\(\int \frac {x^5}{(c+d x)^3 (a+b x^2)^3} \, dx\) [268]

Optimal result

Integrand size = 20, antiderivative size = 393 \[ \int \frac {x^5}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\frac {c^5}{2 \left (b c^2+a d^2\right )^3 (c+d x)^2}+\frac {c^4 \left (b c^2-5 a d^2\right )}{\left (b c^2+a d^2\right )^4 (c+d x)}-\frac {a^2 \left (c \left (b c^2-3 a d^2\right )-d \left (3 b c^2-a d^2\right ) x\right )}{4 b \left (b c^2+a d^2\right )^3 \left (a+b x^2\right )^2}+\frac {a \left (8 b c^3 \left (b c^2-5 a d^2\right )-d \left (27 b^2 c^4-22 a b c^2 d^2-a^2 d^4\right ) x\right )}{8 b \left (b c^2+a d^2\right )^4 \left (a+b x^2\right )}+\frac {\sqrt {a} d \left (45 b^3 c^6-125 a b^2 c^4 d^2+23 a^2 b c^2 d^4+a^3 d^6\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{3/2} \left (b c^2+a d^2\right )^5}-\frac {c^3 \left (b^2 c^4-13 a b c^2 d^2+10 a^2 d^4\right ) \log (c+d x)}{\left (b c^2+a d^2\right )^5}+\frac {c^3 \left (b^2 c^4-13 a b c^2 d^2+10 a^2 d^4\right ) \log \left (a+b x^2\right )}{2 \left (b c^2+a d^2\right )^5} \] Output:

1/2*c^5/(a*d^2+b*c^2)^3/(d*x+c)^2+c^4*(-5*a*d^2+b*c^2)/(a*d^2+b*c^2)^4/(d* 
x+c)-1/4*a^2*(c*(-3*a*d^2+b*c^2)-d*(-a*d^2+3*b*c^2)*x)/b/(a*d^2+b*c^2)^3/( 
b*x^2+a)^2+1/8*a*(8*b*c^3*(-5*a*d^2+b*c^2)-d*(-a^2*d^4-22*a*b*c^2*d^2+27*b 
^2*c^4)*x)/b/(a*d^2+b*c^2)^4/(b*x^2+a)+1/8*a^(1/2)*d*(a^3*d^6+23*a^2*b*c^2 
*d^4-125*a*b^2*c^4*d^2+45*b^3*c^6)*arctan(b^(1/2)*x/a^(1/2))/b^(3/2)/(a*d^ 
2+b*c^2)^5-c^3*(10*a^2*d^4-13*a*b*c^2*d^2+b^2*c^4)*ln(d*x+c)/(a*d^2+b*c^2) 
^5+1/2*c^3*(10*a^2*d^4-13*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 = 0.50 (sec) , antiderivative size = 340, normalized size of antiderivative = 0.87 \[ \int \frac {x^5}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\frac {\frac {4 c^5 \left (b c^2+a d^2\right )^2}{(c+d x)^2}+\frac {8 \left (b c^2+a d^2\right ) \left (b c^6-5 a c^4 d^2\right )}{c+d x}-\frac {2 a^2 \left (b c^2+a d^2\right )^2 \left (b c^2 (c-3 d x)+a d^2 (-3 c+d x)\right )}{b \left (a+b x^2\right )^2}+\frac {a \left (b c^2+a d^2\right ) \left (a^2 d^5 x+b^2 c^4 (8 c-27 d x)+2 a b c^2 d^2 (-20 c+11 d x)\right )}{b \left (a+b x^2\right )}+\frac {\sqrt {a} d \left (45 b^3 c^6-125 a b^2 c^4 d^2+23 a^2 b c^2 d^4+a^3 d^6\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}}-8 \left (b^2 c^7-13 a b c^5 d^2+10 a^2 c^3 d^4\right ) \log (c+d x)+4 \left (b^2 c^7-13 a b c^5 d^2+10 a^2 c^3 d^4\right ) \log \left (a+b x^2\right )}{8 \left (b c^2+a d^2\right )^5} \] Input:

Integrate[x^5/((c + d*x)^3*(a + b*x^2)^3),x]
 

Output:

((4*c^5*(b*c^2 + a*d^2)^2)/(c + d*x)^2 + (8*(b*c^2 + a*d^2)*(b*c^6 - 5*a*c 
^4*d^2))/(c + d*x) - (2*a^2*(b*c^2 + a*d^2)^2*(b*c^2*(c - 3*d*x) + a*d^2*( 
-3*c + d*x)))/(b*(a + b*x^2)^2) + (a*(b*c^2 + a*d^2)*(a^2*d^5*x + b^2*c^4* 
(8*c - 27*d*x) + 2*a*b*c^2*d^2*(-20*c + 11*d*x)))/(b*(a + b*x^2)) + (Sqrt[ 
a]*d*(45*b^3*c^6 - 125*a*b^2*c^4*d^2 + 23*a^2*b*c^2*d^4 + a^3*d^6)*ArcTan[ 
(Sqrt[b]*x)/Sqrt[a]])/b^(3/2) - 8*(b^2*c^7 - 13*a*b*c^5*d^2 + 10*a^2*c^3*d 
^4)*Log[c + d*x] + 4*(b^2*c^7 - 13*a*b*c^5*d^2 + 10*a^2*c^3*d^4)*Log[a + b 
*x^2])/(8*(b*c^2 + a*d^2)^5)
 

Rubi [A] (verified)

Time = 3.51 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.08, 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.

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

\(\Big \downarrow \) 601

\(\displaystyle -\frac {\int \frac {\frac {a^3 d \left (3 b c^2-a d^2\right ) c^3}{b \left (b c^2+a d^2\right )^3}+\frac {a^2 \left (4 b^2 c^4-3 a b d^2 c^2-3 a^2 d^4\right ) x c^2}{b \left (b c^2+a d^2\right )^3}-\frac {a^3 d^3 \left (23 b c^2+3 a d^2\right ) x^2 c}{b \left (b c^2+a d^2\right )^3}-\frac {a \left (4 b^3 c^6+12 a b^2 d^2 c^4+21 a^2 b d^4 c^2+a^3 d^6\right ) x^3}{b \left (b c^2+a d^2\right )^3}}{(c+d x)^3 \left (b x^2+a\right )^2}dx}{4 a}-\frac {a^2 \left (c \left (b c^2-3 a d^2\right )-d x \left (3 b c^2-a d^2\right )\right )}{4 b \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 2178

\(\displaystyle -\frac {-\frac {\int \frac {-\frac {a^3 \left (27 b^2 c^4-22 a b d^2 c^2-a^2 d^4\right ) x^3 d^4}{\left (b c^2+a d^2\right )^4}-\frac {a^3 c \left (65 b^2 c^4+14 a b d^2 c^2-3 a^2 d^4\right ) x^2 d^3}{\left (b c^2+a d^2\right )^4}+\frac {a^3 c^3 \left (21 b^2 c^4-26 a b d^2 c^2+a^2 d^4\right ) d}{\left (b c^2+a d^2\right )^4}+\frac {a^2 c^2 \left (8 b^3 c^6-a b^2 d^2 c^4-54 a^2 b d^4 c^2+3 a^3 d^6\right ) x}{\left (b c^2+a d^2\right )^4}}{(c+d x)^3 \left (b x^2+a\right )}dx}{2 a b}-\frac {a^2 \left (8 b c^3 \left (b c^2-5 a d^2\right )-d x \left (-a^2 d^4-22 a b c^2 d^2+27 b^2 c^4\right )\right )}{2 b \left (a+b x^2\right ) \left (a d^2+b c^2\right )^4}}{4 a}-\frac {a^2 \left (c \left (b c^2-3 a d^2\right )-d x \left (3 b c^2-a d^2\right )\right )}{4 b \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 2160

