\(\int \frac {x^4}{(c+d x)^3 (a+b x^2)^3} \, dx\) [269]

Optimal result

Integrand size = 20, antiderivative size = 406 \[ \int \frac {x^4}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=-\frac {c^4 d}{2 \left (b c^2+a d^2\right )^3 (c+d x)^2}-\frac {2 c^3 d \left (b c^2-2 a d^2\right )}{\left (b c^2+a d^2\right )^4 (c+d x)}+\frac {a \left (a d \left (3 b c^2-a d^2\right )+b c \left (b c^2-3 a d^2\right ) x\right )}{4 b \left (b c^2+a d^2\right )^3 \left (a+b x^2\right )^2}-\frac {c \left (24 a c d \left (b^2 c^4-a^2 d^4\right )+\left (b c^2+a d^2\right ) \left (5 b^2 c^4-34 a b c^2 d^2+9 a^2 d^4\right ) x\right )}{8 \left (b c^2+a d^2\right )^5 \left (a+b x^2\right )}+\frac {3 c \left (b^3 c^6-25 a b^2 c^4 d^2+35 a^2 b c^2 d^4-3 a^3 d^6\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 \sqrt {a} \sqrt {b} \left (b c^2+a d^2\right )^5}+\frac {3 c^2 d \left (b^2 c^4-5 a b c^2 d^2+2 a^2 d^4\right ) \log (c+d x)}{\left (b c^2+a d^2\right )^5}-\frac {3 c^2 d \left (b^2 c^4-5 a b c^2 d^2+2 a^2 d^4\right ) \log \left (a+b x^2\right )}{2 \left (b c^2+a d^2\right )^5} \] Output:

-1/2*c^4*d/(a*d^2+b*c^2)^3/(d*x+c)^2-2*c^3*d*(-2*a*d^2+b*c^2)/(a*d^2+b*c^2 
)^4/(d*x+c)+1/4*a*(a*d*(-a*d^2+3*b*c^2)+b*c*(-3*a*d^2+b*c^2)*x)/b/(a*d^2+b 
*c^2)^3/(b*x^2+a)^2-1/8*c*(24*a*c*d*(-a^2*d^4+b^2*c^4)+(a*d^2+b*c^2)*(9*a^ 
2*d^4-34*a*b*c^2*d^2+5*b^2*c^4)*x)/(a*d^2+b*c^2)^5/(b*x^2+a)+3/8*c*(-3*a^3 
*d^6+35*a^2*b*c^2*d^4-25*a*b^2*c^4*d^2+b^3*c^6)*arctan(b^(1/2)*x/a^(1/2))/ 
a^(1/2)/b^(1/2)/(a*d^2+b*c^2)^5+3*c^2*d*(2*a^2*d^4-5*a*b*c^2*d^2+b^2*c^4)* 
ln(d*x+c)/(a*d^2+b*c^2)^5-3/2*c^2*d*(2*a^2*d^4-5*a*b*c^2*d^2+b^2*c^4)*ln(b 
*x^2+a)/(a*d^2+b*c^2)^5
 

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 346, normalized size of antiderivative = 0.85 \[ \int \frac {x^4}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=-\frac {\frac {4 c^4 d \left (b c^2+a d^2\right )^2}{(c+d x)^2}+\frac {16 c^3 d \left (b c^2-2 a d^2\right ) \left (b c^2+a d^2\right )}{c+d x}-\frac {2 a \left (b c^2+a d^2\right )^2 \left (-a^2 d^3+b^2 c^3 x+3 a b c d (c-d x)\right )}{b \left (a+b x^2\right )^2}+\frac {\left (b c^2+a d^2\right ) \left (5 b^2 c^5 x+2 a b c^3 d (12 c-17 d x)+3 a^2 c d^3 (-8 c+3 d x)\right )}{a+b x^2}+\frac {3 c \left (-b^3 c^6+25 a b^2 c^4 d^2-35 a^2 b c^2 d^4+3 a^3 d^6\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}-24 \left (b^2 c^6 d-5 a b c^4 d^3+2 a^2 c^2 d^5\right ) \log (c+d x)+12 \left (b^2 c^6 d-5 a b c^4 d^3+2 a^2 c^2 d^5\right ) \log \left (a+b x^2\right )}{8 \left (b c^2+a d^2\right )^5} \] Input:

Integrate[x^4/((c + d*x)^3*(a + b*x^2)^3),x]
 

Output:

-1/8*((4*c^4*d*(b*c^2 + a*d^2)^2)/(c + d*x)^2 + (16*c^3*d*(b*c^2 - 2*a*d^2 
)*(b*c^2 + a*d^2))/(c + d*x) - (2*a*(b*c^2 + a*d^2)^2*(-(a^2*d^3) + b^2*c^ 
3*x + 3*a*b*c*d*(c - d*x)))/(b*(a + b*x^2)^2) + ((b*c^2 + a*d^2)*(5*b^2*c^ 
5*x + 2*a*b*c^3*d*(12*c - 17*d*x) + 3*a^2*c*d^3*(-8*c + 3*d*x)))/(a + b*x^ 
2) + (3*c*(-(b^3*c^6) + 25*a*b^2*c^4*d^2 - 35*a^2*b*c^2*d^4 + 3*a^3*d^6)*A 
rcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]) - 24*(b^2*c^6*d - 5*a*b*c^4* 
d^3 + 2*a^2*c^2*d^5)*Log[c + d*x] + 12*(b^2*c^6*d - 5*a*b*c^4*d^3 + 2*a^2* 
c^2*d^5)*Log[a + b*x^2])/(b*c^2 + a*d^2)^5
 

Rubi [A] (verified)

Time = 3.79 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {601, 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.

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

\(\Big \downarrow \) 601

\(\displaystyle \frac {a \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{4 b \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^3}-\frac {\int \frac {\frac {a^2 \left (b c^2-3 a d^2\right ) c^4}{\left (b c^2+a d^2\right )^3}-\frac {a^2 d \left (9 b c^2+5 a d^2\right ) x c^3}{\left (b c^2+a d^2\right )^3}-\frac {a \left (4 b^2 c^4+21 a b d^2 c^2-3 a^2 d^4\right ) x^2 c^2}{\left (b c^2+a d^2\right )^3}-\frac {3 a^2 d^3 \left (b c^2-3 a d^2\right ) x^3 c}{\left (b c^2+a d^2\right )^3}}{(c+d x)^3 \left (b x^2+a\right )^2}dx}{4 a}\)

\(\Big \downarrow \) 2178

\(\displaystyle \frac {a \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{4 b \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^3}-\frac {\frac {a c \left (x \left (9 a^2 d^4-34 a b c^2 d^2+5 b^2 c^4\right )+24 a c d \left (b c^2-a d^2\right )\right )}{2 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^4}-\frac {\int \frac {\frac {3 a^2 b \left (b^2 c^4-10 a b d^2 c^2+5 a^2 d^4\right ) c^4}{\left (b c^2+a d^2\right )^4}-\frac {a^2 b d \left (15 b^2 c^4+26 a b d^2 c^2-37 a^2 d^4\right ) x c^3}{\left (b c^2+a d^2\right )^4}-\frac {3 a^2 b d^2 \left (5 b^2 c^4-18 a b d^2 c^2-7 a^2 d^4\right ) x^2 c^2}{\left (b c^2+a d^2\right )^4}-\frac {a^2 b d^3 \left (5 b^2 c^4-34 a b d^2 c^2+9 a^2 d^4\right ) x^3 c}{\left (b c^2+a d^2\right )^4}}{(c+d x)^3 \left (b x^2+a\right )}dx}{2 a b}}{4 a}\)

