3.2530 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx\)

Optimal. Leaf size=151 \[ \frac{4 (5 x+3)^{7/2}}{77 \sqrt{1-2 x} (3 x+2)^3}-\frac{\sqrt{1-2 x} (5 x+3)^{5/2}}{77 (3 x+2)^3}-\frac{5 \sqrt{1-2 x} (5 x+3)^{3/2}}{196 (3 x+2)^2}-\frac{165 \sqrt{1-2 x} \sqrt{5 x+3}}{2744 (3 x+2)}-\frac{1815 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]

[Out]

(-165*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2744*(2 + 3*x)) - (5*Sqrt[1 - 2*x]*(3 + 5*x)
^(3/2))/(196*(2 + 3*x)^2) - (Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(77*(2 + 3*x)^3) + (
4*(3 + 5*x)^(7/2))/(77*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (1815*ArcTan[Sqrt[1 - 2*x]/(
Sqrt[7]*Sqrt[3 + 5*x])])/(2744*Sqrt[7])

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Rubi [A]  time = 0.218719, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{4 (5 x+3)^{7/2}}{77 \sqrt{1-2 x} (3 x+2)^3}-\frac{\sqrt{1-2 x} (5 x+3)^{5/2}}{77 (3 x+2)^3}-\frac{5 \sqrt{1-2 x} (5 x+3)^{3/2}}{196 (3 x+2)^2}-\frac{165 \sqrt{1-2 x} \sqrt{5 x+3}}{2744 (3 x+2)}-\frac{1815 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-165*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2744*(2 + 3*x)) - (5*Sqrt[1 - 2*x]*(3 + 5*x)
^(3/2))/(196*(2 + 3*x)^2) - (Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(77*(2 + 3*x)^3) + (
4*(3 + 5*x)^(7/2))/(77*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (1815*ArcTan[Sqrt[1 - 2*x]/(
Sqrt[7]*Sqrt[3 + 5*x])])/(2744*Sqrt[7])

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Rubi in Sympy [A]  time = 17.1911, size = 134, normalized size = 0.89 \[ - \frac{165 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2744 \left (3 x + 2\right )} - \frac{5 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{196 \left (3 x + 2\right )^{2}} - \frac{1815 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{19208} - \frac{\left (5 x + 3\right )^{\frac{5}{2}}}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}} + \frac{\left (5 x + 3\right )^{\frac{7}{2}}}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-165*sqrt(-2*x + 1)*sqrt(5*x + 3)/(2744*(3*x + 2)) - 5*sqrt(-2*x + 1)*(5*x + 3)*
*(3/2)/(196*(3*x + 2)**2) - 1815*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x
+ 3)))/19208 - (5*x + 3)**(5/2)/(7*sqrt(-2*x + 1)*(3*x + 2)**2) + (5*x + 3)**(7
/2)/(7*sqrt(-2*x + 1)*(3*x + 2)**3)

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Mathematica [A]  time = 0.126639, size = 82, normalized size = 0.54 \[ \frac{\frac{14 \sqrt{5 x+3} \left (24670 x^3+37405 x^2+17666 x+2448\right )}{\sqrt{1-2 x} (3 x+2)^3}-1815 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{38416} \]

Antiderivative was successfully verified.

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

[Out]

((14*Sqrt[3 + 5*x]*(2448 + 17666*x + 37405*x^2 + 24670*x^3))/(Sqrt[1 - 2*x]*(2 +
3*x)^3) - 1815*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/3
8416

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Maple [B]  time = 0.022, size = 257, normalized size = 1.7 \[{\frac{1}{38416\, \left ( 2+3\,x \right ) ^{3} \left ( -1+2\,x \right ) } \left ( 98010\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+147015\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+32670\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-345380\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-36300\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-523670\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-14520\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -247324\,x\sqrt{-10\,{x}^{2}-x+3}-34272\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/38416*(98010*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+14
7015*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+32670*7^(1/2
)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-345380*x^3*(-10*x^2-x+3
)^(1/2)-36300*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-52367
0*x^2*(-10*x^2-x+3)^(1/2)-14520*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x
+3)^(1/2))-247324*x*(-10*x^2-x+3)^(1/2)-34272*(-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 = 1.49981, size = 285, normalized size = 1.89 \[ \frac{1815}{38416} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{61675 \, x}{37044 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{14335}{74088 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1}{567 \,{\left (27 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt{-10 \, x^{2} - x + 3} x + 8 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} - \frac{83}{2268 \,{\left (9 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt{-10 \, x^{2} - x + 3} x + 4 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{3175}{10584 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1815/38416*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 61675/370
44*x/sqrt(-10*x^2 - x + 3) + 14335/74088/sqrt(-10*x^2 - x + 3) + 1/567/(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)) - 83/2268/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10
*x^2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3)) + 3175/10584/(3*sqrt(-10*x^2 - x + 3)
*x + 2*sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 0.23997, size = 147, normalized size = 0.97 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (24670 \, x^{3} + 37405 \, x^{2} + 17666 \, x + 2448\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 1815 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{38416 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/38416*sqrt(7)*(2*sqrt(7)*(24670*x^3 + 37405*x^2 + 17666*x + 2448)*sqrt(5*x +
3)*sqrt(-2*x + 1) - 1815*(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(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.494541, size = 464, normalized size = 3.07 \[ \frac{363}{76832} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{484 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{12005 \,{\left (2 \, x - 1\right )}} - \frac{121 \,{\left (137 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 105280 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 25636800 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{9604 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

363/76832*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)))) - 484/12005*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) - 121/9604*(1
37*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 + 105280*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 + 25636800*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