3.2139 \(\int \frac{(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^5} \, dx\)

Optimal. Leaf size=148 \[ -\frac{20465 \sqrt{1-2 x}}{134456 (3 x+2)}-\frac{20465 \sqrt{1-2 x}}{57624 (3 x+2)^2}-\frac{4093 \sqrt{1-2 x}}{4116 (3 x+2)^3}+\frac{4093}{2058 \sqrt{1-2 x} (3 x+2)^3}-\frac{727}{588 \sqrt{1-2 x} (3 x+2)^4}+\frac{121}{42 (1-2 x)^{3/2} (3 x+2)^4}-\frac{20465 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{67228 \sqrt{21}} \]

[Out]

121/(42*(1 - 2*x)^(3/2)*(2 + 3*x)^4) - 727/(588*Sqrt[1 - 2*x]*(2 + 3*x)^4) + 409
3/(2058*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (4093*Sqrt[1 - 2*x])/(4116*(2 + 3*x)^3) - (
20465*Sqrt[1 - 2*x])/(57624*(2 + 3*x)^2) - (20465*Sqrt[1 - 2*x])/(134456*(2 + 3*
x)) - (20465*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(67228*Sqrt[21])

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Rubi [A]  time = 0.189457, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{20465 \sqrt{1-2 x}}{134456 (3 x+2)}-\frac{20465 \sqrt{1-2 x}}{57624 (3 x+2)^2}-\frac{4093 \sqrt{1-2 x}}{4116 (3 x+2)^3}+\frac{4093}{2058 \sqrt{1-2 x} (3 x+2)^3}-\frac{727}{588 \sqrt{1-2 x} (3 x+2)^4}+\frac{121}{42 (1-2 x)^{3/2} (3 x+2)^4}-\frac{20465 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{67228 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

121/(42*(1 - 2*x)^(3/2)*(2 + 3*x)^4) - 727/(588*Sqrt[1 - 2*x]*(2 + 3*x)^4) + 409
3/(2058*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (4093*Sqrt[1 - 2*x])/(4116*(2 + 3*x)^3) - (
20465*Sqrt[1 - 2*x])/(57624*(2 + 3*x)^2) - (20465*Sqrt[1 - 2*x])/(134456*(2 + 3*
x)) - (20465*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(67228*Sqrt[21])

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Rubi in Sympy [A]  time = 16.323, size = 133, normalized size = 0.9 \[ - \frac{20465 \sqrt{- 2 x + 1}}{134456 \left (3 x + 2\right )} - \frac{20465 \sqrt{- 2 x + 1}}{57624 \left (3 x + 2\right )^{2}} - \frac{4093 \sqrt{- 2 x + 1}}{4116 \left (3 x + 2\right )^{3}} - \frac{20465 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{1411788} + \frac{4093}{2058 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}} - \frac{727}{588 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{4}} + \frac{121}{42 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-20465*sqrt(-2*x + 1)/(134456*(3*x + 2)) - 20465*sqrt(-2*x + 1)/(57624*(3*x + 2)
**2) - 4093*sqrt(-2*x + 1)/(4116*(3*x + 2)**3) - 20465*sqrt(21)*atanh(sqrt(21)*s
qrt(-2*x + 1)/7)/1411788 + 4093/(2058*sqrt(-2*x + 1)*(3*x + 2)**3) - 727/(588*sq
rt(-2*x + 1)*(3*x + 2)**4) + 121/(42*(-2*x + 1)**(3/2)*(3*x + 2)**4)

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Mathematica [A]  time = 0.191161, size = 73, normalized size = 0.49 \[ \frac{-\frac{7 \left (6630660 x^5+11787840 x^4+3769653 x^3-3646863 x^2-2528226 x-401410\right )}{(1-2 x)^{3/2} (3 x+2)^4}-40930 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2823576} \]

Antiderivative was successfully verified.

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

[Out]

((-7*(-401410 - 2528226*x - 3646863*x^2 + 3769653*x^3 + 11787840*x^4 + 6630660*x
^5))/((1 - 2*x)^(3/2)*(2 + 3*x)^4) - 40930*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2
*x]])/2823576

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Maple [A]  time = 0.024, size = 84, normalized size = 0.6 \[{\frac{968}{50421} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{8360}{117649}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{648}{117649\, \left ( -4-6\,x \right ) ^{4}} \left ({\frac{42935}{96} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{2847691}{864} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{20832595}{2592} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{5609765}{864}\sqrt{1-2\,x}} \right ) }-{\frac{20465\,\sqrt{21}}{1411788}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

968/50421/(1-2*x)^(3/2)+8360/117649/(1-2*x)^(1/2)+648/117649*(42935/96*(1-2*x)^(
7/2)-2847691/864*(1-2*x)^(5/2)[phone]/2592*(1-2*x)^(3/2)-5609765/864*(1-2*x)^(
1/2))/(-4-6*x)^4-20465/1411788*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.49474, size = 173, normalized size = 1.17 \[ \frac{20465}{2823576} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{1657665 \,{\left (2 \, x - 1\right )}^{5} + 14182245 \,{\left (2 \, x - 1\right )}^{4} + 43921983 \,{\left (2 \, x - 1\right )}^{3} + 55955403 \,{\left (2 \, x - 1\right )}^{2} + 36945216 \, x - 27769280}{201684 \,{\left (81 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - 756 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + 2646 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 4116 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 2401 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

20465/2823576*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*
x + 1))) - 1/201684*(1657665*(2*x - 1)^5 + 14182245*(2*x - 1)^4 + 43921983*(2*x
- 1)^3 + 55955403*(2*x - 1)^2 + 36945216*x - 27769280)/(81*(-2*x + 1)^(11/2) - 7
56*(-2*x + 1)^(9/2) + 2646*(-2*x + 1)^(7/2) - 4116*(-2*x + 1)^(5/2) + 2401*(-2*x
+ 1)^(3/2))

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Fricas [A]  time = 0.218124, size = 177, normalized size = 1.2 \[ \frac{\sqrt{21}{\left (61395 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{21}{\left (6630660 \, x^{5} + 11787840 \, x^{4} + 3769653 \, x^{3} - 3646863 \, x^{2} - 2528226 \, x - 401410\right )}\right )}}{8470728 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8470728*sqrt(21)*(61395*(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)*sqr
t(-2*x + 1)*log((sqrt(21)*(3*x - 5) + 21*sqrt(-2*x + 1))/(3*x + 2)) + sqrt(21)*(
6630660*x^5 + 11787840*x^4 + 3769653*x^3 - 3646863*x^2 - 2528226*x - 401410))/((
162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)*sqrt(-2*x + 1))

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.219062, size = 163, normalized size = 1.1 \[ \frac{20465}{2823576} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{176 \,{\left (285 \, x - 181\right )}}{352947 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{1159245 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 8543073 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 20832595 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 16829295 \, \sqrt{-2 \, x + 1}}{7529536 \,{\left (3 \, x + 2\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

20465/2823576*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*
sqrt(-2*x + 1))) + 176/352947*(285*x - 181)/((2*x - 1)*sqrt(-2*x + 1)) - 1/75295
36*(1159245*(2*x - 1)^3*sqrt(-2*x + 1) + 8543073*(2*x - 1)^2*sqrt(-2*x + 1) - 20
832595*(-2*x + 1)^(3/2) + 16829295*sqrt(-2*x + 1))/(3*x + 2)^4