\(\Big \downarrow \) 2160

\(\displaystyle \frac {a \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{4 b \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^3}-\frac {\frac {a c \left (x \left (9 a^2 d^4-34 a b c^2 d^2+5 b^2 c^4\right )+24 a c d \left (b c^2-a d^2\right )\right )}{2 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^4}-\frac {\int \left (\frac {8 a^2 b d^2 c^4}{\left (b c^2+a d^2\right )^3 (c+d x)^3}-\frac {16 a^2 b d^2 \left (2 a d^2-b c^2\right ) c^3}{\left (b c^2+a d^2\right )^4 (c+d x)^2}+\frac {24 a^2 b d^2 \left (b^2 c^4-5 a b d^2 c^2+2 a^2 d^4\right ) c^2}{\left (b c^2+a d^2\right )^5 (c+d x)}+\frac {3 a^2 b \left (b^3 c^6-25 a b^2 d^2 c^4+35 a^2 b d^4 c^2-8 b d \left (b^2 c^4-5 a b d^2 c^2+2 a^2 d^4\right ) x c-3 a^3 d^6\right ) c}{\left (b c^2+a d^2\right )^5 \left (b x^2+a\right )}\right )dx}{2 a b}}{4 a}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{4 b \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^3}-\frac {\frac {a c \left (x \left (9 a^2 d^4-34 a b c^2 d^2+5 b^2 c^4\right )+24 a c d \left (b c^2-a d^2\right )\right )}{2 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^4}-\frac {-\frac {12 a^2 b c^2 d \left (2 a^2 d^4-5 a b c^2 d^2+b^2 c^4\right ) \log \left (a+b x^2\right )}{\left (a d^2+b c^2\right )^5}+\frac {24 a^2 b c^2 d \left (2 a^2 d^4-5 a b c^2 d^2+b^2 c^4\right ) \log (c+d x)}{\left (a d^2+b c^2\right )^5}-\frac {4 a^2 b c^4 d}{(c+d x)^2 \left (a d^2+b c^2\right )^3}-\frac {16 a^2 b c^3 d \left (b c^2-2 a d^2\right )}{(c+d x) \left (a d^2+b c^2\right )^4}+\frac {3 a^{3/2} \sqrt {b} c \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-3 a^3 d^6+35 a^2 b c^2 d^4-25 a b^2 c^4 d^2+b^3 c^6\right )}{\left (a d^2+b c^2\right )^5}}{2 a b}}{4 a}\)

Input:

Int[x^4/((c + d*x)^3*(a + b*x^2)^3),x]
 

Output:

(a*(a*d*(3*b*c^2 - a*d^2) + b*c*(b*c^2 - 3*a*d^2)*x))/(4*b*(b*c^2 + a*d^2) 
^3*(a + b*x^2)^2) - ((a*c*(24*a*c*d*(b*c^2 - a*d^2) + (5*b^2*c^4 - 34*a*b* 
c^2*d^2 + 9*a^2*d^4)*x))/(2*(b*c^2 + a*d^2)^4*(a + b*x^2)) - ((-4*a^2*b*c^ 
4*d)/((b*c^2 + a*d^2)^3*(c + d*x)^2) - (16*a^2*b*c^3*d*(b*c^2 - 2*a*d^2))/ 
((b*c^2 + a*d^2)^4*(c + d*x)) + (3*a^(3/2)*Sqrt[b]*c*(b^3*c^6 - 25*a*b^2*c 
^4*d^2 + 35*a^2*b*c^2*d^4 - 3*a^3*d^6)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(b*c^2 
 + a*d^2)^5 + (24*a^2*b*c^2*d*(b^2*c^4 - 5*a*b*c^2*d^2 + 2*a^2*d^4)*Log[c 
+ d*x])/(b*c^2 + a*d^2)^5 - (12*a^2*b*c^2*d*(b^2*c^4 - 5*a*b*c^2*d^2 + 2*a 
^2*d^4)*Log[a + b*x^2])/(b*c^2 + a*d^2)^5)/(2*a*b))/(4*a)
 