\(\displaystyle -\frac {-\frac {\int \left (-\frac {8 a^2 b d c^5}{\left (b c^2+a d^2\right )^3 (c+d x)^3}+\frac {8 a^2 b d \left (5 a d^2-b c^2\right ) c^4}{\left (b c^2+a d^2\right )^4 (c+d x)^2}-\frac {8 a^2 b d \left (b^2 c^4-13 a b d^2 c^2+10 a^2 d^4\right ) c^3}{\left (b c^2+a d^2\right )^5 (c+d x)}+\frac {a^2 \left (8 b^2 \left (b^2 c^4-13 a b d^2 c^2+10 a^2 d^4\right ) x c^3+a d \left (45 b^3 c^6-125 a b^2 d^2 c^4+23 a^2 b d^4 c^2+a^3 d^6\right )\right )}{\left (b c^2+a d^2\right )^5 \left (b x^2+a\right )}\right )dx}{2 a b}-\frac {a^2 \left (8 b c^3 \left (b c^2-5 a d^2\right )-d x \left (-a^2 d^4-22 a b c^2 d^2+27 b^2 c^4\right )\right )}{2 b \left (a+b x^2\right ) \left (a d^2+b c^2\right )^4}}{4 a}-\frac {a^2 \left (c \left (b c^2-3 a d^2\right )-d x \left (3 b c^2-a d^2\right )\right )}{4 b \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {a^2 \left (c \left (b c^2-3 a d^2\right )-d x \left (3 b c^2-a d^2\right )\right )}{4 b \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^3}-\frac {-\frac {a^2 \left (8 b c^3 \left (b c^2-5 a d^2\right )-d x \left (-a^2 d^4-22 a b c^2 d^2+27 b^2 c^4\right )\right )}{2 b \left (a+b x^2\right ) \left (a d^2+b c^2\right )^4}-\frac {\frac {4 a^2 b c^3 \left (10 a^2 d^4-13 a b c^2 d^2+b^2 c^4\right ) \log \left (a+b x^2\right )}{\left (a d^2+b c^2\right )^5}-\frac {8 a^2 b c^3 \left (10 a^2 d^4-13 a b c^2 d^2+b^2 c^4\right ) \log (c+d x)}{\left (a d^2+b c^2\right )^5}+\frac {4 a^2 b c^5}{(c+d x)^2 \left (a d^2+b c^2\right )^3}+\frac {8 a^2 b c^4 \left (b c^2-5 a d^2\right )}{(c+d x) \left (a d^2+b c^2\right )^4}+\frac {a^{5/2} d \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 d^6+23 a^2 b c^2 d^4-125 a b^2 c^4 d^2+45 b^3 c^6\right )}{\sqrt {b} \left (a d^2+b c^2\right )^5}}{2 a b}}{4 a}\)

Input:

Int[x^5/((c + d*x)^3*(a + b*x^2)^3),x]
 

Output:

-1/4*(a^2*(c*(b*c^2 - 3*a*d^2) - d*(3*b*c^2 - a*d^2)*x))/(b*(b*c^2 + a*d^2 
)^3*(a + b*x^2)^2) - (-1/2*(a^2*(8*b*c^3*(b*c^2 - 5*a*d^2) - d*(27*b^2*c^4 
 - 22*a*b*c^2*d^2 - a^2*d^4)*x))/(b*(b*c^2 + a*d^2)^4*(a + b*x^2)) - ((4*a 
^2*b*c^5)/((b*c^2 + a*d^2)^3*(c + d*x)^2) + (8*a^2*b*c^4*(b*c^2 - 5*a*d^2) 
)/((b*c^2 + a*d^2)^4*(c + d*x)) + (a^(5/2)*d*(45*b^3*c^6 - 125*a*b^2*c^4*d 
^2 + 23*a^2*b*c^2*d^4 + a^3*d^6)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*(b* 
c^2 + a*d^2)^5) - (8*a^2*b*c^3*(b^2*c^4 - 13*a*b*c^2*d^2 + 10*a^2*d^4)*Log 
[c + d*x])/(b*c^2 + a*d^2)^5 + (4*a^2*b*c^3*(b^2*c^4 - 13*a*b*c^2*d^2 + 10 
*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 = 0.47 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.12

