3.2017 \(\int \frac{(3+5 x)^3}{\sqrt{1-2 x} (2+3 x)^6} \, dx\)

Optimal. Leaf size=120 \[ \frac{\sqrt{1-2 x} (5 x+3)^2}{105 (3 x+2)^5}+\frac{\sqrt{1-2 x} (1971 x+1255)}{6615 (3 x+2)^4}-\frac{5293 \sqrt{1-2 x}}{43218 (3 x+2)}-\frac{5293 \sqrt{1-2 x}}{18522 (3 x+2)^2}-\frac{5293 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21609 \sqrt{21}} \]

[Out]

(-5293*Sqrt[1 - 2*x])/(18522*(2 + 3*x)^2) - (5293*Sqrt[1 - 2*x])/(43218*(2 + 3*x
)) + (Sqrt[1 - 2*x]*(3 + 5*x)^2)/(105*(2 + 3*x)^5) + (Sqrt[1 - 2*x]*(1255 + 1971
*x))/(6615*(2 + 3*x)^4) - (5293*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(21609*Sqrt[21
])

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Rubi [A]  time = 0.151539, antiderivative size = 120, 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 \[ \frac{\sqrt{1-2 x} (5 x+3)^2}{105 (3 x+2)^5}+\frac{\sqrt{1-2 x} (1971 x+1255)}{6615 (3 x+2)^4}-\frac{5293 \sqrt{1-2 x}}{43218 (3 x+2)}-\frac{5293 \sqrt{1-2 x}}{18522 (3 x+2)^2}-\frac{5293 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21609 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(-5293*Sqrt[1 - 2*x])/(18522*(2 + 3*x)^2) - (5293*Sqrt[1 - 2*x])/(43218*(2 + 3*x
)) + (Sqrt[1 - 2*x]*(3 + 5*x)^2)/(105*(2 + 3*x)^5) + (Sqrt[1 - 2*x]*(1255 + 1971
*x))/(6615*(2 + 3*x)^4) - (5293*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(21609*Sqrt[21
])

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Rubi in Sympy [A]  time = 15.7059, size = 104, normalized size = 0.87 \[ - \frac{5293 \sqrt{- 2 x + 1}}{43218 \left (3 x + 2\right )} - \frac{5293 \sqrt{- 2 x + 1}}{18522 \left (3 x + 2\right )^{2}} + \frac{\sqrt{- 2 x + 1} \left (165564 x + 105420\right )}{555660 \left (3 x + 2\right )^{4}} + \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{2}}{105 \left (3 x + 2\right )^{5}} - \frac{5293 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{453789} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5293*sqrt(-2*x + 1)/(43218*(3*x + 2)) - 5293*sqrt(-2*x + 1)/(18522*(3*x + 2)**2
) + sqrt(-2*x + 1)*(165564*x + 105420)/(555660*(3*x + 2)**4) + sqrt(-2*x + 1)*(5
*x + 3)**2/(105*(3*x + 2)**5) - 5293*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/4
53789

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Mathematica [A]  time = 0.136995, size = 68, normalized size = 0.57 \[ \frac{-\frac{21 \sqrt{1-2 x} \left (2143665 x^4+7383735 x^3+8806422 x^2+4450198 x+816938\right )}{(3 x+2)^5}-52930 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{4537890} \]

Antiderivative was successfully verified.

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

[Out]

((-21*Sqrt[1 - 2*x]*(816938 + 4450198*x + 8806422*x^2 + 7383735*x^3 + 2143665*x^
4))/(2 + 3*x)^5 - 52930*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/4537890

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Maple [A]  time = 0.017, size = 75, normalized size = 0.6 \[ 1944\,{\frac{1}{ \left ( -4-6\,x \right ) ^{5}} \left ({\frac{5293\, \left ( 1-2\,x \right ) ^{9/2}}{518616}}-{\frac{5293\, \left ( 1-2\,x \right ) ^{7/2}}{47628}}+{\frac{78563\, \left ( 1-2\,x \right ) ^{5/2}}{178605}}-{\frac{324347\, \left ( 1-2\,x \right ) ^{3/2}}{428652}}+{\frac{58781\,\sqrt{1-2\,x}}{122472}} \right ) }-{\frac{5293\,\sqrt{21}}{453789}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1944*(5293/518616*(1-2*x)^(9/2)-5293/47628*(1-2*x)^(7/2)+78563/178605*(1-2*x)^(5
/2)-324347/428652*(1-2*x)^(3/2)+58781/122472*(1-2*x)^(1/2))/(-4-6*x)^5-5293/4537
89*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.51553, size = 173, normalized size = 1.44 \[ \frac{5293}{907578} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2143665 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 23342130 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 92390088 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 158930030 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 100809415 \, \sqrt{-2 \, x + 1}}{108045 \,{\left (243 \,{\left (2 \, x - 1\right )}^{5} + 2835 \,{\left (2 \, x - 1\right )}^{4} + 13230 \,{\left (2 \, x - 1\right )}^{3} + 30870 \,{\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5293/907578*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) - 1/108045*(2143665*(-2*x + 1)^(9/2) - 23342130*(-2*x + 1)^(7/2) + 923900
88*(-2*x + 1)^(5/2) - 158930030*(-2*x + 1)^(3/2) + 100809415*sqrt(-2*x + 1))/(24
3*(2*x - 1)^5 + 2835*(2*x - 1)^4 + 13230*(2*x - 1)^3 + 30870*(2*x - 1)^2 + 72030
*x - 19208)

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Fricas [A]  time = 0.250714, size = 161, normalized size = 1.34 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (2143665 \, x^{4} + 7383735 \, x^{3} + 8806422 \, x^{2} + 4450198 \, x + 816938\right )} \sqrt{-2 \, x + 1} - 26465 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{4537890 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4537890*sqrt(21)*(sqrt(21)*(2143665*x^4 + 7383735*x^3 + 8806422*x^2 + 4450198
*x + 816938)*sqrt(-2*x + 1) - 26465*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 24
0*x + 32)*log((sqrt(21)*(3*x - 5) + 21*sqrt(-2*x + 1))/(3*x + 2)))/(243*x^5 + 81
0*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.219115, size = 157, normalized size = 1.31 \[ \frac{5293}{907578} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{2143665 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 23342130 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 92390088 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 158930030 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 100809415 \, \sqrt{-2 \, x + 1}}{3457440 \,{\left (3 \, x + 2\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5293/907578*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sq
rt(-2*x + 1))) - 1/3457440*(2143665*(2*x - 1)^4*sqrt(-2*x + 1) + 23342130*(2*x -
1)^3*sqrt(-2*x + 1) + 92390088*(2*x - 1)^2*sqrt(-2*x + 1) - 158930030*(-2*x + 1
)^(3/2) + 100809415*sqrt(-2*x + 1))/(3*x + 2)^5