\(\int \frac {x^5}{(c+d x)^2 (a+b x^2)^3} \, dx\) [255]

Optimal result

Integrand size = 20, antiderivative size = 298 \[ \int \frac {x^5}{(c+d x)^2 \left (a+b x^2\right )^3} \, dx=\frac {c^5}{\left (b c^2+a d^2\right )^3 (c+d x)}-\frac {a^2 \left (b c^2-a d^2-2 b c d x\right )}{4 b^2 \left (b c^2+a d^2\right )^2 \left (a+b x^2\right )^2}+\frac {a \left (2 \left (2 b^2 c^4-3 a b c^2 d^2-a^2 d^4\right )-b c d \left (9 b c^2+a d^2\right ) x\right )}{4 b^2 \left (b c^2+a d^2\right )^3 \left (a+b x^2\right )}+\frac {\sqrt {a} c d \left (15 b^2 c^4-10 a b c^2 d^2-a^2 d^4\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{4 b^{3/2} \left (b c^2+a d^2\right )^4}-\frac {c^4 \left (b c^2-5 a d^2\right ) \log (c+d x)}{\left (b c^2+a d^2\right )^4}+\frac {c^4 \left (b c^2-5 a d^2\right ) \log \left (a+b x^2\right )}{2 \left (b c^2+a d^2\right )^4} \] Output:

c^5/(a*d^2+b*c^2)^3/(d*x+c)-1/4*a^2*(-2*b*c*d*x-a*d^2+b*c^2)/b^2/(a*d^2+b* 
c^2)^2/(b*x^2+a)^2+1/4*a*(-2*a^2*d^4-6*a*b*c^2*d^2+4*b^2*c^4-b*c*d*(a*d^2+ 
9*b*c^2)*x)/b^2/(a*d^2+b*c^2)^3/(b*x^2+a)+1/4*a^(1/2)*c*d*(-a^2*d^4-10*a*b 
*c^2*d^2+15*b^2*c^4)*arctan(b^(1/2)*x/a^(1/2))/b^(3/2)/(a*d^2+b*c^2)^4-c^4 
*(-5*a*d^2+b*c^2)*ln(d*x+c)/(a*d^2+b*c^2)^4+1/2*c^4*(-5*a*d^2+b*c^2)*ln(b* 
x^2+a)/(a*d^2+b*c^2)^4
 

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.84 \[ \int \frac {x^5}{(c+d x)^2 \left (a+b x^2\right )^3} \, dx=\frac {\frac {4 c^5 \left (b c^2+a d^2\right )}{c+d x}+\frac {a^2 \left (b c^2+a d^2\right )^2 \left (a d^2-b c (c-2 d x)\right )}{b^2 \left (a+b x^2\right )^2}-\frac {a \left (b c^2+a d^2\right ) \left (2 a^2 d^4+a b c d^2 (6 c+d x)+b^2 c^3 (-4 c+9 d x)\right )}{b^2 \left (a+b x^2\right )}-\frac {\sqrt {a} c d \left (-15 b^2 c^4+10 a b c^2 d^2+a^2 d^4\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}}-4 \left (b c^6-5 a c^4 d^2\right ) \log (c+d x)+2 \left (b c^6-5 a c^4 d^2\right ) \log \left (a+b x^2\right )}{4 \left (b c^2+a d^2\right )^4} \] Input:

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

Output:

((4*c^5*(b*c^2 + a*d^2))/(c + d*x) + (a^2*(b*c^2 + a*d^2)^2*(a*d^2 - b*c*( 
c - 2*d*x)))/(b^2*(a + b*x^2)^2) - (a*(b*c^2 + a*d^2)*(2*a^2*d^4 + a*b*c*d 
^2*(6*c + d*x) + b^2*c^3*(-4*c + 9*d*x)))/(b^2*(a + b*x^2)) - (Sqrt[a]*c*d 
*(-15*b^2*c^4 + 10*a*b*c^2*d^2 + a^2*d^4)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^( 
3/2) - 4*(b*c^6 - 5*a*c^4*d^2)*Log[c + d*x] + 2*(b*c^6 - 5*a*c^4*d^2)*Log[ 
a + b*x^2])/(4*(b*c^2 + a*d^2)^4)
 

Rubi [A] (verified)

Time = 2.31 (sec) , antiderivative size = 327, 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.