Defintions of rubi rules used

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 = 0.45 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.03

method	result	size
default	\(-\frac {\frac {\left (\frac {9}{8} d^{6} c \,a^{3} b -\frac {25}{8} a^{2} c^{3} d^{4} b^{2}-\frac {29}{8} a \,c^{5} d^{2} b^{3}+\frac {5}{8} c^{7} b^{4}\right ) x^{3}+\left (-3 a^{3} b \,c^{2} d^{5}+3 a \,b^{3} c^{6} d \right ) x^{2}+\frac {3 a c \left (5 a^{3} d^{6}-5 a^{2} b \,c^{2} d^{4}-9 a \,b^{2} c^{4} d^{2}+b^{3} c^{6}\right ) x}{8}+\frac {a^{2} d \left (a^{3} d^{6}-13 a^{2} b \,c^{2} d^{4}-5 a \,b^{2} c^{4} d^{2}+9 b^{3} c^{6}\right )}{4 b}}{\left (b \,x^{2}+a \right )^{2}}+\frac {3 c \left (\frac {\left (16 b c \,d^{5} a^{2}-40 a \,b^{2} c^{3} d^{3}+8 c^{5} d \,b^{3}\right ) \ln \left (b \,x^{2}+a \right )}{2 b}+\frac {\left (3 a^{3} d^{6}-35 a^{2} b \,c^{2} d^{4}+25 a \,b^{2} c^{4} d^{2}-b^{3} c^{6}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )}{8}}{\left (a \,d^{2}+b \,c^{2}\right )^{5}}-\frac {c^{4} d}{2 \left (a \,d^{2}+b \,c^{2}\right )^{3} \left (d x +c \right )^{2}}+\frac {3 c^{2} d \left (2 a^{2} d^{4}-5 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \ln \left (d x +c \right )}{\left (a \,d^{2}+b \,c^{2}\right )^{5}}+\frac {2 d \,c^{3} \left (2 a \,d^{2}-b \,c^{2}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{4} \left (d x +c \right )}\)	\(420\)
risch	\(\text {Expression too large to display}\)	\(7449\)

Input:

int(x^4/(d*x+c)^3/(b*x^2+a)^3,x,method=_RETURNVERBOSE)
 

Output:

-1/(a*d^2+b*c^2)^5*(((9/8*d^6*c*a^3*b-25/8*a^2*c^3*d^4*b^2-29/8*a*c^5*d^2* 
b^3+5/8*c^7*b^4)*x^3+(-3*a^3*b*c^2*d^5+3*a*b^3*c^6*d)*x^2+3/8*a*c*(5*a^3*d 
^6-5*a^2*b*c^2*d^4-9*a*b^2*c^4*d^2+b^3*c^6)*x+1/4*a^2*d*(a^3*d^6-13*a^2*b* 
c^2*d^4-5*a*b^2*c^4*d^2+9*b^3*c^6)/b)/(b*x^2+a)^2+3/8*c*(1/2*(16*a^2*b*c*d 
^5-40*a*b^2*c^3*d^3+8*b^3*c^5*d)/b*ln(b*x^2+a)+(3*a^3*d^6-35*a^2*b*c^2*d^4 
+25*a*b^2*c^4*d^2-b^3*c^6)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))))-1/2*c^4*d 
/(a*d^2+b*c^2)^3/(d*x+c)^2+3*c^2*d*(2*a^2*d^4-5*a*b*c^2*d^2+b^2*c^4)*ln(d* 
x+c)/(a*d^2+b*c^2)^5+2*d*c^3*(2*a*d^2-b*c^2)/(a*d^2+b*c^2)^4/(d*x+c)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2041 vs. \(2 (388) = 776\).

Time = 10.41 (sec) , antiderivative size = 4104, normalized size of antiderivative = 10.11 \[ \int \frac {x^4}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(x^4/(d*x+c)^3/(b*x^2+a)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^4}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(x**4/(d*x+c)**3/(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. 1092 vs. \(2 (388) = 776\).

