3.1839 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^4 (3+5 x)^3} \, dx\)

Optimal. Leaf size=180 \[ \frac{4031135 \sqrt{1-2 x}}{1078 (5 x+3)}-\frac{182335 \sqrt{1-2 x}}{294 (5 x+3)^2}+\frac{4042 \sqrt{1-2 x}}{49 (3 x+2) (5 x+3)^2}+\frac{29 \sqrt{1-2 x}}{7 (3 x+2)^2 (5 x+3)^2}+\frac{\sqrt{1-2 x}}{3 (3 x+2)^3 (5 x+3)^2}+\frac{2528082}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{551075}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-182335*Sqrt[1 - 2*x])/(294*(3 + 5*x)^2) + Sqrt[1 - 2*x]/(3*(2 + 3*x)^3*(3 + 5*
x)^2) + (29*Sqrt[1 - 2*x])/(7*(2 + 3*x)^2*(3 + 5*x)^2) + (4042*Sqrt[1 - 2*x])/(4
9*(2 + 3*x)*(3 + 5*x)^2) + (4031135*Sqrt[1 - 2*x])/(1078*(3 + 5*x)) + (2528082*S
qrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 - (551075*Sqrt[5/11]*ArcTanh[Sqrt[
5/11]*Sqrt[1 - 2*x]])/11

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Rubi [A]  time = 0.369008, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{4031135 \sqrt{1-2 x}}{1078 (5 x+3)}-\frac{182335 \sqrt{1-2 x}}{294 (5 x+3)^2}+\frac{4042 \sqrt{1-2 x}}{49 (3 x+2) (5 x+3)^2}+\frac{29 \sqrt{1-2 x}}{7 (3 x+2)^2 (5 x+3)^2}+\frac{\sqrt{1-2 x}}{3 (3 x+2)^3 (5 x+3)^2}+\frac{2528082}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{551075}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - 2*x]/((2 + 3*x)^4*(3 + 5*x)^3),x]

[Out]

(-182335*Sqrt[1 - 2*x])/(294*(3 + 5*x)^2) + Sqrt[1 - 2*x]/(3*(2 + 3*x)^3*(3 + 5*
x)^2) + (29*Sqrt[1 - 2*x])/(7*(2 + 3*x)^2*(3 + 5*x)^2) + (4042*Sqrt[1 - 2*x])/(4
9*(2 + 3*x)*(3 + 5*x)^2) + (4031135*Sqrt[1 - 2*x])/(1078*(3 + 5*x)) + (2528082*S
qrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 - (551075*Sqrt[5/11]*ArcTanh[Sqrt[
5/11]*Sqrt[1 - 2*x]])/11

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Rubi in Sympy [A]  time = 42.9798, size = 160, normalized size = 0.89 \[ \frac{2418681 \sqrt{- 2 x + 1}}{1078 \left (3 x + 2\right )} + \frac{28935 \sqrt{- 2 x + 1}}{77 \left (3 x + 2\right ) \left (5 x + 3\right )} - \frac{1745 \sqrt{- 2 x + 1}}{42 \left (3 x + 2\right ) \left (5 x + 3\right )^{2}} + \frac{29 \sqrt{- 2 x + 1}}{7 \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{2}} + \frac{\sqrt{- 2 x + 1}}{3 \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{2}} + \frac{2528082 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{343} - \frac{551075 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{121} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(1/2)/(2+3*x)**4/(3+5*x)**3,x)

[Out]

2418681*sqrt(-2*x + 1)/(1078*(3*x + 2)) + 28935*sqrt(-2*x + 1)/(77*(3*x + 2)*(5*
x + 3)) - 1745*sqrt(-2*x + 1)/(42*(3*x + 2)*(5*x + 3)**2) + 29*sqrt(-2*x + 1)/(7
*(3*x + 2)**2*(5*x + 3)**2) + sqrt(-2*x + 1)/(3*(3*x + 2)**3*(5*x + 3)**2) + 252
8082*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/343 - 551075*sqrt(55)*atanh(sqrt(
55)*sqrt(-2*x + 1)/11)/121

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Mathematica [A]  time = 0.173552, size = 106, normalized size = 0.59 \[ \frac{\sqrt{1-2 x} \left (544203225 x^4+1396877220 x^3+1343346156 x^2+573620246 x+91763734\right )}{1078 (3 x+2)^3 (5 x+3)^2}+\frac{2528082}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{551075}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 - 2*x]/((2 + 3*x)^4*(3 + 5*x)^3),x]