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

\(\Big \downarrow \) 601

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2178

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

\(\Big \downarrow \) 2160

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {a^2 \left (-a d^2+b c^2-2 b c d x\right )}{4 b^2 \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^2}-\frac {-\frac {a^2 \left (2 \left (-a^2 d^4-3 a b c^2 d^2+2 b^2 c^4\right )-b c d x \left (a d^2+9 b c^2\right )\right )}{2 b^2 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^3}-\frac {\frac {4 a^2 b c^5}{(c+d x) \left (a d^2+b c^2\right )^3}+\frac {2 a^2 b c^4 \left (b c^2-5 a d^2\right ) \log \left (a+b x^2\right )}{\left (a d^2+b c^2\right )^4}-\frac {4 a^2 b c^4 \left (b c^2-5 a d^2\right ) \log (c+d x)}{\left (a d^2+b c^2\right )^4}+\frac {a^{5/2} c d \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-a^2 d^4-10 a b c^2 d^2+15 b^2 c^4\right )}{\sqrt {b} \left (a d^2+b c^2\right )^4}}{2 a b}}{2 a}\)

Input:

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

Output:

-1/4*(a^2*(b*c^2 - a*d^2 - 2*b*c*d*x))/(b^2*(b*c^2 + a*d^2)^2*(a + b*x^2)^ 
2) - (-1/2*(a^2*(2*(2*b^2*c^4 - 3*a*b*c^2*d^2 - a^2*d^4) - b*c*d*(9*b*c^2 
+ a*d^2)*x))/(b^2*(b*c^2 + a*d^2)^3*(a + b*x^2)) - ((4*a^2*b*c^5)/((b*c^2 
+ a*d^2)^3*(c + d*x)) + (a^(5/2)*c*d*(15*b^2*c^4 - 10*a*b*c^2*d^2 - a^2*d^ 
4)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*(b*c^2 + a*d^2)^4) - (4*a^2*b*c^4 
*(b*c^2 - 5*a*d^2)*Log[c + d*x])/(b*c^2 + a*d^2)^4 + (2*a^2*b*c^4*(b*c^2 - 
 5*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 = 0.34 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.15

method	result	size
default	\(-\frac {\frac {\left (\frac {1}{4} a^{3} c \,d^{5}+\frac {5}{2} a^{2} b \,c^{3} d^{3}+\frac {9}{4} a \,b^{2} c^{5} d \right ) x^{3}+\frac {a \left (a^{3} d^{6}+4 a^{2} b \,c^{2} d^{4}+a \,b^{2} c^{4} d^{2}-2 b^{3} c^{6}\right ) x^{2}}{2 b}-\frac {a^{2} c d \left (a^{2} d^{4}-6 b \,c^{2} d^{2} a -7 b^{2} c^{4}\right ) x}{4 b}+\frac {a^{2} \left (a^{3} d^{6}+7 a^{2} b \,c^{2} d^{4}+3 a \,b^{2} c^{4} d^{2}-3 b^{3} c^{6}\right )}{4 b^{2}}}{\left (b \,x^{2}+a \right )^{2}}+\frac {c \left (\frac {\left (20 a \,b^{2} c^{3} d^{2}-4 b^{3} c^{5}\right ) \ln \left (b \,x^{2}+a \right )}{2 b}+\frac {\left (a^{3} d^{5}+10 a^{2} b \,c^{2} d^{3}-15 a \,b^{2} c^{4} d \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )}{4 b}}{\left (a \,d^{2}+b \,c^{2}\right )^{4}}+\frac {c^{4} \left (5 a \,d^{2}-b \,c^{2}\right ) \ln \left (d x +c \right )}{\left (a \,d^{2}+b \,c^{2}\right )^{4}}+\frac {c^{5}}{\left (a \,d^{2}+b \,c^{2}\right )^{3} \left (d x +c \right )}\)	\(344\)
risch	\(\text {Expression too large to display}\)	\(1056\)

