\(\int \frac {x^6}{(c+d x)^3 (a+b x^2)^2} \, dx\) [209]

Optimal result

Integrand size = 20, antiderivative size = 315 \[ \int \frac {x^6}{(c+d x)^3 \left (a+b x^2\right )^2} \, dx=-\frac {c^6}{2 d^3 \left (b c^2+a d^2\right )^2 (c+d x)^2}+\frac {2 c^5 \left (b c^2+3 a d^2\right )}{d^3 \left (b c^2+a d^2\right )^3 (c+d x)}-\frac {a^2 \left (a d \left (3 b c^2-a d^2\right )+b c \left (b c^2-3 a d^2\right ) x\right )}{2 b^2 \left (b c^2+a d^2\right )^3 \left (a+b x^2\right )}+\frac {a^{3/2} c \left (5 b^2 c^4-22 a b c^2 d^2-3 a^2 d^4\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{3/2} \left (b c^2+a d^2\right )^4}+\frac {c^4 \left (b^2 c^4+4 a b c^2 d^2+15 a^2 d^4\right ) \log (c+d x)}{d^3 \left (b c^2+a d^2\right )^4}-\frac {a^2 d \left (9 b^2 c^4-4 a b c^2 d^2-a^2 d^4\right ) \log \left (a+b x^2\right )}{2 b^2 \left (b c^2+a d^2\right )^4} \] Output:

-1/2*c^6/d^3/(a*d^2+b*c^2)^2/(d*x+c)^2+2*c^5*(3*a*d^2+b*c^2)/d^3/(a*d^2+b* 
c^2)^3/(d*x+c)-1/2*a^2*(a*d*(-a*d^2+3*b*c^2)+b*c*(-3*a*d^2+b*c^2)*x)/b^2/( 
a*d^2+b*c^2)^3/(b*x^2+a)+1/2*a^(3/2)*c*(-3*a^2*d^4-22*a*b*c^2*d^2+5*b^2*c^ 
4)*arctan(b^(1/2)*x/a^(1/2))/b^(3/2)/(a*d^2+b*c^2)^4+c^4*(15*a^2*d^4+4*a*b 
*c^2*d^2+b^2*c^4)*ln(d*x+c)/d^3/(a*d^2+b*c^2)^4-1/2*a^2*d*(-a^2*d^4-4*a*b* 
c^2*d^2+9*b^2*c^4)*ln(b*x^2+a)/b^2/(a*d^2+b*c^2)^4
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.87 \[ \int \frac {x^6}{(c+d x)^3 \left (a+b x^2\right )^2} \, dx=\frac {-\frac {c^6 \left (b c^2+a d^2\right )^2}{d^3 (c+d x)^2}+\frac {4 c^5 \left (b c^2+a d^2\right ) \left (b c^2+3 a d^2\right )}{d^3 (c+d x)}+\frac {a^2 \left (b c^2+a d^2\right ) \left (a^2 d^3-b^2 c^3 x-3 a b c d (c-d x)\right )}{b^2 \left (a+b x^2\right )}-\frac {a^{3/2} c \left (-5 b^2 c^4+22 a b c^2 d^2+3 a^2 d^4\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}}+\frac {2 \left (b^2 c^8+4 a b c^6 d^2+15 a^2 c^4 d^4\right ) \log (c+d x)}{d^3}+\frac {a^2 d \left (-9 b^2 c^4+4 a b c^2 d^2+a^2 d^4\right ) \log \left (a+b x^2\right )}{b^2}}{2 \left (b c^2+a d^2\right )^4} \] Input:

Integrate[x^6/((c + d*x)^3*(a + b*x^2)^2),x]
 

Output:

(-((c^6*(b*c^2 + a*d^2)^2)/(d^3*(c + d*x)^2)) + (4*c^5*(b*c^2 + a*d^2)*(b* 
c^2 + 3*a*d^2))/(d^3*(c + d*x)) + (a^2*(b*c^2 + a*d^2)*(a^2*d^3 - b^2*c^3* 
x - 3*a*b*c*d*(c - d*x)))/(b^2*(a + b*x^2)) - (a^(3/2)*c*(-5*b^2*c^4 + 22* 
a*b*c^2*d^2 + 3*a^2*d^4)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(3/2) + (2*(b^2*c^ 
8 + 4*a*b*c^6*d^2 + 15*a^2*c^4*d^4)*Log[c + d*x])/d^3 + (a^2*d*(-9*b^2*c^4 
 + 4*a*b*c^2*d^2 + a^2*d^4)*Log[a + b*x^2])/b^2)/(2*(b*c^2 + a*d^2)^4)
 

Rubi [A] (verified)

Time = 2.65 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {601, 25, 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.

