\(\int \frac {(b d+2 c d x)^{15/2}}{(a+b x+c x^2)^2} \, dx\) [1295]

Optimal result

Integrand size = 26, antiderivative size = 210 \[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^2} \, dx=52 c \left (b^2-4 a c\right )^2 d^7 \sqrt {b d+2 c d x}+\frac {52}{5} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{5/2}+\frac {52}{9} c d^3 (b d+2 c d x)^{9/2}-\frac {d (b d+2 c d x)^{13/2}}{a+b x+c x^2}-26 c \left (b^2-4 a c\right )^{9/4} d^{15/2} \arctan \left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-26 c \left (b^2-4 a c\right )^{9/4} d^{15/2} \text {arctanh}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ) \]

[Out]

52/5*c*(-4*a*c+b^2)*d^5*(2*c*d*x+b*d)^(5/2)+52/9*c*d^3*(2*c*d*x+b*d)^(9/2)-d*(2*c*d*x+b*d)^(13/2)/(c*x^2+b*x+a
)-26*c*(-4*a*c+b^2)^(9/4)*d^(15/2)*arctan((d*(2*c*x+b))^(1/2)/(-4*a*c+b^2)^(1/4)/d^(1/2))-26*c*(-4*a*c+b^2)^(9
/4)*d^(15/2)*arctanh((d*(2*c*x+b))^(1/2)/(-4*a*c+b^2)^(1/4)/d^(1/2))+52*c*(-4*a*c+b^2)^2*d^7*(2*c*d*x+b*d)^(1/
2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {700, 706, 708, 335, 218, 212, 209} \[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^2} \, dx=-26 c d^{15/2} \left (b^2-4 a c\right )^{9/4} \arctan \left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )-26 c d^{15/2} \left (b^2-4 a c\right )^{9/4} \text {arctanh}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )+52 c d^7 \left (b^2-4 a c\right )^2 \sqrt {b d+2 c d x}+\frac {52}{5} c d^5 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}-\frac {d (b d+2 c d x)^{13/2}}{a+b x+c x^2}+\frac {52}{9} c d^3 (b d+2 c d x)^{9/2} \]

[In]

Int[(b*d + 2*c*d*x)^(15/2)/(a + b*x + c*x^2)^2,x]

[Out]

52*c*(b^2 - 4*a*c)^2*d^7*Sqrt[b*d + 2*c*d*x] + (52*c*(b^2 - 4*a*c)*d^5*(b*d + 2*c*d*x)^(5/2))/5 + (52*c*d^3*(b
*d + 2*c*d*x)^(9/2))/9 - (d*(b*d + 2*c*d*x)^(13/2))/(a + b*x + c*x^2) - 26*c*(b^2 - 4*a*c)^(9/4)*d^(15/2)*ArcT
an[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])] - 26*c*(b^2 - 4*a*c)^(9/4)*d^(15/2)*ArcTanh[Sqrt[d*(b +
2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 700

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

Rule 706

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2*d*(d + e*x)^(m - 1
)*((a + b*x + c*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] + Dist[d^2*(m - 1)*((b^2 - 4*a*c)/(b^2*(m + 2*p + 1))), In
t[(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] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] &
& RationalQ[p]) || OddQ[m])

