Integrand size = 22, antiderivative size = 373 \[ \int x^2 (c+d x)^{3/2} \left (a+b x^2\right )^p \, dx=\frac {2 (c+d x)^{5/2} \left (a+b x^2\right )^{1+p}}{b d (9+4 p)}-\frac {2 \left (5 a d^2-4 b c^2 (1+p)\right ) (c+d x)^{5/2} \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \operatorname {AppellF1}\left (\frac {5}{2},-p,-p,\frac {7}{2},\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{5 b d^3 (9+4 p)}-\frac {8 c (1+p) (c+d x)^{7/2} \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \operatorname {AppellF1}\left (\frac {7}{2},-p,-p,\frac {9}{2},\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{7 d^3 (9+4 p)} \] Output:
2*(d*x+c)^(5/2)*(b*x^2+a)^(p+1)/b/d/(9+4*p)-2/5*(5*a*d^2-4*b*c^2*(p+1))*(d *x+c)^(5/2)*(b*x^2+a)^p*AppellF1(5/2,-p,-p,7/2,(d*x+c)/(c-(-a)^(1/2)*d/b^( 1/2)),(d*x+c)/(c+(-a)^(1/2)*d/b^(1/2)))/b/d^3/(9+4*p)/((1-(d*x+c)/(c-(-a)^ (1/2)*d/b^(1/2)))^p)/((1-(d*x+c)/(c+(-a)^(1/2)*d/b^(1/2)))^p)-8/7*c*(p+1)* (d*x+c)^(7/2)*(b*x^2+a)^p*AppellF1(7/2,-p,-p,9/2,(d*x+c)/(c-(-a)^(1/2)*d/b ^(1/2)),(d*x+c)/(c+(-a)^(1/2)*d/b^(1/2)))/d^3/(9+4*p)/((1-(d*x+c)/(c-(-a)^ (1/2)*d/b^(1/2)))^p)/((1-(d*x+c)/(c+(-a)^(1/2)*d/b^(1/2)))^p)
Time = 2.47 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.87 \[ \int x^2 (c+d x)^{3/2} \left (a+b x^2\right )^p \, dx=\frac {2 \left (\frac {d \left (\sqrt {-\frac {a b}{d^2}} d-b x\right )}{b c+\sqrt {-\frac {a b}{d^2}} d^2}\right )^{-p} (c+d x)^{5/2} \left (\frac {a-\sqrt {-\frac {a b}{d^2}} d x}{a+c \sqrt {-\frac {a b}{d^2}}}\right )^{-p} \left (a+b x^2\right )^p \left (63 c^2 \operatorname {AppellF1}\left (\frac {5}{2},-p,-p,\frac {7}{2},\frac {c+d x}{c+\frac {a}{\sqrt {-\frac {a b}{d^2}}}},\frac {b (c+d x)}{b c+\sqrt {-\frac {a b}{d^2}} d^2}\right )-5 (c+d x) \left (18 c \operatorname {AppellF1}\left (\frac {7}{2},-p,-p,\frac {9}{2},\frac {c+d x}{c+\frac {a}{\sqrt {-\frac {a b}{d^2}}}},\frac {b (c+d x)}{b c+\sqrt {-\frac {a b}{d^2}} d^2}\right )-7 (c+d x) \operatorname {AppellF1}\left (\frac {9}{2},-p,-p,\frac {11}{2},\frac {c+d x}{c+\frac {a}{\sqrt {-\frac {a b}{d^2}}}},\frac {b (c+d x)}{b c+\sqrt {-\frac {a b}{d^2}} d^2}\right )\right )\right )}{315 d^3} \] Input:
Integrate[x^2*(c + d*x)^(3/2)*(a + b*x^2)^p,x]
Output:
(2*(c + d*x)^(5/2)*(a + b*x^2)^p*(63*c^2*AppellF1[5/2, -p, -p, 7/2, (c + d *x)/(c + a/Sqrt[-((a*b)/d^2)]), (b*(c + d*x))/(b*c + Sqrt[-((a*b)/d^2)]*d^ 2)] - 5*(c + d*x)*(18*c*AppellF1[7/2, -p, -p, 9/2, (c + d*x)/(c + a/Sqrt[- ((a*b)/d^2)]), (b*(c + d*x))/(b*c + Sqrt[-((a*b)/d^2)]*d^2)] - 7*(c + d*x) *AppellF1[9/2, -p, -p, 11/2, (c + d*x)/(c + a/Sqrt[-((a*b)/d^2)]), (b*(c + d*x))/(b*c + Sqrt[-((a*b)/d^2)]*d^2)])))/(315*d^3*((d*(Sqrt[-((a*b)/d^2)] *d - b*x))/(b*c + Sqrt[-((a*b)/d^2)]*d^2))^p*((a - Sqrt[-((a*b)/d^2)]*d*x) /(a + c*Sqrt[-((a*b)/d^2)]))^p)
Time = 1.21 (sec) , antiderivative size = 729, normalized size of antiderivative = 1.95, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {624, 596, 719, 514, 150}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^2 (c+d x)^{3/2} \left (a+b x^2\right )^p \, dx\) |
