3.1937 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^8} \, dx\)

Optimal. Leaf size=168 \[ \frac{23 (1-2 x)^{7/2}}{882 (3 x+2)^6}-\frac{(1-2 x)^{7/2}}{441 (3 x+2)^7}-\frac{467 (1-2 x)^{5/2}}{2646 (3 x+2)^5}+\frac{2335 (1-2 x)^{3/2}}{31752 (3 x+2)^4}+\frac{2335 \sqrt{1-2 x}}{3111696 (3 x+2)}+\frac{2335 \sqrt{1-2 x}}{1333584 (3 x+2)^2}-\frac{2335 \sqrt{1-2 x}}{95256 (3 x+2)^3}+\frac{2335 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1555848 \sqrt{21}} \]

[Out]

-(1 - 2*x)^(7/2)/(441*(2 + 3*x)^7) + (23*(1 - 2*x)^(7/2))/(882*(2 + 3*x)^6) - (4
67*(1 - 2*x)^(5/2))/(2646*(2 + 3*x)^5) + (2335*(1 - 2*x)^(3/2))/(31752*(2 + 3*x)
^4) - (2335*Sqrt[1 - 2*x])/(95256*(2 + 3*x)^3) + (2335*Sqrt[1 - 2*x])/(1333584*(
2 + 3*x)^2) + (2335*Sqrt[1 - 2*x])/(3111696*(2 + 3*x)) + (2335*ArcTanh[Sqrt[3/7]
*Sqrt[1 - 2*x]])/(1555848*Sqrt[21])

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Rubi [A]  time = 0.208574, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{23 (1-2 x)^{7/2}}{882 (3 x+2)^6}-\frac{(1-2 x)^{7/2}}{441 (3 x+2)^7}-\frac{467 (1-2 x)^{5/2}}{2646 (3 x+2)^5}+\frac{2335 (1-2 x)^{3/2}}{31752 (3 x+2)^4}+\frac{2335 \sqrt{1-2 x}}{3111696 (3 x+2)}+\frac{2335 \sqrt{1-2 x}}{1333584 (3 x+2)^2}-\frac{2335 \sqrt{1-2 x}}{95256 (3 x+2)^3}+\frac{2335 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1555848 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(5/2)*(3 + 5*x)^2)/(2 + 3*x)^8,x]

[Out]

-(1 - 2*x)^(7/2)/(441*(2 + 3*x)^7) + (23*(1 - 2*x)^(7/2))/(882*(2 + 3*x)^6) - (4
67*(1 - 2*x)^(5/2))/(2646*(2 + 3*x)^5) + (2335*(1 - 2*x)^(3/2))/(31752*(2 + 3*x)
^4) - (2335*Sqrt[1 - 2*x])/(95256*(2 + 3*x)^3) + (2335*Sqrt[1 - 2*x])/(1333584*(
2 + 3*x)^2) + (2335*Sqrt[1 - 2*x])/(3111696*(2 + 3*x)) + (2335*ArcTanh[Sqrt[3/7]
*Sqrt[1 - 2*x]])/(1555848*Sqrt[21])

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Rubi in Sympy [A]  time = 19.5954, size = 150, normalized size = 0.89 \[ \frac{23 \left (- 2 x + 1\right )^{\frac{7}{2}}}{882 \left (3 x + 2\right )^{6}} - \frac{\left (- 2 x + 1\right )^{\frac{7}{2}}}{441 \left (3 x + 2\right )^{7}} - \frac{467 \left (- 2 x + 1\right )^{\frac{5}{2}}}{2646 \left (3 x + 2\right )^{5}} + \frac{2335 \left (- 2 x + 1\right )^{\frac{3}{2}}}{31752 \left (3 x + 2\right )^{4}} + \frac{2335 \sqrt{- 2 x + 1}}{3111696 \left (3 x + 2\right )} + \frac{2335 \sqrt{- 2 x + 1}}{1333584 \left (3 x + 2\right )^{2}} - \frac{2335 \sqrt{- 2 x + 1}}{95256 \left (3 x + 2\right )^{3}} + \frac{2335 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{32672808} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)*(3+5*x)**2/(2+3*x)**8,x)

[Out]

23*(-2*x + 1)**(7/2)/(882*(3*x + 2)**6) - (-2*x + 1)**(7/2)/(441*(3*x + 2)**7) -
467*(-2*x + 1)**(5/2)/(2646*(3*x + 2)**5) + 2335*(-2*x + 1)**(3/2)/(31752*(3*x
+ 2)**4) + 2335*sqrt(-2*x + 1)/(3111696*(3*x + 2)) + 2335*sqrt(-2*x + 1)/(133358
4*(3*x + 2)**2) - 2335*sqrt(-2*x + 1)/(95256*(3*x + 2)**3) + 2335*sqrt(21)*atanh
(sqrt(21)*sqrt(-2*x + 1)/7)/32672808

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Mathematica [A]  time = 0.137874, size = 78, normalized size = 0.46 \[ \frac{\frac{21 \sqrt{1-2 x} \left (1702215 x^6+8132805 x^5-24492348 x^4-23950566 x^3+1405308 x^2+1415408 x-1107536\right )}{(3 x+2)^7}+4670 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{65345616} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^2)/(2 + 3*x)^8,x]

[Out]

