3.2051 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^4 (3+5 x)^3} \, dx\)

Optimal. Leaf size=180 \[ \frac{137735775 \sqrt{1-2 x}}{83006 (5 x+3)}-\frac{2076675 \sqrt{1-2 x}}{7546 (5 x+3)^2}+\frac{12555 \sqrt{1-2 x}}{343 (3 x+2) (5 x+3)^2}+\frac{90 \sqrt{1-2 x}}{49 (3 x+2)^2 (5 x+3)^2}+\frac{\sqrt{1-2 x}}{7 (3 x+2)^3 (5 x+3)^2}+\frac{7852680}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{2689875}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-2076675*Sqrt[1 - 2*x])/(7546*(3 + 5*x)^2) + Sqrt[1 - 2*x]/(7*(2 + 3*x)^3*(3 +
5*x)^2) + (90*Sqrt[1 - 2*x])/(49*(2 + 3*x)^2*(3 + 5*x)^2) + (12555*Sqrt[1 - 2*x]
)/(343*(2 + 3*x)*(3 + 5*x)^2) + (137735775*Sqrt[1 - 2*x])/(83006*(3 + 5*x)) + (7
852680*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (2689875*Sqrt[5/11]*Arc
Tanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Rubi [A]  time = 0.401716, 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 \[ \frac{137735775 \sqrt{1-2 x}}{83006 (5 x+3)}-\frac{2076675 \sqrt{1-2 x}}{7546 (5 x+3)^2}+\frac{12555 \sqrt{1-2 x}}{343 (3 x+2) (5 x+3)^2}+\frac{90 \sqrt{1-2 x}}{49 (3 x+2)^2 (5 x+3)^2}+\frac{\sqrt{1-2 x}}{7 (3 x+2)^3 (5 x+3)^2}+\frac{7852680}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{2689875}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2076675*Sqrt[1 - 2*x])/(7546*(3 + 5*x)^2) + Sqrt[1 - 2*x]/(7*(2 + 3*x)^3*(3 +
5*x)^2) + (90*Sqrt[1 - 2*x])/(49*(2 + 3*x)^2*(3 + 5*x)^2) + (12555*Sqrt[1 - 2*x]
)/(343*(2 + 3*x)*(3 + 5*x)^2) + (137735775*Sqrt[1 - 2*x])/(83006*(3 + 5*x)) + (7
852680*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (2689875*Sqrt[5/11]*Arc
Tanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Rubi in Sympy [A]  time = 45.839, size = 151, normalized size = 0.84 \[ \frac{82641465 \sqrt{- 2 x + 1}}{83006 \left (3 x + 2\right )} + \frac{593190 \sqrt{- 2 x + 1}}{5929 \left (3 x + 2\right )^{2}} + \frac{22653 \sqrt{- 2 x + 1}}{1694 \left (3 x + 2\right )^{3}} + \frac{405 \sqrt{- 2 x + 1}}{121 \left (3 x + 2\right )^{3} \left (5 x + 3\right )} - \frac{5 \sqrt{- 2 x + 1}}{22 \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{2}} + \frac{7852680 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{2401} - \frac{2689875 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{1331} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

82641465*sqrt(-2*x + 1)/(83006*(3*x + 2)) + 593190*sqrt(-2*x + 1)/(5929*(3*x + 2
)**2) + 22653*sqrt(-2*x + 1)/(1694*(3*x + 2)**3) + 405*sqrt(-2*x + 1)/(121*(3*x
+ 2)**3*(5*x + 3)) - 5*sqrt(-2*x + 1)/(22*(3*x + 2)**3*(5*x + 3)**2) + 7852680*s
qrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/2401 - 2689875*sqrt(55)*atanh(sqrt(55)*
sqrt(-2*x + 1)/11)/1331

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Mathematica [A]  time = 0.175722, size = 106, normalized size = 0.59 \[ \frac{\sqrt{1-2 x} \left (18594329625 x^4+47728484550 x^3+45899434890 x^2+19599448500 x+3135381218\right )}{83006 (3 x+2)^3 (5 x+3)^2}+\frac{7852680}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{2689875}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(3135381218 + 19599448500*x + 45899434890*x^2 + 47728484550*x^3 +
18594329625*x^4))/(83006*(2 + 3*x)^3*(3 + 5*x)^2) + (7852680*Sqrt[3/7]*ArcTanh[
Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (2689875*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 -
2*x]])/121

