\(\int \frac {(a+b x+c x^2)^{3/2}}{(b d+2 c d x)^9} \, dx\) [1215]

Optimal result

Integrand size = 26, antiderivative size = 207 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^9} \, dx=-\frac {\sqrt {a+b x+c x^2}}{128 c^2 d^9 (b+2 c x)^6}+\frac {\sqrt {a+b x+c x^2}}{512 c^2 \left (b^2-4 a c\right ) d^9 (b+2 c x)^4}+\frac {3 \sqrt {a+b x+c x^2}}{1024 c^2 \left (b^2-4 a c\right )^2 d^9 (b+2 c x)^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{16 c d^9 (b+2 c x)^8}+\frac {3 \arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{2048 c^{5/2} \left (b^2-4 a c\right )^{5/2} d^9} \]

[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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {698, 707, 702, 211} \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^9} \, dx=\frac {3 \arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{2048 c^{5/2} d^9 \left (b^2-4 a c\right )^{5/2}}+\frac {3 \sqrt {a+b x+c x^2}}{1024 c^2 d^9 \left (b^2-4 a c\right )^2 (b+2 c x)^2}+\frac {\sqrt {a+b x+c x^2}}{512 c^2 d^9 \left (b^2-4 a c\right ) (b+2 c x)^4}-\frac {\sqrt {a+b x+c x^2}}{128 c^2 d^9 (b+2 c x)^6}-\frac {\left (a+b x+c x^2\right )^{3/2}}{16 c d^9 (b+2 c x)^8} \]

[In]

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

[Out]

-1/128*Sqrt[a + b*x + c*x^2]/(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)/(1
6*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 211

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

Rule 698

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 702

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 707

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*} \text {integral}& = -\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 \text {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*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 10.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.30 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^9} \, dx=\frac {2 (a+x (b+c x))^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},5,\frac {7}{2},\frac {4 c (a+x (b+c x))}{-b^2+4 a c}\right )}{5 \left (b^2-4 a c\right )^5 d^9} \]

[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)

Maple [A] (verified)

Time = 2.73 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.94

method	result	size
pseudoelliptic	\(-\frac {\frac {3 \left (2 c x +b \right )^{8} \operatorname {arctanh}\left (\frac {2 c \sqrt {c \,x^{2}+b x +a}}{\sqrt {4 c^{2} a -b^{2} c}}\right )}{2048}+\left (\frac {c^{2} x^{2}}{2}+\left (\frac {b x}{2}+a \right ) c -\frac {b^{2}}{8}\right ) \left (-\frac {3 c^{4} x^{4}}{8}+x^{2} \left (-\frac {3 b x}{4}+a \right ) c^{3}+\left (-\frac {13}{16} b^{2} x^{2}+a b x +a^{2}\right ) c^{2}-\frac {b^{2} \left (\frac {7 b x}{4}+a \right ) c}{4}-\frac {3 b^{4}}{128}\right ) \sqrt {4 c^{2} a -b^{2} c}\, \sqrt {c \,x^{2}+b x +a}}{16 \sqrt {4 c^{2} a -b^{2} c}\, d^{9} \left (2 c x +b \right )^{8} c^{2} \left (-\frac {b^{2}}{4}+a c \right )^{2}}\)	\(194\)
default	\(\frac {-\frac {c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{2 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{8}}-\frac {3 c^{2} \left (-\frac {2 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{3 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{6}}-\frac {2 c^{2} \left (-\frac {c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{4}}+\frac {c^{2} \left (-\frac {2 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{2}}+\frac {6 c^{2} \left (\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{3}+\frac {\left (4 a c -b^{2}\right ) \left (\frac {\sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}-\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{2 c \sqrt {\frac {4 a c -b^{2}}{c}}}\right )}{4 c}\right )}{4 a c -b^{2}}\right )}{4 a c -b^{2}}\right )}{3 \left (4 a c -b^{2}\right )}\right )}{2 \left (4 a c -b^{2}\right )}}{512 d^{9} c^{9}}\)	\(488\)

[In]

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

[Out]

-1/16/(4*a*c^2-b^2*c)^(1/2)*(3/2048*(2*c*x+b)^8*arctanh(2*c*(c*x^2+b*x+a)^(1/2)/(4*a*c^2-b^2*c)^(1/2))+(1/2*c^
2*x^2+(1/2*b*x+a)*c-1/8*b^2)*(-3/8*c^4*x^4+x^2*(-3/4*b*x+a)*c^3+(-13/16*b^2*x^2+a*b*x+a^2)*c^2-1/4*b^2*(7/4*b*
x+a)*c-3/128*b^4)*(4*a*c^2-b^2*c)^(1/2)*(c*x^2+b*x+a)^(1/2))/d^9/(2*c*x+b)^8/c^2/(-1/4*b^2+a*c)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 808 vs. \(2 (179) = 358\).

