3.2464 \(\int \frac{(5-x) (2+5 x+3 x^2)^{7/2}}{(3+2 x)^{13}} \, dx\)
Optimal. Leaf size=259 \[ -\frac{6379 \left (3 x^2+5 x+2\right )^{9/2}}{41250 (2 x+3)^9}-\frac{2067 \left (3 x^2+5 x+2\right )^{9/2}}{11000 (2 x+3)^{10}}-\frac{12 \left (3 x^2+5 x+2\right )^{9/2}}{55 (2 x+3)^{11}}-\frac{13 \left (3 x^2+5 x+2\right )^{9/2}}{60 (2 x+3)^{12}}+\frac{25017 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{800000 (2 x+3)^8}-\frac{58373 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{32000000 (2 x+3)^6}+\frac{58373 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{512000000 (2 x+3)^4}-\frac{175119 (8 x+7) \sqrt{3 x^2+5 x+2}}{20480000000 (2 x+3)^2}+\frac{175119 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{40960000000 \sqrt{5}} \]
[Out]
(-175119*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(20480000000*(3 + 2*x)^2) + (58373*(7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2)
)/(512000000*(3 + 2*x)^4) - (58373*(7 + 8*x)*(2 + 5*x + 3*x^2)^(5/2))/(32000000*(3 + 2*x)^6) + (25017*(7 + 8*x
)*(2 + 5*x + 3*x^2)^(7/2))/(800000*(3 + 2*x)^8) - (13*(2 + 5*x + 3*x^2)^(9/2))/(60*(3 + 2*x)^12) - (12*(2 + 5*
x + 3*x^2)^(9/2))/(55*(3 + 2*x)^11) - (2067*(2 + 5*x + 3*x^2)^(9/2))/(11000*(3 + 2*x)^10) - (6379*(2 + 5*x + 3
*x^2)^(9/2))/(41250*(3 + 2*x)^9) + (175119*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(40960000000*
Sqrt[5])
________________________________________________________________________________________
Rubi [A] time = 0.182152, antiderivative size = 259, normalized size of antiderivative = 1.,
number of steps used = 10, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used
= {834, 806, 720, 724, 206} \[ -\frac{6379 \left (3 x^2+5 x+2\right )^{9/2}}{41250 (2 x+3)^9}-\frac{2067 \left (3 x^2+5 x+2\right )^{9/2}}{11000 (2 x+3)^{10}}-\frac{12 \left (3 x^2+5 x+2\right )^{9/2}}{55 (2 x+3)^{11}}-\frac{13 \left (3 x^2+5 x+2\right )^{9/2}}{60 (2 x+3)^{12}}+\frac{25017 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{800000 (2 x+3)^8}-\frac{58373 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{32000000 (2 x+3)^6}+\frac{58373 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{512000000 (2 x+3)^4}-\frac{175119 (8 x+7) \sqrt{3 x^2+5 x+2}}{20480000000 (2 x+3)^2}+\frac{175119 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{40960000000 \sqrt{5}} \]
Antiderivative was successfully verified.
