3.1950 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^8} \, dx\)

Optimal. Leaf size=181 \[ \frac{11 \sqrt{1-2 x} (5 x+3)^3}{7 (3 x+2)^5}+\frac{55 (1-2 x)^{3/2} (5 x+3)^3}{189 (3 x+2)^6}-\frac{(1-2 x)^{5/2} (5 x+3)^3}{21 (3 x+2)^7}-\frac{3223 \sqrt{1-2 x} (5 x+3)^2}{2646 (3 x+2)^4}-\frac{11 \sqrt{1-2 x} (301765 x+187704)}{333396 (3 x+2)^3}+\frac{33935 \sqrt{1-2 x}}{2333772 (3 x+2)}+\frac{33935 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1166886 \sqrt{21}} \]

[Out]

(33935*Sqrt[1 - 2*x])/(2333772*(2 + 3*x)) - (3223*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(26
46*(2 + 3*x)^4) - ((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(21*(2 + 3*x)^7) + (55*(1 - 2*x)
^(3/2)*(3 + 5*x)^3)/(189*(2 + 3*x)^6) + (11*Sqrt[1 - 2*x]*(3 + 5*x)^3)/(7*(2 + 3
*x)^5) - (11*Sqrt[1 - 2*x]*(187704 + 301765*x))/(333396*(2 + 3*x)^3) + (33935*Ar
cTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1166886*Sqrt[21])

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Rubi [A]  time = 0.332418, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{11 \sqrt{1-2 x} (5 x+3)^3}{7 (3 x+2)^5}+\frac{55 (1-2 x)^{3/2} (5 x+3)^3}{189 (3 x+2)^6}-\frac{(1-2 x)^{5/2} (5 x+3)^3}{21 (3 x+2)^7}-\frac{3223 \sqrt{1-2 x} (5 x+3)^2}{2646 (3 x+2)^4}-\frac{11 \sqrt{1-2 x} (301765 x+187704)}{333396 (3 x+2)^3}+\frac{33935 \sqrt{1-2 x}}{2333772 (3 x+2)}+\frac{33935 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1166886 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(33935*Sqrt[1 - 2*x])/(2333772*(2 + 3*x)) - (3223*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(26
46*(2 + 3*x)^4) - ((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(21*(2 + 3*x)^7) + (55*(1 - 2*x)
^(3/2)*(3 + 5*x)^3)/(189*(2 + 3*x)^6) + (11*Sqrt[1 - 2*x]*(3 + 5*x)^3)/(7*(2 + 3
*x)^5) - (11*Sqrt[1 - 2*x]*(187704 + 301765*x))/(333396*(2 + 3*x)^3) + (33935*Ar
cTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1166886*Sqrt[21])

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Rubi in Sympy [A]  time = 24.732, size = 150, normalized size = 0.83 \[ - \frac{11 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (104580 x + 65016\right )}{4667544 \left (3 x + 2\right )^{5}} - \frac{55 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{2}}{1323 \left (3 x + 2\right )^{6}} - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{3}}{21 \left (3 x + 2\right )^{7}} + \frac{33935 \left (- 2 x + 1\right )^{\frac{3}{2}}}{166698 \left (3 x + 2\right )^{3}} + \frac{33935 \sqrt{- 2 x + 1}}{2333772 \left (3 x + 2\right )} - \frac{33935 \sqrt{- 2 x + 1}}{333396 \left (3 x + 2\right )^{2}} + \frac{33935 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{24504606} \]

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)**8,x)

[Out]

-11*(-2*x + 1)**(5/2)*(104580*x + 65016)/(4667544*(3*x + 2)**5) - 55*(-2*x + 1)*
*(5/2)*(5*x + 3)**2/(1323*(3*x + 2)**6) - (-2*x + 1)**(5/2)*(5*x + 3)**3/(21*(3*
x + 2)**7) + 33935*(-2*x + 1)**(3/2)/(166698*(3*x + 2)**3) + 33935*sqrt(-2*x + 1
)/(2333772*(3*x + 2)) - 33935*sqrt(-2*x + 1)/(333396*(3*x + 2)**2) + 33935*sqrt(
21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/24504606

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Mathematica [A]  time = 0.157527, size = 78, normalized size = 0.43 \[ \frac{\frac{63 \sqrt{1-2 x} \left (24738615 x^6-141112395 x^5-283697388 x^4-164222766 x^3-39606312 x^2-12384752 x-4005436\right )}{(3 x+2)^7}+203610 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{147027636} \]

Antiderivative was successfully verified.

