3.1215 \(\int \frac {(a+b x+c x^2)^{3/2}}{(b d+2 c d x)^9} \, dx\)

Optimal. Leaf size=207 \[ \frac {3 \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{2048 c^{5/2} d^9 \left (b^2-4 a c\right )^{5/2}}+\frac {3 \sqrt {a+b x+c x^2}}{1024 c^2 d^9 \left (b^2-4 a c\right )^2 (b+2 c x)^2}+\frac {\sqrt {a+b x+c x^2}}{512 c^2 d^9 \left (b^2-4 a c\right ) (b+2 c x)^4}-\frac {\sqrt {a+b x+c x^2}}{128 c^2 d^9 (b+2 c x)^6}-\frac {\left (a+b x+c x^2\right )^{3/2}}{16 c d^9 (b+2 c x)^8} \]

[Out]

-1/16*(c*x^2+b*x+a)^(3/2)/c/d^9/(2*c*x+b)^8+3/2048*arctan(2*c^(1/2)*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^(1/2))/c^
(5/2)/(-4*a*c+b^2)^(5/2)/d^9-1/128*(c*x^2+b*x+a)^(1/2)/c^2/d^9/(2*c*x+b)^6+1/512*(c*x^2+b*x+a)^(1/2)/c^2/(-4*a
*c+b^2)/d^9/(2*c*x+b)^4+3/1024*(c*x^2+b*x+a)^(1/2)/c^2/(-4*a*c+b^2)^2/d^9/(2*c*x+b)^2

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Rubi [A]  time = 0.15, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {684, 693, 688, 205} \[ \frac {3 \sqrt {a+b x+c x^2}}{1024 c^2 d^9 \left (b^2-4 a c\right )^2 (b+2 c x)^2}+\frac {\sqrt {a+b x+c x^2}}{512 c^2 d^9 \left (b^2-4 a c\right ) (b+2 c x)^4}+\frac {3 \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{2048 c^{5/2} d^9 \left (b^2-4 a c\right )^{5/2}}-\frac {\sqrt {a+b x+c x^2}}{128 c^2 d^9 (b+2 c x)^6}-\frac {\left (a+b x+c x^2\right )^{3/2}}{16 c d^9 (b+2 c x)^8} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^9,x]

[Out]

-Sqrt[a + b*x + c*x^2]/(128*c^2*d^9*(b + 2*c*x)^6) + Sqrt[a + b*x + c*x^2]/(512*c^2*(b^2 - 4*a*c)*d^9*(b + 2*c
*x)^4) + (3*Sqrt[a + b*x + c*x^2])/(1024*c^2*(b^2 - 4*a*c)^2*d^9*(b + 2*c*x)^2) - (a + b*x + c*x^2)^(3/2)/(16*
c*d^9*(b + 2*c*x)^8) + (3*ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]])/(2048*c^(5/2)*(b^2 - 4*
a*c)^(5/2)*d^9)

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 684

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + 1)), x] - Dist[(b*p)/(d*e*(m + 1)), 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] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] &&
GtQ[p, 0] && LtQ[m, -1] &&  !(IntegerQ[m/2] && LtQ[m + 2*p + 3, 0]) && IntegerQ[2*p]

Rule 688

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[4*c, Subst[Int[1/(b^2*e
- 4*a*c*e + 4*c*e*x^2), x], x, Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0]
&& EqQ[2*c*d - b*e, 0]

Rule 693

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-2*b*d*(d + e*x)^(m
+ 1)*(a + b*x + c*x^2)^(p + 1))/(d^2*(m + 1)*(b^2 - 4*a*c)), x] + Dist[(b^2*(m + 2*p + 3))/(d^2*(m + 1)*(b^2
- 4*a*c)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*
c, 0] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[m, -1] && (IntegerQ[2*p] || (IntegerQ[m] && Rationa
lQ[p]) || IntegerQ[(m + 2*p + 3)/2])

