\(\int \frac {x^3}{(c+d x)^2 (a+b x^2)^3} \, dx\) [257]

Optimal result

Integrand size = 20, antiderivative size = 285 \[ \int \frac {x^3}{(c+d x)^2 \left (a+b x^2\right )^3} \, dx=\frac {c^3 d^2}{\left (b c^2+a d^2\right )^3 (c+d x)}+\frac {a \left (b c^2-a d^2-2 b c d x\right )}{4 b \left (b c^2+a d^2\right )^2 \left (a+b x^2\right )^2}-\frac {c \left (2 c \left (b c^2-3 a d^2\right )-d \left (5 b c^2-3 a d^2\right ) x\right )}{4 \left (b c^2+a d^2\right )^3 \left (a+b x^2\right )}-\frac {3 c d \left (b^2 c^4-6 a b c^2 d^2+a^2 d^4\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{4 \sqrt {a} \sqrt {b} \left (b c^2+a d^2\right )^4}-\frac {3 c^2 d^2 \left (b c^2-a d^2\right ) \log (c+d x)}{\left (b c^2+a d^2\right )^4}+\frac {3 c^2 d^2 \left (b c^2-a d^2\right ) \log \left (a+b x^2\right )}{2 \left (b c^2+a d^2\right )^4} \] Output:

c^3*d^2/(a*d^2+b*c^2)^3/(d*x+c)+1/4*a*(-2*b*c*d*x-a*d^2+b*c^2)/b/(a*d^2+b* 
c^2)^2/(b*x^2+a)^2-1/4*c*(2*c*(-3*a*d^2+b*c^2)-d*(-3*a*d^2+5*b*c^2)*x)/(a* 
d^2+b*c^2)^3/(b*x^2+a)-3/4*c*d*(a^2*d^4-6*a*b*c^2*d^2+b^2*c^4)*arctan(b^(1 
/2)*x/a^(1/2))/a^(1/2)/b^(1/2)/(a*d^2+b*c^2)^4-3*c^2*d^2*(-a*d^2+b*c^2)*ln 
(d*x+c)/(a*d^2+b*c^2)^4+3/2*c^2*d^2*(-a*d^2+b*c^2)*ln(b*x^2+a)/(a*d^2+b*c^ 
2)^4
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.85 \[ \int \frac {x^3}{(c+d x)^2 \left (a+b x^2\right )^3} \, dx=\frac {\frac {4 c^3 d^2 \left (b c^2+a d^2\right )}{c+d x}+\frac {a \left (b c^2+a d^2\right )^2 \left (-a d^2+b c (c-2 d x)\right )}{b \left (a+b x^2\right )^2}-\frac {\left (b c^2+a d^2\right ) \left (b c^3 (2 c-5 d x)+3 a c d^2 (-2 c+d x)\right )}{a+b x^2}-\frac {3 c d \left (b^2 c^4-6 a b c^2 d^2+a^2 d^4\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}-12 \left (b c^4 d^2-a c^2 d^4\right ) \log (c+d x)+6 \left (b c^4 d^2-a c^2 d^4\right ) \log \left (a+b x^2\right )}{4 \left (b c^2+a d^2\right )^4} \] Input:

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

Output:

((4*c^3*d^2*(b*c^2 + a*d^2))/(c + d*x) + (a*(b*c^2 + a*d^2)^2*(-(a*d^2) + 
b*c*(c - 2*d*x)))/(b*(a + b*x^2)^2) - ((b*c^2 + a*d^2)*(b*c^3*(2*c - 5*d*x 
) + 3*a*c*d^2*(-2*c + d*x)))/(a + b*x^2) - (3*c*d*(b^2*c^4 - 6*a*b*c^2*d^2 
 + a^2*d^4)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]) - 12*(b*c^4*d^2 
 - a*c^2*d^4)*Log[c + d*x] + 6*(b*c^4*d^2 - a*c^2*d^4)*Log[a + b*x^2])/(4* 
(b*c^2 + a*d^2)^4)
 

Rubi [A] (verified)

Time = 2.07 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.10, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {601, 27, 2178, 2160, 2009}

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.

