\(\int \frac {x^7}{(c+d x)^3 (a+b x^2)^3} \, dx\) [266]

Optimal result

Integrand size = 20, antiderivative size = 388 \[ \int \frac {x^7}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\frac {c^7}{2 d^2 \left (b c^2+a d^2\right )^3 (c+d x)^2}-\frac {c^6 \left (b c^2+7 a d^2\right )}{d^2 \left (b c^2+a d^2\right )^4 (c+d x)}+\frac {a^3 \left (c \left (b c^2-3 a d^2\right )-d \left (3 b c^2-a d^2\right ) x\right )}{4 b^2 \left (b c^2+a d^2\right )^3 \left (a+b x^2\right )^2}-\frac {a^2 \left (12 c \left (b^2 c^4-4 a b c^2 d^2-a^2 d^4\right )-d \left (39 b^2 c^4-14 a b c^2 d^2-5 a^2 d^4\right ) x\right )}{8 b^2 \left (b c^2+a d^2\right )^4 \left (a+b x^2\right )}-\frac {3 a^{3/2} d \left (35 b^3 c^6-35 a b^2 c^4 d^2-7 a^2 b c^2 d^4-a^3 d^6\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{5/2} \left (b c^2+a d^2\right )^5}+\frac {3 a c^5 \left (b c^2-7 a d^2\right ) \log (c+d x)}{\left (b c^2+a d^2\right )^5}-\frac {3 a c^5 \left (b c^2-7 a d^2\right ) \log \left (a+b x^2\right )}{2 \left (b c^2+a d^2\right )^5} \] Output:

1/2*c^7/d^2/(a*d^2+b*c^2)^3/(d*x+c)^2-c^6*(7*a*d^2+b*c^2)/d^2/(a*d^2+b*c^2 
)^4/(d*x+c)+1/4*a^3*(c*(-3*a*d^2+b*c^2)-d*(-a*d^2+3*b*c^2)*x)/b^2/(a*d^2+b 
*c^2)^3/(b*x^2+a)^2-1/8*a^2*(12*c*(-a^2*d^4-4*a*b*c^2*d^2+b^2*c^4)-d*(-5*a 
^2*d^4-14*a*b*c^2*d^2+39*b^2*c^4)*x)/b^2/(a*d^2+b*c^2)^4/(b*x^2+a)-3/8*a^( 
3/2)*d*(-a^3*d^6-7*a^2*b*c^2*d^4-35*a*b^2*c^4*d^2+35*b^3*c^6)*arctan(b^(1/ 
2)*x/a^(1/2))/b^(5/2)/(a*d^2+b*c^2)^5+3*a*c^5*(-7*a*d^2+b*c^2)*ln(d*x+c)/( 
a*d^2+b*c^2)^5-3/2*a*c^5*(-7*a*d^2+b*c^2)*ln(b*x^2+a)/(a*d^2+b*c^2)^5
 

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.86 \[ \int \frac {x^7}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\frac {\frac {4 c^7 \left (b c^2+a d^2\right )^2}{d^2 (c+d x)^2}-\frac {8 \left (b c^2+a d^2\right ) \left (b c^8+7 a c^6 d^2\right )}{d^2 (c+d x)}+\frac {2 a^3 \left (b c^2+a d^2\right )^2 \left (b c^2 (c-3 d x)+a d^2 (-3 c+d x)\right )}{b^2 \left (a+b x^2\right )^2}+\frac {a^2 \left (b c^2+a d^2\right ) \left (2 a b c^2 d^2 (24 c-7 d x)+a^2 d^4 (12 c-5 d x)+3 b^2 c^4 (-4 c+13 d x)\right )}{b^2 \left (a+b x^2\right )}+\frac {3 a^{3/2} d \left (-35 b^3 c^6+35 a b^2 c^4 d^2+7 a^2 b c^2 d^4+a^3 d^6\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{5/2}}+24 a c^5 \left (b c^2-7 a d^2\right ) \log (c+d x)+12 a c^5 \left (-b c^2+7 a d^2\right ) \log \left (a+b x^2\right )}{8 \left (b c^2+a d^2\right )^5} \] Input:

Integrate[x^7/((c + d*x)^3*(a + b*x^2)^3),x]
 

Output:

((4*c^7*(b*c^2 + a*d^2)^2)/(d^2*(c + d*x)^2) - (8*(b*c^2 + a*d^2)*(b*c^8 + 
 7*a*c^6*d^2))/(d^2*(c + d*x)) + (2*a^3*(b*c^2 + a*d^2)^2*(b*c^2*(c - 3*d* 
x) + a*d^2*(-3*c + d*x)))/(b^2*(a + b*x^2)^2) + (a^2*(b*c^2 + a*d^2)*(2*a* 
b*c^2*d^2*(24*c - 7*d*x) + a^2*d^4*(12*c - 5*d*x) + 3*b^2*c^4*(-4*c + 13*d 
*x)))/(b^2*(a + b*x^2)) + (3*a^(3/2)*d*(-35*b^3*c^6 + 35*a*b^2*c^4*d^2 + 7 
*a^2*b*c^2*d^4 + a^3*d^6)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(5/2) + 24*a*c^5* 
(b*c^2 - 7*a*d^2)*Log[c + d*x] + 12*a*c^5*(-(b*c^2) + 7*a*d^2)*Log[a + b*x 
^2])/(8*(b*c^2 + a*d^2)^5)
 

Rubi [A] (verified)

Time = 4.47 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {601, 25, 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.

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

\(\Big \downarrow \) 601

\(\displaystyle \frac {a^3 \left (c \left (b c^2-3 a d^2\right )-d x \left (3 b c^2-a d^2\right )\right )}{4 b^2 \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^3}-\frac {\int -\frac {\frac {4 a x^5}{b}-\frac {a^2 \left (4 b^3 c^6+12 a b^2 d^2 c^4+21 a^2 b d^4 c^2+a^3 d^6\right ) x^3}{b^2 \left (b c^2+a d^2\right )^3}-\frac {a^4 c d^3 \left (23 b c^2+3 a d^2\right ) x^2}{b^2 \left (b c^2+a d^2\right )^3}+\frac {a^3 c^2 \left (4 b^2 c^4-3 a b d^2 c^2-3 a^2 d^4\right ) x}{b^2 \left (b c^2+a d^2\right )^3}+\frac {a^4 c^3 d \left (3 b c^2-a d^2\right )}{b^2 \left (b c^2+a d^2\right )^3}}{(c+d x)^3 \left (b x^2+a\right )^2}dx}{4 a}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\frac {4 a x^5}{b}-\frac {a^2 \left (4 b^3 c^6+12 a b^2 d^2 c^4+21 a^2 b d^4 c^2+a^3 d^6\right ) x^3}{b^2 \left (b c^2+a d^2\right )^3}-\frac {a^4 c d^3 \left (23 b c^2+3 a d^2\right ) x^2}{b^2 \left (b c^2+a d^2\right )^3}+\frac {a^3 c^2 \left (4 b^2 c^4-3 a b d^2 c^2-3 a^2 d^4\right ) x}{b^2 \left (b c^2+a d^2\right )^3}+\frac {a^4 c^3 d \left (3 b c^2-a d^2\right )}{b^2 \left (b c^2+a d^2\right )^3}}{(c+d x)^3 \left (b x^2+a\right )^2}dx}{4 a}+\frac {a^3 \left (c \left (b c^2-3 a d^2\right )-d x \left (3 b c^2-a d^2\right )\right )}{4 b^2 \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 2178