Time = 0.14 (sec) , antiderivative size = 1092, normalized size of antiderivative = 2.69 \[ \int \frac {x^4}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

integrate(x^4/(d*x+c)^3/(b*x^2+a)^3,x, algorithm="maxima")
 

Output:

-3/2*(b^2*c^6*d - 5*a*b*c^4*d^3 + 2*a^2*c^2*d^5)*log(b*x^2 + a)/(b^5*c^10 
+ 5*a*b^4*c^8*d^2 + 10*a^2*b^3*c^6*d^4 + 10*a^3*b^2*c^4*d^6 + 5*a^4*b*c^2* 
d^8 + a^5*d^10) + 3*(b^2*c^6*d - 5*a*b*c^4*d^3 + 2*a^2*c^2*d^5)*log(d*x + 
c)/(b^5*c^10 + 5*a*b^4*c^8*d^2 + 10*a^2*b^3*c^6*d^4 + 10*a^3*b^2*c^4*d^6 + 
 5*a^4*b*c^2*d^8 + a^5*d^10) + 3/8*(b^3*c^7 - 25*a*b^2*c^5*d^2 + 35*a^2*b* 
c^3*d^4 - 3*a^3*c*d^6)*arctan(b*x/sqrt(a*b))/((b^5*c^10 + 5*a*b^4*c^8*d^2 
+ 10*a^2*b^3*c^6*d^4 + 10*a^3*b^2*c^4*d^6 + 5*a^4*b*c^2*d^8 + a^5*d^10)*sq 
rt(a*b)) - 1/8*(38*a^2*b^2*c^6*d - 56*a^3*b*c^4*d^3 + 2*a^4*c^2*d^5 + 3*(7 
*b^4*c^5*d^2 - 22*a*b^3*c^3*d^4 + 3*a^2*b^2*c*d^6)*x^5 + 6*(5*b^4*c^6*d - 
12*a*b^3*c^4*d^3 - a^2*b^2*c^2*d^5)*x^4 + (5*b^4*c^7 + 49*a*b^3*c^5*d^2 - 
133*a^2*b^2*c^3*d^4 + 15*a^3*b*c*d^6)*x^3 + 2*(35*a*b^3*c^6*d - 61*a^2*b^2 
*c^4*d^3 + a^3*b*c^2*d^5 + a^4*d^7)*x^2 + (3*a*b^3*c^7 + 22*a^2*b^2*c^5*d^ 
2 - 73*a^3*b*c^3*d^4 + 4*a^4*c*d^6)*x)/(a^2*b^5*c^10 + 4*a^3*b^4*c^8*d^2 + 
 6*a^4*b^3*c^6*d^4 + 4*a^5*b^2*c^4*d^6 + a^6*b*c^2*d^8 + (b^7*c^8*d^2 + 4* 
a*b^6*c^6*d^4 + 6*a^2*b^5*c^4*d^6 + 4*a^3*b^4*c^2*d^8 + a^4*b^3*d^10)*x^6 
+ 2*(b^7*c^9*d + 4*a*b^6*c^7*d^3 + 6*a^2*b^5*c^5*d^5 + 4*a^3*b^4*c^3*d^7 + 
 a^4*b^3*c*d^9)*x^5 + (b^7*c^10 + 6*a*b^6*c^8*d^2 + 14*a^2*b^5*c^6*d^4 + 1 
6*a^3*b^4*c^4*d^6 + 9*a^4*b^3*c^2*d^8 + 2*a^5*b^2*d^10)*x^4 + 4*(a*b^6*c^9 
*d + 4*a^2*b^5*c^7*d^3 + 6*a^3*b^4*c^5*d^5 + 4*a^4*b^3*c^3*d^7 + a^5*b^2*c 
*d^9)*x^3 + (2*a*b^6*c^10 + 9*a^2*b^5*c^8*d^2 + 16*a^3*b^4*c^6*d^4 + 14...