[In]
Int[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^13,x]
[Out]
(-175119*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(20480000000*(3 + 2*x)^2) + (58373*(7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2)
)/(512000000*(3 + 2*x)^4) - (58373*(7 + 8*x)*(2 + 5*x + 3*x^2)^(5/2))/(32000000*(3 + 2*x)^6) + (25017*(7 + 8*x
)*(2 + 5*x + 3*x^2)^(7/2))/(800000*(3 + 2*x)^8) - (13*(2 + 5*x + 3*x^2)^(9/2))/(60*(3 + 2*x)^12) - (12*(2 + 5*
x + 3*x^2)^(9/2))/(55*(3 + 2*x)^11) - (2067*(2 + 5*x + 3*x^2)^(9/2))/(11000*(3 + 2*x)^10) - (6379*(2 + 5*x + 3
*x^2)^(9/2))/(41250*(3 + 2*x)^9) + (175119*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(40960000000*
Sqrt[5])
Rule 834
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])
Rule 806
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]
Rule 720
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*
(d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^p)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(p*(b^2 -
4*a*c))/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[
{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m +
2*p + 2, 0] && GtQ[p, 0]
Rule 724
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rubi steps
\begin{align*} \int \frac{(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{13}} \, dx &=-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac{1}{60} \int \frac{\left (-\frac{369}{2}+117 x\right ) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{12}} \, dx\\ &=-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac{12 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}+\frac{\int \frac{\left (\frac{18045}{2}-4320 x\right ) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{11}} \, dx}{3300}\\ &=-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac{12 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac{2067 \left (2+5 x+3 x^2\right )^{9/2}}{11000 (3+2 x)^{10}}-\frac{\int \frac{\left (-\frac{869175}{2}+93015 x\right ) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{10}} \, dx}{165000}\\ &=-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac{12 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac{2067 \left (2+5 x+3 x^2\right )^{9/2}}{11000 (3+2 x)^{10}}-\frac{6379 \left (2+5 x+3 x^2\right )^{9/2}}{41250 (3+2 x)^9}+\frac{25017 \int \frac{\left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^9} \, dx}{10000}\\ &=\frac{25017 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800000 (3+2 x)^8}-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac{12 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac{2067 \left (2+5 x+3 x^2\right )^{9/2}}{11000 (3+2 x)^{10}}-\frac{6379 \left (2+5 x+3 x^2\right )^{9/2}}{41250 (3+2 x)^9}-\frac{175119 \int \frac{\left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx}{1600000}\\ &=-\frac{58373 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{32000000 (3+2 x)^6}+\frac{25017 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800000 (3+2 x)^8}-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac{12 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac{2067 \left (2+5 x+3 x^2\right )^{9/2}}{11000 (3+2 x)^{10}}-\frac{6379 \left (2+5 x+3 x^2\right )^{9/2}}{41250 (3+2 x)^9}+\frac{58373 \int \frac{\left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{12800000}\\ &=\frac{58373 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{512000000 (3+2 x)^4}-\frac{58373 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{32000000 (3+2 x)^6}+\frac{25017 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800000 (3+2 x)^8}-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac{12 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac{2067 \left (2+5 x+3 x^2\right )^{9/2}}{11000 (3+2 x)^{10}}-\frac{6379 \left (2+5 x+3 x^2\right )^{9/2}}{41250 (3+2 x)^9}-\frac{175119 \int \frac{\sqrt{2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{1024000000}\\ &=-\frac{175119 (7+8 x) \sqrt{2+5 x+3 x^2}}{20480000000 (3+2 x)^2}+\frac{58373 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{512000000 (3+2 x)^4}-\frac{58373 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{32000000 (3+2 x)^6}+\frac{25017 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800000 (3+2 x)^8}-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac{12 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac{2067 \left (2+5 x+3 x^2\right )^{9/2}}{11000 (3+2 x)^{10}}-\frac{6379 \left (2+5 x+3 x^2\right )^{9/2}}{41250 (3+2 x)^9}+\frac{175119 \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{40960000000}\\ &=-\frac{175119 (7+8 x) \sqrt{2+5 x+3 x^2}}{20480000000 (3+2 x)^2}+\frac{58373 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{512000000 (3+2 x)^4}-\frac{58373 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{32000000 (3+2 x)^6}+\frac{25017 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800000 (3+2 x)^8}-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac{12 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac{2067 \left (2+5 x+3 x^2\right )^{9/2}}{11000 (3+2 x)^{10}}-\frac{6379 \left (2+5 x+3 x^2\right )^{9/2}}{41250 (3+2 x)^9}-\frac{175119 \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )}{20480000000}\\ &=-\frac{175119 (7+8 x) \sqrt{2+5 x+3 x^2}}{20480000000 (3+2 x)^2}+\frac{58373 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{512000000 (3+2 x)^4}-\frac{58373 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{32000000 (3+2 x)^6}+\frac{25017 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800000 (3+2 x)^8}-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac{12 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac{2067 \left (2+5 x+3 x^2\right )^{9/2}}{11000 (3+2 x)^{10}}-\frac{6379 \left (2+5 x+3 x^2\right )^{9/2}}{41250 (3+2 x)^9}+\frac{175119 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{40960000000 \sqrt{5}}\\ \end{align*}
Mathematica [A] time = 0.25891, size = 262, normalized size = 1.01 \[ \frac{-\frac{12758 \left (3 x^2+5 x+2\right )^{9/2}}{25 (2 x+3)^9}-\frac{6201 \left (3 x^2+5 x+2\right )^{9/2}}{10 (2 x+3)^{10}}-\frac{720 \left (3 x^2+5 x+2\right )^{9/2}}{(2 x+3)^{11}}-\frac{715 \left (3 x^2+5 x+2\right )^{9/2}}{(2 x+3)^{12}}+\frac{825561 \left (\frac{2 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{(2 x+3)^8}-\frac{7 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}+\frac{7 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{960 (2 x+3)^4}-\frac{7 \left (\frac{10 \sqrt{3 x^2+5 x+2} (8 x+7)}{(2 x+3)^2}+\sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )\right )}{128000}\right )}{16000}}{3300} \]
Antiderivative was successfully verified.
[In]
Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^13,x]
[Out]
((-715*(2 + 5*x + 3*x^2)^(9/2))/(3 + 2*x)^12 - (720*(2 + 5*x + 3*x^2)^(9/2))/(3 + 2*x)^11 - (6201*(2 + 5*x + 3
*x^2)^(9/2))/(10*(3 + 2*x)^10) - (12758*(2 + 5*x + 3*x^2)^(9/2))/(25*(3 + 2*x)^9) + (825561*((7*(7 + 8*x)*(2 +
5*x + 3*x^2)^(3/2))/(960*(3 + 2*x)^4) - (7*(7 + 8*x)*(2 + 5*x + 3*x^2)^(5/2))/(60*(3 + 2*x)^6) + (2*(7 + 8*x)
*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^8 - (7*((10*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^2 + Sqrt[5]*ArcTanh
[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])]))/128000))/16000)/3300
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Maple [A] time = 0.124, size = 432, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((5-x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^13,x)
[Out]
-775527/400000000/(x+3/2)^5*(3*(x+3/2)^2-4*x-19/4)^(9/2)-48057657/16000000000/(x+3/2)^4*(3*(x+3/2)^2-4*x-19/4)
^(9/2)-46022941/10000000000/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(9/2)-1395223107/200000000000/(x+3/2)^2*(3*(x+3/2
