3.1874 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=131 \[ \frac{\sqrt{1-2 x} (5 x+3)^3}{(3 x+2)^3}-\frac{(1-2 x)^{3/2} (5 x+3)^3}{12 (3 x+2)^4}+\frac{13 \sqrt{1-2 x} (5 x+3)^2}{56 (3 x+2)^2}-\frac{\sqrt{1-2 x} (26775 x+18187)}{1176 (3 x+2)}+\frac{13243 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{588 \sqrt{21}} \]

[Out]

(13*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(56*(2 + 3*x)^2) - ((1 - 2*x)^(3/2)*(3 + 5*x)^3)/
(12*(2 + 3*x)^4) + (Sqrt[1 - 2*x]*(3 + 5*x)^3)/(2 + 3*x)^3 - (Sqrt[1 - 2*x]*(181
87 + 26775*x))/(1176*(2 + 3*x)) + (13243*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(588*
Sqrt[21])

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Rubi [A]  time = 0.211923, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{\sqrt{1-2 x} (5 x+3)^3}{(3 x+2)^3}-\frac{(1-2 x)^{3/2} (5 x+3)^3}{12 (3 x+2)^4}+\frac{13 \sqrt{1-2 x} (5 x+3)^2}{56 (3 x+2)^2}-\frac{\sqrt{1-2 x} (26775 x+18187)}{1176 (3 x+2)}+\frac{13243 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{588 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(13*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(56*(2 + 3*x)^2) - ((1 - 2*x)^(3/2)*(3 + 5*x)^3)/
(12*(2 + 3*x)^4) + (Sqrt[1 - 2*x]*(3 + 5*x)^3)/(2 + 3*x)^3 - (Sqrt[1 - 2*x]*(181
87 + 26775*x))/(1176*(2 + 3*x)) + (13243*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(588*
Sqrt[21])

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Rubi in Sympy [A]  time = 20.2025, size = 105, normalized size = 0.8 \[ - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (1914597 x + 1219131\right )}{666792 \left (3 x + 2\right )^{2}} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{2}}{7 \left (3 x + 2\right )^{3}} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{3}}{12 \left (3 x + 2\right )^{4}} - \frac{13243 \sqrt{- 2 x + 1}}{4116} + \frac{13243 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{12348} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(3/2)*(1914597*x + 1219131)/(666792*(3*x + 2)**2) - (-2*x + 1)**(3/
2)*(5*x + 3)**2/(7*(3*x + 2)**3) - (-2*x + 1)**(3/2)*(5*x + 3)**3/(12*(3*x + 2)*
*4) - 13243*sqrt(-2*x + 1)/4116 + 13243*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7
)/12348

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Mathematica [A]  time = 0.133585, size = 68, normalized size = 0.52 \[ \frac{26486 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{21 \sqrt{1-2 x} \left (196000 x^4+661639 x^3+788415 x^2+401850 x+74810\right )}{(3 x+2)^4}}{24696} \]

Antiderivative was successfully verified.

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

[Out]

((-21*Sqrt[1 - 2*x]*(74810 + 401850*x + 788415*x^2 + 661639*x^3 + 196000*x^4))/(
2 + 3*x)^4 + 26486*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/24696

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Maple [A]  time = 0.017, size = 75, normalized size = 0.6 \[ -{\frac{500}{243}\sqrt{1-2\,x}}-{\frac{4}{3\, \left ( -4-6\,x \right ) ^{4}} \left ( -{\frac{416917}{2352} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{406463}{336} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{1189171}{432} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2706781}{1296}\sqrt{1-2\,x}} \right ) }+{\frac{13243\,\sqrt{21}}{12348}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-500/243*(1-2*x)^(1/2)-4/3*(-416917/2352*(1-2*x)^(7/2)+406463/336*(1-2*x)^(5/2)-
1189171/432*(1-2*x)^(3/2)+2706781/1296*(1-2*x)^(1/2))/(-4-6*x)^4+13243/12348*arc
tanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.52601, size = 161, normalized size = 1.23 \[ -\frac{13243}{24696} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{500}{243} \, \sqrt{-2 \, x + 1} + \frac{11256759 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 76821507 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 174808137 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 132632269 \, \sqrt{-2 \, x + 1}}{47628 \,{\left (81 \,{\left (2 \, x - 1\right )}^{4} + 756 \,{\left (2 \, x - 1\right )}^{3} + 2646 \,{\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13243/24696*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) - 500/243*sqrt(-2*x + 1) + 1/47628*(11256759*(-2*x + 1)^(7/2) - 76821507
*(-2*x + 1)^(5/2) + 174808137*(-2*x + 1)^(3/2) - 132632269*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 = 0.212153, size = 147, normalized size = 1.12 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (196000 \, x^{4} + 661639 \, x^{3} + 788415 \, x^{2} + 401850 \, x + 74810\right )} \sqrt{-2 \, x + 1} - 13243 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{24696 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24696*sqrt(21)*(sqrt(21)*(196000*x^4 + 661639*x^3 + 788415*x^2 + 401850*x + 7
4810)*sqrt(-2*x + 1) - 13243*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((sqrt(
21)*(3*x - 5) - 21*sqrt(-2*x + 1))/(3*x + 2)))/(81*x^4 + 216*x^3 + 216*x^2 + 96*
x + 16)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.215981, size = 147, normalized size = 1.12 \[ -\frac{13243}{24696} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{500}{243} \, \sqrt{-2 \, x + 1} - \frac{11256759 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 76821507 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 174808137 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 132632269 \, \sqrt{-2 \, x + 1}}{762048 \,{\left (3 \, x + 2\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13243/24696*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*s
qrt(-2*x + 1))) - 500/243*sqrt(-2*x + 1) - 1/762048*(11256759*(2*x - 1)^3*sqrt(-
2*x + 1) + 76821507*(2*x - 1)^2*sqrt(-2*x + 1) - 174808137*(-2*x + 1)^(3/2) + 13
2632269*sqrt(-2*x + 1))/(3*x + 2)^4