Input:

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

Output:

-1/(a*d^2+b*c^2)^4*(((1/4*a^3*c*d^5+5/2*a^2*b*c^3*d^3+9/4*a*b^2*c^5*d)*x^3 
+1/2*a*(a^3*d^6+4*a^2*b*c^2*d^4+a*b^2*c^4*d^2-2*b^3*c^6)/b*x^2-1/4*a^2*c*d 
*(a^2*d^4-6*a*b*c^2*d^2-7*b^2*c^4)/b*x+1/4*a^2*(a^3*d^6+7*a^2*b*c^2*d^4+3* 
a*b^2*c^4*d^2-3*b^3*c^6)/b^2)/(b*x^2+a)^2+1/4/b*c*(1/2*(20*a*b^2*c^3*d^2-4 
*b^3*c^5)/b*ln(b*x^2+a)+(a^3*d^5+10*a^2*b*c^2*d^3-15*a*b^2*c^4*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*ln(d*x+c 
)+c^5/(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. 1251 vs. \(2 (279) = 558\).

Time = 2.51 (sec) , antiderivative size = 2526, normalized size of antiderivative = 8.48 \[ \int \frac {x^5}{(c+d x)^2 \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/8*(14*a^2*b^3*c^7 + 2*a^3*b^2*c^5*d^2 - 14*a^4*b*c^3*d^4 - 2*a^5*c*d^6 
+ 2*(4*b^5*c^7 - 5*a*b^4*c^5*d^2 - 10*a^2*b^3*c^3*d^4 - a^3*b^2*c*d^6)*x^4 
 - 2*(5*a*b^4*c^6*d + 12*a^2*b^3*c^4*d^3 + 9*a^3*b^2*c^2*d^5 + 2*a^4*b*d^7 
)*x^3 + 2*(12*a*b^4*c^7 - a^2*b^3*c^5*d^2 - 14*a^3*b^2*c^3*d^4 - a^4*b*c*d 
^6)*x^2 - (15*a^2*b^3*c^6*d - 10*a^3*b^2*c^4*d^3 - a^4*b*c^2*d^5 + (15*b^5 
*c^5*d^2 - 10*a*b^4*c^3*d^4 - a^2*b^3*c*d^6)*x^5 + (15*b^5*c^6*d - 10*a*b^ 
4*c^4*d^3 - a^2*b^3*c^2*d^5)*x^4 + 2*(15*a*b^4*c^5*d^2 - 10*a^2*b^3*c^3*d^ 
4 - a^3*b^2*c*d^6)*x^3 + 2*(15*a*b^4*c^6*d - 10*a^2*b^3*c^4*d^3 - a^3*b^2* 
c^2*d^5)*x^2 + (15*a^2*b^3*c^5*d^2 - 10*a^3*b^2*c^3*d^4 - a^4*b*c*d^6)*x)* 
sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) - 2*(4*a^2*b^3* 
c^6*d + 9*a^3*b^2*c^4*d^3 + 6*a^4*b*c^2*d^5 + a^5*d^7)*x + 4*(a^2*b^3*c^7 
- 5*a^3*b^2*c^5*d^2 + (b^5*c^6*d - 5*a*b^4*c^4*d^3)*x^5 + (b^5*c^7 - 5*a*b 
^4*c^5*d^2)*x^4 + 2*(a*b^4*c^6*d - 5*a^2*b^3*c^4*d^3)*x^3 + 2*(a*b^4*c^7 - 
 5*a^2*b^3*c^5*d^2)*x^2 + (a^2*b^3*c^6*d - 5*a^3*b^2*c^4*d^3)*x)*log(b*x^2 
 + a) - 8*(a^2*b^3*c^7 - 5*a^3*b^2*c^5*d^2 + (b^5*c^6*d - 5*a*b^4*c^4*d^3) 
*x^5 + (b^5*c^7 - 5*a*b^4*c^5*d^2)*x^4 + 2*(a*b^4*c^6*d - 5*a^2*b^3*c^4*d^ 
3)*x^3 + 2*(a*b^4*c^7 - 5*a^2*b^3*c^5*d^2)*x^2 + (a^2*b^3*c^6*d - 5*a^3*b^ 
2*c^4*d^3)*x)*log(d*x + c))/(a^2*b^6*c^9 + 4*a^3*b^5*c^7*d^2 + 6*a^4*b^4*c 
^5*d^4 + 4*a^5*b^3*c^3*d^6 + a^6*b^2*c*d^8 + (b^8*c^8*d + 4*a*b^7*c^6*d^3 
+ 6*a^2*b^6*c^4*d^5 + 4*a^3*b^5*c^2*d^7 + a^4*b^4*d^9)*x^5 + (b^8*c^9 +...
 