[Out]

(Sqrt[1 - 2*x]*(91763734 + 573620246*x + 1343346156*x^2 + 1396877220*x^3 + 54420
3225*x^4))/(1078*(2 + 3*x)^3*(3 + 5*x)^2) + (2528082*Sqrt[3/7]*ArcTanh[Sqrt[3/7]
*Sqrt[1 - 2*x]])/49 - (551075*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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Maple [A]  time = 0.022, size = 103, normalized size = 0.6 \[ -972\,{\frac{1}{ \left ( -4-6\,x \right ) ^{3}} \left ({\frac{7297\, \left ( 1-2\,x \right ) ^{5/2}}{294}}-{\frac{22048\, \left ( 1-2\,x \right ) ^{3/2}}{189}}+{\frac{7403\,\sqrt{1-2\,x}}{54}} \right ) }+{\frac{2528082\,\sqrt{21}}{343}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+62500\,{\frac{1}{ \left ( -6-10\,x \right ) ^{2}} \left ( -{\frac{53\, \left ( 1-2\,x \right ) ^{3/2}}{220}}+{\frac{263\,\sqrt{1-2\,x}}{500}} \right ) }-{\frac{551075\,\sqrt{55}}{121}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(1/2)/(2+3*x)^4/(3+5*x)^3,x)

[Out]

-972*(7297/294*(1-2*x)^(5/2)-22048/189*(1-2*x)^(3/2)+7403/54*(1-2*x)^(1/2))/(-4-
6*x)^3+2528082/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+62500*(-53/220*(
1-2*x)^(3/2)+263/500*(1-2*x)^(1/2))/(-6-10*x)^2-551075/121*arctanh(1/11*55^(1/2)
*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.53046, size = 221, normalized size = 1.23 \[ \frac{551075}{242} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{1264041}{343} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{544203225 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 4970567340 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 17019867294 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 25893807436 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 14768524001 \, \sqrt{-2 \, x + 1}}{539 \,{\left (675 \,{\left (2 \, x - 1\right )}^{5} + 7695 \,{\left (2 \, x - 1\right )}^{4} + 35082 \,{\left (2 \, x - 1\right )}^{3} + 79954 \,{\left (2 \, x - 1\right )}^{2} + 182182 \, x - 49588\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-2*x + 1)/((5*x + 3)^3*(3*x + 2)^4),x, algorithm="maxima")

[Out]

551075/242*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x +
1))) - 1264041/343*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sq
rt(-2*x + 1))) + 1/539*(544203225*(-2*x + 1)^(9/2) - 4970567340*(-2*x + 1)^(7/2)
+ 17019867294*(-2*x + 1)^(5/2) - 25893807436*(-2*x + 1)^(3/2) + 14768524001*sqr
t(-2*x + 1))/(675*(2*x - 1)^5 + 7695*(2*x - 1)^4 + 35082*(2*x - 1)^3 + 79954*(2*
x - 1)^2 + 182182*x - 49588)

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Fricas [A]  time = 0.222448, size = 267, normalized size = 1.48 \[ \frac{\sqrt{11} \sqrt{7}{\left (27002675 \, \sqrt{7} \sqrt{5}{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 27808902 \, \sqrt{11} \sqrt{3}{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{11} \sqrt{7}{\left (544203225 \, x^{4} + 1396877220 \, x^{3} + 1343346156 \, x^{2} + 573620246 \, x + 91763734\right )} \sqrt{-2 \, x + 1}\right )}}{83006 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-2*x + 1)/((5*x + 3)^3*(3*x + 2)^4),x, algorithm="fricas")

[Out]

1/83006*sqrt(11)*sqrt(7)*(27002675*sqrt(7)*sqrt(5)*(675*x^5 + 2160*x^4 + 2763*x^
3 + 1766*x^2 + 564*x + 72)*log((sqrt(11)*(5*x - 8) + 11*sqrt(5)*sqrt(-2*x + 1))/
(5*x + 3)) + 27808902*sqrt(11)*sqrt(3)*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2
+ 564*x + 72)*log((sqrt(7)*(3*x - 5) - 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)) + s
qrt(11)*sqrt(7)*(544203225*x^4 + 1396877220*x^3 + 1343346156*x^2 + 573620246*x +
91763734)*sqrt(-2*x + 1))/(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 7
2)

