3.1960 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)} \, dx\)

Optimal. Leaf size=133 \[ \frac{7 (1-2 x)^{3/2}}{12 (3 x+2)^4}+\frac{100145 \sqrt{1-2 x}}{168 (3 x+2)}+\frac{4313 \sqrt{1-2 x}}{72 (3 x+2)^2}+\frac{301 \sqrt{1-2 x}}{36 (3 x+2)^3}+\frac{3454265 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{84 \sqrt{21}}-1210 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(7*(1 - 2*x)^(3/2))/(12*(2 + 3*x)^4) + (301*Sqrt[1 - 2*x])/(36*(2 + 3*x)^3) + (4
313*Sqrt[1 - 2*x])/(72*(2 + 3*x)^2) + (100145*Sqrt[1 - 2*x])/(168*(2 + 3*x)) + (
3454265*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(84*Sqrt[21]) - 1210*Sqrt[55]*ArcTanh[
Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.313929, antiderivative size = 133, 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 \[ \frac{7 (1-2 x)^{3/2}}{12 (3 x+2)^4}+\frac{100145 \sqrt{1-2 x}}{168 (3 x+2)}+\frac{4313 \sqrt{1-2 x}}{72 (3 x+2)^2}+\frac{301 \sqrt{1-2 x}}{36 (3 x+2)^3}+\frac{3454265 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{84 \sqrt{21}}-1210 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(7*(1 - 2*x)^(3/2))/(12*(2 + 3*x)^4) + (301*Sqrt[1 - 2*x])/(36*(2 + 3*x)^3) + (4
313*Sqrt[1 - 2*x])/(72*(2 + 3*x)^2) + (100145*Sqrt[1 - 2*x])/(168*(2 + 3*x)) + (
3454265*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(84*Sqrt[21]) - 1210*Sqrt[55]*ArcTanh[
Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi in Sympy [A]  time = 34.6217, size = 119, normalized size = 0.89 \[ \frac{7 \left (- 2 x + 1\right )^{\frac{3}{2}}}{12 \left (3 x + 2\right )^{4}} + \frac{100145 \sqrt{- 2 x + 1}}{168 \left (3 x + 2\right )} + \frac{4313 \sqrt{- 2 x + 1}}{72 \left (3 x + 2\right )^{2}} + \frac{301 \sqrt{- 2 x + 1}}{36 \left (3 x + 2\right )^{3}} + \frac{3454265 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{1764} - 1210 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7*(-2*x + 1)**(3/2)/(12*(3*x + 2)**4) + 100145*sqrt(-2*x + 1)/(168*(3*x + 2)) +
4313*sqrt(-2*x + 1)/(72*(3*x + 2)**2) + 301*sqrt(-2*x + 1)/(36*(3*x + 2)**3) + 3
454265*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/1764 - 1210*sqrt(55)*atanh(sqrt
(55)*sqrt(-2*x + 1)/11)

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Mathematica [A]  time = 0.171652, size = 88, normalized size = 0.66 \[ \frac{\sqrt{1-2 x} \left (2703915 x^3+5498403 x^2+3730002 x+844322\right )}{168 (3 x+2)^4}+\frac{3454265 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{84 \sqrt{21}}-1210 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(844322 + 3730002*x + 5498403*x^2 + 2703915*x^3))/(168*(2 + 3*x)^
4) + (3454265*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(84*Sqrt[21]) - 1210*Sqrt[55]*Ar
cTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.019, size = 84, normalized size = 0.6 \[ -162\,{\frac{1}{ \left ( -4-6\,x \right ) ^{4}} \left ({\frac{100145\, \left ( 1-2\,x \right ) ^{7/2}}{504}}-{\frac{909931\, \left ( 1-2\,x \right ) ^{5/2}}{648}}+{\frac{2144065\, \left ( 1-2\,x \right ) ^{3/2}}{648}}-{\frac{5053615\,\sqrt{1-2\,x}}{1944}} \right ) }+{\frac{3454265\,\sqrt{21}}{1764}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-1210\,{\it Artanh} \left ( 1/11\,\sqrt{55}\sqrt{1-2\,x} \right ) \sqrt{55} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-162*(100145/504*(1-2*x)^(7/2)-909931/648*(1-2*x)^(5/2)+2144065/648*(1-2*x)^(3/2
)-5053615/1944*(1-2*x)^(1/2))/(-4-6*x)^4+3454265/1764*arctanh(1/7*21^(1/2)*(1-2*
x)^(1/2))*21^(1/2)-1210*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50777, size = 197, normalized size = 1.48 \[ 605 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{3454265}{3528} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2703915 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 19108551 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 45025365 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 35375305 \, \sqrt{-2 \, x + 1}}{84 \,{\left (81 \,{\left (2 \, x - 1\right )}^{4} + 756 \,{\left (2 \, x - 1\right )}^{3} + 2646 \,{\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

605*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) -
3454265/3528*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*
x + 1))) - 1/84*(2703915*(-2*x + 1)^(7/2) - 19108551*(-2*x + 1)^(5/2) + 45025365
*(-2*x + 1)^(3/2) - 35375305*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 756*(2*x - 1)^3 +
2646*(2*x - 1)^2 + 8232*x - 1715)

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Fricas [A]  time = 0.2177, size = 212, normalized size = 1.59 \[ \frac{\sqrt{21}{\left (101640 \, \sqrt{55} \sqrt{21}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + \sqrt{21}{\left (2703915 \, x^{3} + 5498403 \, x^{2} + 3730002 \, x + 844322\right )} \sqrt{-2 \, x + 1} + 3454265 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{3528 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3528*sqrt(21)*(101640*sqrt(55)*sqrt(21)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 1
6)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + sqrt(21)*(2703915*x^3 +
5498403*x^2 + 3730002*x + 844322)*sqrt(-2*x + 1) + 3454265*(81*x^4 + 216*x^3 + 2
16*x^2 + 96*x + 16)*log((sqrt(21)*(3*x - 5) - 21*sqrt(-2*x + 1))/(3*x + 2)))/(81
*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.215943, size = 188, normalized size = 1.41 \[ 605 \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{3454265}{3528} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{2703915 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 19108551 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 45025365 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 35375305 \, \sqrt{-2 \, x + 1}}{1344 \,{\left (3 \, x + 2\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

605*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x
+ 1))) - 3454265/3528*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt
(21) + 3*sqrt(-2*x + 1))) + 1/1344*(2703915*(2*x - 1)^3*sqrt(-2*x + 1) + 1910855
1*(2*x - 1)^2*sqrt(-2*x + 1) - 45025365*(-2*x + 1)^(3/2) + 35375305*sqrt(-2*x +
1))/(3*x + 2)^4