\(\Big \downarrow \) 624 |
\(\displaystyle \frac {\int x (c+d x)^{5/2} \left (b x^2+a\right )^pdx}{d}-\frac {c \int x (c+d x)^{3/2} \left (b x^2+a\right )^pdx}{d}\) |
\(\Big \downarrow \) 596 |
\(\displaystyle \frac {\frac {2 (c+d x)^{5/2} \left (a+b x^2\right )^{p+1}}{b (4 p+9)}-\frac {5 \int (a d-b c x) (c+d x)^{3/2} \left (b x^2+a\right )^pdx}{b (4 p+9)}}{d}-\frac {c \left (\frac {2 (c+d x)^{3/2} \left (a+b x^2\right )^{p+1}}{b (4 p+7)}-\frac {3 \int (a d-b c x) \sqrt {c+d x} \left (b x^2+a\right )^pdx}{b (4 p+7)}\right )}{d}\) |
\(\Big \downarrow \) 719 |
\(\displaystyle \frac {\frac {2 (c+d x)^{5/2} \left (a+b x^2\right )^{p+1}}{b (4 p+9)}-\frac {5 \left (\frac {\left (a d^2+b c^2\right ) \int (c+d x)^{3/2} \left (b x^2+a\right )^pdx}{d}-\frac {b c \int (c+d x)^{5/2} \left (b x^2+a\right )^pdx}{d}\right )}{b (4 p+9)}}{d}-\frac {c \left (\frac {2 (c+d x)^{3/2} \left (a+b x^2\right )^{p+1}}{b (4 p+7)}-\frac {3 \left (\frac {\left (a d^2+b c^2\right ) \int \sqrt {c+d x} \left (b x^2+a\right )^pdx}{d}-\frac {b c \int (c+d x)^{3/2} \left (b x^2+a\right )^pdx}{d}\right )}{b (4 p+7)}\right )}{d}\) |
\(\Big \downarrow \) 514 |
\(\displaystyle \frac {\frac {2 (c+d x)^{5/2} \left (a+b x^2\right )^{p+1}}{b (4 p+9)}-\frac {5 \left (\frac {\left (a d^2+b c^2\right ) \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \int (c+d x)^{3/2} \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^p \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^pd(c+d x)}{d^2}-\frac {b c \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \int (c+d x)^{5/2} \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^p \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^pd(c+d x)}{d^2}\right )}{b (4 p+9)}}{d}-\frac {c \left (\frac {2 (c+d x)^{3/2} \left (a+b x^2\right )^{p+1}}{b (4 p+7)}-\frac {3 \left (\frac {\left (a d^2+b c^2\right ) \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \int \sqrt {c+d x} \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^p \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^pd(c+d x)}{d^2}-\frac {b c \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \int (c+d x)^{3/2} \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^p \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^pd(c+d x)}{d^2}\right )}{b (4 p+7)}\right )}{d}\) |
\(\Big \downarrow \) 150 |
\(\displaystyle \frac {\frac {2 (c+d x)^{5/2} \left (a+b x^2\right )^{p+1}}{b (4 p+9)}-\frac {5 \left (\frac {2 (c+d x)^{5/2} \left (a d^2+b c^2\right ) \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \operatorname {AppellF1}\left (\frac {5}{2},-p,-p,\frac {7}{2},\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{5 d^2}-\frac {2 b c (c+d x)^{7/2} \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \operatorname {AppellF1}\left (\frac {7}{2},-p,-p,\frac {9}{2},\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{7 d^2}\right )}{b (4 p+9)}}{d}-\frac {c \left (\frac {2 (c+d x)^{3/2} \left (a+b x^2\right )^{p+1}}{b (4 p+7)}-\frac {3 \left (\frac {2 (c+d x)^{3/2} \left (a d^2+b c^2\right ) \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{2},-p,-p,\frac {5}{2},\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{3 d^2}-\frac {2 b c (c+d x)^{5/2} \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \operatorname {AppellF1}\left (\frac {5}{2},-p,-p,\frac {7}{2},\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{5 d^2}\right )}{b (4 p+7)}\right )}{d}\) |
Input:
Int[x^2*(c + d*x)^(3/2)*(a + b*x^2)^p,x]
Output:
-((c*((2*(c + d*x)^(3/2)*(a + b*x^2)^(1 + p))/(b*(7 + 4*p)) - (3*((2*(b*c^ 2 + a*d^2)*(c + d*x)^(3/2)*(a + b*x^2)^p*AppellF1[3/2, -p, -p, 5/2, (c + d *x)/(c - (Sqrt[-a]*d)/Sqrt[b]), (c + d*x)/(c + (Sqrt[-a]*d)/Sqrt[b])])/(3* d^2*(1 - (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]))^p*(1 - (c + d*x)/(c + (Sqrt [-a]*d)/Sqrt[b]))^p) - (2*b*c*(c + d*x)^(5/2)*(a + b*x^2)^p*AppellF1[5/2, -p, -p, 7/2, (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]), (c + d*x)/(c + (Sqrt[-a ]*d)/Sqrt[b])])/(5*d^2*(1 - (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]))^p*(1 - ( c + d*x)/(c + (Sqrt[-a]*d)/Sqrt[b]))^p)))/(b*(7 + 4*p))))/d) + ((2*(c + d* x)^(5/2)*(a + b*x^2)^(1 + p))/(b*(9 + 4*p)) - (5*((2*(b*c^2 + a*d^2)*(c + d*x)^(5/2)*(a + b*x^2)^p*AppellF1[5/2, -p, -p, 7/2, (c + d*x)/(c - (Sqrt[- a]*d)/Sqrt[b]), (c + d*x)/(c + (Sqrt[-a]*d)/Sqrt[b])])/(5*d^2*(1 - (c + d* x)/(c - (Sqrt[-a]*d)/Sqrt[b]))^p*(1 - (c + d*x)/(c + (Sqrt[-a]*d)/Sqrt[b]) )^p) - (2*b*c*(c + d*x)^(7/2)*(a + b*x^2)^p*AppellF1[7/2, -p, -p, 9/2, (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]), (c + d*x)/(c + (Sqrt[-a]*d)/Sqrt[b])])/ (7*d^2*(1 - (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]))^p*(1 - (c + d*x)/(c + (S qrt[-a]*d)/Sqrt[b]))^p)))/(b*(9 + 4*p)))/d
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ ] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 , (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] && !In tegerQ[m] && !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {q = Rt[-a/b, 2]}, Simp[(a + b*x^2)^p/(d*(1 - (c + d*x)/(c - d*q))^p*(1 - ( c + d*x)/(c + d*q))^p) Subst[Int[x^n*Simp[1 - x/(c + d*q), x]^p*Simp[1 - x/(c - d*q), x]^p, x], x, c + d*x], x]] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c^2 + a*d^2, 0]
Int[(x_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c + d*x)^n*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 2))), x] - Simp[n/(b*(n + 2*p + 2)) Int[(c + d*x)^(n - 1)*(a + b*x^2)^p*(a*d - b*c*x), x], x] /; FreeQ[{a, b, c, d, p}, x] && GtQ[n, 0] && NeQ[n + 2*p + 2, 0]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Simp[1/d Int[x^(m - 1)*(c + d*x)^(n + 1)*(a + b*x^2)^p, x], x] - Si mp[c/d Int[x^(m - 1)*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n, p}, x] && IGtQ[m, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
\[\int x^{2} \left (d x +c \right )^{\frac {3}{2}} \left (b \,x^{2}+a \right )^{p}d x\]
Input:
int(x^2*(d*x+c)^(3/2)*(b*x^2+a)^p,x)
Output:
int(x^2*(d*x+c)^(3/2)*(b*x^2+a)^p,x)
