\(\int (c+d x)^{-3-2 p} (a+b x^2)^p (A+B x+C x^2+D x^3) \, dx\) [168]

Optimal result

Integrand size = 36, antiderivative size = 573 \[ \int (c+d x)^{-3-2 p} \left (a+b x^2\right )^p \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {D (c+d x)^{-1-2 p} \left (a+b x^2\right )^{1+p}}{b d^2}-\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) (c+d x)^{-2 (1+p)} \left (a+b x^2\right )^{1+p}}{2 d^2 \left (b c^2+a d^2\right ) (1+p)}-\frac {(C d-c D (3+2 p)) (c+d x)^{-2 p} \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{2 d^4 p}-\frac {\left (a^2 d^4 D (1+2 p)-b^2 \left (c^3 C d-A c d^3-c^4 D (3+2 p)\right )-a b d^2 \left (2 c C d-B d^2-c^2 D (5+4 p)\right )\right ) \left (\sqrt {-a}-\sqrt {b} x\right ) \left (-\frac {\left (\sqrt {b} c+\sqrt {-a} d\right ) \left (\sqrt {-a}+\sqrt {b} x\right )}{\left (\sqrt {b} c-\sqrt {-a} d\right ) \left (\sqrt {-a}-\sqrt {b} x\right )}\right )^{-p} (c+d x)^{-1-2 p} \left (a+b x^2\right )^p \operatorname {Hypergeometric2F1}\left (-1-2 p,-p,-2 p,\frac {2 \sqrt {-a} \sqrt {b} (c+d x)}{\left (\sqrt {b} c-\sqrt {-a} d\right ) \left (\sqrt {-a}-\sqrt {b} x\right )}\right )}{b d^3 \left (\sqrt {b} c+\sqrt {-a} d\right ) \left (b c^2+a d^2\right ) (1+2 p)} \] Output:

D*(d*x+c)^(-1-2*p)*(b*x^2+a)^(p+1)/b/d^2-1/2*(A*d^3-B*c*d^2+C*c^2*d-D*c^3) 
*(b*x^2+a)^(p+1)/d^2/(a*d^2+b*c^2)/(p+1)/((d*x+c)^(2*p+2))-1/2*(C*d-c*D*(3 
+2*p))*(b*x^2+a)^p*AppellF1(-2*p,-p,-p,1-2*p,(d*x+c)/(c-(-a)^(1/2)*d/b^(1/ 
2)),(d*x+c)/(c+(-a)^(1/2)*d/b^(1/2)))/d^4/p/((d*x+c)^(2*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)-(a^2*d 
^4*D*(1+2*p)-b^2*(C*c^3*d-A*c*d^3-c^4*D*(3+2*p))-a*b*d^2*(2*C*c*d-B*d^2-c^ 
2*D*(5+4*p)))*((-a)^(1/2)-b^(1/2)*x)*(d*x+c)^(-1-2*p)*(b*x^2+a)^p*hypergeo 
m([-p, -1-2*p],[-2*p],2*(-a)^(1/2)*b^(1/2)*(d*x+c)/(b^(1/2)*c-(-a)^(1/2)*d 
)/((-a)^(1/2)-b^(1/2)*x))/b/d^3/(b^(1/2)*c+(-a)^(1/2)*d)/(a*d^2+b*c^2)/(1+ 
2*p)/((-(b^(1/2)*c+(-a)^(1/2)*d)*((-a)^(1/2)+b^(1/2)*x)/(b^(1/2)*c-(-a)^(1 
/2)*d)/((-a)^(1/2)-b^(1/2)*x))^p)
 

Mathematica [F]

\[ \int (c+d x)^{-3-2 p} \left (a+b x^2\right )^p \left (A+B x+C x^2+D x^3\right ) \, dx=\int (c+d x)^{-3-2 p} \left (a+b x^2\right )^p \left (A+B x+C x^2+D x^3\right ) \, dx \] Input:

Integrate[(c + d*x)^(-3 - 2*p)*(a + b*x^2)^p*(A + B*x + C*x^2 + D*x^3),x]
 

Output:

Integrate[(c + d*x)^(-3 - 2*p)*(a + b*x^2)^p*(A + B*x + C*x^2 + D*x^3), x]
 

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 575, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2185, 2187, 27, 514, 150, 679, 489}

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.