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

\(\Big \downarrow \) 601

\(\displaystyle -\frac {\int -\frac {\frac {a^3 \left (b c^2-3 a d^2\right ) c^4}{b \left (b c^2+a d^2\right )^3}-\frac {a^3 d \left (3 b c^2+7 a d^2\right ) x c^3}{b \left (b c^2+a d^2\right )^3}-\frac {a^2 \left (2 b^2 c^4+9 a b d^2 c^2+3 a^2 d^4\right ) x^2 c^2}{b \left (b c^2+a d^2\right )^3}-\frac {a^3 d^3 \left (b c^2-3 a d^2\right ) x^3 c}{b \left (b c^2+a d^2\right )^3}+\frac {2 a x^4}{b}}{(c+d x)^3 \left (b x^2+a\right )}dx}{2 a}-\frac {a^2 \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{2 b^2 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\frac {a^3 \left (b c^2-3 a d^2\right ) c^4}{b \left (b c^2+a d^2\right )^3}-\frac {a^3 d \left (3 b c^2+7 a d^2\right ) x c^3}{b \left (b c^2+a d^2\right )^3}-\frac {a^2 \left (2 b^2 c^4+9 a b d^2 c^2+3 a^2 d^4\right ) x^2 c^2}{b \left (b c^2+a d^2\right )^3}-\frac {a^3 d^3 \left (b c^2-3 a d^2\right ) x^3 c}{b \left (b c^2+a d^2\right )^3}+\frac {2 a x^4}{b}}{(c+d x)^3 \left (b x^2+a\right )}dx}{2 a}-\frac {a^2 \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{2 b^2 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 2160

\(\displaystyle \frac {\int \left (\frac {2 a c^6}{d^2 \left (b c^2+a d^2\right )^2 (c+d x)^3}-\frac {4 a \left (b c^2+3 a d^2\right ) c^5}{d^2 \left (b c^2+a d^2\right )^3 (c+d x)^2}+\frac {2 a \left (b^2 c^4+4 a b d^2 c^2+15 a^2 d^4\right ) c^4}{d^2 \left (b c^2+a d^2\right )^4 (c+d x)}+\frac {a^3 \left (c \left (5 b^2 c^4-22 a b d^2 c^2-3 a^2 d^4\right )-2 d \left (9 b^2 c^4-4 a b d^2 c^2-a^2 d^4\right ) x\right )}{b \left (b c^2+a d^2\right )^4 \left (b x^2+a\right )}\right )dx}{2 a}-\frac {a^2 \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{2 b^2 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {2 a c^4 \left (15 a^2 d^4+4 a b c^2 d^2+b^2 c^4\right ) \log (c+d x)}{d^3 \left (a d^2+b c^2\right )^4}+\frac {a^{5/2} c \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-3 a^2 d^4-22 a b c^2 d^2+5 b^2 c^4\right )}{b^{3/2} \left (a d^2+b c^2\right )^4}-\frac {a^3 d \left (-a^2 d^4-4 a b c^2 d^2+9 b^2 c^4\right ) \log \left (a+b x^2\right )}{b^2 \left (a d^2+b c^2\right )^4}-\frac {a c^6}{d^3 (c+d x)^2 \left (a d^2+b c^2\right )^2}+\frac {4 a c^5 \left (3 a d^2+b c^2\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )^3}}{2 a}-\frac {a^2 \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{2 b^2 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^3}\)

Input:

Int[x^6/((c + d*x)^3*(a + b*x^2)^2),x]
 

Output:

-1/2*(a^2*(a*d*(3*b*c^2 - a*d^2) + b*c*(b*c^2 - 3*a*d^2)*x))/(b^2*(b*c^2 + 
 a*d^2)^3*(a + b*x^2)) + (-((a*c^6)/(d^3*(b*c^2 + a*d^2)^2*(c + d*x)^2)) + 
 (4*a*c^5*(b*c^2 + 3*a*d^2))/(d^3*(b*c^2 + a*d^2)^3*(c + d*x)) + (a^(5/2)* 
c*(5*b^2*c^4 - 22*a*b*c^2*d^2 - 3*a^2*d^4)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(b 
^(3/2)*(b*c^2 + a*d^2)^4) + (2*a*c^4*(b^2*c^4 + 4*a*b*c^2*d^2 + 15*a^2*d^4 
)*Log[c + d*x])/(d^3*(b*c^2 + a*d^2)^4) - (a^3*d*(9*b^2*c^4 - 4*a*b*c^2*d^ 
2 - a^2*d^4)*Log[a + b*x^2])/(b^2*(b*c^2 + a*d^2)^4))/(2*a)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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]
 

