3.1852 \(\int \frac{(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=128 \[ \frac{(1-2 x)^{5/2}}{105 (3 x+2)^5}-\frac{17 (1-2 x)^{3/2}}{126 (3 x+2)^4}-\frac{17 \sqrt{1-2 x}}{12348 (3 x+2)}-\frac{17 \sqrt{1-2 x}}{5292 (3 x+2)^2}+\frac{17 \sqrt{1-2 x}}{378 (3 x+2)^3}-\frac{17 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{6174 \sqrt{21}} \]

[Out]

(1 - 2*x)^(5/2)/(105*(2 + 3*x)^5) - (17*(1 - 2*x)^(3/2))/(126*(2 + 3*x)^4) + (17
*Sqrt[1 - 2*x])/(378*(2 + 3*x)^3) - (17*Sqrt[1 - 2*x])/(5292*(2 + 3*x)^2) - (17*
Sqrt[1 - 2*x])/(12348*(2 + 3*x)) - (17*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(6174*S
qrt[21])

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Rubi [A]  time = 0.122894, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{(1-2 x)^{5/2}}{105 (3 x+2)^5}-\frac{17 (1-2 x)^{3/2}}{126 (3 x+2)^4}-\frac{17 \sqrt{1-2 x}}{12348 (3 x+2)}-\frac{17 \sqrt{1-2 x}}{5292 (3 x+2)^2}+\frac{17 \sqrt{1-2 x}}{378 (3 x+2)^3}-\frac{17 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{6174 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(3/2)*(3 + 5*x))/(2 + 3*x)^6,x]

[Out]

(1 - 2*x)^(5/2)/(105*(2 + 3*x)^5) - (17*(1 - 2*x)^(3/2))/(126*(2 + 3*x)^4) + (17
*Sqrt[1 - 2*x])/(378*(2 + 3*x)^3) - (17*Sqrt[1 - 2*x])/(5292*(2 + 3*x)^2) - (17*
Sqrt[1 - 2*x])/(12348*(2 + 3*x)) - (17*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(6174*S
qrt[21])

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Rubi in Sympy [A]  time = 13.7819, size = 112, normalized size = 0.88 \[ \frac{\left (- 2 x + 1\right )^{\frac{5}{2}}}{105 \left (3 x + 2\right )^{5}} - \frac{17 \left (- 2 x + 1\right )^{\frac{3}{2}}}{126 \left (3 x + 2\right )^{4}} - \frac{17 \sqrt{- 2 x + 1}}{12348 \left (3 x + 2\right )} - \frac{17 \sqrt{- 2 x + 1}}{5292 \left (3 x + 2\right )^{2}} + \frac{17 \sqrt{- 2 x + 1}}{378 \left (3 x + 2\right )^{3}} - \frac{17 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{129654} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(3/2)*(3+5*x)/(2+3*x)**6,x)

[Out]

(-2*x + 1)**(5/2)/(105*(3*x + 2)**5) - 17*(-2*x + 1)**(3/2)/(126*(3*x + 2)**4) -
17*sqrt(-2*x + 1)/(12348*(3*x + 2)) - 17*sqrt(-2*x + 1)/(5292*(3*x + 2)**2) + 1
7*sqrt(-2*x + 1)/(378*(3*x + 2)**3) - 17*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/
7)/129654

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Mathematica [A]  time = 0.106587, size = 68, normalized size = 0.53 \[ \frac{-\frac{21 \sqrt{1-2 x} \left (6885 x^4+23715 x^3-48252 x^2-23998 x+7912\right )}{(3 x+2)^5}-170 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1296540} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(3/2)*(3 + 5*x))/(2 + 3*x)^6,x]

[Out]

((-21*Sqrt[1 - 2*x]*(7912 - 23998*x - 48252*x^2 + 23715*x^3 + 6885*x^4))/(2 + 3*
x)^5 - 170*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/1296540

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Maple [A]  time = 0.017, size = 75, normalized size = 0.6 \[ 7776\,{\frac{1}{ \left ( -4-6\,x \right ) ^{5}} \left ({\frac{17\, \left ( 1-2\,x \right ) ^{9/2}}{592704}}-{\frac{17\, \left ( 1-2\,x \right ) ^{7/2}}{54432}}-{\frac{ \left ( 1-2\,x \right ) ^{5/2}}{25515}}+{\frac{119\, \left ( 1-2\,x \right ) ^{3/2}}{69984}}-{\frac{119\,\sqrt{1-2\,x}}{139968}} \right ) }-{\frac{17\,\sqrt{21}}{129654}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(3/2)*(3+5*x)/(2+3*x)^6,x)

[Out]

7776*(17/592704*(1-2*x)^(9/2)-17/54432*(1-2*x)^(7/2)-1/25515*(1-2*x)^(5/2)+119/6
9984*(1-2*x)^(3/2)-119/139968*(1-2*x)^(1/2))/(-4-6*x)^5-17/129654*arctanh(1/7*21
^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.52539, size = 173, normalized size = 1.35 \[ \frac{17}{259308} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{6885 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 74970 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 9408 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 408170 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 204085 \, \sqrt{-2 \, x + 1}}{30870 \,{\left (243 \,{\left (2 \, x - 1\right )}^{5} + 2835 \,{\left (2 \, x - 1\right )}^{4} + 13230 \,{\left (2 \, x - 1\right )}^{3} + 30870 \,{\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)*(-2*x + 1)^(3/2)/(3*x + 2)^6,x, algorithm="maxima")

[Out]

17/259308*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x +
1))) - 1/30870*(6885*(-2*x + 1)^(9/2) - 74970*(-2*x + 1)^(7/2) - 9408*(-2*x + 1)
^(5/2) + 408170*(-2*x + 1)^(3/2) - 204085*sqrt(-2*x + 1))/(243*(2*x - 1)^5 + 283
5*(2*x - 1)^4 + 13230*(2*x - 1)^3 + 30870*(2*x - 1)^2 + 72030*x - 19208)

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Fricas [A]  time = 0.214984, size = 161, normalized size = 1.26 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (6885 \, x^{4} + 23715 \, x^{3} - 48252 \, x^{2} - 23998 \, x + 7912\right )} \sqrt{-2 \, x + 1} - 85 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{1296540 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)*(-2*x + 1)^(3/2)/(3*x + 2)^6,x, algorithm="fricas")

[Out]

-1/1296540*sqrt(21)*(sqrt(21)*(6885*x^4 + 23715*x^3 - 48252*x^2 - 23998*x + 7912
)*sqrt(-2*x + 1) - 85*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log(
(sqrt(21)*(3*x - 5) + 21*sqrt(-2*x + 1))/(3*x + 2)))/(243*x^5 + 810*x^4 + 1080*x
^3 + 720*x^2 + 240*x + 32)

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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)**(3/2)*(3+5*x)/(2+3*x)**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216358, size = 157, normalized size = 1.23 \[ \frac{17}{259308} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{6885 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 74970 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - 9408 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + 408170 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 204085 \, \sqrt{-2 \, x + 1}}{987840 \,{\left (3 \, x + 2\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)*(-2*x + 1)^(3/2)/(3*x + 2)^6,x, algorithm="giac")

[Out]

17/259308*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt
(-2*x + 1))) - 1/987840*(6885*(2*x - 1)^4*sqrt(-2*x + 1) + 74970*(2*x - 1)^3*sqr
t(-2*x + 1) - 9408*(2*x - 1)^2*sqrt(-2*x + 1) + 408170*(-2*x + 1)^(3/2) - 204085
*sqrt(-2*x + 1))/(3*x + 2)^5