\(\displaystyle \int \left (a+b x^2\right )^p (c+d x)^{-2 p-3} \left (A+B x+C x^2+D x^3\right ) \, dx\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\int (c+d x)^{-2 p-3} \left (b x^2+a\right )^p \left (b (C d-c D (2 p+3)) x^2 d^2+(A b d+a c D (2 p+1)) d^2+\left (-2 b D (p+1) c^2+b B d^2+a d^2 D (2 p+1)\right ) x d\right )dx}{b d^3}+\frac {D \left (a+b x^2\right )^{p+1} (c+d x)^{-2 p-1}}{b d^2}\)

\(\Big \downarrow \) 2187

\(\displaystyle \frac {\frac {\int d^2 (c+d x)^{-2 p-3} \left (a c D (2 p+1) d^2+\left (a d^2 D (2 p+1)-b \left (-2 D (p+2) c^2+2 C d c-B d^2\right )\right ) x d-b \left (-D (2 p+3) c^3+C d c^2-A d^3\right )\right ) \left (b x^2+a\right )^pdx}{d^2}+b (C d-c D (2 p+3)) \int (c+d x)^{-2 p-1} \left (b x^2+a\right )^pdx}{b d^3}+\frac {D \left (a+b x^2\right )^{p+1} (c+d x)^{-2 p-1}}{b d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int (c+d x)^{-2 p-3} \left (a c D (2 p+1) d^2+\left (a d^2 D (2 p+1)-b \left (-2 D (p+2) c^2+2 C d c-B d^2\right )\right ) x d-b \left (-D (2 p+3) c^3+C d c^2-A d^3\right )\right ) \left (b x^2+a\right )^pdx+b (C d-c D (2 p+3)) \int (c+d x)^{-2 p-1} \left (b x^2+a\right )^pdx}{b d^3}+\frac {D \left (a+b x^2\right )^{p+1} (c+d x)^{-2 p-1}}{b d^2}\)

\(\Big \downarrow \) 514

\(\displaystyle \frac {\int (c+d x)^{-2 p-3} \left (a c D (2 p+1) d^2+\left (a d^2 D (2 p+1)-b \left (-2 D (p+2) c^2+2 C d c-B d^2\right )\right ) x d-b \left (-D (2 p+3) c^3+C d c^2-A d^3\right )\right ) \left (b x^2+a\right )^pdx+\frac {b \left (a+b x^2\right )^p (C d-c D (2 p+3)) \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \int (c+d x)^{-2 p-1} \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^p \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^pd(c+d x)}{d}}{b d^3}+\frac {D \left (a+b x^2\right )^{p+1} (c+d x)^{-2 p-1}}{b d^2}\)

\(\Big \downarrow \) 150

\(\displaystyle \frac {\int (c+d x)^{-2 p-3} \left (a c D (2 p+1) d^2+\left (a d^2 D (2 p+1)-b \left (-2 D (p+2) c^2+2 C d c-B d^2\right )\right ) x d-b \left (-D (2 p+3) c^3+C d c^2-A d^3\right )\right ) \left (b x^2+a\right )^pdx-\frac {b \left (a+b x^2\right )^p (c+d x)^{-2 p} (C d-c D (2 p+3)) \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{2 d p}}{b d^3}+\frac {D \left (a+b x^2\right )^{p+1} (c+d x)^{-2 p-1}}{b d^2}\)

