3.2041 \(\int \frac{(2+3 x)^6}{\sqrt{1-2 x} (3+5 x)^3} \, dx\)

Optimal. Leaf size=140 \[ -\frac{\sqrt{1-2 x} (3 x+2)^5}{110 (5 x+3)^2}-\frac{117 \sqrt{1-2 x} (3 x+2)^4}{3025 (5 x+3)}-\frac{927 \sqrt{1-2 x} (3 x+2)^3}{211750}-\frac{56556 \sqrt{1-2 x} (3 x+2)^2}{378125}-\frac{9 \sqrt{1-2 x} (934875 x+2815648)}{3781250}-\frac{33069 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1890625 \sqrt{55}} \]

[Out]

(-56556*Sqrt[1 - 2*x]*(2 + 3*x)^2)/378125 - (927*Sqrt[1 - 2*x]*(2 + 3*x)^3)/2117
50 - (Sqrt[1 - 2*x]*(2 + 3*x)^5)/(110*(3 + 5*x)^2) - (117*Sqrt[1 - 2*x]*(2 + 3*x
)^4)/(3025*(3 + 5*x)) - (9*Sqrt[1 - 2*x]*(2815648 + 934875*x))/3781250 - (33069*
ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1890625*Sqrt[55])

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Rubi [A]  time = 0.283717, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\sqrt{1-2 x} (3 x+2)^5}{110 (5 x+3)^2}-\frac{117 \sqrt{1-2 x} (3 x+2)^4}{3025 (5 x+3)}-\frac{927 \sqrt{1-2 x} (3 x+2)^3}{211750}-\frac{56556 \sqrt{1-2 x} (3 x+2)^2}{378125}-\frac{9 \sqrt{1-2 x} (934875 x+2815648)}{3781250}-\frac{33069 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1890625 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 3*x)^6/(Sqrt[1 - 2*x]*(3 + 5*x)^3),x]

[Out]

(-56556*Sqrt[1 - 2*x]*(2 + 3*x)^2)/378125 - (927*Sqrt[1 - 2*x]*(2 + 3*x)^3)/2117
50 - (Sqrt[1 - 2*x]*(2 + 3*x)^5)/(110*(3 + 5*x)^2) - (117*Sqrt[1 - 2*x]*(2 + 3*x
)^4)/(3025*(3 + 5*x)) - (9*Sqrt[1 - 2*x]*(2815648 + 934875*x))/3781250 - (33069*
ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1890625*Sqrt[55])

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Rubi in Sympy [A]  time = 33.1343, size = 124, normalized size = 0.89 \[ - \frac{\sqrt{- 2 x + 1} \left (3 x + 2\right )^{5}}{110 \left (5 x + 3\right )^{2}} - \frac{117 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{4}}{3025 \left (5 x + 3\right )} - \frac{927 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}}{211750} - \frac{56556 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{378125} - \frac{\sqrt{- 2 x + 1} \left (883456875 x + 2660787360\right )}{397031250} - \frac{33069 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{103984375} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**6/(3+5*x)**3/(1-2*x)**(1/2),x)

[Out]

-sqrt(-2*x + 1)*(3*x + 2)**5/(110*(5*x + 3)**2) - 117*sqrt(-2*x + 1)*(3*x + 2)**
4/(3025*(5*x + 3)) - 927*sqrt(-2*x + 1)*(3*x + 2)**3/211750 - 56556*sqrt(-2*x +
1)*(3*x + 2)**2/378125 - sqrt(-2*x + 1)*(883456875*x + 2660787360)/397031250 - 3
3069*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/103984375

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Mathematica [A]  time = 0.14224, size = 73, normalized size = 0.52 \[ \frac{-\frac{55 \sqrt{1-2 x} \left (551306250 x^5+2690374500 x^4+6078090150 x^3+9876010320 x^2+7254126105 x+1804176536\right )}{(5 x+3)^2}-462966 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1455781250} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + 3*x)^6/(Sqrt[1 - 2*x]*(3 + 5*x)^3),x]

[Out]

