac{239 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1764 \left (3 x + 2\right )} + \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{42 \left (3 x + 2\right )^{2}} + \frac{25 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{27} + \frac{17687 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{37044} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

239*sqrt(-2*x + 1)*sqrt(5*x + 3)/(1764*(3*x + 2)) + sqrt(-2*x + 1)*(5*x + 3)**(3
/2)/(42*(3*x + 2)**2) + 25*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/27 + 17687*s
qrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/37044

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Mathematica [A]  time = 0.179509, size = 107, normalized size = 0.89 \[ \frac{\frac{42 \sqrt{1-2 x} \sqrt{5 x+3} (927 x+604)}{(3 x+2)^2}+17687 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+34300 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{74088} \]

Antiderivative was successfully verified.

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

[Out]

((42*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(604 + 927*x))/(2 + 3*x)^2 + 17687*Sqrt[7]*ArcT
an[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] + 34300*Sqrt[10]*ArcTan[(1 + 2
0*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/74088

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Maple [B]  time = 0.02, size = 191, normalized size = 1.6 \[ -{\frac{1}{74088\, \left ( 2+3\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 159183\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-308700\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+212244\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-411600\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+70748\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -137200\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -38934\,x\sqrt{-10\,{x}^{2}-x+3}-25368\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/74088*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(159183*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/
2)/(-10*x^2-x+3)^(1/2))*x^2-308700*10^(1/2)*arcsin(20/11*x+1/11)*x^2+212244*7^(1
/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-411600*10^(1/2)*arcsin(
20/11*x+1/11)*x+70748*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))
-137200*10^(1/2)*arcsin(20/11*x+1/11)-38934*x*(-10*x^2-x+3)^(1/2)-25368*(-10*x^2
-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^2

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Maxima [A]  time = 1.50838, size = 117, normalized size = 0.98 \[ \frac{25}{54} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{17687}{74088} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{\sqrt{-10 \, x^{2} - x + 3}}{126 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{103 \, \sqrt{-10 \, x^{2} - x + 3}}{588 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/54*sqrt(10)*arcsin(20/11*x + 1/11) - 17687/74088*sqrt(7)*arcsin(37/11*x/abs(3
*x + 2) + 20/11/abs(3*x + 2)) - 1/126*sqrt(-10*x^2 - x + 3)/(9*x^2 + 12*x + 4) +
103/588*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.228817, size = 165, normalized size = 1.38 \[ \frac{\sqrt{7}{\left (4900 \, \sqrt{10} \sqrt{7}{\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 6 \, \sqrt{7}{\left (927 \, x + 604\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 17687 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{74088 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/74088*sqrt(7)*(4900*sqrt(10)*sqrt(7)*(9*x^2 + 12*x + 4)*arctan(1/20*sqrt(10)*(
20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) + 6*sqrt(7)*(927*x + 604)*sqrt(5*x + 3
)*sqrt(-2*x + 1) - 17687*(9*x^2 + 12*x + 4)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqr
t(5*x + 3)*sqrt(-2*x + 1))))/(9*x^2 + 12*x + 4)

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.355943, size = 437, normalized size = 3.64 \[ -\frac{17687}{740880} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{25}{54} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{11 \,{\left (239 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 85400 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{882 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-17687/740880*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((s
qrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - s
qrt(22)))) + 25/54*sqrt(10)*(pi + 2*arctan(-1/4*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)))) + 11
/882*(239*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 + 85400*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)^2