Rule 708

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

Rubi steps \begin{align*} \text {integral}& = -\frac {d (b d+2 c d x)^{13/2}}{a+b x+c x^2}+\left (13 c d^2\right ) \int \frac {(b d+2 c d x)^{11/2}}{a+b x+c x^2} \, dx \\ & = \frac {52}{9} c d^3 (b d+2 c d x)^{9/2}-\frac {d (b d+2 c d x)^{13/2}}{a+b x+c x^2}+\left (13 c \left (b^2-4 a c\right ) d^4\right ) \int \frac {(b d+2 c d x)^{7/2}}{a+b x+c x^2} \, dx \\ & = \frac {52}{5} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{5/2}+\frac {52}{9} c d^3 (b d+2 c d x)^{9/2}-\frac {d (b d+2 c d x)^{13/2}}{a+b x+c x^2}+\left (13 c \left (b^2-4 a c\right )^2 d^6\right ) \int \frac {(b d+2 c d x)^{3/2}}{a+b x+c x^2} \, dx \\ & = 52 c \left (b^2-4 a c\right )^2 d^7 \sqrt {b d+2 c d x}+\frac {52}{5} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{5/2}+\frac {52}{9} c d^3 (b d+2 c d x)^{9/2}-\frac {d (b d+2 c d x)^{13/2}}{a+b x+c x^2}+\left (13 c \left (b^2-4 a c\right )^3 d^8\right ) \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )} \, dx \\ & = 52 c \left (b^2-4 a c\right )^2 d^7 \sqrt {b d+2 c d x}+\frac {52}{5} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{5/2}+\frac {52}{9} c d^3 (b d+2 c d x)^{9/2}-\frac {d (b d+2 c d x)^{13/2}}{a+b x+c x^2}+\frac {1}{2} \left (13 \left (b^2-4 a c\right )^3 d^7\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (a-\frac {b^2}{4 c}+\frac {x^2}{4 c d^2}\right )} \, dx,x,b d+2 c d x\right ) \\ & = 52 c \left (b^2-4 a c\right )^2 d^7 \sqrt {b d+2 c d x}+\frac {52}{5} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{5/2}+\frac {52}{9} c d^3 (b d+2 c d x)^{9/2}-\frac {d (b d+2 c d x)^{13/2}}{a+b x+c x^2}+\left (13 \left (b^2-4 a c\right )^3 d^7\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b^2}{4 c}+\frac {x^4}{4 c d^2}} \, dx,x,\sqrt {d (b+2 c x)}\right ) \\ & = 52 c \left (b^2-4 a c\right )^2 d^7 \sqrt {b d+2 c d x}+\frac {52}{5} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{5/2}+\frac {52}{9} c d^3 (b d+2 c d x)^{9/2}-\frac {d (b d+2 c d x)^{13/2}}{a+b x+c x^2}-\left (26 c \left (b^2-4 a c\right )^{5/2} d^8\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c} d-x^2} \, dx,x,\sqrt {d (b+2 c x)}\right )-\left (26 c \left (b^2-4 a c\right )^{5/2} d^8\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c} d+x^2} \, dx,x,\sqrt {d (b+2 c x)}\right ) \\ & = 52 c \left (b^2-4 a c\right )^2 d^7 \sqrt {b d+2 c d x}+\frac {52}{5} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{5/2}+\frac {52}{9} c d^3 (b d+2 c d x)^{9/2}-\frac {d (b d+2 c d x)^{13/2}}{a+b x+c x^2}-26 c \left (b^2-4 a c\right )^{9/4} d^{15/2} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-26 c \left (b^2-4 a c\right )^{9/4} d^{15/2} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ) \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.12 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.67 \[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^2} \, dx=\left (\frac {1}{45}+\frac {i}{45}\right ) c (d (b+2 c x))^{15/2} \left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (-585 b^6+7020 a b^4 c-28080 a^2 b^2 c^2+37440 a^3 c^3+468 b^4 (b+2 c x)^2-3744 a b^2 c (b+2 c x)^2+7488 a^2 c^2 (b+2 c x)^2+52 b^2 (b+2 c x)^4-208 a c (b+2 c x)^4+20 (b+2 c x)^6\right )}{c (b+2 c x)^7 (a+x (b+c x))}-\frac {585 i \left (b^2-4 a c\right )^{9/4} \arctan \left (1-\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{15/2}}+\frac {585 i \left (b^2-4 a c\right )^{9/4} \arctan \left (1+\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{15/2}}+\frac {585 i \left (b^2-4 a c\right )^{9/4} \text {arctanh}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b+2 c x}}{\sqrt {b^2-4 a c}+i (b+2 c x)}\right )}{(b+2 c x)^{15/2}}\right ) \]

[In]

Integrate[(b*d + 2*c*d*x)^(15/2)/(a + b*x + c*x^2)^2,x]

[Out]

