3.2040 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^4 (3+5 x)^2} \, dx\)

Optimal. Leaf size=160 \[ -\frac{1676975 \sqrt{1-2 x}}{7546 (5 x+3)}+\frac{7585 \sqrt{1-2 x}}{343 (3 x+2) (5 x+3)}+\frac{145 \sqrt{1-2 x}}{98 (3 x+2)^2 (5 x+3)}+\frac{\sqrt{1-2 x}}{7 (3 x+2)^3 (5 x+3)}-\frac{1051695}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{32750}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-1676975*Sqrt[1 - 2*x])/(7546*(3 + 5*x)) + Sqrt[1 - 2*x]/(7*(2 + 3*x)^3*(3 + 5*
x)) + (145*Sqrt[1 - 2*x])/(98*(2 + 3*x)^2*(3 + 5*x)) + (7585*Sqrt[1 - 2*x])/(343
*(2 + 3*x)*(3 + 5*x)) - (1051695*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343
+ (32750*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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Rubi [A]  time = 0.331328, antiderivative size = 160, 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 \[ -\frac{1676975 \sqrt{1-2 x}}{7546 (5 x+3)}+\frac{7585 \sqrt{1-2 x}}{343 (3 x+2) (5 x+3)}+\frac{145 \sqrt{1-2 x}}{98 (3 x+2)^2 (5 x+3)}+\frac{\sqrt{1-2 x}}{7 (3 x+2)^3 (5 x+3)}-\frac{1051695}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{32750}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-1676975*Sqrt[1 - 2*x])/(7546*(3 + 5*x)) + Sqrt[1 - 2*x]/(7*(2 + 3*x)^3*(3 + 5*
x)) + (145*Sqrt[1 - 2*x])/(98*(2 + 3*x)^2*(3 + 5*x)) + (7585*Sqrt[1 - 2*x])/(343
*(2 + 3*x)*(3 + 5*x)) - (1051695*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343
+ (32750*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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Rubi in Sympy [A]  time = 36.2403, size = 126, normalized size = 0.79 \[ - \frac{1006185 \sqrt{- 2 x + 1}}{7546 \left (3 x + 2\right )} - \frac{14445 \sqrt{- 2 x + 1}}{1078 \left (3 x + 2\right )^{2}} - \frac{138 \sqrt{- 2 x + 1}}{77 \left (3 x + 2\right )^{3}} - \frac{5 \sqrt{- 2 x + 1}}{11 \left (3 x + 2\right )^{3} \left (5 x + 3\right )} - \frac{1051695 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{2401} + \frac{32750 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{121} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1006185*sqrt(-2*x + 1)/(7546*(3*x + 2)) - 14445*sqrt(-2*x + 1)/(1078*(3*x + 2)*
*2) - 138*sqrt(-2*x + 1)/(77*(3*x + 2)**3) - 5*sqrt(-2*x + 1)/(11*(3*x + 2)**3*(
5*x + 3)) - 1051695*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/2401 + 32750*sqrt(
55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/121

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Mathematica [A]  time = 0.164622, size = 101, normalized size = 0.63 \[ -\frac{\sqrt{1-2 x} \left (45278325 x^3+89054820 x^2+58335165 x+12724912\right )}{7546 (3 x+2)^3 (5 x+3)}-\frac{1051695}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{32750}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(12724912 + 58335165*x + 89054820*x^2 + 45278325*x^3))/(7546*(2
+ 3*x)^3*(3 + 5*x)) - (1051695*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 +
(32750*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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Maple [A]  time = 0.021, size = 91, normalized size = 0.6 \[ 972\,{\frac{1}{ \left ( -4-6\,x \right ) ^{3}} \left ({\frac{7565\, \left ( 1-2\,x \right ) ^{5/2}}{4116}}-{\frac{11455\, \left ( 1-2\,x \right ) ^{3/2}}{1323}}+{\frac{7711\,\sqrt{1-2\,x}}{756}} \right ) }-{\frac{1051695\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{250}{11}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}+{\frac{32750\,\sqrt{55}}{121}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

972*(7565/4116*(1-2*x)^(5/2)-11455/1323*(1-2*x)^(3/2)+7711/756*(1-2*x)^(1/2))/(-
4-6*x)^3-1051695/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+250/11*(1-2*x
)^(1/2)/(-6/5-2*x)+32750/121*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50284, size = 197, normalized size = 1.23 \[ -\frac{16375}{121} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{1051695}{4802} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{45278325 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 313944615 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 725394915 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 558527921 \, \sqrt{-2 \, x + 1}}{3773 \,{\left (135 \,{\left (2 \, x - 1\right )}^{4} + 1242 \,{\left (2 \, x - 1\right )}^{3} + 4284 \,{\left (2 \, x - 1\right )}^{2} + 13132 \, x - 2793\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-16375/121*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x +
1))) + 1051695/4802*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*s
qrt(-2*x + 1))) + 1/3773*(45278325*(-2*x + 1)^(7/2) - 313944615*(-2*x + 1)^(5/2)
+ 725394915*(-2*x + 1)^(3/2) - 558527921*sqrt(-2*x + 1))/(135*(2*x - 1)^4 + 124
2*(2*x - 1)^3 + 4284*(2*x - 1)^2 + 13132*x - 2793)

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Fricas [A]  time = 0.220915, size = 242, normalized size = 1.51 \[ \frac{\sqrt{11} \sqrt{7}{\left (11233250 \, \sqrt{7} \sqrt{5}{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} - 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 11568645 \, \sqrt{11} \sqrt{3}{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} + 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{11} \sqrt{7}{\left (45278325 \, x^{3} + 89054820 \, x^{2} + 58335165 \, x + 12724912\right )} \sqrt{-2 \, x + 1}\right )}}{581042 \,{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/581042*sqrt(11)*sqrt(7)*(11233250*sqrt(7)*sqrt(5)*(135*x^4 + 351*x^3 + 342*x^2
+ 148*x + 24)*log((sqrt(11)*(5*x - 8) - 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) +
11568645*sqrt(11)*sqrt(3)*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*log((sqrt(
7)*(3*x - 5) + 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)) - sqrt(11)*sqrt(7)*(45278325
*x^3 + 89054820*x^2 + 58335165*x + 12724912)*sqrt(-2*x + 1))/(135*x^4 + 351*x^3
+ 342*x^2 + 148*x + 24)

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.220194, size = 188, normalized size = 1.18 \[ -\frac{16375}{121} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{1051695}{4802} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{625 \, \sqrt{-2 \, x + 1}}{11 \,{\left (5 \, x + 3\right )}} - \frac{9 \,{\left (68085 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 320740 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 377839 \, \sqrt{-2 \, x + 1}\right )}}{2744 \,{\left (3 \, x + 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-16375/121*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sq
rt(-2*x + 1))) + 1051695/4802*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1)
)/(sqrt(21) + 3*sqrt(-2*x + 1))) - 625/11*sqrt(-2*x + 1)/(5*x + 3) - 9/2744*(680
85*(2*x - 1)^2*sqrt(-2*x + 1) - 320740*(-2*x + 1)^(3/2) + 377839*sqrt(-2*x + 1))
/(3*x + 2)^3