Rubi steps

\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^9} \, dx &=-\frac {\left (a+b x+c x^2\right )^{3/2}}{16 c d^9 (b+2 c x)^8}+\frac {3 \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^7} \, dx}{32 c d^2}\\ &=-\frac {\sqrt {a+b x+c x^2}}{128 c^2 d^9 (b+2 c x)^6}-\frac {\left (a+b x+c x^2\right )^{3/2}}{16 c d^9 (b+2 c x)^8}+\frac {\int \frac {1}{(b d+2 c d x)^5 \sqrt {a+b x+c x^2}} \, dx}{256 c^2 d^4}\\ &=-\frac {\sqrt {a+b x+c x^2}}{128 c^2 d^9 (b+2 c x)^6}+\frac {\sqrt {a+b x+c x^2}}{512 c^2 \left (b^2-4 a c\right ) d^9 (b+2 c x)^4}-\frac {\left (a+b x+c x^2\right )^{3/2}}{16 c d^9 (b+2 c x)^8}+\frac {3 \int \frac {1}{(b d+2 c d x)^3 \sqrt {a+b x+c x^2}} \, dx}{1024 c^2 \left (b^2-4 a c\right ) d^6}\\ &=-\frac {\sqrt {a+b x+c x^2}}{128 c^2 d^9 (b+2 c x)^6}+\frac {\sqrt {a+b x+c x^2}}{512 c^2 \left (b^2-4 a c\right ) d^9 (b+2 c x)^4}+\frac {3 \sqrt {a+b x+c x^2}}{1024 c^2 \left (b^2-4 a c\right )^2 d^9 (b+2 c x)^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{16 c d^9 (b+2 c x)^8}+\frac {3 \int \frac {1}{(b d+2 c d x) \sqrt {a+b x+c x^2}} \, dx}{2048 c^2 \left (b^2-4 a c\right )^2 d^8}\\ &=-\frac {\sqrt {a+b x+c x^2}}{128 c^2 d^9 (b+2 c x)^6}+\frac {\sqrt {a+b x+c x^2}}{512 c^2 \left (b^2-4 a c\right ) d^9 (b+2 c x)^4}+\frac {3 \sqrt {a+b x+c x^2}}{1024 c^2 \left (b^2-4 a c\right )^2 d^9 (b+2 c x)^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{16 c d^9 (b+2 c x)^8}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{2 b^2 c d-8 a c^2 d+8 c^2 d x^2} \, dx,x,\sqrt {a+b x+c x^2}\right )}{512 c \left (b^2-4 a c\right )^2 d^8}\\ &=-\frac {\sqrt {a+b x+c x^2}}{128 c^2 d^9 (b+2 c x)^6}+\frac {\sqrt {a+b x+c x^2}}{512 c^2 \left (b^2-4 a c\right ) d^9 (b+2 c x)^4}+\frac {3 \sqrt {a+b x+c x^2}}{1024 c^2 \left (b^2-4 a c\right )^2 d^9 (b+2 c x)^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{16 c d^9 (b+2 c x)^8}+\frac {3 \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{2048 c^{5/2} \left (b^2-4 a c\right )^{5/2} d^9}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 62, normalized size = 0.30 \[ \frac {2 (a+x (b+c x))^{5/2} \, _2F_1\left (\frac {5}{2},5;\frac {7}{2};\frac {4 c (a+x (b+c x))}{4 a c-b^2}\right )}{5 d^9 \left (b^2-4 a c\right )^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^9,x]

[Out]

(2*(a + x*(b + c*x))^(5/2)*Hypergeometric2F1[5/2, 5, 7/2, (4*c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])/(5*(b^2 - 4
*a*c)^5*d^9)

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fricas [B]  time = 35.65, size = 1646, normalized size = 7.95 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^9,x, algorithm="fricas")

[Out]