((21*Sqrt[1 - 2*x]*(-1107536 + 1415408*x + 1405308*x^2 - 23950566*x^3 - 24492348
*x^4 + 8132805*x^5 + 1702215*x^6))/(2 + 3*x)^7 + 4670*Sqrt[21]*ArcTanh[Sqrt[3/7]
*Sqrt[1 - 2*x]])/65345616

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Maple [A]  time = 0.02, size = 93, normalized size = 0.6 \[ -139968\,{\frac{1}{ \left ( -4-6\,x \right ) ^{7}} \left ({\frac{2335\, \left ( 1-2\,x \right ) ^{13/2}}{298722816}}-{\frac{11675\, \left ( 1-2\,x \right ) ^{11/2}}{96018048}}+{\frac{6721\, \left ( 1-2\,x \right ) ^{9/2}}{164602368}}+{\frac{571\, \left ( 1-2\,x \right ) ^{7/2}}{321489}}-{\frac{132161\, \left ( 1-2\,x \right ) ^{5/2}}{30233088}}+{\frac{81725\, \left ( 1-2\,x \right ) ^{3/2}}{22674816}}-{\frac{114415\,\sqrt{1-2\,x}}{90699264}} \right ) }+{\frac{2335\,\sqrt{21}}{32672808}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)*(3+5*x)^2/(2+3*x)^8,x)

[Out]

-139968*(2335/298722816*(1-2*x)^(13/2)-11675/96018048*(1-2*x)^(11/2)+6721/164602
368*(1-2*x)^(9/2)+571/321489*(1-2*x)^(7/2)-132161/30233088*(1-2*x)^(5/2)+81725/2
2674816*(1-2*x)^(3/2)-114415/90699264*(1-2*x)^(1/2))/(-4-6*x)^7+2335/32672808*ar
ctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.50551, size = 221, normalized size = 1.32 \[ -\frac{2335}{65345616} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{1702215 \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} - 26478900 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + 8891883 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + 386781696 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 951955683 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 784886900 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 274710415 \, \sqrt{-2 \, x + 1}}{1555848 \,{\left (2187 \,{\left (2 \, x - 1\right )}^{7} + 35721 \,{\left (2 \, x - 1\right )}^{6} + 250047 \,{\left (2 \, x - 1\right )}^{5} + 972405 \,{\left (2 \, x - 1\right )}^{4} + 2268945 \,{\left (2 \, x - 1\right )}^{3} + 3176523 \,{\left (2 \, x - 1\right )}^{2} + 4941258 \, x - 1647086\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^2*(-2*x + 1)^(5/2)/(3*x + 2)^8,x, algorithm="maxima")

[Out]

-2335/65345616*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2
*x + 1))) + 1/1555848*(1702215*(-2*x + 1)^(13/2) - 26478900*(-2*x + 1)^(11/2) +
8891883*(-2*x + 1)^(9/2) + 386781696*(-2*x + 1)^(7/2) - 951955683*(-2*x + 1)^(5/
2) + 784886900*(-2*x + 1)^(3/2) - 274710415*sqrt(-2*x + 1))/(2187*(2*x - 1)^7 +
35721*(2*x - 1)^6 + 250047*(2*x - 1)^5 + 972405*(2*x - 1)^4 + 2268945*(2*x - 1)^
3 + 3176523*(2*x - 1)^2 + 4941258*x - 1647086)

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Fricas [A]  time = 0.221586, size = 201, normalized size = 1.2 \[ \frac{\sqrt{21}{\left (\sqrt{21}{\left (1702215 \, x^{6} + 8132805 \, x^{5} - 24492348 \, x^{4} - 23950566 \, x^{3} + 1405308 \, x^{2} + 1415408 \, x - 1107536\right )} \sqrt{-2 \, x + 1} + 2335 \,{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{65345616 \,{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^2*(-2*x + 1)^(5/2)/(3*x + 2)^8,x, algorithm="fricas")

[Out]

1/65345616*sqrt(21)*(sqrt(21)*(1702215*x^6 + 8132805*x^5 - 24492348*x^4 - 239505
66*x^3 + 1405308*x^2 + 1415408*x - 1107536)*sqrt(-2*x + 1) + 2335*(2187*x^7 + 10
206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)*log((sqrt
(21)*(3*x - 5) - 21*sqrt(-2*x + 1))/(3*x + 2)))/(2187*x^7 + 10206*x^6 + 20412*x^
5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)

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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*x)**(5/2)*(3+5*x)**2/(2+3*x)**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.230304, size = 200, normalized size = 1.19 \[ -\frac{2335}{65345616} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{1702215 \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} + 26478900 \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + 8891883 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - 386781696 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - 951955683 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + 784886900 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 274710415 \, \sqrt{-2 \, x + 1}}{199148544 \,{\left (3 \, x + 2\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^2*(-2*x + 1)^(5/2)/(3*x + 2)^8,x, algorithm="giac")

[Out]

-2335/65345616*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3
*sqrt(-2*x + 1))) + 1/199148544*(1702215*(2*x - 1)^6*sqrt(-2*x + 1) + 26478900*(
2*x - 1)^5*sqrt(-2*x + 1) + 8891883*(2*x - 1)^4*sqrt(-2*x + 1) - 386781696*(2*x
- 1)^3*sqrt(-2*x + 1) - 951955683*(2*x - 1)^2*sqrt(-2*x + 1) + 784886900*(-2*x +
1)^(3/2) - 274710415*sqrt(-2*x + 1))/(3*x + 2)^7