Sympy [F(-1)]

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

integrate(x**5/(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. 757 vs. \(2 (279) = 558\).

Time = 0.13 (sec) , antiderivative size = 757, normalized size of antiderivative = 2.54 \[ \int \frac {x^5}{(c+d x)^2 \left (a+b x^2\right )^3} \, dx=\frac {{\left (b c^{6} - 5 \, a c^{4} d^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{4} c^{8} + 4 \, a b^{3} c^{6} d^{2} + 6 \, a^{2} b^{2} c^{4} d^{4} + 4 \, a^{3} b c^{2} d^{6} + a^{4} d^{8}\right )}} - \frac {{\left (b c^{6} - 5 \, a c^{4} d^{2}\right )} \log \left (d x + c\right )}{b^{4} c^{8} + 4 \, a b^{3} c^{6} d^{2} + 6 \, a^{2} b^{2} c^{4} d^{4} + 4 \, a^{3} b c^{2} d^{6} + a^{4} d^{8}} + \frac {{\left (15 \, a b^{2} c^{5} d - 10 \, a^{2} b c^{3} d^{3} - a^{3} c d^{5}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{4 \, {\left (b^{5} c^{8} + 4 \, a b^{4} c^{6} d^{2} + 6 \, a^{2} b^{3} c^{4} d^{4} + 4 \, a^{3} b^{2} c^{2} d^{6} + a^{4} b d^{8}\right )} \sqrt {a b}} + \frac {7 \, a^{2} b^{2} c^{5} - 6 \, a^{3} b c^{3} d^{2} - a^{4} c d^{4} + {\left (4 \, b^{4} c^{5} - 9 \, a b^{3} c^{3} d^{2} - a^{2} b^{2} c d^{4}\right )} x^{4} - {\left (5 \, a b^{3} c^{4} d + 7 \, a^{2} b^{2} c^{2} d^{3} + 2 \, a^{3} b d^{5}\right )} x^{3} + {\left (12 \, a b^{3} c^{5} - 13 \, a^{2} b^{2} c^{3} d^{2} - a^{3} b c d^{4}\right )} x^{2} - {\left (4 \, a^{2} b^{2} c^{4} d + 5 \, a^{3} b c^{2} d^{3} + a^{4} d^{5}\right )} x}{4 \, {\left (a^{2} b^{5} c^{7} + 3 \, a^{3} b^{4} c^{5} d^{2} + 3 \, a^{4} b^{3} c^{3} d^{4} + a^{5} b^{2} c d^{6} + {\left (b^{7} c^{6} d + 3 \, a b^{6} c^{4} d^{3} + 3 \, a^{2} b^{5} c^{2} d^{5} + a^{3} b^{4} d^{7}\right )} x^{5} + {\left (b^{7} c^{7} + 3 \, a b^{6} c^{5} d^{2} + 3 \, a^{2} b^{5} c^{3} d^{4} + a^{3} b^{4} c d^{6}\right )} x^{4} + 2 \, {\left (a b^{6} c^{6} d + 3 \, a^{2} b^{5} c^{4} d^{3} + 3 \, a^{3} b^{4} c^{2} d^{5} + a^{4} b^{3} d^{7}\right )} x^{3} + 2 \, {\left (a b^{6} c^{7} + 3 \, a^{2} b^{5} c^{5} d^{2} + 3 \, a^{3} b^{4} c^{3} d^{4} + a^{4} b^{3} c d^{6}\right )} x^{2} + {\left (a^{2} b^{5} c^{6} d + 3 \, a^{3} b^{4} c^{4} d^{3} + 3 \, a^{4} b^{3} c^{2} d^{5} + a^{5} b^{2} d^{7}\right )} x\right )}} \] Input:

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

Output:

1/2*(b*c^6 - 5*a*c^4*d^2)*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) - (b*c^6 - 5*a*c^4*d^2)*log(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) + 1/4*(15*a*b^2*c^5*d - 10*a^2*b*c^3*d^3 - a^3*c*d^5)*arctan(b*x/s 
qrt(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)) + 1/4*(7*a^2*b^2*c^5 - 6*a^3*b*c^3*d^2 - a^4*c 
*d^4 + (4*b^4*c^5 - 9*a*b^3*c^3*d^2 - a^2*b^2*c*d^4)*x^4 - (5*a*b^3*c^4*d 
+ 7*a^2*b^2*c^2*d^3 + 2*a^3*b*d^5)*x^3 + (12*a*b^3*c^5 - 13*a^2*b^2*c^3*d^ 
2 - a^3*b*c*d^4)*x^2 - (4*a^2*b^2*c^4*d + 5*a^3*b*c^2*d^3 + a^4*d^5)*x)/(a 
^2*b^5*c^7 + 3*a^3*b^4*c^5*d^2 + 3*a^4*b^3*c^3*d^4 + a^5*b^2*c*d^6 + (b^7* 
c^6*d + 3*a*b^6*c^4*d^3 + 3*a^2*b^5*c^2*d^5 + a^3*b^4*d^7)*x^5 + (b^7*c^7 
+ 3*a*b^6*c^5*d^2 + 3*a^2*b^5*c^3*d^4 + a^3*b^4*c*d^6)*x^4 + 2*(a*b^6*c^6* 
d + 3*a^2*b^5*c^4*d^3 + 3*a^3*b^4*c^2*d^5 + a^4*b^3*d^7)*x^3 + 2*(a*b^6*c^ 
7 + 3*a^2*b^5*c^5*d^2 + 3*a^3*b^4*c^3*d^4 + a^4*b^3*c*d^6)*x^2 + (a^2*b^5* 
c^6*d + 3*a^3*b^4*c^4*d^3 + 3*a^4*b^3*c^2*d^5 + a^5*b^2*d^7)*x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 562 vs. \(2 (279) = 558\).