[-1/4096*(3*(256*c^8*x^8 + 1024*b*c^7*x^7 + 1792*b^2*c^6*x^6 + 1792*b^3*c^5*x^5 + 1120*b^4*c^4*x^4 + 448*b^5*c
^3*x^3 + 112*b^6*c^2*x^2 + 16*b^7*c*x + b^8)*sqrt(-b^2*c + 4*a*c^2)*log(-(4*c^2*x^2 + 4*b*c*x - b^2 + 8*a*c -
4*sqrt(-b^2*c + 4*a*c^2)*sqrt(c*x^2 + b*x + a))/(4*c^2*x^2 + 4*b*c*x + b^2)) + 4*(3*b^8*c - 4*a*b^6*c^2 - 416*
a^2*b^4*c^3 + 2560*a^3*b^2*c^4 - 4096*a^4*c^5 - 192*(b^2*c^7 - 4*a*c^8)*x^6 - 576*(b^3*c^6 - 4*a*b*c^7)*x^5 -
16*(47*b^4*c^5 - 196*a*b^2*c^6 + 32*a^2*c^7)*x^4 - 32*(17*b^5*c^4 - 76*a*b^3*c^5 + 32*a^2*b*c^6)*x^3 - 12*(11*
b^6*c^3 + 4*a*b^4*c^4 - 320*a^2*b^2*c^5 + 512*a^3*c^6)*x^2 + 4*(11*b^7*c^2 - 220*a*b^5*c^3 + 1088*a^2*b^3*c^4
- 1536*a^3*b*c^5)*x)*sqrt(c*x^2 + b*x + a))/(256*(b^6*c^11 - 12*a*b^4*c^12 + 48*a^2*b^2*c^13 - 64*a^3*c^14)*d^
9*x^8 + 1024*(b^7*c^10 - 12*a*b^5*c^11 + 48*a^2*b^3*c^12 - 64*a^3*b*c^13)*d^9*x^7 + 1792*(b^8*c^9 - 12*a*b^6*c
^10 + 48*a^2*b^4*c^11 - 64*a^3*b^2*c^12)*d^9*x^6 + 1792*(b^9*c^8 - 12*a*b^7*c^9 + 48*a^2*b^5*c^10 - 64*a^3*b^3
*c^11)*d^9*x^5 + 1120*(b^10*c^7 - 12*a*b^8*c^8 + 48*a^2*b^6*c^9 - 64*a^3*b^4*c^10)*d^9*x^4 + 448*(b^11*c^6 - 1
2*a*b^9*c^7 + 48*a^2*b^7*c^8 - 64*a^3*b^5*c^9)*d^9*x^3 + 112*(b^12*c^5 - 12*a*b^10*c^6 + 48*a^2*b^8*c^7 - 64*a
^3*b^6*c^8)*d^9*x^2 + 16*(b^13*c^4 - 12*a*b^11*c^5 + 48*a^2*b^9*c^6 - 64*a^3*b^7*c^7)*d^9*x + (b^14*c^3 - 12*a
*b^12*c^4 + 48*a^2*b^10*c^5 - 64*a^3*b^8*c^6)*d^9), -1/2048*(3*(256*c^8*x^8 + 1024*b*c^7*x^7 + 1792*b^2*c^6*x^
6 + 1792*b^3*c^5*x^5 + 1120*b^4*c^4*x^4 + 448*b^5*c^3*x^3 + 112*b^6*c^2*x^2 + 16*b^7*c*x + b^8)*sqrt(b^2*c - 4
*a*c^2)*arctan(1/2*sqrt(b^2*c - 4*a*c^2)*sqrt(c*x^2 + b*x + a)/(c^2*x^2 + b*c*x + a*c)) + 2*(3*b^8*c - 4*a*b^6
*c^2 - 416*a^2*b^4*c^3 + 2560*a^3*b^2*c^4 - 4096*a^4*c^5 - 192*(b^2*c^7 - 4*a*c^8)*x^6 - 576*(b^3*c^6 - 4*a*b*
c^7)*x^5 - 16*(47*b^4*c^5 - 196*a*b^2*c^6 + 32*a^2*c^7)*x^4 - 32*(17*b^5*c^4 - 76*a*b^3*c^5 + 32*a^2*b*c^6)*x^
3 - 12*(11*b^6*c^3 + 4*a*b^4*c^4 - 320*a^2*b^2*c^5 + 512*a^3*c^6)*x^2 + 4*(11*b^7*c^2 - 220*a*b^5*c^3 + 1088*a
^2*b^3*c^4 - 1536*a^3*b*c^5)*x)*sqrt(c*x^2 + b*x + a))/(256*(b^6*c^11 - 12*a*b^4*c^12 + 48*a^2*b^2*c^13 - 64*a
^3*c^14)*d^9*x^8 + 1024*(b^7*c^10 - 12*a*b^5*c^11 + 48*a^2*b^3*c^12 - 64*a^3*b*c^13)*d^9*x^7 + 1792*(b^8*c^9 -
12*a*b^6*c^10 + 48*a^2*b^4*c^11 - 64*a^3*b^2*c^12)*d^9*x^6 + 1792*(b^9*c^8 - 12*a*b^7*c^9 + 48*a^2*b^5*c^10 -
64*a^3*b^3*c^11)*d^9*x^5 + 1120*(b^10*c^7 - 12*a*b^8*c^8 + 48*a^2*b^6*c^9 - 64*a^3*b^4*c^10)*d^9*x^4 + 448*(b
^11*c^6 - 12*a*b^9*c^7 + 48*a^2*b^7*c^8 - 64*a^3*b^5*c^9)*d^9*x^3 + 112*(b^12*c^5 - 12*a*b^10*c^6 + 48*a^2*b^8
*c^7 - 64*a^3*b^6*c^8)*d^9*x^2 + 16*(b^13*c^4 - 12*a*b^11*c^5 + 48*a^2*b^9*c^6 - 64*a^3*b^7*c^7)*d^9*x + (b^14
*c^3 - 12*a*b^12*c^4 + 48*a^2*b^10*c^5 - 64*a^3*b^8*c^6)*d^9)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^9,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to divide, perhaps due to rounding error%%%{%%%{512,[9]%%%},[18,9,0,0]%%%}+%%%{%%{[%%%{-4608,[8]%%%},0]:[1,
0,%%%{-1,[1]%%%}]%%},[17,9,1,0]%%%}+%%%{%%%{20736,[8]%%%},[16,9,2,0]%%%}+%%%{%%%{-4608,[9]%%%},[16,9,0,1]%%%}+
%%%{%%{[%%%{-61440,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[15,9,3,0]%%%}+%%%{%%{[%%%{36864,[8]%%%},0]:[1,0,%%%{-1,
[1]%%%}]%%},[15,9,1,1]%%%}+%%%{%%%{133632,[7]%%%},[14,9,4,0]%%%}+%%%{%%%{-147456,[8]%%%},[14,9,2,1]%%%}+%%%{%%
%{18432,[9]%%%},[14,9,0,2]%%%}+%%%{%%{[%%%{-225792,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[13,9,5,0]%%%}+%%%{%%{[%
%%{387072,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[13,9,3,1]%%%}+%%%{%%{[%%%{-129024,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}
]%%},[13,9,1,2]%%%}+%%%{%%%{306432,[6]%%%},[12,9,6,0]%%%}+%%%{%%%{-741888,[7]%%%},[12,9,4,1]%%%}+%%%{%%%{45158
4,[8]%%%},[12,9,2,2]%%%}+%%%{%%%{-43008,[9]%%%},[12,9,0,3]%%%}+%%%{%%{[%%%{-340992,[5]%%%},0]:[1,0,%%%{-1,[1]%
%%}]%%},[11,9,7,0]%%%}+%%%{%%{[%%%{1096704,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[11,9,5,1]%%%}+%%%{%%{[%%%{-1032
192,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[11,9,3,2]%%%}+%%%{%%{[%%%{258048,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1
1,9,1,3]%%%}+%%%{%%%{315072,[5]%%%},[10,9,8,0]%%%}+%%%{%%%{-1290240,[6]%%%},[10,9,6,1]%%%}+%%%{%%%{1709568,[7]
%%%},[10,9,4,2]%%%}+%%%{%%%{-774144,[8]%%%},[10,9,2,3]%%%}+%%%{%%%{64512,[9]%%%},[10,9,0,4]%%%}+%%%{%%{[%%%{-2
43392,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[9,9,9,0]%%%}+%%%{%%{[%%%{1230336,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},
[9,9,7,1]%%%}+%%%{%%{[%%%{-2161152,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[9,9,5,2]%%%}+%%%{%%{[%%%{1505280,[7]%%%
},0]:[1,0,%%%{-1,[1]%%%}]%%},[9,9,3,3]%%%}+%%%{%%{[%%%{-322560,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[9,9,1,4]%%%
}+%%%{%%%{157536,[4]%%%},[8,9,10,0]%%%}+%%%{%%%{-960192,[5]%%%},[8,9,8,1]%%%}+%%%{%%%{2145024,[6]%%%},[8,9,6,2
]%%%}+%%%{%%%{-2096640,[7]%%%},[8,9,4,3]%%%}+%%%{%%%{806400,[8]%%%},[8,9,2,4]%%%}+%%%{%%%{-64512,[9]%%%},[8,9,
0,5]%%%}+%%%{%%{[%%%{-85248,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,9,11,0]%%%}+%%%{%%{[%%%{615168,[4]%%%},0]:[1