\(\displaystyle \int \frac {x^3}{\left (a+b x^2\right )^3 (c+d x)^2} \, dx\)

\(\Big \downarrow \) 601

\(\displaystyle \frac {a \left (-a d^2+b c^2-2 b c d x\right )}{4 b \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^2}-\frac {\int -\frac {2 \left (\frac {2 a b x c^4}{\left (b c^2+a d^2\right )^2}+\frac {a^2 d c^3}{\left (b c^2+a d^2\right )^2}-\frac {3 a^2 d^3 x^2 c}{\left (b c^2+a d^2\right )^2}\right )}{(c+d x)^2 \left (b x^2+a\right )^2}dx}{4 a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\frac {2 a b x c^4}{\left (b c^2+a d^2\right )^2}+\frac {a^2 d c^3}{\left (b c^2+a d^2\right )^2}-\frac {3 a^2 d^3 x^2 c}{\left (b c^2+a d^2\right )^2}}{(c+d x)^2 \left (b x^2+a\right )^2}dx}{2 a}+\frac {a \left (-a d^2+b c^2-2 b c d x\right )}{4 b \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^2}\)

\(\Big \downarrow \) 2178

\(\displaystyle \frac {-\frac {\int \frac {\frac {a^2 b d \left (3 b c^2-5 a d^2\right ) c^3}{\left (b c^2+a d^2\right )^3}-\frac {6 a^2 b d^2 x c^2}{\left (b c^2+a d^2\right )^2}-\frac {a^2 b d^3 \left (5 b c^2-3 a d^2\right ) x^2 c}{\left (b c^2+a d^2\right )^3}}{(c+d x)^2 \left (b x^2+a\right )}dx}{2 a b}-\frac {a c \left (2 c \left (b c^2-3 a d^2\right )-d x \left (5 b c^2-3 a d^2\right )\right )}{2 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^3}}{2 a}+\frac {a \left (-a d^2+b c^2-2 b c d x\right )}{4 b \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^2}\)

\(\Big \downarrow \) 2160

\(\displaystyle \frac {-\frac {\int \left (-\frac {12 a^2 b c^2 \left (a d^2-b c^2\right ) d^3}{\left (b c^2+a d^2\right )^4 (c+d x)}+\frac {4 a^2 b c^3 d^3}{\left (b c^2+a d^2\right )^3 (c+d x)^2}+\frac {3 a^2 b c \left (b^2 c^4-6 a b d^2 c^2-4 b d \left (b c^2-a d^2\right ) x c+a^2 d^4\right ) d}{\left (b c^2+a d^2\right )^4 \left (b x^2+a\right )}\right )dx}{2 a b}-\frac {a c \left (2 c \left (b c^2-3 a d^2\right )-d x \left (5 b c^2-3 a d^2\right )\right )}{2 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^3}}{2 a}+\frac {a \left (-a d^2+b c^2-2 b c d x\right )}{4 b \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {-\frac {6 a^2 b c^2 d^2 \left (b c^2-a d^2\right ) \log \left (a+b x^2\right )}{\left (a d^2+b c^2\right )^4}+\frac {12 a^2 b c^2 d^2 \left (b c^2-a d^2\right ) \log (c+d x)}{\left (a d^2+b c^2\right )^4}-\frac {4 a^2 b c^3 d^2}{(c+d x) \left (a d^2+b c^2\right )^3}+\frac {3 a^{3/2} \sqrt {b} c d \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^2 d^4-6 a b c^2 d^2+b^2 c^4\right )}{\left (a d^2+b c^2\right )^4}}{2 a b}-\frac {a c \left (2 c \left (b c^2-3 a d^2\right )-d x \left (5 b c^2-3 a d^2\right )\right )}{2 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^3}}{2 a}+\frac {a \left (-a d^2+b c^2-2 b c d x\right )}{4 b \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^2}\)

Input:

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

Output:

(a*(b*c^2 - a*d^2 - 2*b*c*d*x))/(4*b*(b*c^2 + a*d^2)^2*(a + b*x^2)^2) + (- 
1/2*(a*c*(2*c*(b*c^2 - 3*a*d^2) - d*(5*b*c^2 - 3*a*d^2)*x))/((b*c^2 + a*d^ 
2)^3*(a + b*x^2)) - ((-4*a^2*b*c^3*d^2)/((b*c^2 + a*d^2)^3*(c + d*x)) + (3 
*a^(3/2)*Sqrt[b]*c*d*(b^2*c^4 - 6*a*b*c^2*d^2 + a^2*d^4)*ArcTan[(Sqrt[b]*x 
)/Sqrt[a]])/(b*c^2 + a*d^2)^4 + (12*a^2*b*c^2*d^2*(b*c^2 - a*d^2)*Log[c + 
d*x])/(b*c^2 + a*d^2)^4 - (6*a^2*b*c^2*d^2*(b*c^2 - a*d^2)*Log[a + b*x^2]) 
/(b*c^2 + a*d^2)^4)/(2*a*b))/(2*a)
 

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 601

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe 
ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol 
ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) 
*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[(c 
+ d*x)^n*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Qx)/(c + d*x)^n + (e* 
(2*p + 3))/(c + d*x)^n, x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1] 
 && LtQ[p, -1] && ILtQ[n, 0] && NeQ[b*c^2 + a*d^2, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2160