)^2-4*x-19/4)^(9/2)+261602769/50000000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(7/2)-101744139/200000000000*(5+6*x)*
(3*(x+3/2)^2-4*x-19/4)^(5/2)-261602769/25000000000/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(9/2)+1692817/32000000000*(5
+6*x)*(3*(x+3/2)^2-4*x-19/4)^(3/2)-175119/25600000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)-175119/204800000000
*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))+25017/200000000000*(3*(x+3/2)^2-4*x-19/4
)^(7/2)+175119/800000000000*(3*(x+3/2)^2-4*x-19/4)^(5/2)+58373/128000000000*(3*(x+3/2)^2-4*x-19/4)^(3/2)+17511
9/204800000000*(12*(x+3/2)^2-16*x-19)^(1/2)-13/245760/(x+3/2)^12*(3*(x+3/2)^2-4*x-19/4)^(9/2)-3/28160/(x+3/2)^
11*(3*(x+3/2)^2-4*x-19/4)^(9/2)-2067/11264000/(x+3/2)^10*(3*(x+3/2)^2-4*x-19/4)^(9/2)-6379/21120000/(x+3/2)^9*
(3*(x+3/2)^2-4*x-19/4)^(9/2)-25017/51200000/(x+3/2)^8*(3*(x+3/2)^2-4*x-19/4)^(9/2)-25017/32000000/(x+3/2)^7*(3
*(x+3/2)^2-4*x-19/4)^(9/2)-158441/128000000/(x+3/2)^6*(3*(x+3/2)^2-4*x-19/4)^(9/2)
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Maxima [B] time = 2.07673, size = 980, normalized size = 3.78 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((5-x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^13,x, algorithm="maxima")
[Out]
4185669321/200000000000*(3*x^2 + 5*x + 2)^(7/2) - 13/60*(3*x^2 + 5*x + 2)^(9/2)/(4096*x^12 + 73728*x^11 + 6082
56*x^10 + 3041280*x^9 + 10264320*x^8 + 24634368*x^7 + 43110144*x^6 + 55427328*x^5 + 51963120*x^4 + 34642080*x^
3 + 15588936*x^2 + 4251528*x + 531441) - 12/55*(3*x^2 + 5*x + 2)^(9/2)/(2048*x^11 + 33792*x^10 + 253440*x^9 +
1140480*x^8 + 3421440*x^7 + 7185024*x^6 + 10777536*x^5 + 11547360*x^4 + 8660520*x^3 + 4330260*x^2 + 1299078*x
+ 177147) - 2067/11000*(3*x^2 + 5*x + 2)^(9/2)/(1024*x^10 + 15360*x^9 + 103680*x^8 + 414720*x^7 + 1088640*x^6
+ 1959552*x^5 + 2449440*x^4 + 2099520*x^3 + 1180980*x^2 + 393660*x + 59049) - 6379/41250*(3*x^2 + 5*x + 2)^(9/
2)/(512*x^9 + 6912*x^8 + 41472*x^7 + 145152*x^6 + 326592*x^5 + 489888*x^4 + 489888*x^3 + 314928*x^2 + 118098*x
+ 19683) - 25017/200000*(3*x^2 + 5*x + 2)^(9/2)/(256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4 + 108
864*x^3 + 81648*x^2 + 34992*x + 6561) - 25017/250000*(3*x^2 + 5*x + 2)^(9/2)/(128*x^7 + 1344*x^6 + 6048*x^5 +
15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187) - 158441/2000000*(3*x^2 + 5*x + 2)^(9/2)/(64*x^6 + 576*x^5
+ 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 775527/12500000*(3*x^2 + 5*x + 2)^(9/2)/(32*x^5 + 240*x^4
+ 720*x^3 + 1080*x^2 + 810*x + 243) - 48057657/1000000000*(3*x^2 + 5*x + 2)^(9/2)/(16*x^4 + 96*x^3 + 216*x^2 +
216*x + 81) - 46022941/1250000000*(3*x^2 + 5*x + 2)^(9/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 1395223107/500000000
00*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) - 305232417/100000000000*(3*x^2 + 5*x + 2)^(5/2)*x - 2034707661/
800000000000*(3*x^2 + 5*x + 2)^(5/2) - 261602769/10000000000*(3*x^2 + 5*x + 2)^(7/2)/(2*x + 3) + 5078451/16000
000000*(3*x^2 + 5*x + 2)^(3/2)*x + 33914713/128000000000*(3*x^2 + 5*x + 2)^(3/2) - 525357/12800000000*sqrt(3*x
^2 + 5*x + 2)*x - 175119/204800000000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3
) - 2) - 3327261/102400000000*sqrt(3*x^2 + 5*x + 2)
________________________________________________________________________________________
Fricas [A] time = 1.55091, size = 1026, normalized size = 3.96 \begin{align*} \frac{5778927 \, \sqrt{5}{\left (4096 \, x^{12} + 73728 \, x^{11} + 608256 \, x^{10} + 3041280 \, x^{9} + 10264320 \, x^{8} + 24634368 \, x^{7} + 43110144 \, x^{6} + 55427328 \, x^{5} + 51963120 \, x^{4} + 34642080 \, x^{3} + 15588936 \, x^{2} + 4251528 \, x + 531441\right )} \log \left (\frac{4 \, \sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 20 \,{\left (60734693376 \, x^{11} + 1044584776704 \, x^{10} + 8182662620160 \, x^{9} + 38544695427840 \, x^{8} + 123629135656960 \, x^{7} + 273282692080768 \, x^{6} + 410468875350912 \, x^{5} + 412855931529440 \, x^{4} + 271870111600160 \, x^{3} + 111795175925940 \, x^{2} + 25843081681156 \, x + 2531527640959\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{13516800000000 \,{\left (4096 \, x^{12} + 73728 \, x^{11} + 608256 \, x^{10} + 3041280 \, x^{9} + 10264320 \, x^{8} + 24634368 \, x^{7} + 43110144 \, x^{6} + 55427328 \, x^{5} + 51963120 \, x^{4} + 34642080 \, x^{3} + 15588936 \, x^{2} + 4251528 \, x + 531441\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((5-x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^13,x, algorithm="fricas")
[Out]
1/13516800000000*(5778927*sqrt(5)*(4096*x^12 + 73728*x^11 + 608256*x^10 + 3041280*x^9 + 10264320*x^8 + 2463436
8*x^7 + 43110144*x^6 + 55427328*x^5 + 51963120*x^4 + 34642080*x^3 + 15588936*x^2 + 4251528*x + 531441)*log((4*
sqrt(5)*sqrt(3*x^2 + 5*x + 2)*(8*x + 7) + 124*x^2 + 212*x + 89)/(4*x^2 + 12*x + 9)) + 20*(60734693376*x^11 + 1
044584776704*x^10 + 8182662620160*x^9 + 38544695427840*x^8 + 123629135656960*x^7 + 273282692080768*x^6 + 41046
8875350912*x^5 + 412855931529440*x^4 + 271870111600160*x^3 + 111795175925940*x^2 + 25843081681156*x + 25315276
40959)*sqrt(3*x^2 + 5*x + 2))/(4096*x^12 + 73728*x^11 + 608256*x^10 + 3041280*x^9 + 10264320*x^8 + 24634368*x^
7 + 43110144*x^6 + 55427328*x^5 + 51963120*x^4 + 34642080*x^3 + 15588936*x^2 + 4251528*x + 531441)
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((5-x)*(3*x**2+5*x+2)**(7/2)/(3+2*x)**13,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.30083, size = 967, normalized size = 3.73 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((5-x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^13,x, algorithm="giac")
[Out]
175119/204800000000*sqrt(5)*log(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))/abs(-4*sqr
t(3)*x + 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))) - 1/675840000000*(11835242496*(sqrt(3)*x - sqrt(3*x
^2 + 5*x + 2))^23 + 408315866112*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^22 + 20039038086144*(sqrt(3)*x -
sqrt(3*x^2 + 5*x + 2))^21 + 535243596890112*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^20 + 13859706456921600
*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^19 + 31535346744025344*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^18 - 7
89031961976842496*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^17 - 7977976824329385984*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2
+ 5*x + 2))^16 - 113078650509677476096*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^15 - 358779889050339715200*sqrt(3)
*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^14 - 2538162771649151164032*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^13 - 4660
243350382625915904*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^12 - 20499122524155108829248*(sqrt(3)*x - sqrt(
3*x^2 + 5*x + 2))^11 - 24347916060701730772704*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 - 70788415443572
756925600*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 - 56076083911431114398208*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*
x + 2))^8 - 108598043564223524909928*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 - 56663550021725424101412*sqrt(3)*(
sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 - 70668287639831997261828*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 - 2287603
7084903247115200*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 - 16680770211437743348146*(sqrt(3)*x - sqrt(3*x
^2 + 5*x + 2))^3 - 2864949797863813201587*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 - 93027830676920644626
9*sqrt(3)*x - 47729262032858665512*sqrt(3) + 930278306769206446269*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)^12