3.1962 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^7 (3+5 x)} \, dx\)

Optimal. Leaf size=173 \[ \frac{7 (1-2 x)^{3/2}}{18 (3 x+2)^6}+\frac{736065535 \sqrt{1-2 x}}{49392 (3 x+2)}+\frac{31700335 \sqrt{1-2 x}}{21168 (3 x+2)^2}+\frac{302651 \sqrt{1-2 x}}{1512 (3 x+2)^3}+\frac{2165 \sqrt{1-2 x}}{72 (3 x+2)^4}+\frac{91 \sqrt{1-2 x}}{18 (3 x+2)^5}+\frac{25388847535 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{24696 \sqrt{21}}-30250 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(7*(1 - 2*x)^(3/2))/(18*(2 + 3*x)^6) + (91*Sqrt[1 - 2*x])/(18*(2 + 3*x)^5) + (21
65*Sqrt[1 - 2*x])/(72*(2 + 3*x)^4) + (302651*Sqrt[1 - 2*x])/(1512*(2 + 3*x)^3) +
(31700335*Sqrt[1 - 2*x])/(21168*(2 + 3*x)^2) + (736065535*Sqrt[1 - 2*x])/(49392
*(2 + 3*x)) + (25388847535*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(24696*Sqrt[21]) -
30250*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.450076, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{7 (1-2 x)^{3/2}}{18 (3 x+2)^6}+\frac{736065535 \sqrt{1-2 x}}{49392 (3 x+2)}+\frac{31700335 \sqrt{1-2 x}}{21168 (3 x+2)^2}+\frac{302651 \sqrt{1-2 x}}{1512 (3 x+2)^3}+\frac{2165 \sqrt{1-2 x}}{72 (3 x+2)^4}+\frac{91 \sqrt{1-2 x}}{18 (3 x+2)^5}+\frac{25388847535 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{24696 \sqrt{21}}-30250 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(7*(1 - 2*x)^(3/2))/(18*(2 + 3*x)^6) + (91*Sqrt[1 - 2*x])/(18*(2 + 3*x)^5) + (21
65*Sqrt[1 - 2*x])/(72*(2 + 3*x)^4) + (302651*Sqrt[1 - 2*x])/(1512*(2 + 3*x)^3) +
(31700335*Sqrt[1 - 2*x])/(21168*(2 + 3*x)^2) + (736065535*Sqrt[1 - 2*x])/(49392
*(2 + 3*x)) + (25388847535*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(24696*Sqrt[21]) -
30250*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi in Sympy [A]  time = 49.0993, size = 156, normalized size = 0.9 \[ \frac{7 \left (- 2 x + 1\right )^{\frac{3}{2}}}{18 \left (3 x + 2\right )^{6}} + \frac{736065535 \sqrt{- 2 x + 1}}{49392 \left (3 x + 2\right )} + \frac{31700335 \sqrt{- 2 x + 1}}{21168 \left (3 x + 2\right )^{2}} + \frac{302651 \sqrt{- 2 x + 1}}{1512 \left (3 x + 2\right )^{3}} + \frac{2165 \sqrt{- 2 x + 1}}{72 \left (3 x + 2\right )^{4}} + \frac{91 \sqrt{- 2 x + 1}}{18 \left (3 x + 2\right )^{5}} + \frac{25388847535 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{518616} - 30250 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7*(-2*x + 1)**(3/2)/(18*(3*x + 2)**6) + 736065535*sqrt(-2*x + 1)/(49392*(3*x + 2
)) + 31700335*sqrt(-2*x + 1)/(21168*(3*x + 2)**2) + 302651*sqrt(-2*x + 1)/(1512*
(3*x + 2)**3) + 2165*sqrt(-2*x + 1)/(72*(3*x + 2)**4) + 91*sqrt(-2*x + 1)/(18*(3
*x + 2)**5) + 25388847535*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/518616 - 302
50*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)