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] 
:> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, 
 d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
 

rule 2178

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{Qx = PolynomialQuotient[(d + e*x)^m*Pq, a + b*x^2, x], R = Coeff[Po 
lynomialRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 0], S = Coeff[Polynomia 
lRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*S - b*R*x)*((a + 
b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*b*(p + 1))   Int[(d + e*x 
)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*b*(p + 1)*Qx)/(d + e*x)^m + (b*R*( 
2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, d, e}, x] && PolyQ[Pq, x 
] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]
 

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.13

method	result	size
default	\(-\frac {\frac {\left (\frac {3}{4} b c \,d^{5} a^{2}-\frac {1}{2} a \,b^{2} c^{3} d^{3}-\frac {5}{4} c^{5} d \,b^{3}\right ) x^{3}+\left (-\frac {3}{2} a^{2} b \,c^{2} d^{4}-a \,b^{2} c^{4} d^{2}+\frac {1}{2} b^{3} c^{6}\right ) x^{2}+\frac {a d c \left (5 a^{2} d^{4}+2 b \,c^{2} d^{2} a -3 b^{2} c^{4}\right ) x}{4}+\frac {a \left (a^{3} d^{6}-5 a^{2} b \,c^{2} d^{4}-5 a \,b^{2} c^{4} d^{2}+b^{3} c^{6}\right )}{4 b}}{\left (b \,x^{2}+a \right )^{2}}+\frac {3 c d \left (\frac {\left (4 a b c \,d^{3}-4 b^{2} c^{3} d \right ) \ln \left (b \,x^{2}+a \right )}{2 b}+\frac {\left (a^{2} d^{4}-6 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )}{4}}{\left (a \,d^{2}+b \,c^{2}\right )^{4}}+\frac {c^{3} d^{2}}{\left (a \,d^{2}+b \,c^{2}\right )^{3} \left (d x +c \right )}+\frac {3 c^{2} d^{2} \left (a \,d^{2}-b \,c^{2}\right ) \ln \left (d x +c \right )}{\left (a \,d^{2}+b \,c^{2}\right )^{4}}\)	\(322\)
risch	\(\frac {-\frac {3 b \,d^{2} \left (a \,d^{2}-3 b \,c^{2}\right ) c \,x^{4}}{4 \left (a^{3} d^{6}+3 a^{2} b \,c^{2} d^{4}+3 a \,b^{2} c^{4} d^{2}+b^{3} c^{6}\right )}+\frac {3 b \,c^{2} d \,x^{3}}{4 \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}-\frac {c \left (5 a^{2} d^{4}-17 b \,c^{2} d^{2} a +2 b^{2} c^{4}\right ) x^{2}}{4 \left (a^{3} d^{6}+3 a^{2} b \,c^{2} d^{4}+3 a \,b^{2} c^{4} d^{2}+b^{3} c^{6}\right )}-\frac {\left (a \,d^{2}-2 b \,c^{2}\right ) a d x}{4 \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) b}-\frac {a c \left (a^{2} d^{4}-10 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}{4 b \left (a^{3} d^{6}+3 a^{2} b \,c^{2} d^{4}+3 a \,b^{2} c^{4} d^{2}+b^{3} c^{6}\right )}}{\left (d x +c \right ) \left (b \,x^{2}+a \right )^{2}}+\frac {3 c^{2} d^{4} \ln \left (d x +c \right ) a}{a^{4} d^{8}+4 a^{3} b \,c^{2} d^{6}+6 a^{2} b^{2} c^{4} d^{4}+4 a \,b^{3} c^{6} d^{2}+b^{4} c^{8}}-\frac {3 c^{4} d^{2} \ln \left (d x +c \right ) b}{a^{4} d^{8}+4 a^{3} b \,c^{2} d^{6}+6 a^{2} b^{2} c^{4} d^{4}+4 a \,b^{3} c^{6} d^{2}+b^{4} c^{8}}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{5} b \,d^{8}+4 a^{4} c^{2} d^{6} b^{2}+6 a^{3} c^{4} d^{4} b^{3}+4 a^{2} c^{6} d^{2} b^{4}+a \,c^{8} b^{5}\right ) \textit {\_Z}^{2}+\left (8 a^{2} b \,c^{2} d^{4}-8 a \,b^{2} c^{4} d^{2}\right ) \textit {\_Z} +c^{2} d^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (3 a^{7} b \,d^{14}+17 a^{6} b^{2} c^{2} d^{12}+39 a^{5} b^{3} c^{4} d^{10}+45 a^{4} b^{4} c^{6} d^{8}+25 a^{3} b^{5} c^{8} d^{6}+3 a^{2} b^{6} c^{10} d^{4}-3 a \,b^{7} c^{12} d^{2}-b^{8} c^{14}\right ) \textit {\_R}^{2}+\left (10 a^{4} b \,c^{2} d^{10}+24 a^{3} b^{2} c^{4} d^{8}+12 a^{2} b^{3} c^{6} d^{6}-8 a \,b^{4} c^{8} d^{4}-6 b^{5} c^{10} d^{2}\right ) \textit {\_R} +2 a^{2} c^{2} d^{8}-12 a b \,c^{4} d^{6}+18 b^{2} c^{6} d^{4}\right ) x +\left (4 a^{7} b c \,d^{13}+24 a^{6} b^{2} c^{3} d^{11}+60 a^{5} b^{3} c^{5} d^{9}+80 a^{4} b^{4} c^{7} d^{7}+60 a^{3} b^{5} c^{9} d^{5}+24 a^{2} b^{6} c^{11} d^{3}+4 a \,b^{7} c^{13} d \right ) \textit {\_R}^{2}+\left (a^{5} c \,d^{11}+3 a^{4} b \,c^{3} d^{9}+2 a^{3} b^{2} c^{5} d^{7}-2 a^{2} b^{3} c^{7} d^{5}-3 a \,b^{4} c^{9} d^{3}-b^{5} c^{11} d \right ) \textit {\_R} -6 a^{2} c^{3} d^{7}+20 a b \,c^{5} d^{5}-6 b^{2} c^{7} d^{3}\right )\right )}{8}\)	\(966\)

