3.2572 \(\int \frac{5-x}{(3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^3} \, dx\)

Optimal. Leaf size=128 \[ -\frac{3 (47 x+37)}{10 (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^2}+\frac{10551 x+9146}{50 (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )}+\frac{6853}{125 \sqrt{2 x+3}}+\frac{7451}{75 (2 x+3)^{3/2}}+310 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{45603}{125} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

[Out]

7451/(75*(3 + 2*x)^(3/2)) + 6853/(125*Sqrt[3 + 2*x]) - (3*(37 + 47*x))/(10*(3 +
2*x)^(3/2)*(2 + 5*x + 3*x^2)^2) + (9146 + 10551*x)/(50*(3 + 2*x)^(3/2)*(2 + 5*x
+ 3*x^2)) + 310*ArcTanh[Sqrt[3 + 2*x]] - (45603*Sqrt[3/5]*ArcTanh[Sqrt[3/5]*Sqrt
[3 + 2*x]])/125

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Rubi [A]  time = 0.29226, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{3 (47 x+37)}{10 (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^2}+\frac{10551 x+9146}{50 (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )}+\frac{6853}{125 \sqrt{2 x+3}}+\frac{7451}{75 (2 x+3)^{3/2}}+310 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{45603}{125} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

7451/(75*(3 + 2*x)^(3/2)) + 6853/(125*Sqrt[3 + 2*x]) - (3*(37 + 47*x))/(10*(3 +
2*x)^(3/2)*(2 + 5*x + 3*x^2)^2) + (9146 + 10551*x)/(50*(3 + 2*x)^(3/2)*(2 + 5*x
+ 3*x^2)) + 310*ArcTanh[Sqrt[3 + 2*x]] - (45603*Sqrt[3/5]*ArcTanh[Sqrt[3/5]*Sqrt
[3 + 2*x]])/125

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Rubi in Sympy [A]  time = 49.275, size = 112, normalized size = 0.88 \[ - \frac{45603 \sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{625} + 310 \operatorname{atanh}{\left (\sqrt{2 x + 3} \right )} + \frac{6853}{125 \sqrt{2 x + 3}} - \frac{141 x + 111}{10 \left (2 x + 3\right )^{\frac{3}{2}} \left (3 x^{2} + 5 x + 2\right )^{2}} + \frac{10551 x + 9146}{50 \left (2 x + 3\right )^{\frac{3}{2}} \left (3 x^{2} + 5 x + 2\right )} + \frac{7451}{75 \left (2 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-45603*sqrt(15)*atanh(sqrt(15)*sqrt(2*x + 3)/5)/625 + 310*atanh(sqrt(2*x + 3)) +
6853/(125*sqrt(2*x + 3)) - (141*x + 111)/(10*(2*x + 3)**(3/2)*(3*x**2 + 5*x + 2
)**2) + (10551*x + 9146)/(50*(2*x + 3)**(3/2)*(3*x**2 + 5*x + 2)) + 7451/(75*(2*
x + 3)**(3/2))

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Mathematica [A]  time = 0.388504, size = 140, normalized size = 1.09 \[ \frac{-\frac{45 \sqrt{2 x+3} (903 x+653)}{2 \left (3 x^2+5 x+2\right )^2}+\frac{3 \sqrt{2 x+3} (132267 x+116222)}{6 x^2+10 x+4}-\frac{29472}{\sqrt{2 x+3}}-\frac{2080}{(2 x+3)^{3/2}}-290625 \log \left (1-\sqrt{2 x+3}\right )+290625 \log \left (\sqrt{2 x+3}+1\right )-136809 \sqrt{15} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right )}{1875} \]

Antiderivative was successfully verified.

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

[Out]

(-2080/(3 + 2*x)^(3/2) - 29472/Sqrt[3 + 2*x] - (45*Sqrt[3 + 2*x]*(653 + 903*x))/
(2*(2 + 5*x + 3*x^2)^2) + (3*Sqrt[3 + 2*x]*(116222 + 132267*x))/(4 + 10*x + 6*x^
2) - 136809*Sqrt[15]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]] - 290625*Log[1 - Sqrt[3 +
2*x]] + 290625*Log[1 + Sqrt[3 + 2*x]])/1875

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Maple [A]  time = 0.032, size = 142, normalized size = 1.1 \[ 3\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-2}+20\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-1}-155\,\ln \left ( -1+\sqrt{3+2\,x} \right ) +{\frac{4374}{625\, \left ( 4+6\,x \right ) ^{2}} \left ({\frac{707}{18} \left ( 3+2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{1235}{18}\sqrt{3+2\,x}} \right ) }-{\frac{45603\,\sqrt{15}}{625}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }-{\frac{416}{375} \left ( 3+2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{9824}{625}{\frac{1}{\sqrt{3+2\,x}}}}-3\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-2}+20\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-1}+155\,\ln \left ( 1+\sqrt{3+2\,x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/(-1+(3+2*x)^(1/2))^2+20/(-1+(3+2*x)^(1/2))-155*ln(-1+(3+2*x)^(1/2))+4374/625*(
707/18*(3+2*x)^(3/2)-1235/18*(3+2*x)^(1/2))/(4+6*x)^2-45603/625*arctanh(1/5*15^(
1/2)*(3+2*x)^(1/2))*15^(1/2)-416/375/(3+2*x)^(3/2)-9824/625/(3+2*x)^(1/2)-3/(1+(
3+2*x)^(1/2))^2+20/(1+(3+2*x)^(1/2))+155*ln(1+(3+2*x)^(1/2))