(1/45 + I/45)*c*(d*(b + 2*c*x))^(15/2)*(((1/2 - I/2)*(-585*b^6 + 7020*a*b^4*c - 28080*a^2*b^2*c^2 + 37440*a^3*
c^3 + 468*b^4*(b + 2*c*x)^2 - 3744*a*b^2*c*(b + 2*c*x)^2 + 7488*a^2*c^2*(b + 2*c*x)^2 + 52*b^2*(b + 2*c*x)^4 -
208*a*c*(b + 2*c*x)^4 + 20*(b + 2*c*x)^6))/(c*(b + 2*c*x)^7*(a + x*(b + c*x))) - ((585*I)*(b^2 - 4*a*c)^(9/4)
*ArcTan[1 - ((1 + I)*Sqrt[b + 2*c*x])/(b^2 - 4*a*c)^(1/4)])/(b + 2*c*x)^(15/2) + ((585*I)*(b^2 - 4*a*c)^(9/4)*
ArcTan[1 + ((1 + I)*Sqrt[b + 2*c*x])/(b^2 - 4*a*c)^(1/4)])/(b + 2*c*x)^(15/2) + ((585*I)*(b^2 - 4*a*c)^(9/4)*A
rcTanh[((1 + I)*(b^2 - 4*a*c)^(1/4)*Sqrt[b + 2*c*x])/(Sqrt[b^2 - 4*a*c] + I*(b + 2*c*x))])/(b + 2*c*x)^(15/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(438\) vs. \(2(178)=356\).

Time = 2.76 (sec) , antiderivative size = 439, normalized size of antiderivative = 2.09

method	result	size
pseudoelliptic	\(-\frac {416 \left (\frac {4 \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}} d^{2} \left (c \,x^{2}+b x +a \right ) c \left (-\frac {b^{2}}{4}+a c \right ) \left (d \left (2 c x +b \right )\right )^{\frac {5}{2}}}{65}-\frac {\left (d \left (2 c x +b \right )\right )^{\frac {9}{2}} \left (c \,x^{2}+b x +a \right ) c \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}}}{234}+\frac {d^{4} \left (4 a c -b^{2}\right )^{2} \left (-2 \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}} \left (\frac {12 c^{2} x^{2}}{13}+\left (\frac {12 b x}{13}+a \right ) c -\frac {b^{2}}{52}\right ) \sqrt {d \left (2 c x +b \right )}+c \,d^{2} \sqrt {2}\, \left (c \,x^{2}+b x +a \right ) \left (-\frac {b^{2}}{4}+a c \right ) \left (2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right )+\ln \left (\frac {\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+\sqrt {d^{2} \left (4 a c -b^{2}\right )}+d \left (2 c x +b \right )}{\sqrt {d^{2} \left (4 a c -b^{2}\right )}-\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+d \left (2 c x +b \right )}\right )-2 \arctan \left (\frac {-\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right )\right )\right )}{16}\right ) d^{3}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}} \left (c \,x^{2}+b x +a \right )}\)	\(439\)
derivativedivides	\(16 c \,d^{3} \left (48 a^{2} c^{2} d^{4} \sqrt {2 c d x +b d}-24 a \,b^{2} c \,d^{4} \sqrt {2 c d x +b d}-\frac {8 a c \,d^{2} \left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+3 b^{4} d^{4} \sqrt {2 c d x +b d}+\frac {2 b^{2} d^{2} \left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+\frac {\left (2 c d x +b d \right )^{\frac {9}{2}}}{9}-d^{6} \left (\frac {\left (-4 c^{3} a^{3}+3 a^{2} b^{2} c^{2}-\frac {3}{4} a \,b^{4} c +\frac {1}{16} b^{6}\right ) \sqrt {2 c d x +b d}}{a c \,d^{2}-\frac {b^{2} d^{2}}{4}+\frac {\left (2 c d x +b d \right )^{2}}{4}}+\frac {13 \left (16 c^{3} a^{3}-12 a^{2} b^{2} c^{2}+3 a \,b^{4} c -\frac {1}{4} b^{6}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}\right )\right )\)	\(458\)
default	\(16 c \,d^{3} \left (48 a^{2} c^{2} d^{4} \sqrt {2 c d x +b d}-24 a \,b^{2} c \,d^{4} \sqrt {2 c d x +b d}-\frac {8 a c \,d^{2} \left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+3 b^{4} d^{4} \sqrt {2 c d x +b d}+\frac {2 b^{2} d^{2} \left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+\frac {\left (2 c d x +b d \right )^{\frac {9}{2}}}{9}-d^{6} \left (\frac {\left (-4 c^{3} a^{3}+3 a^{2} b^{2} c^{2}-\frac {3}{4} a \,b^{4} c +\frac {1}{16} b^{6}\right ) \sqrt {2 c d x +b d}}{a c \,d^{2}-\frac {b^{2} d^{2}}{4}+\frac {\left (2 c d x +b d \right )^{2}}{4}}+\frac {13 \left (16 c^{3} a^{3}-12 a^{2} b^{2} c^{2}+3 a \,b^{4} c -\frac {1}{4} b^{6}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}\right )\right )\)	\(458\)