,0,%%%{-1,[1]%%%}]%%},[7,9,9,1]%%%}+%%%{%%{[%%%{-1695744,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,9,7,2]%%%}+%%%{
%%{[%%%{2193408,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,9,5,3]%%%}+%%%{%%{[%%%{-1290240,[7]%%%},0]:[1,0,%%%{-1,[
1]%%%}]%%},[7,9,3,4]%%%}+%%%{%%{[%%%{258048,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,9,1,5]%%%}+%%%{%%%{38304,[3]
%%%},[6,9,12,0]%%%}+%%%{%%%{-322560,[4]%%%},[6,9,10,1]%%%}+%%%{%%%{1072512,[5]%%%},[6,9,8,2]%%%}+%%%{%%%{-1763
328,[6]%%%},[6,9,6,3]%%%}+%%%{%%%{1451520,[7]%%%},[6,9,4,4]%%%}+%%%{%%%{-516096,[8]%%%},[6,9,2,5]%%%}+%%%{%%%{
43008,[9]%%%},[6,9,0,6]%%%}+%%%{%%{[%%%{-14112,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,9,13,0]%%%}+%%%{%%{[%%%{1
37088,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,9,11,1]%%%}+%%%{%%{[%%%{-540288,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%}
,[5,9,9,2]%%%}+%%%{%%{[%%%{1096704,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,9,7,3]%%%}+%%%{%%{[%%%{-1193472,[6]%%
%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,9,5,4]%%%}+%%%{%%{[%%%{645120,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,9,3,5]%%%
}+%%%{%%{[%%%{-129024,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,9,1,6]%%%}+%%%{%%%{4176,[2]%%%},[4,9,14,0]%%%}+%%%
{%%%{-46368,[3]%%%},[4,9,12,1]%%%}+%%%{%%%{213696,[4]%%%},[4,9,10,2]%%%}+%%%{%%%{-524160,[5]%%%},[4,9,8,3]%%%}
+%%%{%%%{725760,[6]%%%},[4,9,6,4]%%%}+%%%{%%%{-548352,[7]%%%},[4,9,4,5]%%%}+%%%{%%%{193536,[8]%%%},[4,9,2,6]%%
%}+%%%{%%%{-18432,[9]%%%},[4,9,0,7]%%%}+%%%{%%{[%%%{-960,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,9,15,0]%%%}+%%%
{%%{[%%%{12096,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,9,13,1]%%%}+%%%{%%{[%%%{-64512,[3]%%%},0]:[1,0,%%%{-1,[1]
%%%}]%%},[3,9,11,2]%%%}+%%%{%%{[%%%{188160,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,9,9,3]%%%}+%%%{%%{[%%%{-32256
0,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,9,7,4]%%%}+%%%{%%{[%%%{322560,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,9,
5,5]%%%}+%%%{%%{[%%%{-172032,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,9,3,6]%%%}+%%%{%%{[%%%{36864,[8]%%%},0]:[1,
0,%%%{-1,[1]%%%}]%%},[3,9,1,7]%%%}+%%%{%%%{162,[1]%%%},[2,9,16,0]%%%}+%%%{%%%{-2304,[2]%%%},[2,9,14,1]%%%}+%%%
{%%%{14112,[3]%%%},[2,9,12,2]%%%}+%%%{%%%{-48384,[4]%%%},[2,9,10,3]%%%}+%%%{%%%{100800,[5]%%%},[2,9,8,4]%%%}+%
%%{%%%{-129024,[6]%%%},[2,9,6,5]%%%}+%%%{%%%{96768,[7]%%%},[2,9,4,6]%%%}+%%%{%%%{-36864,[8]%%%},[2,9,2,7]%%%}+
%%%{%%%{4608,[9]%%%},[2,9,0,8]%%%}+%%%{%%{[-18,0]:[1,0,%%%{-1,[1]%%%}]%%},[1,9,17,0]%%%}+%%%{%%{[%%%{288,[1]%%
%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,9,15,1]%%%}+%%%{%%{[%%%{-2016,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,9,13,2]%%
%}+%%%{%%{[%%%{8064,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,9,11,3]%%%}+%%%{%%{[%%%{-20160,[4]%%%},0]:[1,0,%%%{-
1,[1]%%%}]%%},[1,9,9,4]%%%}+%%%{%%{[%%%{32256,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,9,7,5]%%%}+%%%{%%{[%%%{-32
256,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,9,5,6]%%%}+%%%{%%{[%%%{18432,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,9
,3,7]%%%}+%%%{%%{[%%%{-4608,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,9,1,8]%%%}+%%%{1,[0,9,18,0]%%%}+%%%{%%%{-18,
[1]%%%},[0,9,16,1]%%%}+%%%{%%%{144,[2]%%%},[0,9,14,2]%%%}+%%%{%%%{-672,[3]%%%},[0,9,12,3]%%%}+%%%{%%%{2016,[4]
%%%},[0,9,10,4]%%%}+%%%{%%%{-4032,[5]%%%},[0,9,8,5]%%%}+%%%{%%%{5376,[6]%%%},[0,9,6,6]%%%}+%%%{%%%{-4608,[7]%%
%},[0,9,4,7]%%%}+%%%{%%%{2304,[8]%%%},[0,9,2,8]%%%}+%%%{%%%{-512,[9]%%%},[0,9,0,9]%%%} / %%%{%%{poly1[%%%{-512
,[13]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[18,0,0,0]%%%}+%%%{%%%{4608,[13]%%%},[17,0,1,0]%%%}+%%%{%%{poly1[%%%{-207
36,[12]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[16,0,2,0]%%%}+%%%{%%{[%%%{4608,[13]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[16
,0,0,1]%%%}+%%%{%%%{61440,[12]%%%},[15,0,3,0]%%%}+%%%{%%%{-36864,[13]%%%},[15,0,1,1]%%%}+%%%{%%{poly1[%%%{-133
632,[11]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[14,0,4,0]%%%}+%%%{%%{[%%%{147456,[12]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},
[14,0,2,1]%%%}+%%%{%%{poly1[%%%{-18432,[13]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[14,0,0,2]%%%}+%%%{%%%{225792,[11]%
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[%%%{-306432,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[12,0,6,0]%%%}+%%%{%%{[%%%{741888,[11]%%%},0]:[1,0,%%%{-1,[1]
%%%}]%%},[12,0,4,1]%%%}+%%%{%%{poly1[%%%{-451584,[12]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[12,0,2,2]%%%}+%%%{%%{[%%
%{43008,[13]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[12,0,0,3]%%%}+%%%{%%%{340992,[10]%%%},[11,0,7,0]%%%}+%%%{%%%{-109
6704,[11]%%%},[11,0,5,1]%%%}+%%%{%%%{1032192,[12]%%%},[11,0,3,2]%%%}+%%%{%%%{-258048,[13]%%%},[11,0,1,3]%%%}+%
%%{%%{poly1[%%%{-315072,[9]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[10,0,8,0]%%%}+%%%{%%{[%%%{1290240,[10]%%%},0]:[1,0