\(\displaystyle \frac {-\frac {\int \frac {-\frac {3 c d^3 \left (31 b^2 c^4+18 a b d^2 c^2+3 a^2 d^4\right ) x^2 a^4}{b \left (b c^2+a d^2\right )^4}+\frac {3 c^3 d \left (11 b^2 c^4-6 a b d^2 c^2-a^2 d^4\right ) a^4}{b \left (b c^2+a d^2\right )^4}+\frac {c^2 \left (16 b^3 c^6+19 a b^2 d^2 c^4-54 a^2 b d^4 c^2-9 a^3 d^6\right ) x a^3}{b \left (b c^2+a d^2\right )^4}-\frac {\left (8 b^4 c^8+32 a b^3 d^2 c^6+87 a^2 b^2 d^4 c^4+18 a^3 b d^6 c^2+3 a^4 d^8\right ) x^3 a^2}{b \left (b c^2+a d^2\right )^4}}{(c+d x)^3 \left (b x^2+a\right )}dx}{2 a b}-\frac {a^3 \left (12 c \left (-a^2 d^4-4 a b c^2 d^2+b^2 c^4\right )-d x \left (-5 a^2 d^4-14 a b c^2 d^2+39 b^2 c^4\right )\right )}{2 b^2 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^4}}{4 a}+\frac {a^3 \left (c \left (b c^2-3 a d^2\right )-d x \left (3 b c^2-a d^2\right )\right )}{4 b^2 \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 2160

\(\displaystyle \frac {-\frac {\int \left (\frac {8 a^2 b c^7}{d \left (b c^2+a d^2\right )^3 (c+d x)^3}-\frac {8 a^2 b \left (b c^2+7 a d^2\right ) c^6}{d \left (b c^2+a d^2\right )^4 (c+d x)^2}+\frac {24 a^3 b d \left (7 a d^2-b c^2\right ) c^5}{\left (b c^2+a d^2\right )^5 (c+d x)}+\frac {3 a^3 \left (8 b^3 \left (b c^2-7 a d^2\right ) x c^5+a d \left (35 b^3 c^6-35 a b^2 d^2 c^4-7 a^2 b d^4 c^2-a^3 d^6\right )\right )}{b \left (b c^2+a d^2\right )^5 \left (b x^2+a\right )}\right )dx}{2 a b}-\frac {a^3 \left (12 c \left (-a^2 d^4-4 a b c^2 d^2+b^2 c^4\right )-d x \left (-5 a^2 d^4-14 a b c^2 d^2+39 b^2 c^4\right )\right )}{2 b^2 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^4}}{4 a}+\frac {a^3 \left (c \left (b c^2-3 a d^2\right )-d x \left (3 b c^2-a d^2\right )\right )}{4 b^2 \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^3 \left (c \left (b c^2-3 a d^2\right )-d x \left (3 b c^2-a d^2\right )\right )}{4 b^2 \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^3}+\frac {-\frac {a^3 \left (12 c \left (-a^2 d^4-4 a b c^2 d^2+b^2 c^4\right )-d x \left (-5 a^2 d^4-14 a b c^2 d^2+39 b^2 c^4\right )\right )}{2 b^2 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^4}-\frac {\frac {12 a^3 b c^5 \left (b c^2-7 a d^2\right ) \log \left (a+b x^2\right )}{\left (a d^2+b c^2\right )^5}-\frac {24 a^3 b c^5 \left (b c^2-7 a d^2\right ) \log (c+d x)}{\left (a d^2+b c^2\right )^5}-\frac {4 a^2 b c^7}{d^2 (c+d x)^2 \left (a d^2+b c^2\right )^3}+\frac {8 a^2 b c^6 \left (7 a d^2+b c^2\right )}{d^2 (c+d x) \left (a d^2+b c^2\right )^4}+\frac {3 a^{7/2} d \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-a^3 d^6-7 a^2 b c^2 d^4-35 a b^2 c^4 d^2+35 b^3 c^6\right )}{b^{3/2} \left (a d^2+b c^2\right )^5}}{2 a b}}{4 a}\)

Input:

Int[x^7/((c + d*x)^3*(a + b*x^2)^3),x]
 

Output:

(a^3*(c*(b*c^2 - 3*a*d^2) - d*(3*b*c^2 - a*d^2)*x))/(4*b^2*(b*c^2 + a*d^2) 
^3*(a + b*x^2)^2) + (-1/2*(a^3*(12*c*(b^2*c^4 - 4*a*b*c^2*d^2 - a^2*d^4) - 
 d*(39*b^2*c^4 - 14*a*b*c^2*d^2 - 5*a^2*d^4)*x))/(b^2*(b*c^2 + a*d^2)^4*(a 
 + b*x^2)) - ((-4*a^2*b*c^7)/(d^2*(b*c^2 + a*d^2)^3*(c + d*x)^2) + (8*a^2* 
b*c^6*(b*c^2 + 7*a*d^2))/(d^2*(b*c^2 + a*d^2)^4*(c + d*x)) + (3*a^(7/2)*d* 
(35*b^3*c^6 - 35*a*b^2*c^4*d^2 - 7*a^2*b*c^2*d^4 - a^3*d^6)*ArcTan[(Sqrt[b 
]*x)/Sqrt[a]])/(b^(3/2)*(b*c^2 + a*d^2)^5) - (24*a^3*b*c^5*(b*c^2 - 7*a*d^ 
2)*Log[c + d*x])/(b*c^2 + a*d^2)^5 + (12*a^3*b*c^5*(b*c^2 - 7*a*d^2)*Log[a 
 + b*x^2])/(b*c^2 + a*d^2)^5)/(2*a*b))/(4*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]
 

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 = 0.36 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.12

method	result	size
default	\(\frac {a \left (\frac {-\frac {a d \left (5 a^{3} d^{6}+19 a^{2} b \,c^{2} d^{4}-25 a \,b^{2} c^{4} d^{2}-39 b^{3} c^{6}\right ) x^{3}}{8 b}+\frac {3 a c \left (a^{3} d^{6}+5 a^{2} b \,c^{2} d^{4}+3 a \,b^{2} c^{4} d^{2}-b^{3} c^{6}\right ) x^{2}}{2 b}-\frac {3 a^{2} d \left (a^{3} d^{6}+7 a^{2} b \,c^{2} d^{4}-5 a \,b^{2} c^{4} d^{2}-11 b^{3} c^{6}\right ) x}{8 b^{2}}+\frac {a^{2} c \left (3 a^{3} d^{6}+25 a^{2} b \,c^{2} d^{4}+17 a \,b^{2} c^{4} d^{2}-5 b^{3} c^{6}\right )}{4 b^{2}}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\frac {3 \left (56 a \,c^{5} d^{2} b^{3}-8 c^{7} b^{4}\right ) \ln \left (b \,x^{2}+a \right )}{16 b}+\frac {3 \left (a^{4} d^{7}+7 a^{3} b \,c^{2} d^{5}+35 a^{2} b^{2} c^{4} d^{3}-35 a \,b^{3} c^{6} d \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}}}{b^{2}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{5}}-\frac {c^{6} \left (7 a \,d^{2}+b \,c^{2}\right )}{d^{2} \left (a \,d^{2}+b \,c^{2}\right )^{4} \left (d x +c \right )}+\frac {c^{7}}{2 d^{2} \left (a \,d^{2}+b \,c^{2}\right )^{3} \left (d x +c \right )^{2}}-\frac {3 c^{5} a \left (7 a \,d^{2}-b \,c^{2}\right ) \ln \left (d x +c \right )}{\left (a \,d^{2}+b \,c^{2}\right )^{5}}\)	\(435\)
risch	\(\text {Expression too large to display}\)	\(1669\)

Input:

int(x^7/(d*x+c)^3/(b*x^2+a)^3,x,method=_RETURNVERBOSE)
 

Output:

a/(a*d^2+b*c^2)^5*((-1/8*a*d*(5*a^3*d^6+19*a^2*b*c^2*d^4-25*a*b^2*c^4*d^2- 
39*b^3*c^6)/b*x^3+3/2*a*c*(a^3*d^6+5*a^2*b*c^2*d^4+3*a*b^2*c^4*d^2-b^3*c^6 
)/b*x^2-3/8*a^2*d*(a^3*d^6+7*a^2*b*c^2*d^4-5*a*b^2*c^4*d^2-11*b^3*c^6)/b^2 
*x+1/4*a^2*c*(3*a^3*d^6+25*a^2*b*c^2*d^4+17*a*b^2*c^4*d^2-5*b^3*c^6)/b^2)/ 
(b*x^2+a)^2+3/8/b^2*(1/2*(56*a*b^3*c^5*d^2-8*b^4*c^7)/b*ln(b*x^2+a)+(a^4*d 
^7+7*a^3*b*c^2*d^5+35*a^2*b^2*c^4*d^3-35*a*b^3*c^6*d)/(a*b)^(1/2)*arctan(b 
*x/(a*b)^(1/2))))-c^6*(7*a*d^2+b*c^2)/d^2/(a*d^2+b*c^2)^4/(d*x+c)+1/2*c^7/ 
d^2/(a*d^2+b*c^2)^3/(d*x+c)^2-3*c^5*a*(7*a*d^2-b*c^2)/(a*d^2+b*c^2)^5*ln(d 
*x+c)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1977 vs. \(2 (369) = 738\).

Time = 17.24 (sec) , antiderivative size = 3977, normalized size of antiderivative = 10.25 \[ \int \frac {x^7}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(x^7/(d*x+c)^3/(b*x^2+a)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^7}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(x**7/(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. 1184 vs. \(2 (369) = 738\).

Time = 0.17 (sec) , antiderivative size = 1184, normalized size of antiderivative = 3.05 \[ \int \frac {x^7}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

integrate(x^7/(d*x+c)^3/(b*x^2+a)^3,x, algorithm="maxima")
 

Output:

-3/2*(a*b*c^7 - 7*a^2*c^5*d^2)*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*(a*b*c^7 - 7*a^2*c^5*d^2)*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*( 
35*a^2*b^3*c^6*d - 35*a^3*b^2*c^4*d^3 - 7*a^4*b*c^2*d^5 - a^5*d^7)*arctan( 
b*x/sqrt(a*b))/((b^7*c^10 + 5*a*b^6*c^8*d^2 + 10*a^2*b^5*c^6*d^4 + 10*a^3* 
b^4*c^4*d^6 + 5*a^4*b^3*c^2*d^8 + a^5*b^2*d^10)*sqrt(a*b)) - 1/8*(4*a^2*b^ 
3*c^9 + 62*a^3*b^2*c^7*d^2 - 44*a^4*b*c^5*d^4 - 6*a^5*c^3*d^6 + (8*b^5*c^8 
*d + 56*a*b^4*c^6*d^3 - 39*a^2*b^3*c^4*d^5 + 14*a^3*b^2*c^2*d^7 + 5*a^4*b* 
d^9)*x^5 + 2*(2*b^5*c^9 + 26*a*b^4*c^7*d^2 - 33*a^2*b^3*c^5*d^4 - 10*a^3*b 
^2*c^3*d^6 - a^4*b*c*d^8)*x^4 + (16*a*b^4*c^8*d + 97*a^2*b^3*c^6*d^3 - 115 
*a^3*b^2*c^4*d^5 - a^4*b*c^2*d^7 + 3*a^5*d^9)*x^3 + 4*(2*a*b^4*c^9 + 29*a^ 
2*b^3*c^7*d^2 - 26*a^3*b^2*c^5*d^4 - 5*a^4*b*c^3*d^6)*x^2 + (8*a^2*b^3*c^8 
*d + 43*a^3*b^2*c^6*d^3 - 70*a^4*b*c^4*d^5 - 9*a^5*c^2*d^7)*x)/(a^2*b^6*c^ 
10*d^2 + 4*a^3*b^5*c^8*d^4 + 6*a^4*b^4*c^6*d^6 + 4*a^5*b^3*c^4*d^8 + a^6*b 
^2*c^2*d^10 + (b^8*c^8*d^4 + 4*a*b^7*c^6*d^6 + 6*a^2*b^6*c^4*d^8 + 4*a^3*b 
^5*c^2*d^10 + a^4*b^4*d^12)*x^6 + 2*(b^8*c^9*d^3 + 4*a*b^7*c^7*d^5 + 6*a^2 
*b^6*c^5*d^7 + 4*a^3*b^5*c^3*d^9 + a^4*b^4*c*d^11)*x^5 + (b^8*c^10*d^2 + 6 
*a*b^7*c^8*d^4 + 14*a^2*b^6*c^6*d^6 + 16*a^3*b^5*c^4*d^8 + 9*a^4*b^4*c^2*d 
^10 + 2*a^5*b^3*d^12)*x^4 + 4*(a*b^7*c^9*d^3 + 4*a^2*b^6*c^7*d^5 + 6*a^...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 802 vs. \(2 (369) = 738\).