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 309, normalized size of antiderivative = 0.98

method	result	size
default	\(-\frac {a^{2} \left (\frac {-\frac {c \left (3 a^{2} d^{4}+2 b \,c^{2} d^{2} a -b^{2} c^{4}\right ) x}{2 b}-\frac {a d \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a -3 b^{2} c^{4}\right )}{2 b^{2}}}{b \,x^{2}+a}+\frac {\frac {\left (-2 a^{2} d^{5}-8 d^{3} a \,c^{2} b +18 b^{2} c^{4} d \right ) \ln \left (b \,x^{2}+a \right )}{2 b}+\frac {\left (3 a^{2} c \,d^{4}+22 a \,c^{3} d^{2} b -5 c^{5} b^{2}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}}{2 b}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{4}}-\frac {c^{6}}{2 d^{3} \left (a \,d^{2}+b \,c^{2}\right )^{2} \left (d x +c \right )^{2}}+\frac {c^{4} \left (15 a^{2} d^{4}+4 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \ln \left (d x +c \right )}{d^{3} \left (a \,d^{2}+b \,c^{2}\right )^{4}}+\frac {2 c^{5} \left (3 a \,d^{2}+b \,c^{2}\right )}{d^{3} \left (a \,d^{2}+b \,c^{2}\right )^{3} \left (d x +c \right )}\)	\(309\)
risch	\(\text {Expression too large to display}\)	\(1306\)

Input:

int(x^6/(d*x+c)^3/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
 

Output:

-a^2/(a*d^2+b*c^2)^4*((-1/2*c*(3*a^2*d^4+2*a*b*c^2*d^2-b^2*c^4)/b*x-1/2*a* 
d*(a^2*d^4-2*a*b*c^2*d^2-3*b^2*c^4)/b^2)/(b*x^2+a)+1/2/b*(1/2*(-2*a^2*d^5- 
8*a*b*c^2*d^3+18*b^2*c^4*d)/b*ln(b*x^2+a)+(3*a^2*c*d^4+22*a*b*c^3*d^2-5*b^ 
2*c^5)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))))-1/2*c^6/d^3/(a*d^2+b*c^2)^2/( 
d*x+c)^2+c^4*(15*a^2*d^4+4*a*b*c^2*d^2+b^2*c^4)*ln(d*x+c)/d^3/(a*d^2+b*c^2 
)^4+2*c^5*(3*a*d^2+b*c^2)/d^3/(a*d^2+b*c^2)^3/(d*x+c)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1303 vs. \(2 (299) = 598\).

Time = 9.05 (sec) , antiderivative size = 2631, normalized size of antiderivative = 8.35 \[ \int \frac {x^6}{(c+d x)^3 \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(x^6/(d*x+c)^3/(b*x^2+a)^2,x, algorithm="fricas")
 

Output:

[1/4*(6*a*b^4*c^10 + 28*a^2*b^3*c^8*d^2 + 16*a^3*b^2*c^6*d^4 - 4*a^4*b*c^4 
*d^6 + 2*a^5*c^2*d^8 + 2*(4*b^5*c^9*d + 16*a*b^4*c^7*d^3 + 11*a^2*b^3*c^5* 
d^5 + 2*a^3*b^2*c^3*d^7 + 3*a^4*b*c*d^9)*x^3 + 2*(3*b^5*c^10 + 14*a*b^4*c^ 
8*d^2 + 9*a^2*b^3*c^6*d^4 + a^3*b^2*c^4*d^6 + 4*a^4*b*c^2*d^8 + a^5*d^10)* 
x^2 - (5*a^2*b^3*c^7*d^3 - 22*a^3*b^2*c^5*d^5 - 3*a^4*b*c^3*d^7 + (5*a*b^4 
*c^5*d^5 - 22*a^2*b^3*c^3*d^7 - 3*a^3*b^2*c*d^9)*x^4 + 2*(5*a*b^4*c^6*d^4 
- 22*a^2*b^3*c^4*d^6 - 3*a^3*b^2*c^2*d^8)*x^3 + (5*a*b^4*c^7*d^3 - 17*a^2* 
b^3*c^5*d^5 - 25*a^3*b^2*c^3*d^7 - 3*a^4*b*c*d^9)*x^2 + 2*(5*a^2*b^3*c^6*d 
^4 - 22*a^3*b^2*c^4*d^6 - 3*a^4*b*c^2*d^8)*x)*sqrt(-a/b)*log((b*x^2 - 2*b* 
x*sqrt(-a/b) - a)/(b*x^2 + a)) + 2*(4*a*b^4*c^9*d + 15*a^2*b^3*c^7*d^3 + 8 
*a^3*b^2*c^5*d^5 - a^4*b*c^3*d^7 + 2*a^5*c*d^9)*x - 2*(9*a^3*b^2*c^6*d^4 - 
 4*a^4*b*c^4*d^6 - a^5*c^2*d^8 + (9*a^2*b^3*c^4*d^6 - 4*a^3*b^2*c^2*d^8 - 
a^4*b*d^10)*x^4 + 2*(9*a^2*b^3*c^5*d^5 - 4*a^3*b^2*c^3*d^7 - a^4*b*c*d^9)* 
x^3 + (9*a^2*b^3*c^6*d^4 + 5*a^3*b^2*c^4*d^6 - 5*a^4*b*c^2*d^8 - a^5*d^10) 
*x^2 + 2*(9*a^3*b^2*c^5*d^5 - 4*a^4*b*c^3*d^7 - a^5*c*d^9)*x)*log(b*x^2 + 
a) + 4*(a*b^4*c^10 + 4*a^2*b^3*c^8*d^2 + 15*a^3*b^2*c^6*d^4 + (b^5*c^8*d^2 
 + 4*a*b^4*c^6*d^4 + 15*a^2*b^3*c^4*d^6)*x^4 + 2*(b^5*c^9*d + 4*a*b^4*c^7* 
d^3 + 15*a^2*b^3*c^5*d^5)*x^3 + (b^5*c^10 + 5*a*b^4*c^8*d^2 + 19*a^2*b^3*c 
^6*d^4 + 15*a^3*b^2*c^4*d^6)*x^2 + 2*(a*b^4*c^9*d + 4*a^2*b^3*c^7*d^3 + 15 
*a^3*b^2*c^5*d^5)*x)*log(d*x + c))/(a*b^6*c^10*d^3 + 4*a^2*b^5*c^8*d^5 ...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^6}{(c+d x)^3 \left (a+b x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate(x**6/(d*x+c)**3/(b*x**2+a)**2,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 787 vs. \(2 (299) = 598\).

Time = 0.13 (sec) , antiderivative size = 787, normalized size of antiderivative = 2.50 \[ \int \frac {x^6}{(c+d x)^3 \left (a+b x^2\right )^2} \, dx=-\frac {{\left (9 \, a^{2} b^{2} c^{4} d - 4 \, a^{3} b c^{2} d^{3} - a^{4} d^{5}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{6} c^{8} + 4 \, a b^{5} c^{6} d^{2} + 6 \, a^{2} b^{4} c^{4} d^{4} + 4 \, a^{3} b^{3} c^{2} d^{6} + a^{4} b^{2} d^{8}\right )}} + \frac {{\left (b^{2} c^{8} + 4 \, a b c^{6} d^{2} + 15 \, a^{2} c^{4} d^{4}\right )} \log \left (d x + c\right )}{b^{4} c^{8} d^{3} + 4 \, a b^{3} c^{6} d^{5} + 6 \, a^{2} b^{2} c^{4} d^{7} + 4 \, a^{3} b c^{2} d^{9} + a^{4} d^{11}} + \frac {{\left (5 \, a^{2} b^{2} c^{5} - 22 \, a^{3} b c^{3} d^{2} - 3 \, a^{4} c d^{4}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\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 )} \sqrt {a b}} + \frac {3 \, a b^{3} c^{8} + 11 \, a^{2} b^{2} c^{6} d^{2} - 3 \, a^{3} b c^{4} d^{4} + a^{4} c^{2} d^{6} + {\left (4 \, b^{4} c^{7} d + 12 \, a b^{3} c^{5} d^{3} - a^{2} b^{2} c^{3} d^{5} + 3 \, a^{3} b c d^{7}\right )} x^{3} + {\left (3 \, b^{4} c^{8} + 11 \, a b^{3} c^{6} d^{2} - 2 \, a^{2} b^{2} c^{4} d^{4} + 3 \, a^{3} b c^{2} d^{6} + a^{4} d^{8}\right )} x^{2} + {\left (4 \, a b^{3} c^{7} d + 11 \, a^{2} b^{2} c^{5} d^{3} - 3 \, a^{3} b c^{3} d^{5} + 2 \, a^{4} c d^{7}\right )} x}{2 \, {\left (a b^{5} c^{8} d^{3} + 3 \, a^{2} b^{4} c^{6} d^{5} + 3 \, a^{3} b^{3} c^{4} d^{7} + a^{4} b^{2} c^{2} d^{9} + {\left (b^{6} c^{6} d^{5} + 3 \, a b^{5} c^{4} d^{7} + 3 \, a^{2} b^{4} c^{2} d^{9} + a^{3} b^{3} d^{11}\right )} x^{4} + 2 \, {\left (b^{6} c^{7} d^{4} + 3 \, a b^{5} c^{5} d^{6} + 3 \, a^{2} b^{4} c^{3} d^{8} + a^{3} b^{3} c d^{10}\right )} x^{3} + {\left (b^{6} c^{8} d^{3} + 4 \, a b^{5} c^{6} d^{5} + 6 \, a^{2} b^{4} c^{4} d^{7} + 4 \, a^{3} b^{3} c^{2} d^{9} + a^{4} b^{2} d^{11}\right )} x^{2} + 2 \, {\left (a b^{5} c^{7} d^{4} + 3 \, a^{2} b^{4} c^{5} d^{6} + 3 \, a^{3} b^{3} c^{3} d^{8} + a^{4} b^{2} c d^{10}\right )} x\right )}} \] Input:

integrate(x^6/(d*x+c)^3/(b*x^2+a)^2,x, algorithm="maxima")
 

Output:

-1/2*(9*a^2*b^2*c^4*d - 4*a^3*b*c^2*d^3 - a^4*d^5)*log(b*x^2 + a)/(b^6*c^8 
 + 4*a*b^5*c^6*d^2 + 6*a^2*b^4*c^4*d^4 + 4*a^3*b^3*c^2*d^6 + a^4*b^2*d^8) 
+ (b^2*c^8 + 4*a*b*c^6*d^2 + 15*a^2*c^4*d^4)*log(d*x + c)/(b^4*c^8*d^3 + 4 
*a*b^3*c^6*d^5 + 6*a^2*b^2*c^4*d^7 + 4*a^3*b*c^2*d^9 + a^4*d^11) + 1/2*(5* 
a^2*b^2*c^5 - 22*a^3*b*c^3*d^2 - 3*a^4*c*d^4)*arctan(b*x/sqrt(a*b))/((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) 
*sqrt(a*b)) + 1/2*(3*a*b^3*c^8 + 11*a^2*b^2*c^6*d^2 - 3*a^3*b*c^4*d^4 + a^ 
4*c^2*d^6 + (4*b^4*c^7*d + 12*a*b^3*c^5*d^3 - a^2*b^2*c^3*d^5 + 3*a^3*b*c* 
d^7)*x^3 + (3*b^4*c^8 + 11*a*b^3*c^6*d^2 - 2*a^2*b^2*c^4*d^4 + 3*a^3*b*c^2 
*d^6 + a^4*d^8)*x^2 + (4*a*b^3*c^7*d + 11*a^2*b^2*c^5*d^3 - 3*a^3*b*c^3*d^ 
5 + 2*a^4*c*d^7)*x)/(a*b^5*c^8*d^3 + 3*a^2*b^4*c^6*d^5 + 3*a^3*b^3*c^4*d^7 
 + a^4*b^2*c^2*d^9 + (b^6*c^6*d^5 + 3*a*b^5*c^4*d^7 + 3*a^2*b^4*c^2*d^9 + 
a^3*b^3*d^11)*x^4 + 2*(b^6*c^7*d^4 + 3*a*b^5*c^5*d^6 + 3*a^2*b^4*c^3*d^8 + 
 a^3*b^3*c*d^10)*x^3 + (b^6*c^8*d^3 + 4*a*b^5*c^6*d^5 + 6*a^2*b^4*c^4*d^7 
+ 4*a^3*b^3*c^2*d^9 + a^4*b^2*d^11)*x^2 + 2*(a*b^5*c^7*d^4 + 3*a^2*b^4*c^5 
*d^6 + 3*a^3*b^3*c^3*d^8 + a^4*b^2*c*d^10)*x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 612 vs. \(2 (299) = 598\).