,%%%{-1,[1]%%%}]%%},[10,0,6,1]%%%}+%%%{%%{poly1[%%%{-1709568,[11]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[10,0,4,2]%%%
}+%%%{%%{[%%%{774144,[12]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[10,0,2,3]%%%}+%%%{%%{poly1[%%%{-64512,[13]%%%},0]:[1
,0,%%%{-1,[1]%%%}]%%},[10,0,0,4]%%%}+%%%{%%%{243392,[9]%%%},[9,0,9,0]%%%}+%%%{%%%{-1230336,[10]%%%},[9,0,7,1]%
%%}+%%%{%%%{2161152,[11]%%%},[9,0,5,2]%%%}+%%%{%%%{-1505280,[12]%%%},[9,0,3,3]%%%}+%%%{%%%{322560,[13]%%%},[9,
0,1,4]%%%}+%%%{%%{poly1[%%%{-157536,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[8,0,10,0]%%%}+%%%{%%{[%%%{960192,[9]%%
%},0]:[1,0,%%%{-1,[1]%%%}]%%},[8,0,8,1]%%%}+%%%{%%{poly1[%%%{-2145024,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[8,0
,6,2]%%%}+%%%{%%{[%%%{2096640,[11]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[8,0,4,3]%%%}+%%%{%%{poly1[%%%{-806400,[12]%
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%}+%%%{%%%{85248,[8]%%%},[7,0,11,0]%%%}+%%%{%%%{-615168,[9]%%%},[7,0,9,1]%%%}+%%%{%%%{1695744,[10]%%%},[7,0,7,
2]%%%}+%%%{%%%{-2193408,[11]%%%},[7,0,5,3]%%%}+%%%{%%%{1290240,[12]%%%},[7,0,3,4]%%%}+%%%{%%%{-258048,[13]%%%}
,[7,0,1,5]%%%}+%%%{%%{poly1[%%%{-38304,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[6,0,12,0]%%%}+%%%{%%{[%%%{322560,[8
]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[6,0,10,1]%%%}+%%%{%%{poly1[%%%{-1072512,[9]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[
6,0,8,2]%%%}+%%%{%%{[%%%{1763328,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[6,0,6,3]%%%}+%%%{%%{poly1[%%%{-1451520,[
11]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[6,0,4,4]%%%}+%%%{%%{[%%%{516096,[12]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[6,0,2
,5]%%%}+%%%{%%{poly1[%%%{-43008,[13]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[6,0,0,6]%%%}+%%%{%%%{14112,[7]%%%},[5,0,1
3,0]%%%}+%%%{%%%{-137088,[8]%%%},[5,0,11,1]%%%}+%%%{%%%{540288,[9]%%%},[5,0,9,2]%%%}+%%%{%%%{-1096704,[10]%%%}
,[5,0,7,3]%%%}+%%%{%%%{1193472,[11]%%%},[5,0,5,4]%%%}+%%%{%%%{-645120,[12]%%%},[5,0,3,5]%%%}+%%%{%%%{129024,[1
3]%%%},[5,0,1,6]%%%}+%%%{%%{poly1[%%%{-4176,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,14,0]%%%}+%%%{%%{[%%%{4636
8,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,12,1]%%%}+%%%{%%{poly1[%%%{-213696,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%
},[4,0,10,2]%%%}+%%%{%%{[%%%{524160,[9]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,8,3]%%%}+%%%{%%{poly1[%%%{-725760,
[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,6,4]%%%}+%%%{%%{[%%%{548352,[11]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,
4,5]%%%}+%%%{%%{poly1[%%%{-193536,[12]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,2,6]%%%}+%%%{%%{[%%%{18432,[13]%%%}
,0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,0,7]%%%}+%%%{%%%{960,[6]%%%},[3,0,15,0]%%%}+%%%{%%%{-12096,[7]%%%},[3,0,13,1]
%%%}+%%%{%%%{64512,[8]%%%},[3,0,11,2]%%%}+%%%{%%%{-188160,[9]%%%},[3,0,9,3]%%%}+%%%{%%%{322560,[10]%%%},[3,0,7
,4]%%%}+%%%{%%%{-322560,[11]%%%},[3,0,5,5]%%%}+%%%{%%%{172032,[12]%%%},[3,0,3,6]%%%}+%%%{%%%{-36864,[13]%%%},[
3,0,1,7]%%%}+%%%{%%{poly1[%%%{-162,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,16,0]%%%}+%%%{%%{[%%%{2304,[6]%%%},
0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,14,1]%%%}+%%%{%%{poly1[%%%{-14112,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,12,2
]%%%}+%%%{%%{[%%%{48384,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,10,3]%%%}+%%%{%%{poly1[%%%{-100800,[9]%%%},0]:
[1,0,%%%{-1,[1]%%%}]%%},[2,0,8,4]%%%}+%%%{%%{[%%%{129024,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,6,5]%%%}+%%%
{%%{poly1[%%%{-96768,[11]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,4,6]%%%}+%%%{%%{[%%%{36864,[12]%%%},0]:[1,0,%%%{
-1,[1]%%%}]%%},[2,0,2,7]%%%}+%%%{%%{[%%%{-4608,[13]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,0,8]%%%}+%%%{%%%{18,[5
]%%%},[1,0,17,0]%%%}+%%%{%%%{-288,[6]%%%},[1,0,15,1]%%%}+%%%{%%%{2016,[7]%%%},[1,0,13,2]%%%}+%%%{%%%{-8064,[8]
%%%},[1,0,11,3]%%%}+%%%{%%%{20160,[9]%%%},[1,0,9,4]%%%}+%%%{%%%{-32256,[10]%%%},[1,0,7,5]%%%}+%%%{%%%{32256,[1
1]%%%},[1,0,5,6]%%%}+%%%{%%%{-18432,[12]%%%},[1,0,3,7]%%%}+%%%{%%%{4608,[13]%%%},[1,0,1,8]%%%}+%%%{%%{poly1[%%
%{-1,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,18,0]%%%}+%%%{%%{[%%%{18,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,
16,1]%%%}+%%%{%%{poly1[%%%{-144,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,14,2]%%%}+%%%{%%{[%%%{672,[7]%%%},0]:[
1,0,%%%{-1,[1]%%%}]%%},[0,0,12,3]%%%}+%%%{%%{poly1[%%%{-2016,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,10,4]%%%}
+%%%{%%{[%%%{4032,[9]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,8,5]%%%}+%%%{%%{poly1[%%%{-5376,[10]%%%},0]:[1,0,%%%
{-1,[1]%%%}]%%},[0,0,6,6]%%%}+%%%{%%{[%%%{4608,[11]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,4,7]%%%}+%%%{%%{[%%%{-
2304,[12]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,2,8]%%%}+%%%{%%{[%%%{512,[13]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,
0,0,9]%%%} Error: Bad Argument Value