\(\Big \downarrow \) 679

\(\displaystyle \frac {\frac {\left (a^2 d^4 D (2 p+1)-a b d^2 \left (-B d^2+c^2 (-D) (4 p+5)+2 c C d\right )-b^2 \left (-A c d^3+c^4 (-D) (2 p+3)+c^3 C d\right )\right ) \int (c+d x)^{-2 (p+1)} \left (b x^2+a\right )^pdx}{a d^2+b c^2}-\frac {b d \left (a+b x^2\right )^{p+1} (c+d x)^{-2 (p+1)} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 (p+1) \left (a d^2+b c^2\right )}-\frac {b \left (a+b x^2\right )^p (c+d x)^{-2 p} (C d-c D (2 p+3)) \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{2 d p}}{b d^3}+\frac {D \left (a+b x^2\right )^{p+1} (c+d x)^{-2 p-1}}{b d^2}\)

\(\Big \downarrow \) 489

\(\displaystyle \frac {-\frac {\left (\sqrt {-a}-\sqrt {b} x\right ) \left (a+b x^2\right )^p (c+d x)^{-2 p-1} \left (-\frac {\left (\sqrt {-a}+\sqrt {b} x\right ) \left (\sqrt {-a} d+\sqrt {b} c\right )}{\left (\sqrt {-a}-\sqrt {b} x\right ) \left (\sqrt {b} c-\sqrt {-a} d\right )}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-2 p-1,-p,-2 p,\frac {2 \sqrt {-a} \sqrt {b} (c+d x)}{\left (\sqrt {b} c-\sqrt {-a} d\right ) \left (\sqrt {-a}-\sqrt {b} x\right )}\right ) \left (a^2 d^4 D (2 p+1)-a b d^2 \left (-B d^2+c^2 (-D) (4 p+5)+2 c C d\right )-b^2 \left (-A c d^3+c^4 (-D) (2 p+3)+c^3 C d\right )\right )}{(2 p+1) \left (\sqrt {-a} d+\sqrt {b} c\right ) \left (a d^2+b c^2\right )}-\frac {b d \left (a+b x^2\right )^{p+1} (c+d x)^{-2 (p+1)} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 (p+1) \left (a d^2+b c^2\right )}-\frac {b \left (a+b x^2\right )^p (c+d x)^{-2 p} (C d-c D (2 p+3)) \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{2 d p}}{b d^3}+\frac {D \left (a+b x^2\right )^{p+1} (c+d x)^{-2 p-1}}{b d^2}\)

Input:

Int[(c + d*x)^(-3 - 2*p)*(a + b*x^2)^p*(A + B*x + C*x^2 + D*x^3),x]
 

Output:

(D*(c + d*x)^(-1 - 2*p)*(a + b*x^2)^(1 + p))/(b*d^2) + (-1/2*(b*d*(c^2*C*d 
 - B*c*d^2 + A*d^3 - c^3*D)*(a + b*x^2)^(1 + p))/((b*c^2 + a*d^2)*(1 + p)* 
(c + d*x)^(2*(1 + p))) - (b*(C*d - c*D*(3 + 2*p))*(a + b*x^2)^p*AppellF1[- 
2*p, -p, -p, 1 - 2*p, (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]), (c + d*x)/(c + 
 (Sqrt[-a]*d)/Sqrt[b])])/(2*d*p*(c + d*x)^(2*p)*(1 - (c + d*x)/(c - (Sqrt[ 
-a]*d)/Sqrt[b]))^p*(1 - (c + d*x)/(c + (Sqrt[-a]*d)/Sqrt[b]))^p) - ((a^2*d 
^4*D*(1 + 2*p) - b^2*(c^3*C*d - A*c*d^3 - c^4*D*(3 + 2*p)) - a*b*d^2*(2*c* 
C*d - B*d^2 - c^2*D*(5 + 4*p)))*(Sqrt[-a] - Sqrt[b]*x)*(c + d*x)^(-1 - 2*p 
)*(a + b*x^2)^p*Hypergeometric2F1[-1 - 2*p, -p, -2*p, (2*Sqrt[-a]*Sqrt[b]* 
(c + d*x))/((Sqrt[b]*c - Sqrt[-a]*d)*(Sqrt[-a] - Sqrt[b]*x))])/((Sqrt[b]*c 
 + Sqrt[-a]*d)*(b*c^2 + a*d^2)*(1 + 2*p)*(-(((Sqrt[b]*c + Sqrt[-a]*d)*(Sqr 
t[-a] + Sqrt[b]*x))/((Sqrt[b]*c - Sqrt[-a]*d)*(Sqrt[-a] - Sqrt[b]*x))))^p) 
)/(b*d^3)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 150