\[ \int x^2 (c+d x)^{3/2} \left (a+b x^2\right )^p \, dx=\int { {\left (d x + c\right )}^{\frac {3}{2}} {\left (b x^{2} + a\right )}^{p} x^{2} \,d x } \] Input:
integrate(x^2*(d*x+c)^(3/2)*(b*x^2+a)^p,x, algorithm="fricas")
Output:
integral((d*x^3 + c*x^2)*sqrt(d*x + c)*(b*x^2 + a)^p, x)
\[ \int x^2 (c+d x)^{3/2} \left (a+b x^2\right )^p \, dx=\int x^{2} \left (a + b x^{2}\right )^{p} \left (c + d x\right )^{\frac {3}{2}}\, dx \] Input:
integrate(x**2*(d*x+c)**(3/2)*(b*x**2+a)**p,x)
Output:
Integral(x**2*(a + b*x**2)**p*(c + d*x)**(3/2), x)
\[ \int x^2 (c+d x)^{3/2} \left (a+b x^2\right )^p \, dx=\int { {\left (d x + c\right )}^{\frac {3}{2}} {\left (b x^{2} + a\right )}^{p} x^{2} \,d x } \] Input:
integrate(x^2*(d*x+c)^(3/2)*(b*x^2+a)^p,x, algorithm="maxima")
Output:
integrate((d*x + c)^(3/2)*(b*x^2 + a)^p*x^2, x)
\[ \int x^2 (c+d x)^{3/2} \left (a+b x^2\right )^p \, dx=\int { {\left (d x + c\right )}^{\frac {3}{2}} {\left (b x^{2} + a\right )}^{p} x^{2} \,d x } \] Input:
integrate(x^2*(d*x+c)^(3/2)*(b*x^2+a)^p,x, algorithm="giac")
Output:
integrate((d*x + c)^(3/2)*(b*x^2 + a)^p*x^2, x)
Timed out. \[ \int x^2 (c+d x)^{3/2} \left (a+b x^2\right )^p \, dx=\int x^2\,{\left (b\,x^2+a\right )}^p\,{\left (c+d\,x\right )}^{3/2} \,d x \] Input:
int(x^2*(a + b*x^2)^p*(c + d*x)^(3/2),x)
Output:
int(x^2*(a + b*x^2)^p*(c + d*x)^(3/2), x)
\[ \int x^2 (c+d x)^{3/2} \left (a+b x^2\right )^p \, dx=\text {too large to display} \] Input:
int(x^2*(d*x+c)^(3/2)*(b*x^2+a)^p,x)
Output:
( - 224*sqrt(c + d*x)*(a + b*x**2)**p*a**2*d**3*p**2 - 704*sqrt(c + d*x)*( a + b*x**2)**p*a**2*d**3*p - 510*sqrt(c + d*x)*(a + b*x**2)**p*a**2*d**3 - 6*sqrt(c + d*x)*(a + b*x**2)**p*a*b*c**2*d + 128*sqrt(c + d*x)*(a + b*x** 2)**p*a*b*c*d**2*p**3*x + 512*sqrt(c + d*x)*(a + b*x**2)**p*a*b*c*d**2*p** 2*x + 456*sqrt(c + d*x)*(a + b*x**2)**p*a*b*c*d**2*p*x + 128*sqrt(c + d*x) *(a + b*x**2)**p*a*b*d**3*p**3*x**2 + 320*sqrt(c + d*x)*(a + b*x**2)**p*a* b*d**3*p**2*x**2 + 168*sqrt(c + d*x)*(a + b*x**2)**p*a*b*d**3*p*x**2 - 24* sqrt(c + d*x)*(a + b*x**2)**p*b**2*c**3*p*x - 24*sqrt(c + d*x)*(a + b*x**2 )**p*b**2*c**3*x + 24*sqrt(c + d*x)*(a + b*x**2)**p*b**2*c**2*d*p*x**2 + 1 8*sqrt(c + d*x)*(a + b*x**2)**p*b**2*c**2*d*x**2 + 128*sqrt(c + d*x)*(a + b*x**2)**p*b**2*c*d**2*p**3*x**3 + 576*sqrt(c + d*x)*(a + b*x**2)**p*b**2* c*d**2*p**2*x**3 + 760*sqrt(c + d*x)*(a + b*x**2)**p*b**2*c*d**2*p*x**3 + 300*sqrt(c + d*x)*(a + b*x**2)**p*b**2*c*d**2*x**3 + 128*sqrt(c + d*x)*(a + b*x**2)**p*b**2*d**3*p**3*x**4 + 480*sqrt(c + d*x)*(a + b*x**2)**p*b**2* d**3*p**2*x**4 + 568*sqrt(c + d*x)*(a + b*x**2)**p*b**2*d**3*p*x**4 + 210* sqrt(c + d*x)*(a + b*x**2)**p*b**2*d**3*x**4 + 32768*int((sqrt(c + d*x)*(a + b*x**2)**p*x**2)/(256*a*c*p**4 + 1536*a*c*p**3 + 3296*a*c*p**2 + 2976*a *c*p + 945*a*c + 256*a*d*p**4*x + 1536*a*d*p**3*x + 3296*a*d*p**2*x + 2976 *a*d*p*x + 945*a*d*x + 256*b*c*p**4*x**2 + 1536*b*c*p**3*x**2 + 3296*b*c*p **2*x**2 + 2976*b*c*p*x**2 + 945*b*c*x**2 + 256*b*d*p**4*x**3 + 1536*b*...