Time = 0.13 (sec) , antiderivative size = 612, normalized size of antiderivative = 1.94 \[ \int \frac {x^6}{(c+d x)^3 \left (a+b x^2\right )^2} \, dx=-\frac {{\left (9 \, a^{2} b^{2} c^{4} d - 4 \, a^{3} b c^{2} d^{3} - a^{4} d^{5}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{6} c^{8} + 4 \, a b^{5} c^{6} d^{2} + 6 \, a^{2} b^{4} c^{4} d^{4} + 4 \, a^{3} b^{3} c^{2} d^{6} + a^{4} b^{2} d^{8}\right )}} + \frac {{\left (b^{2} c^{8} + 4 \, a b c^{6} d^{2} + 15 \, a^{2} c^{4} d^{4}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{4} c^{8} d^{3} + 4 \, a b^{3} c^{6} d^{5} + 6 \, a^{2} b^{2} c^{4} d^{7} + 4 \, a^{3} b c^{2} d^{9} + a^{4} d^{11}} + \frac {{\left (5 \, a^{2} b^{2} c^{5} - 22 \, a^{3} b c^{3} d^{2} - 3 \, a^{4} c d^{4}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (b^{5} c^{8} + 4 \, a b^{4} c^{6} d^{2} + 6 \, a^{2} b^{3} c^{4} d^{4} + 4 \, a^{3} b^{2} c^{2} d^{6} + a^{4} b d^{8}\right )} \sqrt {a b}} + \frac {{\left (4 \, b^{4} c^{9} + 16 \, a b^{3} c^{7} d^{2} + 11 \, a^{2} b^{2} c^{5} d^{4} + 2 \, a^{3} b c^{3} d^{6} + 3 \, a^{4} c d^{8}\right )} x^{3} + \frac {{\left (4 \, a b^{4} c^{9} + 15 \, a^{2} b^{3} c^{7} d^{2} + 8 \, a^{3} b^{2} c^{5} d^{4} - a^{4} b c^{3} d^{6} + 2 \, a^{5} c d^{8}\right )} x}{b} + \frac {{\left (3 \, b^{5} c^{10} + 14 \, a b^{4} c^{8} d^{2} + 9 \, a^{2} b^{3} c^{6} d^{4} + a^{3} b^{2} c^{4} d^{6} + 4 \, a^{4} b c^{2} d^{8} + a^{5} d^{10}\right )} x^{2}}{b d} + \frac {3 \, a b^{4} c^{10} + 14 \, a^{2} b^{3} c^{8} d^{2} + 8 \, a^{3} b^{2} c^{6} d^{4} - 2 \, a^{4} b c^{4} d^{6} + a^{5} c^{2} d^{8}}{b d}}{2 \, {\left (b c^{2} + a d^{2}\right )}^{4} {\left (b x^{2} + a\right )} {\left (d x + c\right )}^{2} b d^{2}} \] Input:

integrate(x^6/(d*x+c)^3/(b*x^2+a)^2,x, algorithm="giac")
 

Output:

-1/2*(9*a^2*b^2*c^4*d - 4*a^3*b*c^2*d^3 - a^4*d^5)*log(b*x^2 + a)/(b^6*c^8 
 + 4*a*b^5*c^6*d^2 + 6*a^2*b^4*c^4*d^4 + 4*a^3*b^3*c^2*d^6 + a^4*b^2*d^8) 
+ (b^2*c^8 + 4*a*b*c^6*d^2 + 15*a^2*c^4*d^4)*log(abs(d*x + c))/(b^4*c^8*d^ 
3 + 4*a*b^3*c^6*d^5 + 6*a^2*b^2*c^4*d^7 + 4*a^3*b*c^2*d^9 + a^4*d^11) + 1/ 
2*(5*a^2*b^2*c^5 - 22*a^3*b*c^3*d^2 - 3*a^4*c*d^4)*arctan(b*x/sqrt(a*b))/( 
(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)*sqrt(a*b)) + 1/2*((4*b^4*c^9 + 16*a*b^3*c^7*d^2 + 11*a^2*b^2*c^5*d^4 
 + 2*a^3*b*c^3*d^6 + 3*a^4*c*d^8)*x^3 + (4*a*b^4*c^9 + 15*a^2*b^3*c^7*d^2 
+ 8*a^3*b^2*c^5*d^4 - a^4*b*c^3*d^6 + 2*a^5*c*d^8)*x/b + (3*b^5*c^10 + 14* 
a*b^4*c^8*d^2 + 9*a^2*b^3*c^6*d^4 + a^3*b^2*c^4*d^6 + 4*a^4*b*c^2*d^8 + a^ 
5*d^10)*x^2/(b*d) + (3*a*b^4*c^10 + 14*a^2*b^3*c^8*d^2 + 8*a^3*b^2*c^6*d^4 
 - 2*a^4*b*c^4*d^6 + a^5*c^2*d^8)/(b*d))/((b*c^2 + a*d^2)^4*(b*x^2 + a)*(d 
*x + c)^2*b*d^2)
 

Mupad [B] (verification not implemented)

