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3.2416 \(\int \frac{(5-x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^7} \, dx\)

Optimal. Leaf size=169 \[ -\frac{2237 \left (3 x^2+5 x+2\right )^{3/2}}{3750 (2 x+3)^3}-\frac{3113 \left (3 x^2+5 x+2\right )^{3/2}}{5000 (2 x+3)^4}-\frac{73 \left (3 x^2+5 x+2\right )^{3/2}}{125 (2 x+3)^5}-\frac{13 \left (3 x^2+5 x+2\right )^{3/2}}{30 (2 x+3)^6}+\frac{26453 (8 x+7) \sqrt{3 x^2+5 x+2}}{200000 (2 x+3)^2}-\frac{26453 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{400000 \sqrt{5}} \]

[Out]

(26453*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(200000*(3 + 2*x)^2) - (13*(2 + 5*x + 3*
x^2)^(3/2))/(30*(3 + 2*x)^6) - (73*(2 + 5*x + 3*x^2)^(3/2))/(125*(3 + 2*x)^5) -
(3113*(2 + 5*x + 3*x^2)^(3/2))/(5000*(3 + 2*x)^4) - (2237*(2 + 5*x + 3*x^2)^(3/2
))/(3750*(3 + 2*x)^3) - (26453*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2
])])/(400000*Sqrt[5])

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Rubi [A]  time = 0.300815, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{2237 \left (3 x^2+5 x+2\right )^{3/2}}{3750 (2 x+3)^3}-\frac{3113 \left (3 x^2+5 x+2\right )^{3/2}}{5000 (2 x+3)^4}-\frac{73 \left (3 x^2+5 x+2\right )^{3/2}}{125 (2 x+3)^5}-\frac{13 \left (3 x^2+5 x+2\right )^{3/2}}{30 (2 x+3)^6}+\frac{26453 (8 x+7) \sqrt{3 x^2+5 x+2}}{200000 (2 x+3)^2}-\frac{26453 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{400000 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^7,x]

[Out]

(26453*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(200000*(3 + 2*x)^2) - (13*(2 + 5*x + 3*
x^2)^(3/2))/(30*(3 + 2*x)^6) - (73*(2 + 5*x + 3*x^2)^(3/2))/(125*(3 + 2*x)^5) -
(3113*(2 + 5*x + 3*x^2)^(3/2))/(5000*(3 + 2*x)^4) - (2237*(2 + 5*x + 3*x^2)^(3/2
))/(3750*(3 + 2*x)^3) - (26453*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2
])])/(400000*Sqrt[5])

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Rubi in Sympy [A]  time = 46.0077, size = 160, normalized size = 0.95 \[ \frac{26453 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{2000000} + \frac{26453 \left (8 x + 7\right ) \sqrt{3 x^{2} + 5 x + 2}}{200000 \left (2 x + 3\right )^{2}} - \frac{2237 \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{3750 \left (2 x + 3\right )^{3}} - \frac{3113 \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{5000 \left (2 x + 3\right )^{4}} - \frac{73 \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{125 \left (2 x + 3\right )^{5}} - \frac{13 \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{30 \left (2 x + 3\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3*x**2+5*x+2)**(1/2)/(3+2*x)**7,x)

[Out]

26453*sqrt(5)*atanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2)))/2000000 + 26
453*(8*x + 7)*sqrt(3*x**2 + 5*x + 2)/(200000*(2*x + 3)**2) - 2237*(3*x**2 + 5*x
+ 2)**(3/2)/(3750*(2*x + 3)**3) - 3113*(3*x**2 + 5*x + 2)**(3/2)/(5000*(2*x + 3)
**4) - 73*(3*x**2 + 5*x + 2)**(3/2)/(125*(2*x + 3)**5) - 13*(3*x**2 + 5*x + 2)**
(3/2)/(30*(2*x + 3)**6)

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Mathematica [A]  time = 0.180808, size = 100, normalized size = 0.59 \[ \frac{79359 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )+\frac{10 \sqrt{3 x^2+5 x+2} \left (1567872 x^5+12381040 x^4+39304480 x^3+62797200 x^2+50707640 x+16322393\right )}{(2 x+3)^6}-79359 \sqrt{5} \log (2 x+3)}{6000000} \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^7,x]

[Out]

((10*Sqrt[2 + 5*x + 3*x^2]*(16322393 + 50707640*x + 62797200*x^2 + 39304480*x^3
+ 12381040*x^4 + 1567872*x^5))/(3 + 2*x)^6 - 79359*Sqrt[5]*Log[3 + 2*x] + 79359*
Sqrt[5]*Log[-7 - 8*x + 2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]])/6000000

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Maple [A]  time = 0.019, size = 195, normalized size = 1.2 \[ -{\frac{13}{1920} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-6}}-{\frac{73}{4000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-5}}-{\frac{3113}{80000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{2237}{30000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{26453}{200000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{26453}{125000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{26453}{2000000}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}+{\frac{26453\,\sqrt{5}}{2000000}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) }+{\frac{132265+158718\,x}{250000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^7,x)

[Out]