Input:

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

Output:

-1/(a*d^2+b*c^2)^4*(((3/4*b*c*d^5*a^2-1/2*a*b^2*c^3*d^3-5/4*c^5*d*b^3)*x^3 
+(-3/2*a^2*b*c^2*d^4-a*b^2*c^4*d^2+1/2*b^3*c^6)*x^2+1/4*a*d*c*(5*a^2*d^4+2 
*a*b*c^2*d^2-3*b^2*c^4)*x+1/4*a*(a^3*d^6-5*a^2*b*c^2*d^4-5*a*b^2*c^4*d^2+b 
^3*c^6)/b)/(b*x^2+a)^2+3/4*c*d*(1/2*(4*a*b*c*d^3-4*b^2*c^3*d)/b*ln(b*x^2+a 
)+(a^2*d^4-6*a*b*c^2*d^2+b^2*c^4)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))))+c^ 
3*d^2/(a*d^2+b*c^2)^3/(d*x+c)+3*c^2*d^2*(a*d^2-b*c^2)/(a*d^2+b*c^2)^4*ln(d 
*x+c)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1243 vs. \(2 (268) = 536\).

Time = 2.06 (sec) , antiderivative size = 2509, normalized size of antiderivative = 8.80 \[ \int \frac {x^3}{(c+d x)^2 \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/8*(2*a^2*b^3*c^7 - 18*a^3*b^2*c^5*d^2 - 18*a^4*b*c^3*d^4 + 2*a^5*c*d^6 
 - 6*(3*a*b^4*c^5*d^2 + 2*a^2*b^3*c^3*d^4 - a^3*b^2*c*d^6)*x^4 - 6*(a*b^4* 
c^6*d + 2*a^2*b^3*c^4*d^3 + a^3*b^2*c^2*d^5)*x^3 + 2*(2*a*b^4*c^7 - 15*a^2 
*b^3*c^5*d^2 - 12*a^3*b^2*c^3*d^4 + 5*a^4*b*c*d^6)*x^2 + 3*(a^2*b^2*c^6*d 
- 6*a^3*b*c^4*d^3 + a^4*c^2*d^5 + (b^4*c^5*d^2 - 6*a*b^3*c^3*d^4 + a^2*b^2 
*c*d^6)*x^5 + (b^4*c^6*d - 6*a*b^3*c^4*d^3 + a^2*b^2*c^2*d^5)*x^4 + 2*(a*b 
^3*c^5*d^2 - 6*a^2*b^2*c^3*d^4 + a^3*b*c*d^6)*x^3 + 2*(a*b^3*c^6*d - 6*a^2 
*b^2*c^4*d^3 + a^3*b*c^2*d^5)*x^2 + (a^2*b^2*c^5*d^2 - 6*a^3*b*c^3*d^4 + a 
^4*c*d^6)*x)*sqrt(-a*b)*log((b*x^2 + 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) - 2* 
(2*a^2*b^3*c^6*d + 3*a^3*b^2*c^4*d^3 - a^5*d^7)*x - 12*(a^3*b^2*c^5*d^2 - 
a^4*b*c^3*d^4 + (a*b^4*c^4*d^3 - a^2*b^3*c^2*d^5)*x^5 + (a*b^4*c^5*d^2 - a 
^2*b^3*c^3*d^4)*x^4 + 2*(a^2*b^3*c^4*d^3 - a^3*b^2*c^2*d^5)*x^3 + 2*(a^2*b 
^3*c^5*d^2 - a^3*b^2*c^3*d^4)*x^2 + (a^3*b^2*c^4*d^3 - a^4*b*c^2*d^5)*x)*l 
og(b*x^2 + a) + 24*(a^3*b^2*c^5*d^2 - a^4*b*c^3*d^4 + (a*b^4*c^4*d^3 - a^2 
*b^3*c^2*d^5)*x^5 + (a*b^4*c^5*d^2 - a^2*b^3*c^3*d^4)*x^4 + 2*(a^2*b^3*c^4 
*d^3 - a^3*b^2*c^2*d^5)*x^3 + 2*(a^2*b^3*c^5*d^2 - a^3*b^2*c^3*d^4)*x^2 + 
(a^3*b^2*c^4*d^3 - a^4*b*c^2*d^5)*x)*log(d*x + c))/(a^3*b^5*c^9 + 4*a^4*b^ 
4*c^7*d^2 + 6*a^5*b^3*c^5*d^4 + 4*a^6*b^2*c^3*d^6 + a^7*b*c*d^8 + (a*b^7*c 
^8*d + 4*a^2*b^6*c^6*d^3 + 6*a^3*b^5*c^4*d^5 + 4*a^4*b^4*c^2*d^7 + a^5*b^3 
*d^9)*x^5 + (a*b^7*c^9 + 4*a^2*b^6*c^7*d^2 + 6*a^3*b^5*c^5*d^4 + 4*a^4*...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 718 vs. \(2 (268) = 536\).