Time = 0.14 (sec) , antiderivative size = 562, normalized size of antiderivative = 1.89 \[ \int \frac {x^5}{(c+d x)^2 \left (a+b x^2\right )^3} \, dx=\frac {c^{5} d^{6}}{{\left (b^{3} c^{6} d^{6} + 3 \, a b^{2} c^{4} d^{8} + 3 \, a^{2} b c^{2} d^{10} + a^{3} d^{12}\right )} {\left (d x + c\right )}} + \frac {{\left (b c^{6} - 5 \, a c^{4} d^{2}\right )} \log \left (b - \frac {2 \, b c}{d x + c} + \frac {b c^{2}}{{\left (d x + c\right )}^{2}} + \frac {a d^{2}}{{\left (d x + c\right )}^{2}}\right )}{2 \, {\left (b^{4} c^{8} + 4 \, a b^{3} c^{6} d^{2} + 6 \, a^{2} b^{2} c^{4} d^{4} + 4 \, a^{3} b c^{2} d^{6} + a^{4} d^{8}\right )}} + \frac {{\left (15 \, a b^{2} c^{5} d^{3} - 10 \, a^{2} b c^{3} d^{5} - a^{3} c d^{7}\right )} \arctan \left (\frac {b c - \frac {b c^{2}}{d x + c} - \frac {a d^{2}}{d x + c}}{\sqrt {a b} d}\right )}{4 \, {\left (b^{5} c^{8} + 4 \, a b^{4} c^{6} d^{2} + 6 \, a^{2} b^{3} c^{4} d^{4} + 4 \, a^{3} b^{2} c^{2} d^{6} + a^{4} b d^{8}\right )} \sqrt {a b} d^{2}} - \frac {13 \, a b^{3} c^{4} d^{2} - 8 \, a^{2} b^{2} c^{2} d^{4} - a^{3} b d^{6} - \frac {43 \, a b^{3} c^{5} d^{3} - 42 \, a^{2} b^{2} c^{3} d^{5} - 5 \, a^{3} b c d^{7}}{{\left (d x + c\right )} d} + \frac {47 \, a b^{3} c^{6} d^{4} - 50 \, a^{2} b^{2} c^{4} d^{6} - 17 \, a^{3} b c^{2} d^{8}}{{\left (d x + c\right )}^{2} d^{2}} - \frac {17 \, a b^{3} c^{7} d^{5} - 13 \, a^{2} b^{2} c^{5} d^{7} - 29 \, a^{3} b c^{3} d^{9} + a^{4} c d^{11}}{{\left (d x + c\right )}^{3} d^{3}}}{4 \, {\left (b c^{2} + a d^{2}\right )}^{4} {\left (b - \frac {2 \, b c}{d x + c} + \frac {b c^{2}}{{\left (d x + c\right )}^{2}} + \frac {a d^{2}}{{\left (d x + c\right )}^{2}}\right )}^{2} b} \] Input:

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

Output:

c^5*d^6/((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)) + 1/2*(b*c^6 - 5*a*c^4*d^2)*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) + 1/4*(15*a*b^2*c^5*d^3 - 10*a^2*b*c^3*d^5 - 
a^3*c*d^7)*arctan((b*c - b*c^2/(d*x + c) - a*d^2/(d*x + c))/(sqrt(a*b)*d)) 
/((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)*d^2) - 1/4*(13*a*b^3*c^4*d^2 - 8*a^2*b^2*c^2*d^4 - a^3*b 
*d^6 - (43*a*b^3*c^5*d^3 - 42*a^2*b^2*c^3*d^5 - 5*a^3*b*c*d^7)/((d*x + c)* 
d) + (47*a*b^3*c^6*d^4 - 50*a^2*b^2*c^4*d^6 - 17*a^3*b*c^2*d^8)/((d*x + c) 
^2*d^2) - (17*a*b^3*c^7*d^5 - 13*a^2*b^2*c^5*d^7 - 29*a^3*b*c^3*d^9 + a^4* 
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*b)
 

Mupad [B] (verification not implemented)