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

[Out]

((63*Sqrt[1 - 2*x]*(-4005436 - 12384752*x - 39606312*x^2 - 164222766*x^3 - 28369
7388*x^4 - 141112395*x^5 + 24738615*x^6))/(2 + 3*x)^7 + 203610*Sqrt[21]*ArcTanh[
Sqrt[3/7]*Sqrt[1 - 2*x]])/147027636

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Maple [A]  time = 0.02, size = 93, normalized size = 0.5 \[ 69984\,{\frac{1}{ \left ( -4-6\,x \right ) ^{7}} \left ( -{\frac{33935\, \left ( 1-2\,x \right ) ^{13/2}}{112021056}}-{\frac{176975\, \left ( 1-2\,x \right ) ^{11/2}}{108020304}}+{\frac{4931597\, \left ( 1-2\,x \right ) ^{9/2}}{185177664}}-{\frac{96613\, \left ( 1-2\,x \right ) ^{7/2}}{964467}}+{\frac{1920721\, \left ( 1-2\,x \right ) ^{5/2}}{11337408}}-{\frac{1187725\, \left ( 1-2\,x \right ) ^{3/2}}{8503056}}+{\frac{1662815\,\sqrt{1-2\,x}}{34012224}} \right ) }+{\frac{33935\,\sqrt{21}}{24504606}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

69984*(-33935/112021056*(1-2*x)^(13/2)-176975/108020304*(1-2*x)^(11/2)+4931597/1
85177664*(1-2*x)^(9/2)-96613/964467*(1-2*x)^(7/2)+1920721/11337408*(1-2*x)^(5/2)
-1187725/8503056*(1-2*x)^(3/2)+1662815/34012224*(1-2*x)^(1/2))/(-4-6*x)^7+33935/
24504606*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.50757, size = 221, normalized size = 1.22 \[ -\frac{33935}{49009212} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{24738615 \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} + 133793100 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - 2174834277 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + 8180415936 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 13834953363 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 11406910900 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 3992418815 \, \sqrt{-2 \, x + 1}}{1166886 \,{\left (2187 \,{\left (2 \, x - 1\right )}^{7} + 35721 \,{\left (2 \, x - 1\right )}^{6} + 250047 \,{\left (2 \, x - 1\right )}^{5} + 972405 \,{\left (2 \, x - 1\right )}^{4} + 2268945 \,{\left (2 \, x - 1\right )}^{3} + 3176523 \,{\left (2 \, x - 1\right )}^{2} + 4941258 \, x - 1647086\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-33935/49009212*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-
2*x + 1))) + 1/1166886*(24738615*(-2*x + 1)^(13/2) + 133793100*(-2*x + 1)^(11/2)
- 2174834277*(-2*x + 1)^(9/2) + 8180415936*(-2*x + 1)^(7/2) - 13834953363*(-2*x
+ 1)^(5/2) + 11406910900*(-2*x + 1)^(3/2) - 3992418815*sqrt(-2*x + 1))/(2187*(2
*x - 1)^7 + 35721*(2*x - 1)^6 + 250047*(2*x - 1)^5 + 972405*(2*x - 1)^4 + 226894
5*(2*x - 1)^3 + 3176523*(2*x - 1)^2 + 4941258*x - 1647086)

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Fricas [A]  time = 0.212073, size = 201, normalized size = 1.11 \[ \frac{\sqrt{21}{\left (\sqrt{21}{\left (24738615 \, x^{6} - 141112395 \, x^{5} - 283697388 \, x^{4} - 164222766 \, x^{3} - 39606312 \, x^{2} - 12384752 \, x - 4005436\right )} \sqrt{-2 \, x + 1} + 33935 \,{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{49009212 \,{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/49009212*sqrt(21)*(sqrt(21)*(24738615*x^6 - 141112395*x^5 - 283697388*x^4 - 16
4222766*x^3 - 39606312*x^2 - 12384752*x - 4005436)*sqrt(-2*x + 1) + 33935*(2187*
x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)*l
og((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)**3/(2+3*x)**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220849, size = 200, normalized size = 1.1 \[ -\frac{33935}{49009212} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{24738615 \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} - 133793100 \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - 2174834277 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - 8180415936 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - 13834953363 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + 11406910900 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 3992418815 \, \sqrt{-2 \, x + 1}}{149361408 \,{\left (3 \, x + 2\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-33935/49009212*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) +
3*sqrt(-2*x + 1))) + 1/149361408*(24738615*(2*x - 1)^6*sqrt(-2*x + 1) - 13379310
0*(2*x - 1)^5*sqrt(-2*x + 1) - 2174834277*(2*x - 1)^4*sqrt(-2*x + 1) - 818041593
6*(2*x - 1)^3*sqrt(-2*x + 1) - 13834953363*(2*x - 1)^2*sqrt(-2*x + 1) + 11406910
900*(-2*x + 1)^(3/2) - 3992418815*sqrt(-2*x + 1))/(3*x + 2)^7