[In]

int((2*c*d*x+b*d)^(15/2)/(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

-416*(4/65*(d^2*(4*a*c-b^2))^(3/4)*d^2*(c*x^2+b*x+a)*c*(-1/4*b^2+a*c)*(d*(2*c*x+b))^(5/2)-1/234*(d*(2*c*x+b))^
(9/2)*(c*x^2+b*x+a)*c*(d^2*(4*a*c-b^2))^(3/4)+1/16*d^4*(4*a*c-b^2)^2*(-2*(d^2*(4*a*c-b^2))^(3/4)*(12/13*c^2*x^
2+(12/13*b*x+a)*c-1/52*b^2)*(d*(2*c*x+b))^(1/2)+c*d^2*2^(1/2)*(c*x^2+b*x+a)*(-1/4*b^2+a*c)*(2*arctan((2^(1/2)*
(d*(2*c*x+b))^(1/2)+(d^2*(4*a*c-b^2))^(1/4))/(d^2*(4*a*c-b^2))^(1/4))+ln(((d^2*(4*a*c-b^2))^(1/4)*(d*(2*c*x+b)
)^(1/2)*2^(1/2)+(d^2*(4*a*c-b^2))^(1/2)+d*(2*c*x+b))/((d^2*(4*a*c-b^2))^(1/2)-(d^2*(4*a*c-b^2))^(1/4)*(d*(2*c*
x+b))^(1/2)*2^(1/2)+d*(2*c*x+b)))-2*arctan((-2^(1/2)*(d*(2*c*x+b))^(1/2)+(d^2*(4*a*c-b^2))^(1/4))/(d^2*(4*a*c-
b^2))^(1/4)))))/(d^2*(4*a*c-b^2))^(3/4)*d^3/(c*x^2+b*x+a)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 1293, normalized size of antiderivative = 6.16 \[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate((2*c*d*x+b*d)^(15/2)/(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