Time = 0.13 (sec) , antiderivative size = 718, normalized size of antiderivative = 2.52 \[ \int \frac {x^3}{(c+d x)^2 \left (a+b x^2\right )^3} \, dx=\frac {3 \, {\left (b c^{4} d^{2} - a c^{2} d^{4}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{4} c^{8} + 4 \, a b^{3} c^{6} d^{2} + 6 \, a^{2} b^{2} c^{4} d^{4} + 4 \, a^{3} b c^{2} d^{6} + a^{4} d^{8}\right )}} - \frac {3 \, {\left (b c^{4} d^{2} - a c^{2} d^{4}\right )} \log \left (d x + c\right )}{b^{4} c^{8} + 4 \, a b^{3} c^{6} d^{2} + 6 \, a^{2} b^{2} c^{4} d^{4} + 4 \, a^{3} b c^{2} d^{6} + a^{4} d^{8}} - \frac {3 \, {\left (b^{2} c^{5} d - 6 \, a b c^{3} d^{3} + a^{2} c d^{5}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{4 \, {\left (b^{4} c^{8} + 4 \, a b^{3} c^{6} d^{2} + 6 \, a^{2} b^{2} c^{4} d^{4} + 4 \, a^{3} b c^{2} d^{6} + a^{4} d^{8}\right )} \sqrt {a b}} - \frac {a b^{2} c^{5} - 10 \, a^{2} b c^{3} d^{2} + a^{3} c d^{4} - 3 \, {\left (3 \, b^{3} c^{3} d^{2} - a b^{2} c d^{4}\right )} x^{4} - 3 \, {\left (b^{3} c^{4} d + a b^{2} c^{2} d^{3}\right )} x^{3} + {\left (2 \, b^{3} c^{5} - 17 \, a b^{2} c^{3} d^{2} + 5 \, a^{2} b c d^{4}\right )} x^{2} - {\left (2 \, a b^{2} c^{4} d + a^{2} b c^{2} d^{3} - a^{3} d^{5}\right )} x}{4 \, {\left (a^{2} b^{4} c^{7} + 3 \, a^{3} b^{3} c^{5} d^{2} + 3 \, a^{4} b^{2} c^{3} d^{4} + a^{5} b c d^{6} + {\left (b^{6} c^{6} d + 3 \, a b^{5} c^{4} d^{3} + 3 \, a^{2} b^{4} c^{2} d^{5} + a^{3} b^{3} d^{7}\right )} x^{5} + {\left (b^{6} c^{7} + 3 \, a b^{5} c^{5} d^{2} + 3 \, a^{2} b^{4} c^{3} d^{4} + a^{3} b^{3} c d^{6}\right )} x^{4} + 2 \, {\left (a b^{5} c^{6} d + 3 \, a^{2} b^{4} c^{4} d^{3} + 3 \, a^{3} b^{3} c^{2} d^{5} + a^{4} b^{2} d^{7}\right )} x^{3} + 2 \, {\left (a b^{5} c^{7} + 3 \, a^{2} b^{4} c^{5} d^{2} + 3 \, a^{3} b^{3} c^{3} d^{4} + a^{4} b^{2} c d^{6}\right )} x^{2} + {\left (a^{2} b^{4} c^{6} d + 3 \, a^{3} b^{3} c^{4} d^{3} + 3 \, a^{4} b^{2} c^{2} d^{5} + a^{5} b d^{7}\right )} x\right )}} \] Input:

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

Output:

3/2*(b*c^4*d^2 - a*c^2*d^4)*log(b*x^2 + a)/(b^4*c^8 + 4*a*b^3*c^6*d^2 + 6* 
a^2*b^2*c^4*d^4 + 4*a^3*b*c^2*d^6 + a^4*d^8) - 3*(b*c^4*d^2 - a*c^2*d^4)*l 
og(d*x + c)/(b^4*c^8 + 4*a*b^3*c^6*d^2 + 6*a^2*b^2*c^4*d^4 + 4*a^3*b*c^2*d 
^6 + a^4*d^8) - 3/4*(b^2*c^5*d - 6*a*b*c^3*d^3 + a^2*c*d^5)*arctan(b*x/sqr 
t(a*b))/((b^4*c^8 + 4*a*b^3*c^6*d^2 + 6*a^2*b^2*c^4*d^4 + 4*a^3*b*c^2*d^6 
+ a^4*d^8)*sqrt(a*b)) - 1/4*(a*b^2*c^5 - 10*a^2*b*c^3*d^2 + a^3*c*d^4 - 3* 
(3*b^3*c^3*d^2 - a*b^2*c*d^4)*x^4 - 3*(b^3*c^4*d + a*b^2*c^2*d^3)*x^3 + (2 
*b^3*c^5 - 17*a*b^2*c^3*d^2 + 5*a^2*b*c*d^4)*x^2 - (2*a*b^2*c^4*d + a^2*b* 
c^2*d^3 - a^3*d^5)*x)/(a^2*b^4*c^7 + 3*a^3*b^3*c^5*d^2 + 3*a^4*b^2*c^3*d^4 
 + a^5*b*c*d^6 + (b^6*c^6*d + 3*a*b^5*c^4*d^3 + 3*a^2*b^4*c^2*d^5 + a^3*b^ 
3*d^7)*x^5 + (b^6*c^7 + 3*a*b^5*c^5*d^2 + 3*a^2*b^4*c^3*d^4 + a^3*b^3*c*d^ 
6)*x^4 + 2*(a*b^5*c^6*d + 3*a^2*b^4*c^4*d^3 + 3*a^3*b^3*c^2*d^5 + a^4*b^2* 
d^7)*x^3 + 2*(a*b^5*c^7 + 3*a^2*b^4*c^5*d^2 + 3*a^3*b^3*c^3*d^4 + a^4*b^2* 
c*d^6)*x^2 + (a^2*b^4*c^6*d + 3*a^3*b^3*c^4*d^3 + 3*a^4*b^2*c^2*d^5 + a^5* 
b*d^7)*x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 550 vs. \(2 (268) = 536\).