method	result	size
default	\(\frac {\frac {\left (\frac {1}{8} a^{4} d^{7}+\frac {23}{8} a^{3} b \,c^{2} d^{5}-\frac {5}{8} a^{2} b^{2} c^{4} d^{3}-\frac {27}{8} a \,b^{3} c^{6} d \right ) x^{3}+\left (-5 a^{3} b \,c^{3} d^{4}-4 a^{2} b^{2} c^{5} d^{2}+a \,b^{3} c^{7}\right ) x^{2}-\frac {a^{2} d \left (a^{3} d^{6}-25 a^{2} b \,c^{2} d^{4}-5 a \,b^{2} c^{4} d^{2}+21 b^{3} c^{6}\right ) x}{8 b}+\frac {3 a^{2} c \left (a^{3} d^{6}-5 a^{2} b \,c^{2} d^{4}-5 a \,b^{2} c^{4} d^{2}+b^{3} c^{6}\right )}{4 b}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\frac {\left (80 a^{2} c^{3} d^{4} b^{2}-104 a \,c^{5} d^{2} b^{3}+8 c^{7} b^{4}\right ) \ln \left (b \,x^{2}+a \right )}{2 b}+\frac {\left (a^{4} d^{7}+23 a^{3} b \,c^{2} d^{5}-125 a^{2} b^{2} c^{4} d^{3}+45 a \,b^{3} c^{6} d \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}}{8 b}}{\left (a \,d^{2}+b \,c^{2}\right )^{5}}-\frac {c^{4} \left (5 a \,d^{2}-b \,c^{2}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{4} \left (d x +c \right )}+\frac {c^{5}}{2 \left (a \,d^{2}+b \,c^{2}\right )^{3} \left (d x +c \right )^{2}}-\frac {c^{3} \left (10 a^{2} d^{4}-13 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \ln \left (d x +c \right )}{\left (a \,d^{2}+b \,c^{2}\right )^{5}}\)	\(440\)
risch	\(\text {Expression too large to display}\)	\(8311\)

Input:

int(x^5/(d*x+c)^3/(b*x^2+a)^3,x,method=_RETURNVERBOSE)
 

Output:

1/(a*d^2+b*c^2)^5*(((1/8*a^4*d^7+23/8*a^3*b*c^2*d^5-5/8*a^2*b^2*c^4*d^3-27 
/8*a*b^3*c^6*d)*x^3+(-5*a^3*b*c^3*d^4-4*a^2*b^2*c^5*d^2+a*b^3*c^7)*x^2-1/8 
*a^2*d*(a^3*d^6-25*a^2*b*c^2*d^4-5*a*b^2*c^4*d^2+21*b^3*c^6)/b*x+3/4*a^2*c 
*(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+1/8/b*(1 
/2*(80*a^2*b^2*c^3*d^4-104*a*b^3*c^5*d^2+8*b^4*c^7)/b*ln(b*x^2+a)+(a^4*d^7 
+23*a^3*b*c^2*d^5-125*a^2*b^2*c^4*d^3+45*a*b^3*c^6*d)/(a*b)^(1/2)*arctan(b 
*x/(a*b)^(1/2))))-c^4*(5*a*d^2-b*c^2)/(a*d^2+b*c^2)^4/(d*x+c)+1/2*c^5/(a*d 
^2+b*c^2)^3/(d*x+c)^2-c^3*(10*a^2*d^4-13*a*b*c^2*d^2+b^2*c^4)*ln(d*x+c)/(a 
*d^2+b*c^2)^5
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1980 vs. \(2 (375) = 750\).

Time = 11.27 (sec) , antiderivative size = 3982, normalized size of antiderivative = 10.13 \[ \int \frac {x^5}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(x^5/(d*x+c)^3/(b*x^2+a)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^5}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(x**5/(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. 1102 vs. \(2 (375) = 750\).