-13/1920/(x+3/2)^6*(3*(x+3/2)^2-4*x-19/4)^(3/2)-73/4000/(x+3/2)^5*(3*(x+3/2)^2-4
*x-19/4)^(3/2)-3113/80000/(x+3/2)^4*(3*(x+3/2)^2-4*x-19/4)^(3/2)-2237/30000/(x+3
/2)^3*(3*(x+3/2)^2-4*x-19/4)^(3/2)-26453/200000/(x+3/2)^2*(3*(x+3/2)^2-4*x-19/4)
^(3/2)-26453/125000/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(3/2)-26453/2000000*(12*(x+3/
2)^2-16*x-19)^(1/2)+26453/2000000*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+
3/2)^2-16*x-19)^(1/2))+26453/250000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)

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Maxima [A]  time = 0.778755, size = 348, normalized size = 2.06 \[ \frac{26453}{2000000} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) + \frac{79359}{200000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}{30 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac{73 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}{125 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{3113 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}{5000 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{2237 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}{3750 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{26453 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}{50000 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{26453 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{50000 \,{\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(3*x^2 + 5*x + 2)*(x - 5)/(2*x + 3)^7,x, algorithm="maxima")

[Out]

26453/2000000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2
*x + 3) - 2) + 79359/200000*sqrt(3*x^2 + 5*x + 2) - 13/30*(3*x^2 + 5*x + 2)^(3/2
)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 73/125*(3
*x^2 + 5*x + 2)^(3/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 31
13/5000*(3*x^2 + 5*x + 2)^(3/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 2237/
3750*(3*x^2 + 5*x + 2)^(3/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 26453/50000*(3*x^2 +
 5*x + 2)^(3/2)/(4*x^2 + 12*x + 9) - 26453/50000*sqrt(3*x^2 + 5*x + 2)/(2*x + 3)

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Fricas [A]  time = 0.281109, size = 216, normalized size = 1.28 \[ \frac{\sqrt{5}{\left (4 \, \sqrt{5}{\left (1567872 \, x^{5} + 12381040 \, x^{4} + 39304480 \, x^{3} + 62797200 \, x^{2} + 50707640 \, x + 16322393\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + 79359 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} \log \left (\frac{\sqrt{5}{\left (124 \, x^{2} + 212 \, x + 89\right )} - 20 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{12000000 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(3*x^2 + 5*x + 2)*(x - 5)/(2*x + 3)^7,x, algorithm="fricas")

[Out]

1/12000000*sqrt(5)*(4*sqrt(5)*(1567872*x^5 + 12381040*x^4 + 39304480*x^3 + 62797
200*x^2 + 50707640*x + 16322393)*sqrt(3*x^2 + 5*x + 2) + 79359*(64*x^6 + 576*x^5
 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729)*log((sqrt(5)*(124*x^2 + 212*x
+ 89) - 20*sqrt(3*x^2 + 5*x + 2)*(8*x + 7))/(4*x^2 + 12*x + 9)))/(64*x^6 + 576*x
^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{5 \sqrt{3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\right )\, dx - \int \frac{x \sqrt{3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)*(3*x**2+5*x+2)**(1/2)/(3+2*x)**7,x)

[Out]

-Integral(-5*sqrt(3*x**2 + 5*x + 2)/(128*x**7 + 1344*x**6 + 6048*x**5 + 15120*x*
*4 + 22680*x**3 + 20412*x**2 + 10206*x + 2187), x) - Integral(x*sqrt(3*x**2 + 5*
x + 2)/(128*x**7 + 1344*x**6 + 6048*x**5 + 15120*x**4 + 22680*x**3 + 20412*x**2
+ 10206*x + 2187), x)

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GIAC/XCAS [A]  time = 0.303537, size = 554, normalized size = 3.28 \[ -\frac{26453}{2000000} \, \sqrt{5}{\rm ln}\left (\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt{3} x + 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac{2539488 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 41901552 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 924796880 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 3988893600 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 33933192480 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 66530947296 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 275158218192 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 265623867480 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 526452161650 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 226453420305 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 171288605499 \, \sqrt{3} x + 19197814536 \, \sqrt{3} - 171288605499 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{600000 \,{\left (2 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(3*x^2 + 5*x + 2)*(x - 5)/(2*x + 3)^7,x, algorithm="giac")

[Out]

-26453/2000000*sqrt(5)*ln(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^
2 + 5*x + 2))/abs(-4*sqrt(3)*x + 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2)
)) + 1/600000*(2539488*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^11 + 41901552*sqrt(3)
*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 + 924796880*(sqrt(3)*x - sqrt(3*x^2 + 5*
x + 2))^9 + 3988893600*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 + 339331924
80*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 + 66530947296*sqrt(3)*(sqrt(3)*x - sqrt
(3*x^2 + 5*x + 2))^6 + 275158218192*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 2656
23867480*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 526452161650*(sqrt(3)*x
 - sqrt(3*x^2 + 5*x + 2))^3 + 226453420305*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x
 + 2))^2 + 171288605499*sqrt(3)*x + 19197814536*sqrt(3) - 171288605499*sqrt(3*x^
2 + 5*x + 2))/(2*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 6*sqrt(3)*(sqrt(3)*x -
sqrt(3*x^2 + 5*x + 2)) + 11)^6

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