3.2449 \(\int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{7/2} \, dx\)

Optimal. Leaf size=154 \[ \frac{1}{810} (265-54 x) \left (3 x^2+5 x+2\right )^{9/2}+\frac{1399 (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}}{8640}-\frac{9793 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{622080}+\frac{9793 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{5971968}-\frac{9793 (6 x+5) \sqrt{3 x^2+5 x+2}}{47775744}+\frac{9793 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{95551488 \sqrt{3}} \]

[Out]

(-9793*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/47775744 + (9793*(5 + 6*x)*(2 + 5*x + 3*
x^2)^(3/2))/5971968 - (9793*(5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/622080 + (1399*(5
+ 6*x)*(2 + 5*x + 3*x^2)^(7/2))/8640 + ((265 - 54*x)*(2 + 5*x + 3*x^2)^(9/2))/8
10 + (9793*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(95551488*Sqrt[
3])

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Rubi [A]  time = 0.139522, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{1}{810} (265-54 x) \left (3 x^2+5 x+2\right )^{9/2}+\frac{1399 (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}}{8640}-\frac{9793 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{622080}+\frac{9793 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{5971968}-\frac{9793 (6 x+5) \sqrt{3 x^2+5 x+2}}{47775744}+\frac{9793 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{95551488 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(-9793*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/47775744 + (9793*(5 + 6*x)*(2 + 5*x + 3*
x^2)^(3/2))/5971968 - (9793*(5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/622080 + (1399*(5
+ 6*x)*(2 + 5*x + 3*x^2)^(7/2))/8640 + ((265 - 54*x)*(2 + 5*x + 3*x^2)^(9/2))/8
10 + (9793*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(95551488*Sqrt[
3])

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Rubi in Sympy [A]  time = 11.9815, size = 143, normalized size = 0.93 \[ \frac{\left (- 54 x + 265\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{9}{2}}}{810} + \frac{1399 \left (6 x + 5\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{8640} - \frac{9793 \left (6 x + 5\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{622080} + \frac{9793 \left (6 x + 5\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{5971968} - \frac{9793 \left (6 x + 5\right ) \sqrt{3 x^{2} + 5 x + 2}}{47775744} + \frac{9793 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{286654464} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-54*x + 265)*(3*x**2 + 5*x + 2)**(9/2)/810 + 1399*(6*x + 5)*(3*x**2 + 5*x + 2)*
*(7/2)/8640 - 9793*(6*x + 5)*(3*x**2 + 5*x + 2)**(5/2)/622080 + 9793*(6*x + 5)*(
3*x**2 + 5*x + 2)**(3/2)/5971968 - 9793*(6*x + 5)*sqrt(3*x**2 + 5*x + 2)/4777574
4 + 9793*sqrt(3)*atanh(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5*x + 2)))/286654464

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Mathematica [A]  time = 0.120333, size = 95, normalized size = 0.62 \[ \frac{48965 \sqrt{3} \log \left (2 \sqrt{9 x^2+15 x+6}+6 x+5\right )-6 \sqrt{3 x^2+5 x+2} \left (1289945088 x^9+2269347840 x^8-23529056256 x^7-117850567680 x^6-250227954432 x^5-302902600320 x^4-224097754320 x^3-100612822920 x^2-25257845290 x-2726071095\right )}{1433272320} \]

Antiderivative was successfully verified.

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-2726071095 - 25257845290*x - 100612822920*x^2 - 2240
97754320*x^3 - 302902600320*x^4 - 250227954432*x^5 - 117850567680*x^6 - 23529056
256*x^7 + 2269347840*x^8 + 1289945088*x^9) + 48965*Sqrt[3]*Log[5 + 6*x + 2*Sqrt[
6 + 15*x + 9*x^2]])/1433272320

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Maple [A]  time = 0.007, size = 136, normalized size = 0.9 \[{\frac{6995+8394\,x}{8640} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}-{\frac{48965+58758\,x}{622080} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{48965+58758\,x}{5971968} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}-{\frac{48965+58758\,x}{47775744}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{9793\,\sqrt{3}}{286654464}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{53}{162} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{9}{2}}}}-{\frac{x}{15} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1399/8640*(5+6*x)*(3*x^2+5*x+2)^(7/2)-9793/622080*(5+6*x)*(3*x^2+5*x+2)^(5/2)+97
93/5971968*(5+6*x)*(3*x^2+5*x+2)^(3/2)-9793/47775744*(5+6*x)*(3*x^2+5*x+2)^(1/2)
+9793/286654464*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)+53/162*(3*
x^2+5*x+2)^(9/2)-1/15*x*(3*x^2+5*x+2)^(9/2)