Time = 0.14 (sec) , antiderivative size = 802, normalized size of antiderivative = 2.07 \[ \int \frac {x^7}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=-\frac {3 \, {\left (a b c^{7} - 7 \, a^{2} c^{5} d^{2}\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 (a b c^{7} d - 7 \, a^{2} c^{5} d^{3}\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 (35 \, a^{2} b^{3} c^{6} d - 35 \, a^{3} b^{2} c^{4} d^{3} - 7 \, a^{4} b c^{2} d^{5} - a^{5} d^{7}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, {\left (b^{7} c^{10} + 5 \, a b^{6} c^{8} d^{2} + 10 \, a^{2} b^{5} c^{6} d^{4} + 10 \, a^{3} b^{4} c^{4} d^{6} + 5 \, a^{4} b^{3} c^{2} d^{8} + a^{5} b^{2} d^{10}\right )} \sqrt {a b}} - \frac {8 \, b^{5} c^{8} d x^{5} + 56 \, a b^{4} c^{6} d^{3} x^{5} - 39 \, a^{2} b^{3} c^{4} d^{5} x^{5} + 14 \, a^{3} b^{2} c^{2} d^{7} x^{5} + 5 \, a^{4} b d^{9} x^{5} + 4 \, b^{5} c^{9} x^{4} + 52 \, a b^{4} c^{7} d^{2} x^{4} - 66 \, a^{2} b^{3} c^{5} d^{4} x^{4} - 20 \, a^{3} b^{2} c^{3} d^{6} x^{4} - 2 \, a^{4} b c d^{8} x^{4} + 16 \, a b^{4} c^{8} d x^{3} + 97 \, a^{2} b^{3} c^{6} d^{3} x^{3} - 115 \, a^{3} b^{2} c^{4} d^{5} x^{3} - a^{4} b c^{2} d^{7} x^{3} + 3 \, a^{5} d^{9} x^{3} + 8 \, a b^{4} c^{9} x^{2} + 116 \, a^{2} b^{3} c^{7} d^{2} x^{2} - 104 \, a^{3} b^{2} c^{5} d^{4} x^{2} - 20 \, a^{4} b c^{3} d^{6} x^{2} + 8 \, a^{2} b^{3} c^{8} d x + 43 \, a^{3} b^{2} c^{6} d^{3} x - 70 \, a^{4} b c^{4} d^{5} x - 9 \, a^{5} c^{2} d^{7} x + 4 \, a^{2} b^{3} c^{9} + 62 \, a^{3} b^{2} c^{7} d^{2} - 44 \, a^{4} b c^{5} d^{4} - 6 \, a^{5} c^{3} d^{6}}{8 \, {\left (b^{6} c^{8} d^{2} + 4 \, a b^{5} c^{6} d^{4} + 6 \, a^{2} b^{4} c^{4} d^{6} + 4 \, a^{3} b^{3} c^{2} d^{8} + a^{4} b^{2} d^{10}\right )} {\left (b d x^{3} + b c x^{2} + a d x + a c\right )}^{2}} \] Input:

integrate(x^7/(d*x+c)^3/(b*x^2+a)^3,x, algorithm="giac")
 

Output:

-3/2*(a*b*c^7 - 7*a^2*c^5*d^2)*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*(a*b*c^7*d - 7*a^2*c^5*d^3)*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*(35*a^2*b^3*c^6*d - 35*a^3*b^2*c^4*d^3 - 7*a^4*b*c^2*d^5 - a^5*d^7 
)*arctan(b*x/sqrt(a*b))/((b^7*c^10 + 5*a*b^6*c^8*d^2 + 10*a^2*b^5*c^6*d^4 
+ 10*a^3*b^4*c^4*d^6 + 5*a^4*b^3*c^2*d^8 + a^5*b^2*d^10)*sqrt(a*b)) - 1/8* 
(8*b^5*c^8*d*x^5 + 56*a*b^4*c^6*d^3*x^5 - 39*a^2*b^3*c^4*d^5*x^5 + 14*a^3* 
b^2*c^2*d^7*x^5 + 5*a^4*b*d^9*x^5 + 4*b^5*c^9*x^4 + 52*a*b^4*c^7*d^2*x^4 - 
 66*a^2*b^3*c^5*d^4*x^4 - 20*a^3*b^2*c^3*d^6*x^4 - 2*a^4*b*c*d^8*x^4 + 16* 
a*b^4*c^8*d*x^3 + 97*a^2*b^3*c^6*d^3*x^3 - 115*a^3*b^2*c^4*d^5*x^3 - a^4*b 
*c^2*d^7*x^3 + 3*a^5*d^9*x^3 + 8*a*b^4*c^9*x^2 + 116*a^2*b^3*c^7*d^2*x^2 - 
 104*a^3*b^2*c^5*d^4*x^2 - 20*a^4*b*c^3*d^6*x^2 + 8*a^2*b^3*c^8*d*x + 43*a 
^3*b^2*c^6*d^3*x - 70*a^4*b*c^4*d^5*x - 9*a^5*c^2*d^7*x + 4*a^2*b^3*c^9 + 
62*a^3*b^2*c^7*d^2 - 44*a^4*b*c^5*d^4 - 6*a^5*c^3*d^6)/((b^6*c^8*d^2 + 4*a 
*b^5*c^6*d^4 + 6*a^2*b^4*c^4*d^6 + 4*a^3*b^3*c^2*d^8 + a^4*b^2*d^10)*(b*d* 
x^3 + b*c*x^2 + a*d*x + a*c)^2)
 

Mupad [B] (verification not implemented)

Time = 8.74 (sec) , antiderivative size = 1978, normalized size of antiderivative = 5.10 \[ \int \frac {x^7}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

int(x^7/((a + b*x^2)^3*(c + d*x)^3),x)
 

Output:

(log(576*a^2*b^14*c^18*x + a^11*b^5*d^18*x + 576*a*b^11*c^18*(-a^3*b^5)^(1 
/2) + a^10*b^2*d^18*(-a^3*b^5)^(1/2) - 672*a^4*c^6*d^12*(-a^3*b^5)^(3/2) - 
 12672*b^4*c^14*d^4*(-a^3*b^5)^(3/2) - 45444*a*b^3*c^12*d^6*(-a^3*b^5)^(3/ 
2) - 1694*a^3*b*c^8*d^10*(-a^3*b^5)^(3/2) - 27664*a^2*b^2*c^10*d^8*(-a^3*b 
^5)^(3/2) - 5687*a^2*b^10*c^16*d^2*(-a^3*b^5)^(1/2) + 148*a^8*b^4*c^4*d^14 
*(-a^3*b^5)^(1/2) + 16*a^9*b^3*c^2*d^16*(-a^3*b^5)^(1/2) - 5687*a^3*b^13*c 
^16*d^2*x + 12672*a^4*b^12*c^14*d^4*x + 45444*a^5*b^11*c^12*d^6*x + 27664* 
a^6*b^10*c^10*d^8*x + 1694*a^7*b^9*c^8*d^10*x + 672*a^8*b^8*c^6*d^12*x + 1 
48*a^9*b^7*c^4*d^14*x + 16*a^10*b^6*c^2*d^16*x)*(a^2*((21*b^5*c^5*d^2)/2 + 
 (21*b*c^2*d^5*(-a^3*b^5)^(1/2))/16) - a*((3*b^6*c^7)/2 - (105*b^2*c^4*d^3 
*(-a^3*b^5)^(1/2))/16) + (3*a^3*d^7*(-a^3*b^5)^(1/2))/16 - (105*b^3*c^6*d* 
(-a^3*b^5)^(1/2))/16))/(b^10*c^10 + a^5*b^5*d^10 + 5*a*b^9*c^8*d^2 + 10*a^ 
2*b^8*c^6*d^4 + 10*a^3*b^7*c^4*d^6 + 5*a^4*b^6*c^2*d^8) - (log(c + d*x)*(2 
1*a^2*c^5*d^2 - 3*a*b*c^7))/(a^5*d^10 + b^5*c^10 + 5*a*b^4*c^8*d^2 + 5*a^4 
*b*c^2*d^8 + 10*a^2*b^3*c^6*d^4 + 10*a^3*b^2*c^4*d^6) - ((x^5*(5*a^4*d^8 + 
 8*b^4*c^8 + 56*a*b^3*c^6*d^2 + 14*a^3*b*c^2*d^6 - 39*a^2*b^2*c^4*d^4))/(8 
*d*(b^5*c^8 + a^4*b*d^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)) + (x^2*(2*a*b^3*c^9 - 5*a^4*c^3*d^6 - 26*a^3*b*c^5*d^4 + 29*a^2* 
b^2*c^7*d^2))/(2*b*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^4*(a^4*c*d^8 - 2*b^4*c^9 - 26*a*b^3*c^7*...
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 2776, normalized size of antiderivative = 7.15 \[ \int \frac {x^7}{(c+d x)^3 \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int(x^7/(d*x+c)^3/(b*x^2+a)^3,x)
 