Time = 0.15 (sec) , antiderivative size = 1102, normalized size of antiderivative = 2.80 \[ \int \frac {x^5}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

integrate(x^5/(d*x+c)^3/(b*x^2+a)^3,x, algorithm="maxima")
 

Output:

1/2*(b^2*c^7 - 13*a*b*c^5*d^2 + 10*a^2*c^3*d^4)*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) - (b^2*c^7 - 13*a*b*c^5*d^2 + 10*a^2*c^3*d^4)*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) + 1/8*(45*a*b^3*c^6*d - 125*a^2*b^2*c^4*d^3 + 23 
*a^3*b*c^2*d^5 + a^4*d^7)*arctan(b*x/sqrt(a*b))/((b^6*c^10 + 5*a*b^5*c^8*d 
^2 + 10*a^2*b^4*c^6*d^4 + 10*a^3*b^3*c^4*d^6 + 5*a^4*b^2*c^2*d^8 + a^5*b*d 
^10)*sqrt(a*b)) + 1/8*(18*a^2*b^2*c^7 - 72*a^3*b*c^5*d^2 + 6*a^4*c^3*d^4 + 
 (8*b^4*c^6*d - 67*a*b^3*c^4*d^3 + 22*a^2*b^2*c^2*d^5 + a^3*b*d^7)*x^5 + 2 
*(6*b^4*c^7 - 41*a*b^3*c^5*d^2 + 2*a^2*b^2*c^3*d^4 + a^3*b*c*d^6)*x^4 + (5 
*a*b^3*c^6*d - 159*a^2*b^2*c^4*d^3 + 27*a^3*b*c^2*d^5 - a^4*d^7)*x^3 + 4*( 
8*a*b^3*c^7 - 37*a^2*b^2*c^5*d^2 + 4*a^3*b*c^3*d^4 + a^4*c*d^6)*x^2 - (a^2 
*b^2*c^6*d + 86*a^3*b*c^4*d^3 - 11*a^4*c^2*d^5)*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 + 16*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*...
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 728, normalized size of antiderivative = 1.85 \[ \int \frac {x^5}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\frac {{\left (b^{2} c^{7} - 13 \, a b c^{5} d^{2} + 10 \, a^{2} c^{3} d^{4}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{5} c^{10} + 5 \, a b^{4} c^{8} d^{2} + 10 \, a^{2} b^{3} c^{6} d^{4} + 10 \, a^{3} b^{2} c^{4} d^{6} + 5 \, a^{4} b c^{2} d^{8} + a^{5} d^{10}\right )}} - \frac {{\left (b^{2} c^{7} d - 13 \, a b c^{5} d^{3} + 10 \, a^{2} c^{3} d^{5}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{5} c^{10} d + 5 \, a b^{4} c^{8} d^{3} + 10 \, a^{2} b^{3} c^{6} d^{5} + 10 \, a^{3} b^{2} c^{4} d^{7} + 5 \, a^{4} b c^{2} d^{9} + a^{5} d^{11}} + \frac {{\left (45 \, a b^{3} c^{6} d - 125 \, a^{2} b^{2} c^{4} d^{3} + 23 \, a^{3} b c^{2} d^{5} + a^{4} d^{7}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, {\left (b^{6} c^{10} + 5 \, a b^{5} c^{8} d^{2} + 10 \, a^{2} b^{4} c^{6} d^{4} + 10 \, a^{3} b^{3} c^{4} d^{6} + 5 \, a^{4} b^{2} c^{2} d^{8} + a^{5} b d^{10}\right )} \sqrt {a b}} + \frac {8 \, b^{4} c^{6} d x^{5} - 67 \, a b^{3} c^{4} d^{3} x^{5} + 22 \, a^{2} b^{2} c^{2} d^{5} x^{5} + a^{3} b d^{7} x^{5} + 12 \, b^{4} c^{7} x^{4} - 82 \, a b^{3} c^{5} d^{2} x^{4} + 4 \, a^{2} b^{2} c^{3} d^{4} x^{4} + 2 \, a^{3} b c d^{6} x^{4} + 5 \, a b^{3} c^{6} d x^{3} - 159 \, a^{2} b^{2} c^{4} d^{3} x^{3} + 27 \, a^{3} b c^{2} d^{5} x^{3} - a^{4} d^{7} x^{3} + 32 \, a b^{3} c^{7} x^{2} - 148 \, a^{2} b^{2} c^{5} d^{2} x^{2} + 16 \, a^{3} b c^{3} d^{4} x^{2} + 4 \, a^{4} c d^{6} x^{2} - a^{2} b^{2} c^{6} d x - 86 \, a^{3} b c^{4} d^{3} x + 11 \, a^{4} c^{2} d^{5} x + 18 \, a^{2} b^{2} c^{7} - 72 \, a^{3} b c^{5} d^{2} + 6 \, a^{4} c^{3} d^{4}}{8 \, {\left (b^{5} c^{8} + 4 \, a b^{4} c^{6} d^{2} + 6 \, a^{2} b^{3} c^{4} d^{4} + 4 \, a^{3} b^{2} c^{2} d^{6} + a^{4} b d^{8}\right )} {\left (b d x^{3} + b c x^{2} + a d x + a c\right )}^{2}} \] Input:

integrate(x^5/(d*x+c)^3/(b*x^2+a)^3,x, algorithm="giac")
 

Output:

1/2*(b^2*c^7 - 13*a*b*c^5*d^2 + 10*a^2*c^3*d^4)*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) - (b^2*c^7*d - 13*a*b*c^5*d^3 + 10*a^2*c^3*d^5)*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) + 1/8*(45*a*b^3*c^6*d - 125*a^2*b^2*c^4 
*d^3 + 23*a^3*b*c^2*d^5 + a^4*d^7)*arctan(b*x/sqrt(a*b))/((b^6*c^10 + 5*a* 
b^5*c^8*d^2 + 10*a^2*b^4*c^6*d^4 + 10*a^3*b^3*c^4*d^6 + 5*a^4*b^2*c^2*d^8 
+ a^5*b*d^10)*sqrt(a*b)) + 1/8*(8*b^4*c^6*d*x^5 - 67*a*b^3*c^4*d^3*x^5 + 2 
2*a^2*b^2*c^2*d^5*x^5 + a^3*b*d^7*x^5 + 12*b^4*c^7*x^4 - 82*a*b^3*c^5*d^2* 
x^4 + 4*a^2*b^2*c^3*d^4*x^4 + 2*a^3*b*c*d^6*x^4 + 5*a*b^3*c^6*d*x^3 - 159* 
a^2*b^2*c^4*d^3*x^3 + 27*a^3*b*c^2*d^5*x^3 - a^4*d^7*x^3 + 32*a*b^3*c^7*x^ 
2 - 148*a^2*b^2*c^5*d^2*x^2 + 16*a^3*b*c^3*d^4*x^2 + 4*a^4*c*d^6*x^2 - a^2 
*b^2*c^6*d*x - 86*a^3*b*c^4*d^3*x + 11*a^4*c^2*d^5*x + 18*a^2*b^2*c^7 - 72 
*a^3*b*c^5*d^2 + 6*a^4*c^3*d^4)/((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 = 8.52 (sec) , antiderivative size = 1838, normalized size of antiderivative = 4.68 \[ \int \frac {x^5}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

int(x^5/((a + b*x^2)^3*(c + d*x)^3),x)
 

Output:

((x^4*(6*b^3*c^7 + a^3*c*d^6 - 41*a*b^2*c^5*d^2 + 2*a^2*b*c^3*d^4))/(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)) 
+ (x^5*(a^3*d^7 + 8*b^3*c^6*d - 67*a*b^2*c^4*d^3 + 22*a^2*b*c^2*d^5))/(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*(8*a*b^3*c^7 + a^4*c*d^6 + 4*a^3*b*c^3*d^4 - 37*a^2*b^2*c^5*d^2)) 
/(2*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)) - (d*x^3*(a^4*d^6 - 5*a*b^3*c^6 - 27*a^3*b*c^2*d^4 + 159*a^2*b^2* 
c^4*d^2))/(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)) + (3*a*c^2*(3*a*b^2*c^5 + a^3*c*d^4 - 12*a^2*b*c^3*d^2)) 
/(4*b*(a*d^2 + b*c^2)*(a^3*d^6 + b^3*c^6 + 3*a*b^2*c^4*d^2 + 3*a^2*b*c^2*d 
^4)) - (a*c*d*x*(a*b^2*c^5 - 11*a^3*c*d^4 + 86*a^2*b*c^3*d^2))/(8*b*(a*d^2 
 + b*c^2)*(a^3*d^6 + b^3*c^6 + 3*a*b^2*c^4*d^2 + 3*a^2*b*c^2*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(576*b^8*c^18*(-a*b^3) 
^(3/2) + 72288*c^14*d^4*(-a*b^3)^(7/2) + a^10*b^4*d^18*x - a^10*b^2*d^18*( 
-a*b^3)^(1/2) + 27906*a^4*c^8*d^10*(-a*b^3)^(5/2) + 48*a^8*c^2*d^16*(-a*b^ 
3)^(3/2) + 11799*b^4*c^16*d^2*(-a*b^3)^(5/2) + 576*a*b^13*c^18*x - 50212*a 
^2*b^2*c^12*d^6*(-a*b^3)^(5/2) + 52544*a^6*b^2*c^6*d^12*(-a*b^3)^(3/2) - 1 
1799*a^2*b^12*c^16*d^2*x + 72288*a^3*b^11*c^14*d^4*x + 50212*a^4*b^10*c^12 
*d^6*x - 114576*a^5*b^9*c^10*d^8*x - 27906*a^6*b^8*c^8*d^10*x + 52544*a...
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 3042, normalized size of antiderivative = 7.74 \[ \int \frac {x^5}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int(x^5/(d*x+c)^3/(b*x^2+a)^3,x)
 

Output:

(2*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**5*c**3*d**7 + 4*sqrt(b 
)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**5*c**2*d**8*x + 2*sqrt(b)*sqrt( 
a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**5*c*d**9*x**2 + 46*sqrt(b)*sqrt(a)*ata 
n((b*x)/(sqrt(b)*sqrt(a)))*a**4*b*c**5*d**5 + 92*sqrt(b)*sqrt(a)*atan((b*x 
)/(sqrt(b)*sqrt(a)))*a**4*b*c**4*d**6*x + 50*sqrt(b)*sqrt(a)*atan((b*x)/(s 
qrt(b)*sqrt(a)))*a**4*b*c**3*d**7*x**2 + 8*sqrt(b)*sqrt(a)*atan((b*x)/(sqr 
t(b)*sqrt(a)))*a**4*b*c**2*d**8*x**3 + 4*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt( 
b)*sqrt(a)))*a**4*b*c*d**9*x**4 - 250*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)* 
sqrt(a)))*a**3*b**2*c**7*d**3 - 500*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sq 
rt(a)))*a**3*b**2*c**6*d**4*x - 158*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sq 
rt(a)))*a**3*b**2*c**5*d**5*x**2 + 184*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b) 
*sqrt(a)))*a**3*b**2*c**4*d**6*x**3 + 94*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt( 
b)*sqrt(a)))*a**3*b**2*c**3*d**7*x**4 + 4*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt 
(b)*sqrt(a)))*a**3*b**2*c**2*d**8*x**5 + 2*sqrt(b)*sqrt(a)*atan((b*x)/(sqr 
t(b)*sqrt(a)))*a**3*b**2*c*d**9*x**6 + 90*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt 
(b)*sqrt(a)))*a**2*b**3*c**9*d + 180*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*s 
qrt(a)))*a**2*b**3*c**8*d**2*x - 410*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*s 
qrt(a)))*a**2*b**3*c**7*d**3*x**2 - 1000*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt( 
b)*sqrt(a)))*a**2*b**3*c**6*d**4*x**3 - 454*sqrt(b)*sqrt(a)*atan((b*x)/(sq 
rt(b)*sqrt(a)))*a**2*b**3*c**5*d**5*x**4 + 92*sqrt(b)*sqrt(a)*atan((b*x...