-1/45*(585*((b^18*c^4 - 36*a*b^16*c^5 + 576*a^2*b^14*c^6 - 5376*a^3*b^12*c^7 + 32256*a^4*b^10*c^8 - 129024*a^5
*b^8*c^9 + 344064*a^6*b^6*c^10 - 589824*a^7*b^4*c^11 + 589824*a^8*b^2*c^12 - 262144*a^9*c^13)*d^30)^(1/4)*(c*x
^2 + b*x + a)*log(13*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*sqrt(2*c*d*x + b*d)*d^7 + 13*((b^18*c^4 - 36*a*b^16*c^
5 + 576*a^2*b^14*c^6 - 5376*a^3*b^12*c^7 + 32256*a^4*b^10*c^8 - 129024*a^5*b^8*c^9 + 344064*a^6*b^6*c^10 - 589
824*a^7*b^4*c^11 + 589824*a^8*b^2*c^12 - 262144*a^9*c^13)*d^30)^(1/4)) + 585*((b^18*c^4 - 36*a*b^16*c^5 + 576*
a^2*b^14*c^6 - 5376*a^3*b^12*c^7 + 32256*a^4*b^10*c^8 - 129024*a^5*b^8*c^9 + 344064*a^6*b^6*c^10 - 589824*a^7*
b^4*c^11 + 589824*a^8*b^2*c^12 - 262144*a^9*c^13)*d^30)^(1/4)*(I*c*x^2 + I*b*x + I*a)*log(13*(b^4*c - 8*a*b^2*
c^2 + 16*a^2*c^3)*sqrt(2*c*d*x + b*d)*d^7 + 13*I*((b^18*c^4 - 36*a*b^16*c^5 + 576*a^2*b^14*c^6 - 5376*a^3*b^12
*c^7 + 32256*a^4*b^10*c^8 - 129024*a^5*b^8*c^9 + 344064*a^6*b^6*c^10 - 589824*a^7*b^4*c^11 + 589824*a^8*b^2*c^
12 - 262144*a^9*c^13)*d^30)^(1/4)) + 585*((b^18*c^4 - 36*a*b^16*c^5 + 576*a^2*b^14*c^6 - 5376*a^3*b^12*c^7 + 3
2256*a^4*b^10*c^8 - 129024*a^5*b^8*c^9 + 344064*a^6*b^6*c^10 - 589824*a^7*b^4*c^11 + 589824*a^8*b^2*c^12 - 262
144*a^9*c^13)*d^30)^(1/4)*(-I*c*x^2 - I*b*x - I*a)*log(13*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*sqrt(2*c*d*x + b*
d)*d^7 - 13*I*((b^18*c^4 - 36*a*b^16*c^5 + 576*a^2*b^14*c^6 - 5376*a^3*b^12*c^7 + 32256*a^4*b^10*c^8 - 129024*
a^5*b^8*c^9 + 344064*a^6*b^6*c^10 - 589824*a^7*b^4*c^11 + 589824*a^8*b^2*c^12 - 262144*a^9*c^13)*d^30)^(1/4))
- 585*((b^18*c^4 - 36*a*b^16*c^5 + 576*a^2*b^14*c^6 - 5376*a^3*b^12*c^7 + 32256*a^4*b^10*c^8 - 129024*a^5*b^8*
c^9 + 344064*a^6*b^6*c^10 - 589824*a^7*b^4*c^11 + 589824*a^8*b^2*c^12 - 262144*a^9*c^13)*d^30)^(1/4)*(c*x^2 +
b*x + a)*log(13*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*sqrt(2*c*d*x + b*d)*d^7 - 13*((b^18*c^4 - 36*a*b^16*c^5 + 5
76*a^2*b^14*c^6 - 5376*a^3*b^12*c^7 + 32256*a^4*b^10*c^8 - 129024*a^5*b^8*c^9 + 344064*a^6*b^6*c^10 - 589824*a
^7*b^4*c^11 + 589824*a^8*b^2*c^12 - 262144*a^9*c^13)*d^30)^(1/4)) - (1280*c^6*d^7*x^6 + 3840*b*c^5*d^7*x^5 + 2
56*(22*b^2*c^4 - 13*a*c^5)*d^7*x^4 + 256*(19*b^3*c^3 - 26*a*b*c^4)*d^7*x^3 + 96*(45*b^4*c^2 - 208*a*b^2*c^3 +
312*a^2*c^4)*d^7*x^2 + 32*(79*b^5*c - 520*a*b^3*c^2 + 936*a^2*b*c^3)*d^7*x - (45*b^6 - 3068*a*b^4*c + 20592*a^
2*b^2*c^2 - 37440*a^3*c^3)*d^7)*sqrt(2*c*d*x + b*d))/(c*x^2 + b*x + a)

Sympy [F(-1)]

Timed out. \[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]

[In]

integrate((2*c*d*x+b*d)**(15/2)/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((2*c*d*x+b*d)^(15/2)/(c*x^2+b*x+a)^2,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. 876 vs. \(2 (178) = 356\).

Time = 0.33 (sec) , antiderivative size = 876, normalized size of antiderivative = 4.17 \[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^2} \, dx=48 \, \sqrt {2 \, c d x + b d} b^{4} c d^{7} - 384 \, \sqrt {2 \, c d x + b d} a b^{2} c^{2} d^{7} + 768 \, \sqrt {2 \, c d x + b d} a^{2} c^{3} d^{7} + \frac {32}{5} \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{2} c d^{5} - \frac {128}{5} \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} a c^{2} d^{5} + \frac {16}{9} \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}} c d^{3} - 13 \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{4} c d^{7} - 8 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a b^{2} c^{2} d^{7} + 16 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a^{2} c^{3} d^{7}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} + 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) - 13 \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{4} c d^{7} - 8 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a b^{2} c^{2} d^{7} + 16 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a^{2} c^{3} d^{7}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} - 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) - \frac {13}{2} \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{4} c d^{7} - 8 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a b^{2} c^{2} d^{7} + 16 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a^{2} c^{3} d^{7}\right )} \log \left (2 \, c d x + b d + \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + \frac {13}{2} \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{4} c d^{7} - 8 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a b^{2} c^{2} d^{7} + 16 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a^{2} c^{3} d^{7}\right )} \log \left (2 \, c d x + b d - \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + \frac {4 \, {\left (\sqrt {2 \, c d x + b d} b^{6} c d^{9} - 12 \, \sqrt {2 \, c d x + b d} a b^{4} c^{2} d^{9} + 48 \, \sqrt {2 \, c d x + b d} a^{2} b^{2} c^{3} d^{9} - 64 \, \sqrt {2 \, c d x + b d} a^{3} c^{4} d^{9}\right )}}{b^{2} d^{2} - 4 \, a c d^{2} - {\left (2 \, c d x + b d\right )}^{2}} \]