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Sympy [A]  time = 141.276, size = 804, normalized size = 4.47 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-2*x)**(1/2)/(2+3*x)**4/(3+5*x)**3,x)

[Out]

36720*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(-2*x + 1)/7 - 1)/4 + log(sqrt(21)*
sqrt(-2*x + 1)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)) - 1/(4*(sqrt(21)
*sqrt(-2*x + 1)/7 - 1)))/147, (x <= 1/2) & (x > -2/3))) - 7416*Piecewise((sqrt(2
1)*(3*log(sqrt(21)*sqrt(-2*x + 1)/7 - 1)/16 - 3*log(sqrt(21)*sqrt(-2*x + 1)/7 +
1)/16 + 3/(16*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)) + 1/(16*(sqrt(21)*sqrt(-2*x + 1)/
7 + 1)**2) + 3/(16*(sqrt(21)*sqrt(-2*x + 1)/7 - 1)) - 1/(16*(sqrt(21)*sqrt(-2*x
+ 1)/7 - 1)**2))/1029, (x <= 1/2) & (x > -2/3))) + 1008*Piecewise((sqrt(21)*(-5*
log(sqrt(21)*sqrt(-2*x + 1)/7 - 1)/32 + 5*log(sqrt(21)*sqrt(-2*x + 1)/7 + 1)/32
- 5/(32*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)) - 1/(16*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)
**2) - 1/(48*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)**3) - 5/(32*(sqrt(21)*sqrt(-2*x + 1
)/7 - 1)) + 1/(16*(sqrt(21)*sqrt(-2*x + 1)/7 - 1)**2) - 1/(48*(sqrt(21)*sqrt(-2*
x + 1)/7 - 1)**3))/7203, (x <= 1/2) & (x > -2/3))) + 67000*Piecewise((sqrt(55)*(
-log(sqrt(55)*sqrt(-2*x + 1)/11 - 1)/4 + log(sqrt(55)*sqrt(-2*x + 1)/11 + 1)/4 -
1/(4*(sqrt(55)*sqrt(-2*x + 1)/11 + 1)) - 1/(4*(sqrt(55)*sqrt(-2*x + 1)/11 - 1))
)/605, (x <= 1/2) & (x > -3/5))) + 11000*Piecewise((sqrt(55)*(3*log(sqrt(55)*sqr
t(-2*x + 1)/11 - 1)/16 - 3*log(sqrt(55)*sqrt(-2*x + 1)/11 + 1)/16 + 3/(16*(sqrt(
55)*sqrt(-2*x + 1)/11 + 1)) + 1/(16*(sqrt(55)*sqrt(-2*x + 1)/11 + 1)**2) + 3/(16
*(sqrt(55)*sqrt(-2*x + 1)/11 - 1)) - 1/(16*(sqrt(55)*sqrt(-2*x + 1)/11 - 1)**2))
/6655, (x <= 1/2) & (x > -3/5))) - 152100*Piecewise((-sqrt(21)*acoth(sqrt(21)*sq
rt(-2*x + 1)/7)/21, -2*x + 1 > 7/3), (-sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)
/21, -2*x + 1 < 7/3)) + 253500*Piecewise((-sqrt(55)*acoth(sqrt(55)*sqrt(-2*x + 1
)/11)/55, -2*x + 1 > 11/5), (-sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/55, -2*
x + 1 < 11/5))

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GIAC/XCAS [A]  time = 0.247513, size = 204, normalized size = 1.13 \[ \frac{551075}{242} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{1264041}{343} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{125 \,{\left (1325 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2893 \, \sqrt{-2 \, x + 1}\right )}}{44 \,{\left (5 \, x + 3\right )}^{2}} + \frac{9 \,{\left (65673 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 308672 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 362747 \, \sqrt{-2 \, x + 1}\right )}}{196 \,{\left (3 \, x + 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-2*x + 1)/((5*x + 3)^3*(3*x + 2)^4),x, algorithm="giac")

[Out]

551075/242*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sq
rt(-2*x + 1))) - 1264041/343*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))
/(sqrt(21) + 3*sqrt(-2*x + 1))) - 125/44*(1325*(-2*x + 1)^(3/2) - 2893*sqrt(-2*x
+ 1))/(5*x + 3)^2 + 9/196*(65673*(2*x - 1)^2*sqrt(-2*x + 1) - 308672*(-2*x + 1)
^(3/2) + 362747*sqrt(-2*x + 1))/(3*x + 2)^3