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maple [B]  time = 0.08, size = 742, normalized size = 3.58 \[ -\frac {3 a^{2} \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{128 \left (4 a c -b^{2}\right )^{4} \sqrt {\frac {4 a c -b^{2}}{c}}\, c \,d^{9}}+\frac {3 a \,b^{2} \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{256 \left (4 a c -b^{2}\right )^{4} \sqrt {\frac {4 a c -b^{2}}{c}}\, c^{2} d^{9}}-\frac {3 b^{4} \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{2048 \left (4 a c -b^{2}\right )^{4} \sqrt {\frac {4 a c -b^{2}}{c}}\, c^{3} d^{9}}+\frac {3 \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}\, a}{512 \left (4 a c -b^{2}\right )^{4} c \,d^{9}}-\frac {3 \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}\, b^{2}}{2048 \left (4 a c -b^{2}\right )^{4} c^{2} d^{9}}+\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{256 \left (4 a c -b^{2}\right )^{4} c \,d^{9}}-\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{256 \left (4 a c -b^{2}\right )^{4} \left (x +\frac {b}{2 c}\right )^{2} c^{2} d^{9}}-\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{512 \left (4 a c -b^{2}\right )^{3} \left (x +\frac {b}{2 c}\right )^{4} c^{4} d^{9}}+\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{512 \left (4 a c -b^{2}\right )^{2} \left (x +\frac {b}{2 c}\right )^{6} c^{6} d^{9}}-\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{1024 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{8} c^{8} d^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^9,x)

