3.2043 \(\int \frac{(2+3 x)^4}{\sqrt{1-2 x} (3+5 x)^3} \, dx\)

Optimal. Leaf size=100 \[ -\frac{\sqrt{1-2 x} (3 x+2)^3}{110 (5 x+3)^2}-\frac{84 \sqrt{1-2 x} (3 x+2)^2}{3025 (5 x+3)}-\frac{63 \sqrt{1-2 x} (75 x+352)}{30250}-\frac{2667 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15125 \sqrt{55}} \]

[Out]

-(Sqrt[1 - 2*x]*(2 + 3*x)^3)/(110*(3 + 5*x)^2) - (84*Sqrt[1 - 2*x]*(2 + 3*x)^2)/
(3025*(3 + 5*x)) - (63*Sqrt[1 - 2*x]*(352 + 75*x))/30250 - (2667*ArcTanh[Sqrt[5/
11]*Sqrt[1 - 2*x]])/(15125*Sqrt[55])

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Rubi [A]  time = 0.166347, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{\sqrt{1-2 x} (3 x+2)^3}{110 (5 x+3)^2}-\frac{84 \sqrt{1-2 x} (3 x+2)^2}{3025 (5 x+3)}-\frac{63 \sqrt{1-2 x} (75 x+352)}{30250}-\frac{2667 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15125 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(2 + 3*x)^3)/(110*(3 + 5*x)^2) - (84*Sqrt[1 - 2*x]*(2 + 3*x)^2)/
(3025*(3 + 5*x)) - (63*Sqrt[1 - 2*x]*(352 + 75*x))/30250 - (2667*ArcTanh[Sqrt[5/
11]*Sqrt[1 - 2*x]])/(15125*Sqrt[55])

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Rubi in Sympy [A]  time = 18.332, size = 87, normalized size = 0.87 \[ - \frac{\sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}}{110 \left (5 x + 3\right )^{2}} - \frac{84 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{3025 \left (5 x + 3\right )} - \frac{\sqrt{- 2 x + 1} \left (70875 x + 332640\right )}{453750} - \frac{2667 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{831875} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-2*x + 1)*(3*x + 2)**3/(110*(5*x + 3)**2) - 84*sqrt(-2*x + 1)*(3*x + 2)**2
/(3025*(5*x + 3)) - sqrt(-2*x + 1)*(70875*x + 332640)/453750 - 2667*sqrt(55)*ata
nh(sqrt(55)*sqrt(-2*x + 1)/11)/831875

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Mathematica [A]  time = 0.11369, size = 63, normalized size = 0.63 \[ \frac{-\frac{55 \sqrt{1-2 x} \left (163350 x^3+784080 x^2+764745 x+211864\right )}{(5 x+3)^2}-5334 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1663750} \]

Antiderivative was successfully verified.

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

[Out]

((-55*Sqrt[1 - 2*x]*(211864 + 764745*x + 784080*x^2 + 163350*x^3))/(3 + 5*x)^2 -
5334*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1663750

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Maple [A]  time = 0.018, size = 66, normalized size = 0.7 \[{\frac{27}{250} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{1107}{1250}\sqrt{1-2\,x}}+{\frac{4}{25\, \left ( -6-10\,x \right ) ^{2}} \left ({\frac{267}{2420} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{269}{1100}\sqrt{1-2\,x}} \right ) }-{\frac{2667\,\sqrt{55}}{831875}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27/250*(1-2*x)^(3/2)-1107/1250*(1-2*x)^(1/2)+4/25*(267/2420*(1-2*x)^(3/2)-269/11
00*(1-2*x)^(1/2))/(-6-10*x)^2-2667/831875*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*5
5^(1/2)

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Maxima [A]  time = 1.50245, size = 124, normalized size = 1.24 \[ \frac{27}{250} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{2667}{1663750} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{1107}{1250} \, \sqrt{-2 \, x + 1} + \frac{1335 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2959 \, \sqrt{-2 \, x + 1}}{75625 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27/250*(-2*x + 1)^(3/2) + 2667/1663750*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1
))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 1107/1250*sqrt(-2*x + 1) + 1/75625*(1335*(-2
*x + 1)^(3/2) - 2959*sqrt(-2*x + 1))/(25*(2*x - 1)^2 + 220*x + 11)

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Fricas [A]  time = 0.248498, size = 113, normalized size = 1.13 \[ -\frac{\sqrt{55}{\left (\sqrt{55}{\left (163350 \, x^{3} + 784080 \, x^{2} + 764745 \, x + 211864\right )} \sqrt{-2 \, x + 1} - 2667 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{1663750 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1663750*sqrt(55)*(sqrt(55)*(163350*x^3 + 784080*x^2 + 764745*x + 211864)*sqrt
(-2*x + 1) - 2667*(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)**4/(3+5*x)**3/(1-2*x)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.230878, size = 116, normalized size = 1.16 \[ \frac{27}{250} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{2667}{1663750} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{1107}{1250} \, \sqrt{-2 \, x + 1} + \frac{1335 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2959 \, \sqrt{-2 \, x + 1}}{302500 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27/250*(-2*x + 1)^(3/2) + 2667/1663750*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt
(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 1107/1250*sqrt(-2*x + 1) + 1/302500
*(1335*(-2*x + 1)^(3/2) - 2959*sqrt(-2*x + 1))/(5*x + 3)^2