3.2050 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)^3} \, dx\)

Optimal. Leaf size=153 \[ \frac{1177080 \sqrt{1-2 x}}{5929 (5 x+3)}-\frac{35495 \sqrt{1-2 x}}{1078 (5 x+3)^2}+\frac{429 \sqrt{1-2 x}}{98 (3 x+2) (5 x+3)^2}+\frac{3 \sqrt{1-2 x}}{14 (3 x+2)^2 (5 x+3)^2}+\frac{134217}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{321825}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-35495*Sqrt[1 - 2*x])/(1078*(3 + 5*x)^2) + (3*Sqrt[1 - 2*x])/(14*(2 + 3*x)^2*(3
+ 5*x)^2) + (429*Sqrt[1 - 2*x])/(98*(2 + 3*x)*(3 + 5*x)^2) + (1177080*Sqrt[1 -
2*x])/(5929*(3 + 5*x)) + (134217*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49
- (321825*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Rubi [A]  time = 0.332625, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{1177080 \sqrt{1-2 x}}{5929 (5 x+3)}-\frac{35495 \sqrt{1-2 x}}{1078 (5 x+3)^2}+\frac{429 \sqrt{1-2 x}}{98 (3 x+2) (5 x+3)^2}+\frac{3 \sqrt{1-2 x}}{14 (3 x+2)^2 (5 x+3)^2}+\frac{134217}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{321825}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-35495*Sqrt[1 - 2*x])/(1078*(3 + 5*x)^2) + (3*Sqrt[1 - 2*x])/(14*(2 + 3*x)^2*(3
+ 5*x)^2) + (429*Sqrt[1 - 2*x])/(98*(2 + 3*x)*(3 + 5*x)^2) + (1177080*Sqrt[1 -
2*x])/(5929*(3 + 5*x)) + (134217*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49
- (321825*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Rubi in Sympy [A]  time = 34.9273, size = 133, normalized size = 0.87 \[ \frac{706248 \sqrt{- 2 x + 1}}{5929 \left (3 x + 2\right )} + \frac{20277 \sqrt{- 2 x + 1}}{1694 \left (3 x + 2\right )^{2}} + \frac{645 \sqrt{- 2 x + 1}}{242 \left (3 x + 2\right )^{2} \left (5 x + 3\right )} - \frac{5 \sqrt{- 2 x + 1}}{22 \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{2}} + \frac{134217 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{343} - \frac{321825 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{1331} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

706248*sqrt(-2*x + 1)/(5929*(3*x + 2)) + 20277*sqrt(-2*x + 1)/(1694*(3*x + 2)**2
) + 645*sqrt(-2*x + 1)/(242*(3*x + 2)**2*(5*x + 3)) - 5*sqrt(-2*x + 1)/(22*(3*x
+ 2)**2*(5*x + 3)**2) + 134217*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/343 - 3
21825*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/1331

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Mathematica [A]  time = 0.159226, size = 101, normalized size = 0.66 \[ \frac{\sqrt{1-2 x} \left (105937200 x^3+201297915 x^2+127303347 x+26794499\right )}{11858 (3 x+2)^2 (5 x+3)^2}+\frac{134217}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{321825}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(26794499 + 127303347*x + 201297915*x^2 + 105937200*x^3))/(11858*
(2 + 3*x)^2*(3 + 5*x)^2) + (134217*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/4
9 - (321825*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Maple [A]  time = 0.022, size = 94, normalized size = 0.6 \[ -972\,{\frac{1}{ \left ( -4-6\,x \right ) ^{2}} \left ({\frac{71\, \left ( 1-2\,x \right ) ^{3/2}}{196}}-{\frac{215\,\sqrt{1-2\,x}}{252}} \right ) }+{\frac{134217\,\sqrt{21}}{343}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+62500\,{\frac{1}{ \left ( -6-10\,x \right ) ^{2}} \left ( -{\frac{39\, \left ( 1-2\,x \right ) ^{3/2}}{2420}}+{\frac{193\,\sqrt{1-2\,x}}{5500}} \right ) }-{\frac{321825\,\sqrt{55}}{1331}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-972*(71/196*(1-2*x)^(3/2)-215/252*(1-2*x)^(1/2))/(-4-6*x)^2+134217/343*arctanh(
1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+62500*(-39/2420*(1-2*x)^(3/2)+193/5500*(1-2
*x)^(1/2))/(-6-10*x)^2-321825/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50778, size = 197, normalized size = 1.29 \[ \frac{321825}{2662} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{134217}{686} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2 \,{\left (52968600 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 360203715 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 816108324 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 616051205 \, \sqrt{-2 \, x + 1}\right )}}{5929 \,{\left (225 \,{\left (2 \, x - 1\right )}^{4} + 2040 \,{\left (2 \, x - 1\right )}^{3} + 6934 \,{\left (2 \, x - 1\right )}^{2} + 20944 \, x - 4543\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

321825/2662*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x
+ 1))) - 134217/686*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sq
rt(-2*x + 1))) - 2/5929*(52968600*(-2*x + 1)^(7/2) - 360203715*(-2*x + 1)^(5/2)
+ 816108324*(-2*x + 1)^(3/2) - 616051205*sqrt(-2*x + 1))/(225*(2*x - 1)^4 + 2040
*(2*x - 1)^3 + 6934*(2*x - 1)^2 + 20944*x - 4543)

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Fricas [A]  time = 0.249351, size = 240, normalized size = 1.57 \[ \frac{\sqrt{11} \sqrt{7}{\left (15769425 \, \sqrt{7} \sqrt{5}{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 16240257 \, \sqrt{11} \sqrt{3}{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{11} \sqrt{7}{\left (105937200 \, x^{3} + 201297915 \, x^{2} + 127303347 \, x + 26794499\right )} \sqrt{-2 \, x + 1}\right )}}{913066 \,{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/913066*sqrt(11)*sqrt(7)*(15769425*sqrt(7)*sqrt(5)*(225*x^4 + 570*x^3 + 541*x^2
+ 228*x + 36)*log((sqrt(11)*(5*x - 8) + 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) +
16240257*sqrt(11)*sqrt(3)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*log((sqrt(
7)*(3*x - 5) - 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)) + sqrt(11)*sqrt(7)*(10593720
0*x^3 + 201297915*x^2 + 127303347*x + 26794499)*sqrt(-2*x + 1))/(225*x^4 + 570*x
^3 + 541*x^2 + 228*x + 36)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.24664, size = 200, normalized size = 1.31 \[ \frac{321825}{2662} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{134217}{686} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{2 \,{\left (52968600 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 360203715 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 816108324 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 616051205 \, \sqrt{-2 \, x + 1}\right )}}{5929 \,{\left (15 \,{\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

321825/2662*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*s
qrt(-2*x + 1))) - 134217/686*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))
/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/5929*(52968600*(2*x - 1)^3*sqrt(-2*x + 1) +
360203715*(2*x - 1)^2*sqrt(-2*x + 1) - 816108324*(-2*x + 1)^(3/2) + 616051205*sq
rt(-2*x + 1))/(15*(2*x - 1)^2 + 136*x + 9)^2