[In]

integrate((2*c*d*x+b*d)^(15/2)/(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

48*sqrt(2*c*d*x + b*d)*b^4*c*d^7 - 384*sqrt(2*c*d*x + b*d)*a*b^2*c^2*d^7 + 768*sqrt(2*c*d*x + b*d)*a^2*c^3*d^7
+ 32/5*(2*c*d*x + b*d)^(5/2)*b^2*c*d^5 - 128/5*(2*c*d*x + b*d)^(5/2)*a*c^2*d^5 + 16/9*(2*c*d*x + b*d)^(9/2)*c
*d^3 - 13*(sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*b^4*c*d^7 - 8*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*a*b^2*c^2*d
^7 + 16*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*a^2*c^3*d^7)*arctan(1/2*sqrt(2)*(sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(
1/4) + 2*sqrt(2*c*d*x + b*d))/(-b^2*d^2 + 4*a*c*d^2)^(1/4)) - 13*(sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*b^4*c*d
^7 - 8*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*a*b^2*c^2*d^7 + 16*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*a^2*c^3*d^
7)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4) - 2*sqrt(2*c*d*x + b*d))/(-b^2*d^2 + 4*a*c*d^2)^(
1/4)) - 13/2*(sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*b^4*c*d^7 - 8*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*a*b^2*c^
2*d^7 + 16*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*a^2*c^3*d^7)*log(2*c*d*x + b*d + sqrt(2)*(-b^2*d^2 + 4*a*c*d^2
)^(1/4)*sqrt(2*c*d*x + b*d) + sqrt(-b^2*d^2 + 4*a*c*d^2)) + 13/2*(sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*b^4*c*d
^7 - 8*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*a*b^2*c^2*d^7 + 16*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*a^2*c^3*d^
7)*log(2*c*d*x + b*d - sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*sqrt(2*c*d*x + b*d) + sqrt(-b^2*d^2 + 4*a*c*d^2))
+ 4*(sqrt(2*c*d*x + b*d)*b^6*c*d^9 - 12*sqrt(2*c*d*x + b*d)*a*b^4*c^2*d^9 + 48*sqrt(2*c*d*x + b*d)*a^2*b^2*c^3
*d^9 - 64*sqrt(2*c*d*x + b*d)*a^3*c^4*d^9)/(b^2*d^2 - 4*a*c*d^2 - (2*c*d*x + b*d)^2)

Mupad [B] (verification not implemented)