[Out]

-1/1024/d^9/c^8/(4*a*c-b^2)/(x+1/2*b/c)^8*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(5/2)+1/512/d^9/c^6/(4*a*c-b^2)^
2/(x+1/2*b/c)^6*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(5/2)-1/512/d^9/c^4/(4*a*c-b^2)^3/(x+1/2*b/c)^4*((x+1/2*b/
c)^2*c+1/4*(4*a*c-b^2)/c)^(5/2)-1/256/d^9/c^2/(4*a*c-b^2)^4/(x+1/2*b/c)^2*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^
(5/2)+1/256/d^9/c/(4*a*c-b^2)^4*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)+3/512/d^9/c/(4*a*c-b^2)^4*(4*(x+1/2*
b/c)^2*c+(4*a*c-b^2)/c)^(1/2)*a-3/2048/d^9/c^2/(4*a*c-b^2)^4*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2)*b^2-3/128
/d^9/c/(4*a*c-b^2)^4/((4*a*c-b^2)/c)^(1/2)*ln((1/2*(4*a*c-b^2)/c+1/2*((4*a*c-b^2)/c)^(1/2)*(4*(x+1/2*b/c)^2*c+
(4*a*c-b^2)/c)^(1/2))/(x+1/2*b/c))*a^2+3/256/d^9/c^2/(4*a*c-b^2)^4/((4*a*c-b^2)/c)^(1/2)*ln((1/2*(4*a*c-b^2)/c
+1/2*((4*a*c-b^2)/c)^(1/2)*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2))/(x+1/2*b/c))*a*b^2-3/2048/d^9/c^3/(4*a*c-b
^2)^4/((4*a*c-b^2)/c)^(1/2)*ln((1/2*(4*a*c-b^2)/c+1/2*((4*a*c-b^2)/c)^(1/2)*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^
(1/2))/(x+1/2*b/c))*b^4