Time = 7.50 (sec) , antiderivative size = 834, normalized size of antiderivative = 2.65 \[ \int \frac {x^6}{(c+d x)^3 \left (a+b x^2\right )^2} \, dx=\frac {\frac {x^2\,\left (a^4\,d^8+3\,a^3\,b\,c^2\,d^6-2\,a^2\,b^2\,c^4\,d^4+11\,a\,b^3\,c^6\,d^2+3\,b^4\,c^8\right )}{2\,b^2\,d^3\,\left (a^3\,d^6+3\,a^2\,b\,c^2\,d^4+3\,a\,b^2\,c^4\,d^2+b^3\,c^6\right )}+\frac {c\,x^3\,\left (3\,a^3\,d^6-a^2\,b\,c^2\,d^4+12\,a\,b^2\,c^4\,d^2+4\,b^3\,c^6\right )}{2\,b\,d^2\,\left (a^3\,d^6+3\,a^2\,b\,c^2\,d^4+3\,a\,b^2\,c^4\,d^2+b^3\,c^6\right )}+\frac {a\,c^2\,\left (a^3\,d^6-3\,a^2\,b\,c^2\,d^4+11\,a\,b^2\,c^4\,d^2+3\,b^3\,c^6\right )}{2\,b^2\,d^3\,\left (b\,c^2+a\,d^2\right )\,\left (a^2\,d^4+2\,a\,b\,c^2\,d^2+b^2\,c^4\right )}+\frac {a\,c\,x\,\left (2\,a^3\,d^6-3\,a^2\,b\,c^2\,d^4+11\,a\,b^2\,c^4\,d^2+4\,b^3\,c^6\right )}{2\,b^2\,d^2\,\left (b\,c^2+a\,d^2\right )\,\left (a^2\,d^4+2\,a\,b\,c^2\,d^2+b^2\,c^4\right )}}{a\,c^2+x^2\,\left (b\,c^2+a\,d^2\right )+b\,d^2\,x^4+2\,b\,c\,d\,x^3+2\,a\,c\,d\,x}-\frac {\ln \left (\sqrt {-a^3\,b^5}+a\,b^3\,x\right )\,\left (\frac {9\,a^2\,b^4\,c^4\,d}{2}-b^2\,\left (\frac {5\,c^5\,\sqrt {-a^3\,b^5}}{4}+\frac {a^4\,d^5}{2}\right )-2\,a^3\,b^3\,c^2\,d^3+\frac {3\,a^2\,c\,d^4\,\sqrt {-a^3\,b^5}}{4}+\frac {11\,a\,b\,c^3\,d^2\,\sqrt {-a^3\,b^5}}{2}\right )}{a^4\,b^4\,d^8+4\,a^3\,b^5\,c^2\,d^6+6\,a^2\,b^6\,c^4\,d^4+4\,a\,b^7\,c^6\,d^2+b^8\,c^8}+\frac {\ln \left (\sqrt {-a^3\,b^5}-a\,b^3\,x\right )\,\left (2\,a^3\,b^3\,c^2\,d^3-\frac {9\,a^2\,b^4\,c^4\,d}{2}-b^2\,\left (\frac {5\,c^5\,\sqrt {-a^3\,b^5}}{4}-\frac {a^4\,d^5}{2}\right )+\frac {3\,a^2\,c\,d^4\,\sqrt {-a^3\,b^5}}{4}+\frac {11\,a\,b\,c^3\,d^2\,\sqrt {-a^3\,b^5}}{2}\right )}{a^4\,b^4\,d^8+4\,a^3\,b^5\,c^2\,d^6+6\,a^2\,b^6\,c^4\,d^4+4\,a\,b^7\,c^6\,d^2+b^8\,c^8}+\frac {c^4\,\ln \left (c+d\,x\right )\,\left (15\,a^2\,d^4+4\,a\,b\,c^2\,d^2+b^2\,c^4\right )}{d^3\,{\left (b\,c^2+a\,d^2\right )}^4} \] Input:

int(x^6/((a + b*x^2)^2*(c + d*x)^3),x)
 

Output:

((x^2*(a^4*d^8 + 3*b^4*c^8 + 11*a*b^3*c^6*d^2 + 3*a^3*b*c^2*d^6 - 2*a^2*b^ 
2*c^4*d^4))/(2*b^2*d^3*(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*x^3*(3*a^3*d^6 + 4*b^3*c^6 + 12*a*b^2*c^4*d^2 - a^2*b*c^2*d^4)) 
/(2*b*d^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^ 
2*(a^3*d^6 + 3*b^3*c^6 + 11*a*b^2*c^4*d^2 - 3*a^2*b*c^2*d^4))/(2*b^2*d^3*( 
a*d^2 + b*c^2)*(a^2*d^4 + b^2*c^4 + 2*a*b*c^2*d^2)) + (a*c*x*(2*a^3*d^6 + 
4*b^3*c^6 + 11*a*b^2*c^4*d^2 - 3*a^2*b*c^2*d^4))/(2*b^2*d^2*(a*d^2 + b*c^2 
)*(a^2*d^4 + b^2*c^4 + 2*a*b*c^2*d^2)))/(a*c^2 + x^2*(a*d^2 + b*c^2) + b*d 
^2*x^4 + 2*b*c*d*x^3 + 2*a*c*d*x) - (log((-a^3*b^5)^(1/2) + a*b^3*x)*((9*a 
^2*b^4*c^4*d)/2 - b^2*((5*c^5*(-a^3*b^5)^(1/2))/4 + (a^4*d^5)/2) - 2*a^3*b 
^3*c^2*d^3 + (3*a^2*c*d^4*(-a^3*b^5)^(1/2))/4 + (11*a*b*c^3*d^2*(-a^3*b^5) 
^(1/2))/2))/(b^8*c^8 + a^4*b^4*d^8 + 4*a*b^7*c^6*d^2 + 6*a^2*b^6*c^4*d^4 + 
 4*a^3*b^5*c^2*d^6) + (log((-a^3*b^5)^(1/2) - a*b^3*x)*(2*a^3*b^3*c^2*d^3 
- (9*a^2*b^4*c^4*d)/2 - b^2*((5*c^5*(-a^3*b^5)^(1/2))/4 - (a^4*d^5)/2) + ( 
3*a^2*c*d^4*(-a^3*b^5)^(1/2))/4 + (11*a*b*c^3*d^2*(-a^3*b^5)^(1/2))/2))/(b 
^8*c^8 + a^4*b^4*d^8 + 4*a*b^7*c^6*d^2 + 6*a^2*b^6*c^4*d^4 + 4*a^3*b^5*c^2 
*d^6) + (c^4*log(c + d*x)*(15*a^2*d^4 + b^2*c^4 + 4*a*b*c^2*d^2))/(d^3*(a* 
d^2 + b*c^2)^4)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 1803, normalized size of antiderivative = 5.72 \[ \int \frac {x^6}{(c+d x)^3 \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int(x^6/(d*x+c)^3/(b*x^2+a)^2,x)
 

Output:

( - 6*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*c**3*d**7 - 12*sq 
rt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*c**2*d**8*x - 6*sqrt(b)*s 
qrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*c*d**9*x**2 - 44*sqrt(b)*sqrt(a) 
*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b*c**5*d**5 - 88*sqrt(b)*sqrt(a)*atan( 
(b*x)/(sqrt(b)*sqrt(a)))*a**3*b*c**4*d**6*x - 50*sqrt(b)*sqrt(a)*atan((b*x 
)/(sqrt(b)*sqrt(a)))*a**3*b*c**3*d**7*x**2 - 12*sqrt(b)*sqrt(a)*atan((b*x) 
/(sqrt(b)*sqrt(a)))*a**3*b*c**2*d**8*x**3 - 6*sqrt(b)*sqrt(a)*atan((b*x)/( 
sqrt(b)*sqrt(a)))*a**3*b*c*d**9*x**4 + 10*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt 
(b)*sqrt(a)))*a**2*b**2*c**7*d**3 + 20*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b) 
*sqrt(a)))*a**2*b**2*c**6*d**4*x - 34*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)* 
sqrt(a)))*a**2*b**2*c**5*d**5*x**2 - 88*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b 
)*sqrt(a)))*a**2*b**2*c**4*d**6*x**3 - 44*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt 
(b)*sqrt(a)))*a**2*b**2*c**3*d**7*x**4 + 10*sqrt(b)*sqrt(a)*atan((b*x)/(sq 
rt(b)*sqrt(a)))*a*b**3*c**7*d**3*x**2 + 20*sqrt(b)*sqrt(a)*atan((b*x)/(sqr 
t(b)*sqrt(a)))*a*b**3*c**6*d**4*x**3 + 10*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt 
(b)*sqrt(a)))*a*b**3*c**5*d**5*x**4 + 2*log(a + b*x**2)*a**5*c**2*d**8 + 4 
*log(a + b*x**2)*a**5*c*d**9*x + 2*log(a + b*x**2)*a**5*d**10*x**2 + 8*log 
(a + b*x**2)*a**4*b*c**4*d**6 + 16*log(a + b*x**2)*a**4*b*c**3*d**7*x + 10 
*log(a + b*x**2)*a**4*b*c**2*d**8*x**2 + 4*log(a + b*x**2)*a**4*b*c*d**9*x 
**3 + 2*log(a + b*x**2)*a**4*b*d**10*x**4 - 18*log(a + b*x**2)*a**3*b**...