((-55*Sqrt[1 - 2*x]*(1804176536 + 7254126105*x + 9876010320*x^2 + 6078090150*x^3
+ 2690374500*x^4 + 551306250*x^5))/(3 + 5*x)^2 - 462966*Sqrt[55]*ArcTanh[Sqrt[5
/11]*Sqrt[1 - 2*x]])/1455781250

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Maple [A]  time = 0.017, size = 84, normalized size = 0.6 \[{\frac{729}{7000} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{26973}{25000} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{111213}{25000} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{276183}{25000}\sqrt{1-2\,x}}+{\frac{2}{125\, \left ( -6-10\,x \right ) ^{2}} \left ({\frac{399}{6050} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{401}{2750}\sqrt{1-2\,x}} \right ) }-{\frac{33069\,\sqrt{55}}{103984375}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^6/(3+5*x)^3/(1-2*x)^(1/2),x)

[Out]

729/7000*(1-2*x)^(7/2)-26973/25000*(1-2*x)^(5/2)+111213/25000*(1-2*x)^(3/2)-2761
83/25000*(1-2*x)^(1/2)+2/125*(399/6050*(1-2*x)^(3/2)-401/2750*(1-2*x)^(1/2))/(-6
-10*x)^2-33069/103984375*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50547, size = 149, normalized size = 1.06 \[ \frac{729}{7000} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{26973}{25000} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{111213}{25000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{33069}{207968750} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{276183}{25000} \, \sqrt{-2 \, x + 1} + \frac{1995 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 4411 \, \sqrt{-2 \, x + 1}}{1890625 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^6/((5*x + 3)^3*sqrt(-2*x + 1)),x, algorithm="maxima")

[Out]

729/7000*(-2*x + 1)^(7/2) - 26973/25000*(-2*x + 1)^(5/2) + 111213/25000*(-2*x +
1)^(3/2) + 33069/207968750*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55)
+ 5*sqrt(-2*x + 1))) - 276183/25000*sqrt(-2*x + 1) + 1/1890625*(1995*(-2*x + 1)
^(3/2) - 4411*sqrt(-2*x + 1))/(25*(2*x - 1)^2 + 220*x + 11)

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Fricas [A]  time = 0.218382, size = 127, normalized size = 0.91 \[ -\frac{\sqrt{55}{\left (\sqrt{55}{\left (551306250 \, x^{5} + 2690374500 \, x^{4} + 6078090150 \, x^{3} + 9876010320 \, x^{2} + 7254126105 \, x + 1804176536\right )} \sqrt{-2 \, x + 1} - 231483 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{1455781250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^6/((5*x + 3)^3*sqrt(-2*x + 1)),x, algorithm="fricas")

[Out]

-1/1455781250*sqrt(55)*(sqrt(55)*(551306250*x^5 + 2690374500*x^4 + 6078090150*x^
3 + 9876010320*x^2 + 7254126105*x + 1804176536)*sqrt(-2*x + 1) - 231483*(25*x^2
+ 30*x + 9)*log((sqrt(55)*(5*x - 8) + 55*sqrt(-2*x + 1))/(5*x + 3)))/(25*x^2 + 3
0*x + 9)

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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((2+3*x)**6/(3+5*x)**3/(1-2*x)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.217377, size = 159, normalized size = 1.14 \[ -\frac{729}{7000} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{26973}{25000} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{111213}{25000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{33069}{207968750} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{276183}{25000} \, \sqrt{-2 \, x + 1} + \frac{1995 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 4411 \, \sqrt{-2 \, x + 1}}{7562500 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^6/((5*x + 3)^3*sqrt(-2*x + 1)),x, algorithm="giac")

[Out]

-729/7000*(2*x - 1)^3*sqrt(-2*x + 1) - 26973/25000*(2*x - 1)^2*sqrt(-2*x + 1) +
111213/25000*(-2*x + 1)^(3/2) + 33069/207968750*sqrt(55)*ln(1/2*abs(-2*sqrt(55)
+ 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 276183/25000*sqrt(-2*x + 1
) + 1/7562500*(1995*(-2*x + 1)^(3/2) - 4411*sqrt(-2*x + 1))/(5*x + 3)^2