3.1949 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=162 \[ \frac{11 (1-2 x)^{3/2} (5 x+3)^3}{27 (3 x+2)^5}-\frac{(1-2 x)^{5/2} (5 x+3)^3}{18 (3 x+2)^6}+\frac{559625 \sqrt{1-2 x}}{1333584 (3 x+2)}-\frac{559625 \sqrt{1-2 x}}{190512 (3 x+2)^2}+\frac{33275 (1-2 x)^{3/2}}{95256 (3 x+2)^3}-\frac{121 (1-2 x)^{3/2}}{4536 (3 x+2)^4}+\frac{559625 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{666792 \sqrt{21}} \]

[Out]

(-121*(1 - 2*x)^(3/2))/(4536*(2 + 3*x)^4) + (33275*(1 - 2*x)^(3/2))/(95256*(2 +
3*x)^3) - (559625*Sqrt[1 - 2*x])/(190512*(2 + 3*x)^2) + (559625*Sqrt[1 - 2*x])/(
1333584*(2 + 3*x)) - ((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(18*(2 + 3*x)^6) + (11*(1 - 2
*x)^(3/2)*(3 + 5*x)^3)/(27*(2 + 3*x)^5) + (559625*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x
]])/(666792*Sqrt[21])

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Rubi [A]  time = 0.258565, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ \frac{11 (1-2 x)^{3/2} (5 x+3)^3}{27 (3 x+2)^5}-\frac{(1-2 x)^{5/2} (5 x+3)^3}{18 (3 x+2)^6}+\frac{559625 \sqrt{1-2 x}}{1333584 (3 x+2)}-\frac{559625 \sqrt{1-2 x}}{190512 (3 x+2)^2}+\frac{33275 (1-2 x)^{3/2}}{95256 (3 x+2)^3}-\frac{121 (1-2 x)^{3/2}}{4536 (3 x+2)^4}+\frac{559625 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{666792 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(2 + 3*x)^7,x]

[Out]

(-121*(1 - 2*x)^(3/2))/(4536*(2 + 3*x)^4) + (33275*(1 - 2*x)^(3/2))/(95256*(2 +
3*x)^3) - (559625*Sqrt[1 - 2*x])/(190512*(2 + 3*x)^2) + (559625*Sqrt[1 - 2*x])/(
1333584*(2 + 3*x)) - ((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(18*(2 + 3*x)^6) + (11*(1 - 2
*x)^(3/2)*(3 + 5*x)^3)/(27*(2 + 3*x)^5) + (559625*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x
]])/(666792*Sqrt[21])

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Rubi in Sympy [A]  time = 22.3947, size = 131, normalized size = 0.81 \[ - \frac{11 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (81675 x + 50985\right )}{2000376 \left (3 x + 2\right )^{4}} - \frac{11 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{2}}{189 \left (3 x + 2\right )^{5}} - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{3}}{18 \left (3 x + 2\right )^{6}} + \frac{559625 \left (- 2 x + 1\right )^{\frac{3}{2}}}{1333584 \left (3 x + 2\right )^{2}} - \frac{559625 \sqrt{- 2 x + 1}}{1333584 \left (3 x + 2\right )} + \frac{559625 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{14002632} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)*(3+5*x)**3/(2+3*x)**7,x)

[Out]

-11*(-2*x + 1)**(5/2)*(81675*x + 50985)/(2000376*(3*x + 2)**4) - 11*(-2*x + 1)**
(5/2)*(5*x + 3)**2/(189*(3*x + 2)**5) - (-2*x + 1)**(5/2)*(5*x + 3)**3/(18*(3*x
+ 2)**6) + 559625*(-2*x + 1)**(3/2)/(1333584*(3*x + 2)**2) - 559625*sqrt(-2*x +
1)/(1333584*(3*x + 2)) + 559625*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/140026
32

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Mathematica [A]  time = 0.14685, size = 73, normalized size = 0.45 \[ \frac{3357750 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{63 \sqrt{1-2 x} \left (308539125 x^5+720187425 x^4+687940758 x^3+352611738 x^2+102558856 x+13847024\right )}{(3 x+2)^6}}{84015792} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(2 + 3*x)^7,x]

[Out]

((-63*Sqrt[1 - 2*x]*(13847024 + 102558856*x + 352611738*x^2 + 687940758*x^3 + 72
0187425*x^4 + 308539125*x^5))/(2 + 3*x)^6 + 3357750*Sqrt[21]*ArcTanh[Sqrt[3/7]*S
qrt[1 - 2*x]])/84015792