Time = 9.37 (sec) , antiderivative size = 1646, normalized size of antiderivative = 7.95 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^9} \, dx=\text {Too large to display} \]

[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)]

Sympy [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^9} \, dx=\frac {\int \frac {a \sqrt {a + b x + c x^{2}}}{b^{9} + 18 b^{8} c x + 144 b^{7} c^{2} x^{2} + 672 b^{6} c^{3} x^{3} + 2016 b^{5} c^{4} x^{4} + 4032 b^{4} c^{5} x^{5} + 5376 b^{3} c^{6} x^{6} + 4608 b^{2} c^{7} x^{7} + 2304 b c^{8} x^{8} + 512 c^{9} x^{9}}\, dx + \int \frac {b x \sqrt {a + b x + c x^{2}}}{b^{9} + 18 b^{8} c x + 144 b^{7} c^{2} x^{2} + 672 b^{6} c^{3} x^{3} + 2016 b^{5} c^{4} x^{4} + 4032 b^{4} c^{5} x^{5} + 5376 b^{3} c^{6} x^{6} + 4608 b^{2} c^{7} x^{7} + 2304 b c^{8} x^{8} + 512 c^{9} x^{9}}\, dx + \int \frac {c x^{2} \sqrt {a + b x + c x^{2}}}{b^{9} + 18 b^{8} c x + 144 b^{7} c^{2} x^{2} + 672 b^{6} c^{3} x^{3} + 2016 b^{5} c^{4} x^{4} + 4032 b^{4} c^{5} x^{5} + 5376 b^{3} c^{6} x^{6} + 4608 b^{2} c^{7} x^{7} + 2304 b c^{8} x^{8} + 512 c^{9} x^{9}}\, dx}{d^{9}} \]

[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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^9} \, dx=\text {Exception raised: ValueError} \]

[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 deta

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2225 vs. \(2 (179) = 358\).

Time = 0.53 (sec) , antiderivative size = 2225, normalized size of antiderivative = 10.75 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^9} \, dx=\text {Too large to display} \]

[In]

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

[Out]

3/1024*arctan(-(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*c + b*sqrt(c))/sqrt(b^2*c - 4*a*c^2))/((b^4*c^2*d^9 - 8*
a*b^2*c^3*d^9 + 16*a^2*c^4*d^9)*sqrt(b^2*c - 4*a*c^2)) - 1/1024*(384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^15*c^
(15/2) + 2880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^14*b*c^7 + 10816*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*b^2*
c^(13/2) - 2944*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*a*c^(15/2) + 26624*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
12*b^3*c^6 - 19136*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*a*b*c^7 + 44448*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
11*b^4*c^(11/2) - 36096*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*a*b^2*c^(13/2) - 42624*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^11*a^2*c^(15/2) + 47696*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*b^5*c^5 + 11968*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))^10*a*b^3*c^6 - 234432*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^2*b*c^7 + 27632*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^9*b^6*c^(9/2) + 145376*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b^4*c^(11/2) - 521664*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^2*b^2*c^(13/2) - 85888*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^3*c^(15/2)
+ 248688*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b^5*c^5 - 589248*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^2*b
^3*c^6 - 386496*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^3*b*c^7 - 13816*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*
b^8*c^(7/2) + 221056*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b^6*c^(9/2) - 331584*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^7*a^2*b^4*c^(11/2) - 687104*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^3*b^2*c^(13/2) - 85888*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^7*a^4*c^(15/2) - 11924*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^9*c^3 + 117920*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))^6*a*b^7*c^4 - 51744*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*b^5*c^5 - 601216*(s
qrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^3*b^3*c^6 - 300608*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^4*b*c^7 - 555
6*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^10*c^(5/2) + 39576*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^8*c^(7/
2) + 37152*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*b^6*c^(9/2) - 252096*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
5*a^3*b^4*c^(11/2) - 397632*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^4*b^2*c^(13/2) - 42624*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))^5*a^5*c^(15/2) - 1664*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^11*c^2 + 8828*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^4*a*b^9*c^3 + 19488*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b^7*c^4 - 29024*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^4*a^3*b^5*c^5 - 242560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^4*b^3*c^6 - 106560*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^4*a^5*b*c^7 - 338*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^12*c^(3/2) + 1456*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^10*c^(5/2) + 3096*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^8*c^(7/2)
+ 9472*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b^6*c^(9/2) - 57440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^
4*b^4*c^(11/2) - 102144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^5*b^2*c^(13/2) - 2944*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^3*a^6*c^(15/2) - 45*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^13*c + 156*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))^2*a*b^11*c^2 + 468*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^9*c^3 + 288*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^2*a^3*b^7*c^4 + 6096*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^4*b^5*c^5 - 46656*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^2*a^5*b^3*c^6 - 4416*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^6*b*c^7 - 3*(sqrt(c)*x - sqrt(c*x^
2 + b*x + a))*b^14*sqrt(c) - 6*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^12*c^(3/2) + 228*(sqrt(c)*x - sqrt(c*x^
2 + b*x + a))*a^2*b^10*c^(5/2) - 1208*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b^8*c^(7/2) + 4976*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))*a^4*b^6*c^(9/2) - 9504*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^5*b^4*c^(11/2) - 2880*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))*a^6*b^2*c^(13/2) + 384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^7*c^(15/2) - 3*a*
b^13*c + 36*a^2*b^11*c^2 - 188*a^3*b^9*c^3 + 544*a^4*b^7*c^4 - 528*a^5*b^5*c^5 - 704*a^6*b^3*c^6 + 192*a^7*b*c
^7)/((b^4*c^(5/2)*d^9 - 8*a*b^2*c^(7/2)*d^9 + 16*a^2*c^(9/2)*d^9)*(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c +
2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*sqrt(c) + b^2 - 2*a*c)^8)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^9} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (b\,d+2\,c\,d\,x\right )}^9} \,d x \]

[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)