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Maxima [A]  time = 0.791399, size = 235, normalized size = 1.53 \[ -\frac{1}{15} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}} x + \frac{53}{162} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}} + \frac{1399}{1440} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} x + \frac{1399}{1728} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} - \frac{9793}{103680} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x - \frac{9793}{124416} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} + \frac{9793}{995328} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{48965}{5971968} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{9793}{7962624} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{9793}{286654464} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac{48965}{47775744} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(3*x^2 + 5*x + 2)^(9/2)*x + 53/162*(3*x^2 + 5*x + 2)^(9/2) + 1399/1440*(3*
x^2 + 5*x + 2)^(7/2)*x + 1399/1728*(3*x^2 + 5*x + 2)^(7/2) - 9793/103680*(3*x^2
+ 5*x + 2)^(5/2)*x - 9793/124416*(3*x^2 + 5*x + 2)^(5/2) + 9793/995328*(3*x^2 +
5*x + 2)^(3/2)*x + 48965/5971968*(3*x^2 + 5*x + 2)^(3/2) - 9793/7962624*sqrt(3*x
^2 + 5*x + 2)*x + 9793/286654464*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6
*x + 5) - 48965/47775744*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.27972, size = 142, normalized size = 0.92 \[ -\frac{1}{2866544640} \, \sqrt{3}{\left (4 \, \sqrt{3}{\left (1289945088 \, x^{9} + 2269347840 \, x^{8} - 23529056256 \, x^{7} - 117850567680 \, x^{6} - 250227954432 \, x^{5} - 302902600320 \, x^{4} - 224097754320 \, x^{3} - 100612822920 \, x^{2} - 25257845290 \, x - 2726071095\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - 48965 \, \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} + 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2866544640*sqrt(3)*(4*sqrt(3)*(1289945088*x^9 + 2269347840*x^8 - 23529056256*
x^7 - 117850567680*x^6 - 250227954432*x^5 - 302902600320*x^4 - 224097754320*x^3
- 100612822920*x^2 - 25257845290*x - 2726071095)*sqrt(3*x^2 + 5*x + 2) - 48965*l
og(sqrt(3)*(72*x^2 + 120*x + 49) + 12*sqrt(3*x^2 + 5*x + 2)*(6*x + 5)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- 956 x \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 3194 x^{2} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 5757 x^{3} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 5948 x^{4} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 3368 x^{5} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 792 x^{6} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int 81 x^{7} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 54 x^{8} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int \left (- 120 \sqrt{3 x^{2} + 5 x + 2}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-956*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-3194*x**2*sqrt(3*x**2 +
5*x + 2), x) - Integral(-5757*x**3*sqrt(3*x**2 + 5*x + 2), x) - Integral(-5948*x
**4*sqrt(3*x**2 + 5*x + 2), x) - Integral(-3368*x**5*sqrt(3*x**2 + 5*x + 2), x)
- Integral(-792*x**6*sqrt(3*x**2 + 5*x + 2), x) - Integral(81*x**7*sqrt(3*x**2 +
5*x + 2), x) - Integral(54*x**8*sqrt(3*x**2 + 5*x + 2), x) - Integral(-120*sqrt
(3*x**2 + 5*x + 2), x)

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GIAC/XCAS [A]  time = 0.268031, size = 127, normalized size = 0.82 \[ -\frac{1}{238878720} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (8 \,{\left (6 \,{\left (36 \,{\left (2 \,{\left (48 \,{\left (54 \, x + 95\right )} x - 47279\right )} x - 473615\right )} x - 36201961\right )} x - 262936285\right )} x - 1556234405\right )} x - 4192200955\right )} x - 12628922645\right )} x - 2726071095\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{9793}{286654464} \, \sqrt{3}{\rm ln}\left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/238878720*(2*(12*(6*(8*(6*(36*(2*(48*(54*x + 95)*x - 47279)*x - 473615)*x - 3
6201961)*x - 262936285)*x - 1556234405)*x - 4192200955)*x - 12628922645)*x - 272
6071095)*sqrt(3*x^2 + 5*x + 2) - 9793/286654464*sqrt(3)*ln(abs(-2*sqrt(3)*(sqrt(
3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))