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 713, normalized size of antiderivative = 1.76 \[ \int \frac {x^4}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=-\frac {3 \, {\left (b^{2} c^{6} d - 5 \, a b c^{4} d^{3} + 2 \, a^{2} c^{2} d^{5}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{5} c^{10} + 5 \, a b^{4} c^{8} d^{2} + 10 \, a^{2} b^{3} c^{6} d^{4} + 10 \, a^{3} b^{2} c^{4} d^{6} + 5 \, a^{4} b c^{2} d^{8} + a^{5} d^{10}\right )}} + \frac {3 \, {\left (b^{2} c^{6} d^{2} - 5 \, a b c^{4} d^{4} + 2 \, a^{2} c^{2} d^{6}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{5} c^{10} d + 5 \, a b^{4} c^{8} d^{3} + 10 \, a^{2} b^{3} c^{6} d^{5} + 10 \, a^{3} b^{2} c^{4} d^{7} + 5 \, a^{4} b c^{2} d^{9} + a^{5} d^{11}} + \frac {3 \, {\left (b^{3} c^{7} - 25 \, a b^{2} c^{5} d^{2} + 35 \, a^{2} b c^{3} d^{4} - 3 \, a^{3} c d^{6}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, {\left (b^{5} c^{10} + 5 \, a b^{4} c^{8} d^{2} + 10 \, a^{2} b^{3} c^{6} d^{4} + 10 \, a^{3} b^{2} c^{4} d^{6} + 5 \, a^{4} b c^{2} d^{8} + a^{5} d^{10}\right )} \sqrt {a b}} - \frac {21 \, b^{4} c^{5} d^{2} x^{5} - 66 \, a b^{3} c^{3} d^{4} x^{5} + 9 \, a^{2} b^{2} c d^{6} x^{5} + 30 \, b^{4} c^{6} d x^{4} - 72 \, a b^{3} c^{4} d^{3} x^{4} - 6 \, a^{2} b^{2} c^{2} d^{5} x^{4} + 5 \, b^{4} c^{7} x^{3} + 49 \, a b^{3} c^{5} d^{2} x^{3} - 133 \, a^{2} b^{2} c^{3} d^{4} x^{3} + 15 \, a^{3} b c d^{6} x^{3} + 70 \, a b^{3} c^{6} d x^{2} - 122 \, a^{2} b^{2} c^{4} d^{3} x^{2} + 2 \, a^{3} b c^{2} d^{5} x^{2} + 2 \, a^{4} d^{7} x^{2} + 3 \, a b^{3} c^{7} x + 22 \, a^{2} b^{2} c^{5} d^{2} x - 73 \, a^{3} b c^{3} d^{4} x + 4 \, a^{4} c d^{6} x + 38 \, a^{2} b^{2} c^{6} d - 56 \, a^{3} b c^{4} d^{3} + 2 \, a^{4} c^{2} d^{5}}{8 \, {\left (b^{5} c^{8} + 4 \, a b^{4} c^{6} d^{2} + 6 \, a^{2} b^{3} c^{4} d^{4} + 4 \, a^{3} b^{2} c^{2} d^{6} + a^{4} b d^{8}\right )} {\left (b d x^{3} + b c x^{2} + a d x + a c\right )}^{2}} \] Input:

integrate(x^4/(d*x+c)^3/(b*x^2+a)^3,x, algorithm="giac")
 

Output:

-3/2*(b^2*c^6*d - 5*a*b*c^4*d^3 + 2*a^2*c^2*d^5)*log(b*x^2 + a)/(b^5*c^10 
+ 5*a*b^4*c^8*d^2 + 10*a^2*b^3*c^6*d^4 + 10*a^3*b^2*c^4*d^6 + 5*a^4*b*c^2* 
d^8 + a^5*d^10) + 3*(b^2*c^6*d^2 - 5*a*b*c^4*d^4 + 2*a^2*c^2*d^6)*log(abs( 
d*x + c))/(b^5*c^10*d + 5*a*b^4*c^8*d^3 + 10*a^2*b^3*c^6*d^5 + 10*a^3*b^2* 
c^4*d^7 + 5*a^4*b*c^2*d^9 + a^5*d^11) + 3/8*(b^3*c^7 - 25*a*b^2*c^5*d^2 + 
35*a^2*b*c^3*d^4 - 3*a^3*c*d^6)*arctan(b*x/sqrt(a*b))/((b^5*c^10 + 5*a*b^4 
*c^8*d^2 + 10*a^2*b^3*c^6*d^4 + 10*a^3*b^2*c^4*d^6 + 5*a^4*b*c^2*d^8 + a^5 
*d^10)*sqrt(a*b)) - 1/8*(21*b^4*c^5*d^2*x^5 - 66*a*b^3*c^3*d^4*x^5 + 9*a^2 
*b^2*c*d^6*x^5 + 30*b^4*c^6*d*x^4 - 72*a*b^3*c^4*d^3*x^4 - 6*a^2*b^2*c^2*d 
^5*x^4 + 5*b^4*c^7*x^3 + 49*a*b^3*c^5*d^2*x^3 - 133*a^2*b^2*c^3*d^4*x^3 + 
15*a^3*b*c*d^6*x^3 + 70*a*b^3*c^6*d*x^2 - 122*a^2*b^2*c^4*d^3*x^2 + 2*a^3* 
b*c^2*d^5*x^2 + 2*a^4*d^7*x^2 + 3*a*b^3*c^7*x + 22*a^2*b^2*c^5*d^2*x - 73* 
a^3*b*c^3*d^4*x + 4*a^4*c*d^6*x + 38*a^2*b^2*c^6*d - 56*a^3*b*c^4*d^3 + 2* 
a^4*c^2*d^5)/((b^5*c^8 + 4*a*b^4*c^6*d^2 + 6*a^2*b^3*c^4*d^4 + 4*a^3*b^2*c 
^2*d^6 + a^4*b*d^8)*(b*d*x^3 + b*c*x^2 + a*d*x + a*c)^2)
 

Mupad [B] (verification not implemented)

Time = 8.50 (sec) , antiderivative size = 1652, normalized size of antiderivative = 4.07 \[ \int \frac {x^4}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

int(x^4/((a + b*x^2)^3*(c + d*x)^3),x)
 

Output:

log(c + d*x)*((6*c^2*d)/(a*d^2 + b*c^2)^3 + (24*b^2*c^6*d)/(a*d^2 + b*c^2) 
^5 - (27*b*c^4*d)/(a*d^2 + b*c^2)^4) - ((x^3*(5*b^3*c^7 + 15*a^3*c*d^6 + 4 
9*a*b^2*c^5*d^2 - 133*a^2*b*c^3*d^4))/(8*(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)) - (3*x^4*(12*a*b^2*c^4*d^3 - 5 
*b^3*c^6*d + a^2*b*c^2*d^5))/(4*(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)) + (3*x^5*(7*b^3*c^5*d^2 - 22*a*b^2*c^3* 
d^4 + 3*a^2*b*c*d^6))/(8*(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*(a^4*d^7 + a^3*b*c^2*d^5 - 61*a^2*b^2*c 
^4*d^3 + 35*a*b^3*c^6*d))/(4*b*(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)) + (a*c^2*(a^3*d^5 - 28*a^2*b*c^2*d^3 + 1 
9*a*b^2*c^4*d))/(4*b*(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)) + (a*c*x*(4*a^3*d^6 + 3*b^3*c^6 + 22*a*b^2*c^4*d^2 
 - 73*a^2*b*c^2*d^4))/(8*b*(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*(a^2*d^2 + 2*a*b*c^2) + x^4*(b^2*c^2 + 
 2*a*b*d^2) + a^2*c^2 + b^2*d^2*x^6 + 2*a^2*c*d*x + 2*b^2*c*d*x^5 + 4*a*b* 
c*d*x^3) + (log(9*a^9*d^16*(-a*b)^(1/2) - b^7*c^16*(-a*b)^(3/2) + 3248*a^3 
*c^6*d^10*(-a*b)^(7/2) - 5948*a^5*c^4*d^12*(-a*b)^(5/2) - 2112*a^7*c^2*d^1 
4*(-a*b)^(3/2) + 4012*b^3*c^12*d^4*(-a*b)^(7/2) + 528*b^5*c^14*d^2*(-a*b)^ 
(5/2) + a*b^9*c^16*x + 9*a^9*b*d^16*x + 14686*a*c^8*d^8*(-a*b)^(9/2) + 534 
4*b*c^10*d^6*(-a*b)^(9/2) + 528*a^2*b^8*c^14*d^2*x - 4012*a^3*b^7*c^12*...
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 3009, normalized size of antiderivative = 7.41 \[ \int \frac {x^4}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int(x^4/(d*x+c)^3/(b*x^2+a)^3,x)
 

