3.2150 \(\int \frac{(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^5} \, dx\)

Optimal. Leaf size=140 \[ \frac{11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^4}+\frac{85754 x+57069}{4116 \sqrt{1-2 x} (3 x+2)^4}-\frac{177635 \sqrt{1-2 x}}{403368 (3 x+2)}-\frac{177635 \sqrt{1-2 x}}{172872 (3 x+2)^2}-\frac{35527 \sqrt{1-2 x}}{12348 (3 x+2)^3}-\frac{177635 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{201684 \sqrt{21}} \]

[Out]

(-35527*Sqrt[1 - 2*x])/(12348*(2 + 3*x)^3) - (177635*Sqrt[1 - 2*x])/(172872*(2 +
3*x)^2) - (177635*Sqrt[1 - 2*x])/(403368*(2 + 3*x)) + (11*(3 + 5*x)^2)/(21*(1 -
2*x)^(3/2)*(2 + 3*x)^4) + (57069 + 85754*x)/(4116*Sqrt[1 - 2*x]*(2 + 3*x)^4) -
(177635*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(201684*Sqrt[21])

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Rubi [A]  time = 0.183674, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^4}+\frac{85754 x+57069}{4116 \sqrt{1-2 x} (3 x+2)^4}-\frac{177635 \sqrt{1-2 x}}{403368 (3 x+2)}-\frac{177635 \sqrt{1-2 x}}{172872 (3 x+2)^2}-\frac{35527 \sqrt{1-2 x}}{12348 (3 x+2)^3}-\frac{177635 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{201684 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(-35527*Sqrt[1 - 2*x])/(12348*(2 + 3*x)^3) - (177635*Sqrt[1 - 2*x])/(172872*(2 +
3*x)^2) - (177635*Sqrt[1 - 2*x])/(403368*(2 + 3*x)) + (11*(3 + 5*x)^2)/(21*(1 -
2*x)^(3/2)*(2 + 3*x)^4) + (57069 + 85754*x)/(4116*Sqrt[1 - 2*x]*(2 + 3*x)^4) -
(177635*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(201684*Sqrt[21])

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Rubi in Sympy [A]  time = 17.9389, size = 124, normalized size = 0.89 \[ - \frac{177635 \sqrt{- 2 x + 1}}{403368 \left (3 x + 2\right )} - \frac{177635 \sqrt{- 2 x + 1}}{172872 \left (3 x + 2\right )^{2}} - \frac{35527 \sqrt{- 2 x + 1}}{12348 \left (3 x + 2\right )^{3}} - \frac{177635 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{4235364} + \frac{257262 x + 171207}{12348 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{4}} + \frac{11 \left (5 x + 3\right )^{2}}{21 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-177635*sqrt(-2*x + 1)/(403368*(3*x + 2)) - 177635*sqrt(-2*x + 1)/(172872*(3*x +
2)**2) - 35527*sqrt(-2*x + 1)/(12348*(3*x + 2)**3) - 177635*sqrt(21)*atanh(sqrt
(21)*sqrt(-2*x + 1)/7)/4235364 + (257262*x + 171207)/(12348*sqrt(-2*x + 1)*(3*x
+ 2)**4) + 11*(5*x + 3)**2/(21*(-2*x + 1)**(3/2)*(3*x + 2)**4)

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Mathematica [A]  time = 0.185884, size = 73, normalized size = 0.52 \[ \frac{-\frac{21 \left (19184580 x^5+34105920 x^4+10906789 x^3-12952519 x^2-10307138 x-2094250\right )}{(1-2 x)^{3/2} (3 x+2)^4}-355270 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{8470728} \]

Antiderivative was successfully verified.

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

[Out]

((-21*(-2094250 - 10307138*x - 12952519*x^2 + 10906789*x^3 + 34105920*x^4 + 1918
4580*x^5))/((1 - 2*x)^(3/2)*(2 + 3*x)^4) - 355270*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqr
t[1 - 2*x]])/8470728

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Maple [A]  time = 0.023, size = 84, normalized size = 0.6 \[{\frac{5324}{50421} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{29040}{117649}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{324}{117649\, \left ( -4-6\,x \right ) ^{4}} \left ({\frac{198005}{144} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{11953249}{1296} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{80180905}{3888} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{59762605}{3888}\sqrt{1-2\,x}} \right ) }-{\frac{177635\,\sqrt{21}}{4235364}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5324/50421/(1-2*x)^(3/2)+29040/117649/(1-2*x)^(1/2)+324/117649*(198005/144*(1-2*
x)^(7/2)-11953249/1296*(1-2*x)^(5/2)[phone]/3888*(1-2*x)^(3/2)-59762605/3888*(
1-2*x)^(1/2))/(-4-6*x)^4-177635/4235364*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(
1/2)

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Maxima [A]  time = 1.49755, size = 173, normalized size = 1.24 \[ \frac{177635}{8470728} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{4796145 \,{\left (2 \, x - 1\right )}^{5} + 41033685 \,{\left (2 \, x - 1\right )}^{4} + 127080079 \,{\left (2 \, x - 1\right )}^{3} + 157094539 \,{\left (2 \, x - 1\right )}^{2} + 63748608 \, x - 83006000}{201684 \,{\left (81 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - 756 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + 2646 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 4116 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 2401 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

177635/8470728*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2
*x + 1))) - 1/201684*(4796145*(2*x - 1)^5 + 41033685*(2*x - 1)^4 + 127080079*(2*
x - 1)^3 + 157094539*(2*x - 1)^2 + 63748608*x - 83006000)/(81*(-2*x + 1)^(11/2)
- 756*(-2*x + 1)^(9/2) + 2646*(-2*x + 1)^(7/2) - 4116*(-2*x + 1)^(5/2) + 2401*(-
2*x + 1)^(3/2))

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Fricas [A]  time = 0.225934, size = 177, normalized size = 1.26 \[ \frac{\sqrt{21}{\left (177635 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{21}{\left (19184580 \, x^{5} + 34105920 \, x^{4} + 10906789 \, x^{3} - 12952519 \, x^{2} - 10307138 \, x - 2094250\right )}\right )}}{8470728 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8470728*sqrt(21)*(177635*(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)*sq
rt(-2*x + 1)*log((sqrt(21)*(3*x - 5) + 21*sqrt(-2*x + 1))/(3*x + 2)) + sqrt(21)*
(19184580*x^5 + 34105920*x^4 + 10906789*x^3 - 12952519*x^2 - 10307138*x - 209425
0))/((162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)*sqrt(-2*x + 1))

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.221569, size = 163, normalized size = 1.16 \[ \frac{177635}{8470728} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{484 \,{\left (360 \, x - 257\right )}}{352947 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{5346135 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 35859747 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 80180905 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 59762605 \, \sqrt{-2 \, x + 1}}{22588608 \,{\left (3 \, x + 2\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

177635/8470728*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3
*sqrt(-2*x + 1))) + 484/352947*(360*x - 257)/((2*x - 1)*sqrt(-2*x + 1)) - 1/2258
8608*(5346135*(2*x - 1)^3*sqrt(-2*x + 1) + 35859747*(2*x - 1)^2*sqrt(-2*x + 1) -
80180905*(-2*x + 1)^(3/2) + 59762605*sqrt(-2*x + 1))/(3*x + 2)^4