Time = 0.14 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.93 \[ \int \frac {x^3}{(c+d x)^2 \left (a+b x^2\right )^3} \, dx=\frac {c^{3} d^{8}}{{\left (b^{3} c^{6} d^{6} + 3 \, a b^{2} c^{4} d^{8} + 3 \, a^{2} b c^{2} d^{10} + a^{3} d^{12}\right )} {\left (d x + c\right )}} + \frac {3 \, {\left (b c^{4} d^{2} - a c^{2} d^{4}\right )} \log \left (b - \frac {2 \, b c}{d x + c} + \frac {b c^{2}}{{\left (d x + c\right )}^{2}} + \frac {a d^{2}}{{\left (d x + c\right )}^{2}}\right )}{2 \, {\left (b^{4} c^{8} + 4 \, a b^{3} c^{6} d^{2} + 6 \, a^{2} b^{2} c^{4} d^{4} + 4 \, a^{3} b c^{2} d^{6} + a^{4} d^{8}\right )}} - \frac {3 \, {\left (b^{2} c^{5} d^{3} - 6 \, a b c^{3} d^{5} + a^{2} c d^{7}\right )} \arctan \left (\frac {b c - \frac {b c^{2}}{d x + c} - \frac {a d^{2}}{d x + c}}{\sqrt {a b} d}\right )}{4 \, {\left (b^{4} c^{8} + 4 \, a b^{3} c^{6} d^{2} + 6 \, a^{2} b^{2} c^{4} d^{4} + 4 \, a^{3} b c^{2} d^{6} + a^{4} d^{8}\right )} \sqrt {a b} d^{2}} + \frac {7 \, b^{3} c^{4} d^{2} - 12 \, a b^{2} c^{2} d^{4} + a^{2} b d^{6} - \frac {23 \, b^{3} c^{5} d^{3} - 50 \, a b^{2} c^{3} d^{5} + 7 \, a^{2} b c d^{7}}{{\left (d x + c\right )} d} + \frac {25 \, b^{3} c^{6} d^{4} - 60 \, a b^{2} c^{4} d^{6} - 3 \, a^{2} b c^{2} d^{8} + 2 \, a^{3} d^{10}}{{\left (d x + c\right )}^{2} d^{2}} - \frac {3 \, {\left (3 \, b^{3} c^{7} d^{5} - 7 \, a b^{2} c^{5} d^{7} - 7 \, a^{2} b c^{3} d^{9} + 3 \, a^{3} c d^{11}\right )}}{{\left (d x + c\right )}^{3} d^{3}}}{4 \, {\left (b c^{2} + a d^{2}\right )}^{4} {\left (b - \frac {2 \, b c}{d x + c} + \frac {b c^{2}}{{\left (d x + c\right )}^{2}} + \frac {a d^{2}}{{\left (d x + c\right )}^{2}}\right )}^{2}} \] Input:

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

Output:

c^3*d^8/((b^3*c^6*d^6 + 3*a*b^2*c^4*d^8 + 3*a^2*b*c^2*d^10 + a^3*d^12)*(d* 
x + c)) + 3/2*(b*c^4*d^2 - a*c^2*d^4)*log(b - 2*b*c/(d*x + c) + b*c^2/(d*x 
 + c)^2 + a*d^2/(d*x + c)^2)/(b^4*c^8 + 4*a*b^3*c^6*d^2 + 6*a^2*b^2*c^4*d^ 
4 + 4*a^3*b*c^2*d^6 + a^4*d^8) - 3/4*(b^2*c^5*d^3 - 6*a*b*c^3*d^5 + a^2*c* 
d^7)*arctan((b*c - b*c^2/(d*x + c) - a*d^2/(d*x + c))/(sqrt(a*b)*d))/((b^4 
*c^8 + 4*a*b^3*c^6*d^2 + 6*a^2*b^2*c^4*d^4 + 4*a^3*b*c^2*d^6 + a^4*d^8)*sq 
rt(a*b)*d^2) + 1/4*(7*b^3*c^4*d^2 - 12*a*b^2*c^2*d^4 + a^2*b*d^6 - (23*b^3 
*c^5*d^3 - 50*a*b^2*c^3*d^5 + 7*a^2*b*c*d^7)/((d*x + c)*d) + (25*b^3*c^6*d 
^4 - 60*a*b^2*c^4*d^6 - 3*a^2*b*c^2*d^8 + 2*a^3*d^10)/((d*x + c)^2*d^2) - 
3*(3*b^3*c^7*d^5 - 7*a*b^2*c^5*d^7 - 7*a^2*b*c^3*d^9 + 3*a^3*c*d^11)/((d*x 
 + c)^3*d^3))/((b*c^2 + a*d^2)^4*(b - 2*b*c/(d*x + c) + b*c^2/(d*x + c)^2 
+ a*d^2/(d*x + c)^2)^2)
 

Mupad [B] (verification not implemented)