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])
 

rule 489

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Rt[(-a)*b, 2]}, Simp[(q - b*x)*(c + d*x)^(n + 1)*((a + b*x^2)^p/((n + 
1)*(b*c + d*q)*((b*c + d*q)*((q + b*x)/((b*c - d*q)*(-q + b*x))))^p))*Hyper 
geometric2F1[n + 1, -p, n + 2, 2*b*q*((c + d*x)/((b*c - d*q)*(q - b*x)))], 
x]] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[n + 2*p + 2, 0]
 

rule 514

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]
 

rule 679

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 
)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) 
 Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, 
 p}, x] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 2185

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) 
^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si 
mp[1/(b*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ 
b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x 
)^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p 
)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d 
, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && 
True) &&  !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 
 1/2, 0]))
 

rule 2187

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] 
:> With[{q = Expon[Pq, x]}, Simp[Coeff[Pq, x, q]/e^q   Int[(d + e*x)^(m + q 
)*(a + b*x^2)^p, x], x] + Simp[1/e^q   Int[(d + e*x)^m*(a + b*x^2)^p*Expand 
ToSum[e^q*Pq - Coeff[Pq, x, q]*(d + e*x)^q, x], x], x]] /; FreeQ[{a, b, d, 
e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] &&  !(IGtQ[m, 0] && R 
ationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
 

Maple [F]

\[\int \left (d x +c \right )^{-3-2 p} \left (b \,x^{2}+a \right )^{p} \left (D x^{3}+C \,x^{2}+B x +A \right )d x\]

Input:

int((d*x+c)^(-3-2*p)*(b*x^2+a)^p*(D*x^3+C*x^2+B*x+A),x)
 

Output:

int((d*x+c)^(-3-2*p)*(b*x^2+a)^p*(D*x^3+C*x^2+B*x+A),x)
 

Fricas [F]

\[ \int (c+d x)^{-3-2 p} \left (a+b x^2\right )^p \left (A+B x+C x^2+D x^3\right ) \, dx=\int { {\left (D x^{3} + C x^{2} + B x + A\right )} {\left (b x^{2} + a\right )}^{p} {\left (d x + c\right )}^{-2 \, p - 3} \,d x } \] Input:

integrate((d*x+c)^(-3-2*p)*(b*x^2+a)^p*(D*x^3+C*x^2+B*x+A),x, algorithm="f 
ricas")
 

Output:

integral((D*x^3 + C*x^2 + B*x + A)*(b*x^2 + a)^p*(d*x + c)^(-2*p - 3), x)
 

Sympy [F(-1)]

Timed out. \[ \int (c+d x)^{-3-2 p} \left (a+b x^2\right )^p \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**(-3-2*p)*(b*x**2+a)**p*(D*x**3+C*x**2+B*x+A),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (c+d x)^{-3-2 p} \left (a+b x^2\right )^p \left (A+B x+C x^2+D x^3\right ) \, dx=\int { {\left (D x^{3} + C x^{2} + B x + A\right )} {\left (b x^{2} + a\right )}^{p} {\left (d x + c\right )}^{-2 \, p - 3} \,d x } \] Input:

integrate((d*x+c)^(-3-2*p)*(b*x^2+a)^p*(D*x^3+C*x^2+B*x+A),x, algorithm="m 
axima")
 

Output:

integrate((D*x^3 + C*x^2 + B*x + A)*(b*x^2 + a)^p*(d*x + c)^(-2*p - 3), x)
 