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Maple [A]  time = 0.02, size = 84, normalized size = 0.5 \[ -11664\,{\frac{1}{ \left ( -4-6\,x \right ) ^{6}} \left ( -{\frac{3809125\, \left ( 1-2\,x \right ) ^{11/2}}{96018048}}+{\frac{47350325\, \left ( 1-2\,x \right ) ^{9/2}}{123451776}}-{\frac{4383467\, \left ( 1-2\,x \right ) ^{7/2}}{2939328}}+{\frac{1231175\, \left ( 1-2\,x \right ) ^{5/2}}{419904}}-{\frac{66595375\, \left ( 1-2\,x \right ) ^{3/2}}{22674816}}+{\frac{27421625\,\sqrt{1-2\,x}}{22674816}} \right ) }+{\frac{559625\,\sqrt{21}}{14002632}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)*(3+5*x)^3/(2+3*x)^7,x)

[Out]

-11664*(-3809125/96018048*(1-2*x)^(11/2)[phone]/123451776*(1-2*x)^(9/2)-438346
7/2939328*(1-2*x)^(7/2)+1231175/419904*(1-2*x)^(5/2)-66595375/22674816*(1-2*x)^(
3/2)[phone]/22674816*(1-2*x)^(1/2))/(-4-6*x)^6+559625/14002632*arctanh(1/7*21^
(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.52216, size = 197, normalized size = 1.22 \[ -\frac{559625}{28005264} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{308539125 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - 2983070475 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + 11598653682 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 22803823350 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 22842213625 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 9405617375 \, \sqrt{-2 \, x + 1}}{666792 \,{\left (729 \,{\left (2 \, x - 1\right )}^{6} + 10206 \,{\left (2 \, x - 1\right )}^{5} + 59535 \,{\left (2 \, x - 1\right )}^{4} + 185220 \,{\left (2 \, x - 1\right )}^{3} + 324135 \,{\left (2 \, x - 1\right )}^{2} + 605052 \, x - 184877\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^3*(-2*x + 1)^(5/2)/(3*x + 2)^7,x, algorithm="maxima")

[Out]

-559625/28005264*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(
-2*x + 1))) + 1/666792*(308539125*(-2*x + 1)^(11/2) - 2983070475*(-2*x + 1)^(9/2
) + 11598653682*(-2*x + 1)^(7/2) - 22803823350*(-2*x + 1)^(5/2) + 22842213625*(-
2*x + 1)^(3/2) - 9405617375*sqrt(-2*x + 1))/(729*(2*x - 1)^6 + 10206*(2*x - 1)^5
+ 59535*(2*x - 1)^4 + 185220*(2*x - 1)^3 + 324135*(2*x - 1)^2 + 605052*x - 1848
77)

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Fricas [A]  time = 0.214057, size = 181, normalized size = 1.12 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (308539125 \, x^{5} + 720187425 \, x^{4} + 687940758 \, x^{3} + 352611738 \, x^{2} + 102558856 \, x + 13847024\right )} \sqrt{-2 \, x + 1} - 559625 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{28005264 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^3*(-2*x + 1)^(5/2)/(3*x + 2)^7,x, algorithm="fricas")

[Out]

-1/28005264*sqrt(21)*(sqrt(21)*(308539125*x^5 + 720187425*x^4 + 687940758*x^3 +
352611738*x^2 + 102558856*x + 13847024)*sqrt(-2*x + 1) - 559625*(729*x^6 + 2916*
x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*log((sqrt(21)*(3*x - 5) - 21*
sqrt(-2*x + 1))/(3*x + 2)))/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2
+ 576*x + 64)

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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)**3/(2+3*x)**7,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.235034, size = 178, normalized size = 1.1 \[ -\frac{559625}{28005264} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{308539125 \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + 2983070475 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 11598653682 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 22803823350 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 22842213625 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 9405617375 \, \sqrt{-2 \, x + 1}}{42674688 \,{\left (3 \, x + 2\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^3*(-2*x + 1)^(5/2)/(3*x + 2)^7,x, algorithm="giac")

[Out]

-559625/28005264*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) +
3*sqrt(-2*x + 1))) - 1/42674688*(308539125*(2*x - 1)^5*sqrt(-2*x + 1) + 2983070
475*(2*x - 1)^4*sqrt(-2*x + 1) + 11598653682*(2*x - 1)^3*sqrt(-2*x + 1) + 228038
23350*(2*x - 1)^2*sqrt(-2*x + 1) - 22842213625*(-2*x + 1)^(3/2) + 9405617375*sqr
t(-2*x + 1))/(3*x + 2)^6