3.2029 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^5 (3+5 x)} \, dx\)

Optimal. Leaf size=137 \[ \frac{3135 \sqrt{1-2 x}}{56 (3 x+2)}+\frac{45 \sqrt{1-2 x}}{8 (3 x+2)^2}+\frac{3 \sqrt{1-2 x}}{4 (3 x+2)^3}+\frac{3 \sqrt{1-2 x}}{28 (3 x+2)^4}+\frac{36045}{28} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-1250 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(3*Sqrt[1 - 2*x])/(28*(2 + 3*x)^4) + (3*Sqrt[1 - 2*x])/(4*(2 + 3*x)^3) + (45*Sqr
t[1 - 2*x])/(8*(2 + 3*x)^2) + (3135*Sqrt[1 - 2*x])/(56*(2 + 3*x)) + (36045*Sqrt[
3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/28 - 1250*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*S
qrt[1 - 2*x]]

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Rubi [A]  time = 0.329373, antiderivative size = 137, 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 \[ \frac{3135 \sqrt{1-2 x}}{56 (3 x+2)}+\frac{45 \sqrt{1-2 x}}{8 (3 x+2)^2}+\frac{3 \sqrt{1-2 x}}{4 (3 x+2)^3}+\frac{3 \sqrt{1-2 x}}{28 (3 x+2)^4}+\frac{36045}{28} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-1250 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(3*Sqrt[1 - 2*x])/(28*(2 + 3*x)^4) + (3*Sqrt[1 - 2*x])/(4*(2 + 3*x)^3) + (45*Sqr
t[1 - 2*x])/(8*(2 + 3*x)^2) + (3135*Sqrt[1 - 2*x])/(56*(2 + 3*x)) + (36045*Sqrt[
3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/28 - 1250*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*S
qrt[1 - 2*x]]

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Rubi in Sympy [A]  time = 34.6293, size = 121, normalized size = 0.88 \[ \frac{3135 \sqrt{- 2 x + 1}}{56 \left (3 x + 2\right )} + \frac{45 \sqrt{- 2 x + 1}}{8 \left (3 x + 2\right )^{2}} + \frac{3 \sqrt{- 2 x + 1}}{4 \left (3 x + 2\right )^{3}} + \frac{3 \sqrt{- 2 x + 1}}{28 \left (3 x + 2\right )^{4}} + \frac{36045 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{196} - \frac{1250 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3135*sqrt(-2*x + 1)/(56*(3*x + 2)) + 45*sqrt(-2*x + 1)/(8*(3*x + 2)**2) + 3*sqrt
(-2*x + 1)/(4*(3*x + 2)**3) + 3*sqrt(-2*x + 1)/(28*(3*x + 2)**4) + 36045*sqrt(21
)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/196 - 1250*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x
+ 1)/11)/11

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Mathematica [A]  time = 0.167286, size = 92, normalized size = 0.67 \[ \frac{3 \sqrt{1-2 x} \left (28215 x^3+57375 x^2+38922 x+8810\right )}{56 (3 x+2)^4}+\frac{36045}{28} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-1250 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(3*Sqrt[1 - 2*x]*(8810 + 38922*x + 57375*x^2 + 28215*x^3))/(56*(2 + 3*x)^4) + (3
6045*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/28 - 1250*Sqrt[5/11]*ArcTanh[Sq
rt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.019, size = 84, normalized size = 0.6 \[ -486\,{\frac{1}{ \left ( -4-6\,x \right ) ^{4}} \left ({\frac{1045\, \left ( 1-2\,x \right ) ^{7/2}}{168}}-{\frac{1055\, \left ( 1-2\,x \right ) ^{5/2}}{24}}+{\frac{22373\, \left ( 1-2\,x \right ) ^{3/2}}{216}}-{\frac{369133\,\sqrt{1-2\,x}}{4536}} \right ) }+{\frac{36045\,\sqrt{21}}{196}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{1250\,\sqrt{55}}{11}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-486*(1045/168*(1-2*x)^(7/2)-1055/24*(1-2*x)^(5/2)+22373/216*(1-2*x)^(3/2)-36913
3/4536*(1-2*x)^(1/2))/(-4-6*x)^4+36045/196*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*2
1^(1/2)-1250/11*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.5022, size = 197, normalized size = 1.44 \[ \frac{625}{11} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{36045}{392} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{3 \,{\left (28215 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 199395 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 469833 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 369133 \, \sqrt{-2 \, x + 1}\right )}}{28 \,{\left (81 \,{\left (2 \, x - 1\right )}^{4} + 756 \,{\left (2 \, x - 1\right )}^{3} + 2646 \,{\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

625/11*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))
) - 36045/392*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*
x + 1))) - 3/28*(28215*(-2*x + 1)^(7/2) - 199395*(-2*x + 1)^(5/2) + 469833*(-2*x
+ 1)^(3/2) - 369133*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 756*(2*x - 1)^3 + 2646*(2
*x - 1)^2 + 8232*x - 1715)

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Fricas [A]  time = 0.223605, size = 242, normalized size = 1.77 \[ \frac{\sqrt{11} \sqrt{7}{\left (35000 \, \sqrt{7} \sqrt{5}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 36045 \, \sqrt{11} \sqrt{3}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + 3 \, \sqrt{11} \sqrt{7}{\left (28215 \, x^{3} + 57375 \, x^{2} + 38922 \, x + 8810\right )} \sqrt{-2 \, x + 1}\right )}}{4312 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4312*sqrt(11)*sqrt(7)*(35000*sqrt(7)*sqrt(5)*(81*x^4 + 216*x^3 + 216*x^2 + 96*
x + 16)*log((sqrt(11)*(5*x - 8) + 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) + 36045*
sqrt(11)*sqrt(3)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((sqrt(7)*(3*x - 5)
- 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)) + 3*sqrt(11)*sqrt(7)*(28215*x^3 + 57375*
x^2 + 38922*x + 8810)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.238248, size = 188, normalized size = 1.37 \[ \frac{625}{11} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{36045}{392} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{3 \,{\left (28215 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 199395 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 469833 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 369133 \, \sqrt{-2 \, x + 1}\right )}}{448 \,{\left (3 \, x + 2\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

625/11*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-
2*x + 1))) - 36045/392*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt
(21) + 3*sqrt(-2*x + 1))) + 3/448*(28215*(2*x - 1)^3*sqrt(-2*x + 1) + 199395*(2*
x - 1)^2*sqrt(-2*x + 1) - 469833*(-2*x + 1)^(3/2) + 369133*sqrt(-2*x + 1))/(3*x
+ 2)^4