Time = 8.36 (sec) , antiderivative size = 1109, normalized size of antiderivative = 3.89 \[ \int \frac {x^3}{(c+d x)^2 \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(log(c + d*x)*(3*a*c^2*d^4 - 3*b*c^4*d^2))/(a^4*d^8 + b^4*c^8 + 4*a*b^3*c^ 
6*d^2 + 4*a^3*b*c^2*d^6 + 6*a^2*b^2*c^4*d^4) - ((x^2*(2*b^2*c^5 + 5*a^2*c* 
d^4 - 17*a*b*c^3*d^2))/(4*(a*d^2 + b*c^2)*(a^2*d^4 + b^2*c^4 + 2*a*b*c^2*d 
^2)) - (3*x^4*(3*b^2*c^3*d^2 - a*b*c*d^4))/(4*(a^3*d^6 + b^3*c^6 + 3*a*b^2 
*c^4*d^2 + 3*a^2*b*c^2*d^4)) + (c*(a^3*d^4 + a*b^2*c^4 - 10*a^2*b*c^2*d^2) 
)/(4*b*(a*d^2 + b*c^2)*(a^2*d^4 + b^2*c^4 + 2*a*b*c^2*d^2)) + (d*x*(a^2*d^ 
2 - 2*a*b*c^2))/(4*b*(a^2*d^4 + b^2*c^4 + 2*a*b*c^2*d^2)) - (3*b*c^2*d*x^3 
)/(4*(a^2*d^4 + b^2*c^4 + 2*a*b*c^2*d^2)))/(a^2*c + b^2*c*x^4 + b^2*d*x^5 
+ a^2*d*x + 2*a*b*c*x^2 + 2*a*b*d*x^3) + (log(b^5*c^12*(-a*b)^(3/2) - a^7* 
d^12*(-a*b)^(1/2) - 15*a^3*c^4*d^8*(-a*b)^(5/2) + 134*a^5*c^2*d^10*(-a*b)^ 
(3/2) - 134*b^3*c^10*d^2*(-a*b)^(5/2) + a*b^7*c^12*x + a^7*b*d^12*x - 236* 
a*c^6*d^6*(-a*b)^(7/2) + 15*b*c^8*d^4*(-a*b)^(7/2) + 134*a^2*b^6*c^10*d^2* 
x + 15*a^3*b^5*c^8*d^4*x - 236*a^4*b^4*c^6*d^6*x + 15*a^5*b^3*c^4*d^8*x + 
134*a^6*b^2*c^2*d^10*x)*(b^2*((3*c^5*d*(-a*b)^(1/2))/8 + (3*a*c^4*d^2)/2) 
- b*((3*a^2*c^2*d^4)/2 + (9*a*c^3*d^3*(-a*b)^(1/2))/4) + (3*a^2*c*d^5*(-a* 
b)^(1/2))/8))/(a*b^5*c^8 + a^5*b*d^8 + 4*a^2*b^4*c^6*d^2 + 6*a^3*b^3*c^4*d 
^4 + 4*a^4*b^2*c^2*d^6) - (log(a^7*d^12*(-a*b)^(1/2) - b^5*c^12*(-a*b)^(3/ 
2) + 15*a^3*c^4*d^8*(-a*b)^(5/2) - 134*a^5*c^2*d^10*(-a*b)^(3/2) + 134*b^3 
*c^10*d^2*(-a*b)^(5/2) + a*b^7*c^12*x + a^7*b*d^12*x + 236*a*c^6*d^6*(-a*b 
)^(7/2) - 15*b*c^8*d^4*(-a*b)^(7/2) + 134*a^2*b^6*c^10*d^2*x + 15*a^3*b...
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 1738, normalized size of antiderivative = 6.10 \[ \int \frac {x^3}{(c+d x)^2 \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 3*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*c**2*d**5 - 3*sqr 
t(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*c*d**6*x + 18*sqrt(b)*sqrt 
(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b*c**4*d**3 + 18*sqrt(b)*sqrt(a)*at 
an((b*x)/(sqrt(b)*sqrt(a)))*a**3*b*c**3*d**4*x - 6*sqrt(b)*sqrt(a)*atan((b 
*x)/(sqrt(b)*sqrt(a)))*a**3*b*c**2*d**5*x**2 - 6*sqrt(b)*sqrt(a)*atan((b*x 
)/(sqrt(b)*sqrt(a)))*a**3*b*c*d**6*x**3 - 3*sqrt(b)*sqrt(a)*atan((b*x)/(sq 
rt(b)*sqrt(a)))*a**2*b**2*c**6*d - 3*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*s 
qrt(a)))*a**2*b**2*c**5*d**2*x + 36*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sq 
rt(a)))*a**2*b**2*c**4*d**3*x**2 + 36*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)* 
sqrt(a)))*a**2*b**2*c**3*d**4*x**3 - 3*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b) 
*sqrt(a)))*a**2*b**2*c**2*d**5*x**4 - 3*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b 
)*sqrt(a)))*a**2*b**2*c*d**6*x**5 - 6*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)* 
sqrt(a)))*a*b**3*c**6*d*x**2 - 6*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt( 
a)))*a*b**3*c**5*d**2*x**3 + 18*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a 
)))*a*b**3*c**4*d**3*x**4 + 18*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a) 
))*a*b**3*c**3*d**4*x**5 - 3*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a))) 
*b**4*c**6*d*x**4 - 3*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**4*c 
**5*d**2*x**5 - 6*log(a + b*x**2)*a**4*b*c**3*d**4 - 6*log(a + b*x**2)*a** 
4*b*c**2*d**5*x + 6*log(a + b*x**2)*a**3*b**2*c**5*d**2 + 6*log(a + b*x**2 
)*a**3*b**2*c**4*d**3*x - 12*log(a + b*x**2)*a**3*b**2*c**3*d**4*x**2 -...