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Maxima [A]  time = 0.788378, size = 205, normalized size = 1.6 \[ \frac{45603}{1250} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) + \frac{185031 \,{\left (2 \, x + 3\right )}^{5} - 651537 \,{\left (2 \, x + 3\right )}^{4} + 619101 \,{\left (2 \, x + 3\right )}^{3} - 10115 \,{\left (2 \, x + 3\right )}^{2} - 228160 \, x - 352640}{375 \,{\left (9 \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} - 48 \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} + 94 \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - 80 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + 25 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}}\right )}} + 155 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 155 \, \log \left (\sqrt{2 \, x + 3} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

45603/1250*sqrt(15)*log(-(sqrt(15) - 3*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3
))) + 1/375*(185031*(2*x + 3)^5 - 651537*(2*x + 3)^4 + 619101*(2*x + 3)^3 - 1011
5*(2*x + 3)^2 - 228160*x - 352640)/(9*(2*x + 3)^(11/2) - 48*(2*x + 3)^(9/2) + 94
*(2*x + 3)^(7/2) - 80*(2*x + 3)^(5/2) + 25*(2*x + 3)^(3/2)) + 155*log(sqrt(2*x +
3) + 1) - 155*log(sqrt(2*x + 3) - 1)

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Fricas [A]  time = 0.292669, size = 312, normalized size = 2.44 \[ \frac{\sqrt{5}{\left (116250 \, \sqrt{5}{\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )} \sqrt{2 \, x + 3} \log \left (\sqrt{2 \, x + 3} + 1\right ) - 116250 \, \sqrt{5}{\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )} \sqrt{2 \, x + 3} \log \left (\sqrt{2 \, x + 3} - 1\right ) + 136809 \, \sqrt{3}{\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )} \sqrt{2 \, x + 3} \log \left (\frac{\sqrt{5}{\left (3 \, x + 7\right )} - 5 \, \sqrt{3} \sqrt{2 \, x + 3}}{3 \, x + 2}\right ) + \sqrt{5}{\left (740124 \, x^{5} + 4247856 \, x^{4} + 9453447 \, x^{3} + 10168583 \, x^{2} + 5278129 \, x + 1057511\right )}\right )}}{3750 \,{\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )} \sqrt{2 \, x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3750*sqrt(5)*(116250*sqrt(5)*(18*x^5 + 87*x^4 + 164*x^3 + 151*x^2 + 68*x + 12)
*sqrt(2*x + 3)*log(sqrt(2*x + 3) + 1) - 116250*sqrt(5)*(18*x^5 + 87*x^4 + 164*x^
3 + 151*x^2 + 68*x + 12)*sqrt(2*x + 3)*log(sqrt(2*x + 3) - 1) + 136809*sqrt(3)*(
18*x^5 + 87*x^4 + 164*x^3 + 151*x^2 + 68*x + 12)*sqrt(2*x + 3)*log((sqrt(5)*(3*x
+ 7) - 5*sqrt(3)*sqrt(2*x + 3))/(3*x + 2)) + sqrt(5)*(740124*x^5 + 4247856*x^4
+ 9453447*x^3 + 10168583*x^2 + 5278129*x + 1057511))/((18*x^5 + 87*x^4 + 164*x^3
+ 151*x^2 + 68*x + 12)*sqrt(2*x + 3))

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.280371, size = 181, normalized size = 1.41 \[ \frac{45603}{1250} \, \sqrt{15}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{15} + 6 \, \sqrt{2 \, x + 3} \right |}}{2 \,{\left (\sqrt{15} + 3 \, \sqrt{2 \, x + 3}\right )}}\right ) - \frac{64 \,{\left (921 \, x + 1414\right )}}{1875 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}}} + \frac{396801 \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - 1551207 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + 1922011 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 737605 \, \sqrt{2 \, x + 3}}{625 \,{\left (3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}^{2}} + 155 \,{\rm ln}\left (\sqrt{2 \, x + 3} + 1\right ) - 155 \,{\rm ln}\left ({\left | \sqrt{2 \, x + 3} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

45603/1250*sqrt(15)*ln(1/2*abs(-2*sqrt(15) + 6*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt
(2*x + 3))) - 64/1875*(921*x + 1414)/(2*x + 3)^(3/2) + 1/625*(396801*(2*x + 3)^(
7/2) - 1551207*(2*x + 3)^(5/2) + 1922011*(2*x + 3)^(3/2) - 737605*sqrt(2*x + 3))
/(3*(2*x + 3)^2 - 16*x - 19)^2 + 155*ln(sqrt(2*x + 3) + 1) - 155*ln(abs(sqrt(2*x
+ 3) - 1))