Giac [F]

\[ \int (c+d x)^{-3-2 p} \left (a+b x^2\right )^p \left (A+B x+C x^2+D x^3\right ) \, dx=\int { {\left (D x^{3} + C x^{2} + B x + A\right )} {\left (b x^{2} + a\right )}^{p} {\left (d x + c\right )}^{-2 \, p - 3} \,d x } \] Input:

integrate((d*x+c)^(-3-2*p)*(b*x^2+a)^p*(D*x^3+C*x^2+B*x+A),x, algorithm="g 
iac")
 

Output:

integrate((D*x^3 + C*x^2 + B*x + A)*(b*x^2 + a)^p*(d*x + c)^(-2*p - 3), x)
 

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^{-3-2 p} \left (a+b x^2\right )^p \left (A+B x+C x^2+D x^3\right ) \, dx=\int \frac {{\left (b\,x^2+a\right )}^p\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^{2\,p+3}} \,d x \] Input:

int(((a + b*x^2)^p*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(2*p + 3),x)
 

Output:

int(((a + b*x^2)^p*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(2*p + 3), x)
 

Reduce [F]

\[ \int (c+d x)^{-3-2 p} \left (a+b x^2\right )^p \left (A+B x+C x^2+D x^3\right ) \, dx=\left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{\left (d x +c \right )^{2 p} c^{3}+3 \left (d x +c \right )^{2 p} c^{2} d x +3 \left (d x +c \right )^{2 p} c \,d^{2} x^{2}+\left (d x +c \right )^{2 p} d^{3} x^{3}}d x \right ) a +\left (\int \frac {\left (b \,x^{2}+a \right )^{p} x^{3}}{\left (d x +c \right )^{2 p} c^{3}+3 \left (d x +c \right )^{2 p} c^{2} d x +3 \left (d x +c \right )^{2 p} c \,d^{2} x^{2}+\left (d x +c \right )^{2 p} d^{3} x^{3}}d x \right ) d +\left (\int \frac {\left (b \,x^{2}+a \right )^{p} x^{2}}{\left (d x +c \right )^{2 p} c^{3}+3 \left (d x +c \right )^{2 p} c^{2} d x +3 \left (d x +c \right )^{2 p} c \,d^{2} x^{2}+\left (d x +c \right )^{2 p} d^{3} x^{3}}d x \right ) c +\left (\int \frac {\left (b \,x^{2}+a \right )^{p} x}{\left (d x +c \right )^{2 p} c^{3}+3 \left (d x +c \right )^{2 p} c^{2} d x +3 \left (d x +c \right )^{2 p} c \,d^{2} x^{2}+\left (d x +c \right )^{2 p} d^{3} x^{3}}d x \right ) b \] Input:

int((d*x+c)^(-3-2*p)*(b*x^2+a)^p*(D*x^3+C*x^2+B*x+A),x)
 

Output:

int((a + b*x**2)**p/((c + d*x)**(2*p)*c**3 + 3*(c + d*x)**(2*p)*c**2*d*x + 
 3*(c + d*x)**(2*p)*c*d**2*x**2 + (c + d*x)**(2*p)*d**3*x**3),x)*a + int(( 
(a + b*x**2)**p*x**3)/((c + d*x)**(2*p)*c**3 + 3*(c + d*x)**(2*p)*c**2*d*x 
 + 3*(c + d*x)**(2*p)*c*d**2*x**2 + (c + d*x)**(2*p)*d**3*x**3),x)*d + int 
(((a + b*x**2)**p*x**2)/((c + d*x)**(2*p)*c**3 + 3*(c + d*x)**(2*p)*c**2*d 
*x + 3*(c + d*x)**(2*p)*c*d**2*x**2 + (c + d*x)**(2*p)*d**3*x**3),x)*c + i 
nt(((a + b*x**2)**p*x)/((c + d*x)**(2*p)*c**3 + 3*(c + d*x)**(2*p)*c**2*d* 
x + 3*(c + d*x)**(2*p)*c*d**2*x**2 + (c + d*x)**(2*p)*d**3*x**3),x)*b