Time = 9.45 (sec) , antiderivative size = 1060, normalized size of antiderivative = 5.05 \[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^2} \, dx=\frac {16\,c\,d^3\,{\left (b\,d+2\,c\,d\,x\right )}^{9/2}}{9}-\frac {\sqrt {b\,d+2\,c\,d\,x}\,\left (-256\,a^3\,c^4\,d^9+192\,a^2\,b^2\,c^3\,d^9-48\,a\,b^4\,c^2\,d^9+4\,b^6\,c\,d^9\right )}{{\left (b\,d+2\,c\,d\,x\right )}^2-b^2\,d^2+4\,a\,c\,d^2}+48\,c\,d^7\,\sqrt {b\,d+2\,c\,d\,x}\,{\left (4\,a\,c-b^2\right )}^2-26\,c\,d^{15/2}\,\mathrm {atan}\left (\frac {b^4\,\sqrt {b\,d+2\,c\,d\,x}+16\,a^2\,c^2\,\sqrt {b\,d+2\,c\,d\,x}-8\,a\,b^2\,c\,\sqrt {b\,d+2\,c\,d\,x}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{9/4}}\right )\,{\left (b^2-4\,a\,c\right )}^{9/4}-\frac {32\,c\,d^5\,{\left (b\,d+2\,c\,d\,x\right )}^{5/2}\,\left (4\,a\,c-b^2\right )}{5}-c\,d^{15/2}\,\mathrm {atan}\left (\frac {c\,d^{15/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (44302336\,a^6\,c^8\,d^{18}-66453504\,a^5\,b^2\,c^7\,d^{18}[phone]\,a^4\,b^4\,c^6\,d^{18}-13844480\,a^3\,b^6\,c^5\,d^{18}+2595840\,a^2\,b^8\,c^4\,d^{18}-259584\,a\,b^{10}\,c^3\,d^{18}+10816\,b^{12}\,c^2\,d^{18}\right )-13\,c\,d^{15/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}\,\left (212992\,a^4\,c^5\,d^{11}-212992\,a^3\,b^2\,c^4\,d^{11}+79872\,a^2\,b^4\,c^3\,d^{11}-13312\,a\,b^6\,c^2\,d^{11}+832\,b^8\,c\,d^{11}\right )\right )\,13{}\mathrm {i}+c\,d^{15/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (44302336\,a^6\,c^8\,d^{18}-66453504\,a^5\,b^2\,c^7\,d^{18}[phone]\,a^4\,b^4\,c^6\,d^{18}-13844480\,a^3\,b^6\,c^5\,d^{18}+2595840\,a^2\,b^8\,c^4\,d^{18}-259584\,a\,b^{10}\,c^3\,d^{18}+10816\,b^{12}\,c^2\,d^{18}\right )+13\,c\,d^{15/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}\,\left (212992\,a^4\,c^5\,d^{11}-212992\,a^3\,b^2\,c^4\,d^{11}+79872\,a^2\,b^4\,c^3\,d^{11}-13312\,a\,b^6\,c^2\,d^{11}+832\,b^8\,c\,d^{11}\right )\right )\,13{}\mathrm {i}}{13\,c\,d^{15/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (44302336\,a^6\,c^8\,d^{18}-66453504\,a^5\,b^2\,c^7\,d^{18}[phone]\,a^4\,b^4\,c^6\,d^{18}-13844480\,a^3\,b^6\,c^5\,d^{18}+2595840\,a^2\,b^8\,c^4\,d^{18}-259584\,a\,b^{10}\,c^3\,d^{18}+10816\,b^{12}\,c^2\,d^{18}\right )-13\,c\,d^{15/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}\,\left (212992\,a^4\,c^5\,d^{11}-212992\,a^3\,b^2\,c^4\,d^{11}+79872\,a^2\,b^4\,c^3\,d^{11}-13312\,a\,b^6\,c^2\,d^{11}+832\,b^8\,c\,d^{11}\right )\right )-13\,c\,d^{15/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (44302336\,a^6\,c^8\,d^{18}-66453504\,a^5\,b^2\,c^7\,d^{18}[phone]\,a^4\,b^4\,c^6\,d^{18}-13844480\,a^3\,b^6\,c^5\,d^{18}+2595840\,a^2\,b^8\,c^4\,d^{18}-259584\,a\,b^{10}\,c^3\,d^{18}+10816\,b^{12}\,c^2\,d^{18}\right )+13\,c\,d^{15/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}\,\left (212992\,a^4\,c^5\,d^{11}-212992\,a^3\,b^2\,c^4\,d^{11}+79872\,a^2\,b^4\,c^3\,d^{11}-13312\,a\,b^6\,c^2\,d^{11}+832\,b^8\,c\,d^{11}\right )\right )}\right )\,{\left (b^2-4\,a\,c\right )}^{9/4}\,26{}\mathrm {i} \]

[In]

int((b*d + 2*c*d*x)^(15/2)/(a + b*x + c*x^2)^2,x)

[Out]

