3.1808 \(\int \frac{\sqrt{1-2 x} (3+5 x)^3}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=127 \[ -\frac{\sqrt{1-2 x} (5 x+3)^3}{15 (3 x+2)^5}-\frac{53 \sqrt{1-2 x} (5 x+3)^2}{630 (3 x+2)^4}-\frac{\sqrt{1-2 x} (59665 x+37224)}{79380 (3 x+2)^3}+\frac{11237 \sqrt{1-2 x}}{111132 (3 x+2)}+\frac{11237 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{55566 \sqrt{21}} \]

[Out]

(11237*Sqrt[1 - 2*x])/(111132*(2 + 3*x)) - (53*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(630*(
2 + 3*x)^4) - (Sqrt[1 - 2*x]*(3 + 5*x)^3)/(15*(2 + 3*x)^5) - (Sqrt[1 - 2*x]*(372
24 + 59665*x))/(79380*(2 + 3*x)^3) + (11237*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(5
5566*Sqrt[21])

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Rubi [A]  time = 0.179008, antiderivative size = 127, 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 \[ -\frac{\sqrt{1-2 x} (5 x+3)^3}{15 (3 x+2)^5}-\frac{53 \sqrt{1-2 x} (5 x+3)^2}{630 (3 x+2)^4}-\frac{\sqrt{1-2 x} (59665 x+37224)}{79380 (3 x+2)^3}+\frac{11237 \sqrt{1-2 x}}{111132 (3 x+2)}+\frac{11237 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{55566 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(11237*Sqrt[1 - 2*x])/(111132*(2 + 3*x)) - (53*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(630*(
2 + 3*x)^4) - (Sqrt[1 - 2*x]*(3 + 5*x)^3)/(15*(2 + 3*x)^5) - (Sqrt[1 - 2*x]*(372
24 + 59665*x))/(79380*(2 + 3*x)^3) + (11237*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(5
5566*Sqrt[21])

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Rubi in Sympy [A]  time = 20.8472, size = 110, normalized size = 0.87 \[ \frac{11237 \sqrt{- 2 x + 1}}{111132 \left (3 x + 2\right )} - \frac{\sqrt{- 2 x + 1} \left (2505930 x + 1563408\right )}{3333960 \left (3 x + 2\right )^{3}} - \frac{53 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{2}}{630 \left (3 x + 2\right )^{4}} - \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{3}}{15 \left (3 x + 2\right )^{5}} + \frac{11237 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{1166886} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11237*sqrt(-2*x + 1)/(111132*(3*x + 2)) - sqrt(-2*x + 1)*(2505930*x + 1563408)/(
3333960*(3*x + 2)**3) - 53*sqrt(-2*x + 1)*(5*x + 3)**2/(630*(3*x + 2)**4) - sqrt
(-2*x + 1)*(5*x + 3)**3/(15*(3*x + 2)**5) + 11237*sqrt(21)*atanh(sqrt(21)*sqrt(-
2*x + 1)/7)/1166886

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Mathematica [A]  time = 0.116667, size = 68, normalized size = 0.54 \[ \frac{\frac{21 \sqrt{1-2 x} \left (4550985 x^4+240615 x^3-10100352 x^2-8471518 x-1984928\right )}{(3 x+2)^5}+112370 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{11668860} \]

Antiderivative was successfully verified.

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

[Out]

((21*Sqrt[1 - 2*x]*(-1984928 - 8471518*x - 10100352*x^2 + 240615*x^3 + 4550985*x
^4))/(2 + 3*x)^5 + 112370*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/11668860

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Maple [A]  time = 0.017, size = 75, normalized size = 0.6 \[ 1944\,{\frac{1}{ \left ( -4-6\,x \right ) ^{5}} \left ( -{\frac{11237\, \left ( 1-2\,x \right ) ^{9/2}}{1333584}}+{\frac{4237\, \left ( 1-2\,x \right ) ^{7/2}}{122472}}+{\frac{4954\, \left ( 1-2\,x \right ) ^{5/2}}{229635}}-{\frac{263117\, \left ( 1-2\,x \right ) ^{3/2}}{1102248}}+{\frac{78659\,\sqrt{1-2\,x}}{314928}} \right ) }+{\frac{11237\,\sqrt{21}}{1166886}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1944*(-11237/1333584*(1-2*x)^(9/2)+4237/122472*(1-2*x)^(7/2)+4954/229635*(1-2*x)
^(5/2)-263117/1102248*(1-2*x)^(3/2)+78659/314928*(1-2*x)^(1/2))/(-4-6*x)^5+11237
/1166886*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.52382, size = 173, normalized size = 1.36 \[ -\frac{11237}{2333772} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{4550985 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 18685170 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 11651808 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 128927330 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 134900185 \, \sqrt{-2 \, x + 1}}{277830 \,{\left (243 \,{\left (2 \, x - 1\right )}^{5} + 2835 \,{\left (2 \, x - 1\right )}^{4} + 13230 \,{\left (2 \, x - 1\right )}^{3} + 30870 \,{\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-11237/2333772*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2
*x + 1))) + 1/277830*(4550985*(-2*x + 1)^(9/2) - 18685170*(-2*x + 1)^(7/2) - 116
51808*(-2*x + 1)^(5/2) + 128927330*(-2*x + 1)^(3/2) - 134900185*sqrt(-2*x + 1))/
(243*(2*x - 1)^5 + 2835*(2*x - 1)^4 + 13230*(2*x - 1)^3 + 30870*(2*x - 1)^2 + 72
030*x - 19208)

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Fricas [A]  time = 0.210595, size = 161, normalized size = 1.27 \[ \frac{\sqrt{21}{\left (\sqrt{21}{\left (4550985 \, x^{4} + 240615 \, x^{3} - 10100352 \, x^{2} - 8471518 \, x - 1984928\right )} \sqrt{-2 \, x + 1} + 56185 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{11668860 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11668860*sqrt(21)*(sqrt(21)*(4550985*x^4 + 240615*x^3 - 10100352*x^2 - 8471518
*x - 1984928)*sqrt(-2*x + 1) + 56185*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 2
40*x + 32)*log((sqrt(21)*(3*x - 5) - 21*sqrt(-2*x + 1))/(3*x + 2)))/(243*x^5 + 8
10*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.236427, size = 157, normalized size = 1.24 \[ -\frac{11237}{2333772} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{4550985 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 18685170 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - 11651808 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + 128927330 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 134900185 \, \sqrt{-2 \, x + 1}}{8890560 \,{\left (3 \, x + 2\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-11237/2333772*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3
*sqrt(-2*x + 1))) + 1/8890560*(4550985*(2*x - 1)^4*sqrt(-2*x + 1) + 18685170*(2*
x - 1)^3*sqrt(-2*x + 1) - 11651808*(2*x - 1)^2*sqrt(-2*x + 1) + 128927330*(-2*x
+ 1)^(3/2) - 134900185*sqrt(-2*x + 1))/(3*x + 2)^5