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Mathematica [A]  time = 0.210102, size = 98, normalized size = 0.57 \[ \frac{\sqrt{1-2 x} \left (178863925005 x^5+602204446665 x^4+811194684822 x^3+546491397114 x^2+184131053992 x+24823128464\right )}{49392 (3 x+2)^6}+\frac{25388847535 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{24696 \sqrt{21}}-30250 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(24823128464 + 184131053992*x + 546491397114*x^2 + 811194684822*x
^3 + 602204446665*x^4 + 178863925005*x^5))/(49392*(2 + 3*x)^6) + (25388847535*Ar
cTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(24696*Sqrt[21]) - 30250*Sqrt[55]*ArcTanh[Sqrt[5
/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.019, size = 102, normalized size = 0.6 \[ -1458\,{\frac{1}{ \left ( -4-6\,x \right ) ^{6}} \left ({\frac{736065535\, \left ( 1-2\,x \right ) ^{11/2}}{148176}}-{\frac{11104383695\, \left ( 1-2\,x \right ) ^{9/2}}{190512}}+{\frac{1240999441\, \left ( 1-2\,x \right ) ^{7/2}}{4536}}-{\frac{3744956269\, \left ( 1-2\,x \right ) ^{5/2}}{5832}}+{\frac{79114433335\, \left ( 1-2\,x \right ) ^{3/2}}{104976}}-{\frac{37144080785\,\sqrt{1-2\,x}}{104976}} \right ) }+{\frac{25388847535\,\sqrt{21}}{518616}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-30250\,{\it Artanh} \left ( 1/11\,\sqrt{55}\sqrt{1-2\,x} \right ) \sqrt{55} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1458*(736065535/148176*(1-2*x)^(11/2)-11104383695/190512*(1-2*x)^(9/2)[phone]
41/4536*(1-2*x)^(7/2)-3744956269/5832*(1-2*x)^(5/2)[phone]/104976*(1-2*x)^(
3/2)-37144080785/104976*(1-2*x)^(1/2))/(-4-6*x)^6+25388847535/518616*arctanh(1/7
*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-30250*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^
(1/2)

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Maxima [A]  time = 1.51115, size = 246, normalized size = 1.42 \[ 15125 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{25388847535}{1037232} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{178863925005 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - 2098728518355 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + 9851053562658 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 23121360004806 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 27136250633905 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 12740419709255 \, \sqrt{-2 \, x + 1}}{24696 \,{\left (729 \,{\left (2 \, x - 1\right )}^{6} + 10206 \,{\left (2 \, x - 1\right )}^{5} + 59535 \,{\left (2 \, x - 1\right )}^{4} + 185220 \,{\left (2 \, x - 1\right )}^{3} + 324135 \,{\left (2 \, x - 1\right )}^{2} + 605052 \, x - 184877\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)))
- 25388847535/1037232*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3
*sqrt(-2*x + 1))) - 1/24696*(178863925005*(-2*x + 1)^(11/2) - 2098728518355*(-2*
x + 1)^(9/2) + 9851053562658*(-2*x + 1)^(7/2) - 23121360004806*(-2*x + 1)^(5/2)
+ 27136250633905*(-2*x + 1)^(3/2) - 12740419709255*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 - 184877)

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Fricas [A]  time = 0.22387, size = 266, normalized size = 1.54 \[ \frac{\sqrt{21}{\left (747054000 \, \sqrt{55} \sqrt{21}{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + \sqrt{21}{\left (178863925005 \, x^{5} + 602204446665 \, x^{4} + 811194684822 \, x^{3} + 546491397114 \, x^{2} + 184131053992 \, x + 24823128464\right )} \sqrt{-2 \, x + 1} + 25388847535 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{1037232 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1037232*sqrt(21)*(747054000*sqrt(55)*sqrt(21)*(729*x^6 + 2916*x^5 + 4860*x^4 +
4320*x^3 + 2160*x^2 + 576*x + 64)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x
+ 3)) + sqrt(21)*(178863925005*x^5 + 602204446665*x^4 + 811194684822*x^3 + 54649
1397114*x^2 + 184131053992*x + 24823128464)*sqrt(-2*x + 1) + 25388847535*(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)/(2+3*x)**7/(3+5*x),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219507, size = 231, normalized size = 1.34 \[ 15125 \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{25388847535}{1037232} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{178863925005 \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + 2098728518355 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 9851053562658 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 23121360004806 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 27136250633905 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 12740419709255 \, \sqrt{-2 \, x + 1}}{1580544 \,{\left (3 \, x + 2\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15125*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2
*x + 1))) - 25388847535/1037232*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x +
1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/1580544*(178863925005*(2*x - 1)^5*sqrt(-2
*x + 1) + 2098728518355*(2*x - 1)^4*sqrt(-2*x + 1) + 9851053562658*(2*x - 1)^3*s
qrt(-2*x + 1) + 23121360004806*(2*x - 1)^2*sqrt(-2*x + 1) - 27136250633905*(-2*x
+ 1)^(3/2) + 12740419709255*sqrt(-2*x + 1))/(3*x + 2)^6