(16*c*d^3*(b*d + 2*c*d*x)^(9/2))/9 - ((b*d + 2*c*d*x)^(1/2)*(4*b^6*c*d^9 - 256*a^3*c^4*d^9 - 48*a*b^4*c^2*d^9
+ 192*a^2*b^2*c^3*d^9))/((b*d + 2*c*d*x)^2 - b^2*d^2 + 4*a*c*d^2) + 48*c*d^7*(b*d + 2*c*d*x)^(1/2)*(4*a*c - b^
2)^2 - 26*c*d^(15/2)*atan((b^4*(b*d + 2*c*d*x)^(1/2) + 16*a^2*c^2*(b*d + 2*c*d*x)^(1/2) - 8*a*b^2*c*(b*d + 2*c
*d*x)^(1/2))/(d^(1/2)*(b^2 - 4*a*c)^(9/4)))*(b^2 - 4*a*c)^(9/4) - c*d^(15/2)*atan((c*d^(15/2)*(b^2 - 4*a*c)^(9
/4)*((b*d + 2*c*d*x)^(1/2)*(44302336*a^6*c^8*d^18 + 10816*b^12*c^2*d^18 - 259584*a*b^10*c^3*d^18 + 2595840*a^2
*b^8*c^4*d^18 - 13844480*a^3*b^6*c^5*d^18 + 41533440*a^4*b^4*c^6*d^18 - 66453504*a^5*b^2*c^7*d^18) - 13*c*d^(1
5/2)*(b^2 - 4*a*c)^(9/4)*(832*b^8*c*d^11 + 212992*a^4*c^5*d^11 - 13312*a*b^6*c^2*d^11 + 79872*a^2*b^4*c^3*d^11
- 212992*a^3*b^2*c^4*d^11))*13i + c*d^(15/2)*(b^2 - 4*a*c)^(9/4)*((b*d + 2*c*d*x)^(1/2)*(44302336*a^6*c^8*d^1
8 + 10816*b^12*c^2*d^18 - 259584*a*b^10*c^3*d^18 + 2595840*a^2*b^8*c^4*d^18 - 13844480*a^3*b^6*c^5*d^18 + 4153
3440*a^4*b^4*c^6*d^18 - 66453504*a^5*b^2*c^7*d^18) + 13*c*d^(15/2)*(b^2 - 4*a*c)^(9/4)*(832*b^8*c*d^11 + 21299
2*a^4*c^5*d^11 - 13312*a*b^6*c^2*d^11 + 79872*a^2*b^4*c^3*d^11 - 212992*a^3*b^2*c^4*d^11))*13i)/(13*c*d^(15/2)
*(b^2 - 4*a*c)^(9/4)*((b*d + 2*c*d*x)^(1/2)*(44302336*a^6*c^8*d^18 + 10816*b^12*c^2*d^18 - 259584*a*b^10*c^3*d
^18 + 2595840*a^2*b^8*c^4*d^18 - 13844480*a^3*b^6*c^5*d^18 + 41533440*a^4*b^4*c^6*d^18 - 66453504*a^5*b^2*c^7*
d^18) - 13*c*d^(15/2)*(b^2 - 4*a*c)^(9/4)*(832*b^8*c*d^11 + 212992*a^4*c^5*d^11 - 13312*a*b^6*c^2*d^11 + 79872
*a^2*b^4*c^3*d^11 - 212992*a^3*b^2*c^4*d^11)) - 13*c*d^(15/2)*(b^2 - 4*a*c)^(9/4)*((b*d + 2*c*d*x)^(1/2)*(4430
2336*a^6*c^8*d^18 + 10816*b^12*c^2*d^18 - 259584*a*b^10*c^3*d^18 + 2595840*a^2*b^8*c^4*d^18 - 13844480*a^3*b^6
*c^5*d^18 + 41533440*a^4*b^4*c^6*d^18 - 66453504*a^5*b^2*c^7*d^18) + 13*c*d^(15/2)*(b^2 - 4*a*c)^(9/4)*(832*b^
8*c*d^11 + 212992*a^4*c^5*d^11 - 13312*a*b^6*c^2*d^11 + 79872*a^2*b^4*c^3*d^11 - 212992*a^3*b^2*c^4*d^11))))*(
b^2 - 4*a*c)^(9/4)*26i - (32*c*d^5*(b*d + 2*c*d*x)^(5/2)*(4*a*c - b^2))/5