Output:

( - 18*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**5*c**3*d**6 - 36*s 
qrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**5*c**2*d**7*x - 18*sqrt(b) 
*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**5*c*d**8*x**2 + 210*sqrt(b)*sqrt 
(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*b*c**5*d**4 + 420*sqrt(b)*sqrt(a)*a 
tan((b*x)/(sqrt(b)*sqrt(a)))*a**4*b*c**4*d**5*x + 174*sqrt(b)*sqrt(a)*atan 
((b*x)/(sqrt(b)*sqrt(a)))*a**4*b*c**3*d**6*x**2 - 72*sqrt(b)*sqrt(a)*atan( 
(b*x)/(sqrt(b)*sqrt(a)))*a**4*b*c**2*d**7*x**3 - 36*sqrt(b)*sqrt(a)*atan(( 
b*x)/(sqrt(b)*sqrt(a)))*a**4*b*c*d**8*x**4 - 150*sqrt(b)*sqrt(a)*atan((b*x 
)/(sqrt(b)*sqrt(a)))*a**3*b**2*c**7*d**2 - 300*sqrt(b)*sqrt(a)*atan((b*x)/ 
(sqrt(b)*sqrt(a)))*a**3*b**2*c**6*d**3*x + 270*sqrt(b)*sqrt(a)*atan((b*x)/ 
(sqrt(b)*sqrt(a)))*a**3*b**2*c**5*d**4*x**2 + 840*sqrt(b)*sqrt(a)*atan((b* 
x)/(sqrt(b)*sqrt(a)))*a**3*b**2*c**4*d**5*x**3 + 402*sqrt(b)*sqrt(a)*atan( 
(b*x)/(sqrt(b)*sqrt(a)))*a**3*b**2*c**3*d**6*x**4 - 36*sqrt(b)*sqrt(a)*ata 
n((b*x)/(sqrt(b)*sqrt(a)))*a**3*b**2*c**2*d**7*x**5 - 18*sqrt(b)*sqrt(a)*a 
tan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b**2*c*d**8*x**6 + 6*sqrt(b)*sqrt(a)*ata 
n((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**3*c**9 + 12*sqrt(b)*sqrt(a)*atan((b*x)/ 
(sqrt(b)*sqrt(a)))*a**2*b**3*c**8*d*x - 294*sqrt(b)*sqrt(a)*atan((b*x)/(sq 
rt(b)*sqrt(a)))*a**2*b**3*c**7*d**2*x**2 - 600*sqrt(b)*sqrt(a)*atan((b*x)/ 
(sqrt(b)*sqrt(a)))*a**2*b**3*c**6*d**3*x**3 - 90*sqrt(b)*sqrt(a)*atan((b*x 
)/(sqrt(b)*sqrt(a)))*a**2*b**3*c**5*d**4*x**4 + 420*sqrt(b)*sqrt(a)*ata...