3.2028 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^4 (3+5 x)} \, dx\)

Optimal. Leaf size=117 \[ \frac{3840 \sqrt{1-2 x}}{343 (3 x+2)}+\frac{55 \sqrt{1-2 x}}{49 (3 x+2)^2}+\frac{\sqrt{1-2 x}}{7 (3 x+2)^3}+\frac{88310}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-250 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

Sqrt[1 - 2*x]/(7*(2 + 3*x)^3) + (55*Sqrt[1 - 2*x])/(49*(2 + 3*x)^2) + (3840*Sqrt
[1 - 2*x])/(343*(2 + 3*x)) + (88310*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/
343 - 250*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.261473, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{3840 \sqrt{1-2 x}}{343 (3 x+2)}+\frac{55 \sqrt{1-2 x}}{49 (3 x+2)^2}+\frac{\sqrt{1-2 x}}{7 (3 x+2)^3}+\frac{88310}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-250 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[1 - 2*x]/(7*(2 + 3*x)^3) + (55*Sqrt[1 - 2*x])/(49*(2 + 3*x)^2) + (3840*Sqrt
[1 - 2*x])/(343*(2 + 3*x)) + (88310*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/
343 - 250*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi in Sympy [A]  time = 27.7662, size = 100, normalized size = 0.85 \[ \frac{3840 \sqrt{- 2 x + 1}}{343 \left (3 x + 2\right )} + \frac{55 \sqrt{- 2 x + 1}}{49 \left (3 x + 2\right )^{2}} + \frac{\sqrt{- 2 x + 1}}{7 \left (3 x + 2\right )^{3}} + \frac{88310 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{2401} - \frac{250 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3840*sqrt(-2*x + 1)/(343*(3*x + 2)) + 55*sqrt(-2*x + 1)/(49*(3*x + 2)**2) + sqrt
(-2*x + 1)/(7*(3*x + 2)**3) + 88310*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/24
01 - 250*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/11

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Mathematica [A]  time = 0.177671, size = 87, normalized size = 0.74 \[ \frac{3 \sqrt{1-2 x} \left (11520 x^2+15745 x+5393\right )}{343 (3 x+2)^3}+\frac{88310}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-250 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(3*Sqrt[1 - 2*x]*(5393 + 15745*x + 11520*x^2))/(343*(2 + 3*x)^3) + (88310*Sqrt[3
/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - 250*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sq
rt[1 - 2*x]]

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Maple [A]  time = 0.017, size = 75, normalized size = 0.6 \[ -162\,{\frac{1}{ \left ( -4-6\,x \right ) ^{3}} \left ({\frac{1280\, \left ( 1-2\,x \right ) ^{5/2}}{1029}}-{\frac{7790\, \left ( 1-2\,x \right ) ^{3/2}}{1323}}+{\frac{1318\,\sqrt{1-2\,x}}{189}} \right ) }+{\frac{88310\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{250\,\sqrt{55}}{11}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-162*(1280/1029*(1-2*x)^(5/2)-7790/1323*(1-2*x)^(3/2)+1318/189*(1-2*x)^(1/2))/(-
4-6*x)^3+88310/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-250/11*arctanh(
1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.49577, size = 173, normalized size = 1.48 \[ \frac{125}{11} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{44155}{2401} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{12 \,{\left (5760 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 27265 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 32291 \, \sqrt{-2 \, x + 1}\right )}}{343 \,{\left (27 \,{\left (2 \, x - 1\right )}^{3} + 189 \,{\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125/11*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))
) - 44155/2401*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2
*x + 1))) + 12/343*(5760*(-2*x + 1)^(5/2) - 27265*(-2*x + 1)^(3/2) + 32291*sqrt(
-2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)

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Fricas [A]  time = 0.230609, size = 215, normalized size = 1.84 \[ \frac{\sqrt{11} \sqrt{7}{\left (42875 \, \sqrt{7} \sqrt{5}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 44155 \, \sqrt{11} \sqrt{3}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + 3 \, \sqrt{11} \sqrt{7}{\left (11520 \, x^{2} + 15745 \, x + 5393\right )} \sqrt{-2 \, x + 1}\right )}}{26411 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/26411*sqrt(11)*sqrt(7)*(42875*sqrt(7)*sqrt(5)*(27*x^3 + 54*x^2 + 36*x + 8)*log
((sqrt(11)*(5*x - 8) + 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) + 44155*sqrt(11)*sq
rt(3)*(27*x^3 + 54*x^2 + 36*x + 8)*log((sqrt(7)*(3*x - 5) - 7*sqrt(3)*sqrt(-2*x
+ 1))/(3*x + 2)) + 3*sqrt(11)*sqrt(7)*(11520*x^2 + 15745*x + 5393)*sqrt(-2*x + 1
))/(27*x^3 + 54*x^2 + 36*x + 8)

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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)/(1-2*x)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.265351, size = 166, normalized size = 1.42 \[ \frac{125}{11} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{44155}{2401} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{3 \,{\left (5760 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 27265 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 32291 \, \sqrt{-2 \, x + 1}\right )}}{686 \,{\left (3 \, x + 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125/11*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-
2*x + 1))) - 44155/2401*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqr
t(21) + 3*sqrt(-2*x + 1))) + 3/686*(5760*(2*x - 1)^2*sqrt(-2*x + 1) - 27265*(-2*
x + 1)^(3/2) + 32291*sqrt(-2*x + 1))/(3*x + 2)^3