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Maple [A]  time = 0.021, size = 103, normalized size = 0.6 \[ -2916\,{\frac{1}{ \left ( -4-6\,x \right ) ^{3}} \left ({\frac{3755\, \left ( 1-2\,x \right ) ^{5/2}}{1029}}-{\frac{22690\, \left ( 1-2\,x \right ) ^{3/2}}{1323}}+{\frac{3809\,\sqrt{1-2\,x}}{189}} \right ) }+{\frac{7852680\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+312500\,{\frac{1}{ \left ( -6-10\,x \right ) ^{2}} \left ( -{\frac{261\, \left ( 1-2\,x \right ) ^{3/2}}{12100}}+{\frac{259\,\sqrt{1-2\,x}}{5500}} \right ) }-{\frac{2689875\,\sqrt{55}}{1331}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2916*(3755/1029*(1-2*x)^(5/2)-22690/1323*(1-2*x)^(3/2)+3809/189*(1-2*x)^(1/2))/
(-4-6*x)^3+7852680/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+312500*(-26
1/12100*(1-2*x)^(3/2)+259/5500*(1-2*x)^(1/2))/(-6-10*x)^2-2689875/1331*arctanh(1
/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.51158, size = 221, normalized size = 1.23 \[ \frac{2689875}{2662} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{3926340}{2401} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{18594329625 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 169834287600 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 581534624610 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 884739292920 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 504610725773 \, \sqrt{-2 \, x + 1}}{41503 \,{\left (675 \,{\left (2 \, x - 1\right )}^{5} + 7695 \,{\left (2 \, x - 1\right )}^{4} + 35082 \,{\left (2 \, x - 1\right )}^{3} + 79954 \,{\left (2 \, x - 1\right )}^{2} + 182182 \, x - 49588\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2689875/2662*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x
+ 1))) - 3926340/2401*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3
*sqrt(-2*x + 1))) + 1/41503*(18594329625*(-2*x + 1)^(9/2) - 169834287600*(-2*x +
1)^(7/2) + 581534624610*(-2*x + 1)^(5/2) - 884739292920*(-2*x + 1)^(3/2) + 5046
10725773*sqrt(-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 = 0.258246, size = 267, normalized size = 1.48 \[ \frac{\sqrt{11} \sqrt{7}{\left (922627125 \, \sqrt{7} \sqrt{5}{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 950174280 \, \sqrt{11} \sqrt{3}{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{11} \sqrt{7}{\left (18594329625 \, x^{4} + 47728484550 \, x^{3} + 45899434890 \, x^{2} + 19599448500 \, x + 3135381218\right )} \sqrt{-2 \, x + 1}\right )}}{6391462 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6391462*sqrt(11)*sqrt(7)*(922627125*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)) + 950174280*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))
+ sqrt(11)*sqrt(7)*(18594329625*x^4 + 47728484550*x^3 + 45899434890*x^2 + 19599
448500*x + 3135381218)*sqrt(-2*x + 1))/(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2
+ 564*x + 72)

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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)**4/(3+5*x)**3/(1-2*x)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.231134, size = 204, normalized size = 1.13 \[ \frac{2689875}{2662} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{3926340}{2401} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{625 \,{\left (1305 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2849 \, \sqrt{-2 \, x + 1}\right )}}{484 \,{\left (5 \, x + 3\right )}^{2}} + \frac{27 \,{\left (33795 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 158830 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 186641 \, \sqrt{-2 \, x + 1}\right )}}{686 \,{\left (3 \, x + 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2689875/2662*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*
sqrt(-2*x + 1))) - 3926340/2401*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x +
1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 625/484*(1305*(-2*x + 1)^(3/2) - 2849*sqrt(
-2*x + 1))/(5*x + 3)^2 + 27/686*(33795*(2*x - 1)^2*sqrt(-2*x + 1) - 158830*(-2*x
+ 1)^(3/2) + 186641*sqrt(-2*x + 1))/(3*x + 2)^3