3.2042 \(\int \frac{(2+3 x)^5}{\sqrt{1-2 x} (3+5 x)^3} \, dx\)

Optimal. Leaf size=120 \[ -\frac{\sqrt{1-2 x} (3 x+2)^4}{110 (5 x+3)^2}-\frac{201 \sqrt{1-2 x} (3 x+2)^3}{6050 (5 x+3)}-\frac{1512 \sqrt{1-2 x} (3 x+2)^2}{75625}-\frac{189 \sqrt{1-2 x} (2875 x+8976)}{756250}-\frac{22113 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{378125 \sqrt{55}} \]

[Out]

(-1512*Sqrt[1 - 2*x]*(2 + 3*x)^2)/75625 - (Sqrt[1 - 2*x]*(2 + 3*x)^4)/(110*(3 +
5*x)^2) - (201*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(6050*(3 + 5*x)) - (189*Sqrt[1 - 2*x]*
(8976 + 2875*x))/756250 - (22113*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(378125*Sqrt
[55])

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Rubi [A]  time = 0.222784, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\sqrt{1-2 x} (3 x+2)^4}{110 (5 x+3)^2}-\frac{201 \sqrt{1-2 x} (3 x+2)^3}{6050 (5 x+3)}-\frac{1512 \sqrt{1-2 x} (3 x+2)^2}{75625}-\frac{189 \sqrt{1-2 x} (2875 x+8976)}{756250}-\frac{22113 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{378125 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(-1512*Sqrt[1 - 2*x]*(2 + 3*x)^2)/75625 - (Sqrt[1 - 2*x]*(2 + 3*x)^4)/(110*(3 +
5*x)^2) - (201*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(6050*(3 + 5*x)) - (189*Sqrt[1 - 2*x]*
(8976 + 2875*x))/756250 - (22113*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(378125*Sqrt
[55])

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Rubi in Sympy [A]  time = 26.118, size = 105, normalized size = 0.88 \[ - \frac{\sqrt{- 2 x + 1} \left (3 x + 2\right )^{4}}{110 \left (5 x + 3\right )^{2}} - \frac{201 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}}{6050 \left (5 x + 3\right )} - \frac{1512 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{75625} - \frac{\sqrt{- 2 x + 1} \left (8150625 x + 25446960\right )}{11343750} - \frac{22113 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{20796875} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-2*x + 1)*(3*x + 2)**4/(110*(5*x + 3)**2) - 201*sqrt(-2*x + 1)*(3*x + 2)**
3/(6050*(5*x + 3)) - 1512*sqrt(-2*x + 1)*(3*x + 2)**2/75625 - sqrt(-2*x + 1)*(81
50625*x + 25446960)/11343750 - 22113*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/
20796875

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Mathematica [A]  time = 0.124935, size = 68, normalized size = 0.57 \[ \frac{-\frac{55 \sqrt{1-2 x} \left (7350750 x^4+32506650 x^3+76970520 x^2+63610155 x+16525496\right )}{(5 x+3)^2}-44226 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{41593750} \]

Antiderivative was successfully verified.

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

[Out]

((-55*Sqrt[1 - 2*x]*(16525496 + 63610155*x + 76970520*x^2 + 32506650*x^3 + 73507
50*x^4))/(3 + 5*x)^2 - 44226*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/4159375
0

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Maple [A]  time = 0.019, size = 75, normalized size = 0.6 \[ -{\frac{243}{2500} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{513}{625} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{39393}{12500}\sqrt{1-2\,x}}+{\frac{4}{125\, \left ( -6-10\,x \right ) ^{2}} \left ({\frac{333}{2420} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{67}{220}\sqrt{1-2\,x}} \right ) }-{\frac{22113\,\sqrt{55}}{20796875}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-243/2500*(1-2*x)^(5/2)+513/625*(1-2*x)^(3/2)-39393/12500*(1-2*x)^(1/2)+4/125*(3
33/2420*(1-2*x)^(3/2)-67/220*(1-2*x)^(1/2))/(-6-10*x)^2-22113/20796875*arctanh(1
/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50301, size = 136, normalized size = 1.13 \[ -\frac{243}{2500} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{513}{625} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{22113}{41593750} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{39393}{12500} \, \sqrt{-2 \, x + 1} + \frac{333 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 737 \, \sqrt{-2 \, x + 1}}{75625 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-243/2500*(-2*x + 1)^(5/2) + 513/625*(-2*x + 1)^(3/2) + 22113/41593750*sqrt(55)*
log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 39393/12500*
sqrt(-2*x + 1) + 1/75625*(333*(-2*x + 1)^(3/2) - 737*sqrt(-2*x + 1))/(25*(2*x -
1)^2 + 220*x + 11)

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Fricas [A]  time = 0.222752, size = 120, normalized size = 1. \[ -\frac{\sqrt{55}{\left (\sqrt{55}{\left (7350750 \, x^{4} + 32506650 \, x^{3} + 76970520 \, x^{2} + 63610155 \, x + 16525496\right )} \sqrt{-2 \, x + 1} - 22113 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{41593750 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/41593750*sqrt(55)*(sqrt(55)*(7350750*x^4 + 32506650*x^3 + 76970520*x^2 + 6361
0155*x + 16525496)*sqrt(-2*x + 1) - 22113*(25*x^2 + 30*x + 9)*log((sqrt(55)*(5*x
- 8) + 55*sqrt(-2*x + 1))/(5*x + 3)))/(25*x^2 + 30*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)**5/(3+5*x)**3/(1-2*x)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.226931, size = 138, normalized size = 1.15 \[ -\frac{243}{2500} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{513}{625} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{22113}{41593750} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{39393}{12500} \, \sqrt{-2 \, x + 1} + \frac{333 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 737 \, \sqrt{-2 \, x + 1}}{302500 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-243/2500*(2*x - 1)^2*sqrt(-2*x + 1) + 513/625*(-2*x + 1)^(3/2) + 22113/41593750
*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x +
1))) - 39393/12500*sqrt(-2*x + 1) + 1/302500*(333*(-2*x + 1)^(3/2) - 737*sqrt(-2
*x + 1))/(5*x + 3)^2