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (b\,d+2\,c\,d\,x\right )}^9} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^9,x)

[Out]

int((a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^9, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {a \sqrt {a + b x + c x^{2}}}{b^{9} + 18 b^{8} c x + 144 b^{7} c^{2} x^{2} + 672 b^{6} c^{3} x^{3} + 2016 b^{5} c^{4} x^{4} + 4032 b^{4} c^{5} x^{5} + 5376 b^{3} c^{6} x^{6} + 4608 b^{2} c^{7} x^{7} + 2304 b c^{8} x^{8} + 512 c^{9} x^{9}}\, dx + \int \frac {b x \sqrt {a + b x + c x^{2}}}{b^{9} + 18 b^{8} c x + 144 b^{7} c^{2} x^{2} + 672 b^{6} c^{3} x^{3} + 2016 b^{5} c^{4} x^{4} + 4032 b^{4} c^{5} x^{5} + 5376 b^{3} c^{6} x^{6} + 4608 b^{2} c^{7} x^{7} + 2304 b c^{8} x^{8} + 512 c^{9} x^{9}}\, dx + \int \frac {c x^{2} \sqrt {a + b x + c x^{2}}}{b^{9} + 18 b^{8} c x + 144 b^{7} c^{2} x^{2} + 672 b^{6} c^{3} x^{3} + 2016 b^{5} c^{4} x^{4} + 4032 b^{4} c^{5} x^{5} + 5376 b^{3} c^{6} x^{6} + 4608 b^{2} c^{7} x^{7} + 2304 b c^{8} x^{8} + 512 c^{9} x^{9}}\, dx}{d^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**(3/2)/(2*c*d*x+b*d)**9,x)

[Out]

(Integral(a*sqrt(a + b*x + c*x**2)/(b**9 + 18*b**8*c*x + 144*b**7*c**2*x**2 + 672*b**6*c**3*x**3 + 2016*b**5*c
**4*x**4 + 4032*b**4*c**5*x**5 + 5376*b**3*c**6*x**6 + 4608*b**2*c**7*x**7 + 2304*b*c**8*x**8 + 512*c**9*x**9)
, x) + Integral(b*x*sqrt(a + b*x + c*x**2)/(b**9 + 18*b**8*c*x + 144*b**7*c**2*x**2 + 672*b**6*c**3*x**3 + 201
6*b**5*c**4*x**4 + 4032*b**4*c**5*x**5 + 5376*b**3*c**6*x**6 + 4608*b**2*c**7*x**7 + 2304*b*c**8*x**8 + 512*c*
*9*x**9), x) + Integral(c*x**2*sqrt(a + b*x + c*x**2)/(b**9 + 18*b**8*c*x + 144*b**7*c**2*x**2 + 672*b**6*c**3
*x**3 + 2016*b**5*c**4*x**4 + 4032*b**4*c**5*x**5 + 5376*b**3*c**6*x**6 + 4608*b**2*c**7*x**7 + 2304*b*c**8*x*
*8 + 512*c**9*x**9), x))/d**9

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