Time = 8.86 (sec) , antiderivative size = 1259, normalized size of antiderivative = 4.22 \[ \int \frac {x^5}{(c+d x)^2 \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(log(a^8*b^4*d^14*x - 144*b^6*c^14*(-a*b^3)^(3/2) + a^8*b^2*d^14*(-a*b^3)^ 
(1/2) + 5455*a^2*c^8*d^6*(-a*b^3)^(5/2) - 22*a^6*c^2*d^12*(-a*b^3)^(3/2) - 
 927*b^2*c^12*d^2*(-a*b^3)^(5/2) + 144*a*b^11*c^14*x - 3460*a^4*b^2*c^6*d^ 
8*(-a*b^3)^(3/2) + 1014*a*b*c^10*d^4*(-a*b^3)^(5/2) - 927*a^2*b^10*c^12*d^ 
2*x + 1014*a^3*b^9*c^10*d^4*x + 5455*a^4*b^8*c^8*d^6*x + 3460*a^5*b^7*c^6* 
d^8*x + 111*a^6*b^6*c^4*d^10*x + 22*a^7*b^5*c^2*d^12*x - 111*a^5*b*c^4*d^1 
0*(-a*b^3)^(3/2))*((b^4*c^6)/2 - a*((5*b^3*c^4*d^2)/2 + (5*b*c^3*d^3*(-a*b 
^3)^(1/2))/4) - (a^2*c*d^5*(-a*b^3)^(1/2))/8 + (15*b^2*c^5*d*(-a*b^3)^(1/2 
))/8))/(b^7*c^8 + a^4*b^3*d^8 + 4*a*b^6*c^6*d^2 + 6*a^2*b^5*c^4*d^4 + 4*a^ 
3*b^4*c^2*d^6) - ((x^4*(a^2*c*d^4 - 4*b^2*c^5 + 9*a*b*c^3*d^2))/(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)) + (d*x^3*(2*a^2*d^2 + 5* 
a*b*c^2))/(4*b*(a^2*d^4 + b^2*c^4 + 2*a*b*c^2*d^2)) + (a*c*(a^3*d^4 - 7*a* 
b^2*c^4 + 6*a^2*b*c^2*d^2))/(4*b^2*(a*d^2 + b*c^2)*(a^2*d^4 + b^2*c^4 + 2* 
a*b*c^2*d^2)) + (a*d*x*(a^2*d^2 + 4*a*b*c^2))/(4*b^2*(a^2*d^4 + b^2*c^4 + 
2*a*b*c^2*d^2)) + (c*x^2*(a^3*d^4 - 12*a*b^2*c^4 + 13*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)))/(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(144*b^6*c^14*(-a 
*b^3)^(3/2) + a^8*b^4*d^14*x - a^8*b^2*d^14*(-a*b^3)^(1/2) - 5455*a^2*c^8* 
d^6*(-a*b^3)^(5/2) + 22*a^6*c^2*d^12*(-a*b^3)^(3/2) + 927*b^2*c^12*d^2*(-a 
*b^3)^(5/2) + 144*a*b^11*c^14*x + 3460*a^4*b^2*c^6*d^8*(-a*b^3)^(3/2) -...
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 1705, normalized size of antiderivative = 5.72 \[ \int \frac {x^5}{(c+d x)^2 \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*c**2*d**5 - sqrt(b) 
*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*c*d**6*x - 10*sqrt(b)*sqrt(a)* 
atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b*c**4*d**3 - 10*sqrt(b)*sqrt(a)*atan(( 
b*x)/(sqrt(b)*sqrt(a)))*a**3*b*c**3*d**4*x - 2*sqrt(b)*sqrt(a)*atan((b*x)/ 
(sqrt(b)*sqrt(a)))*a**3*b*c**2*d**5*x**2 - 2*sqrt(b)*sqrt(a)*atan((b*x)/(s 
qrt(b)*sqrt(a)))*a**3*b*c*d**6*x**3 + 15*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt( 
b)*sqrt(a)))*a**2*b**2*c**6*d + 15*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqr 
t(a)))*a**2*b**2*c**5*d**2*x - 20*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt 
(a)))*a**2*b**2*c**4*d**3*x**2 - 20*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sq 
rt(a)))*a**2*b**2*c**3*d**4*x**3 - sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqr 
t(a)))*a**2*b**2*c**2*d**5*x**4 - sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt 
(a)))*a**2*b**2*c*d**6*x**5 + 30*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt( 
a)))*a*b**3*c**6*d*x**2 + 30*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a))) 
*a*b**3*c**5*d**2*x**3 - 10*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))* 
a*b**3*c**4*d**3*x**4 - 10*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a 
*b**3*c**3*d**4*x**5 + 15*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b* 
*4*c**6*d*x**4 + 15*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**4*c** 
5*d**2*x**5 - 10*log(a + b*x**2)*a**3*b**2*c**5*d**2 - 10*log(a + b*x**2)* 
a**3*b**2*c**4*d**3*x + 2*log(a + b*x**2)*a**2*b**3*c**7 + 2*log(a + b*x** 
2)*a**2*b**3*c**6*d*x - 20*log(a + b*x**2)*a**2*b**3*c**5*d**2*x**2 - 2...