Output:

(6*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**6*c**3*d**7 + 12*sqrt( 
b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**6*c**2*d**8*x + 6*sqrt(b)*sqrt 
(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**6*c*d**9*x**2 + 42*sqrt(b)*sqrt(a)*at 
an((b*x)/(sqrt(b)*sqrt(a)))*a**5*b*c**5*d**5 + 84*sqrt(b)*sqrt(a)*atan((b* 
x)/(sqrt(b)*sqrt(a)))*a**5*b*c**4*d**6*x + 54*sqrt(b)*sqrt(a)*atan((b*x)/( 
sqrt(b)*sqrt(a)))*a**5*b*c**3*d**7*x**2 + 24*sqrt(b)*sqrt(a)*atan((b*x)/(s 
qrt(b)*sqrt(a)))*a**5*b*c**2*d**8*x**3 + 12*sqrt(b)*sqrt(a)*atan((b*x)/(sq 
rt(b)*sqrt(a)))*a**5*b*c*d**9*x**4 + 210*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt( 
b)*sqrt(a)))*a**4*b**2*c**7*d**3 + 420*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b) 
*sqrt(a)))*a**4*b**2*c**6*d**4*x + 294*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b) 
*sqrt(a)))*a**4*b**2*c**5*d**5*x**2 + 168*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt 
(b)*sqrt(a)))*a**4*b**2*c**4*d**6*x**3 + 90*sqrt(b)*sqrt(a)*atan((b*x)/(sq 
rt(b)*sqrt(a)))*a**4*b**2*c**3*d**7*x**4 + 12*sqrt(b)*sqrt(a)*atan((b*x)/( 
sqrt(b)*sqrt(a)))*a**4*b**2*c**2*d**8*x**5 + 6*sqrt(b)*sqrt(a)*atan((b*x)/ 
(sqrt(b)*sqrt(a)))*a**4*b**2*c*d**9*x**6 - 210*sqrt(b)*sqrt(a)*atan((b*x)/ 
(sqrt(b)*sqrt(a)))*a**3*b**3*c**9*d - 420*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt 
(b)*sqrt(a)))*a**3*b**3*c**8*d**2*x + 210*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt 
(b)*sqrt(a)))*a**3*b**3*c**7*d**3*x**2 + 840*sqrt(b)*sqrt(a)*atan((b*x)/(s 
qrt(b)*sqrt(a)))*a**3*b**3*c**6*d**4*x**3 + 462*sqrt(b)*sqrt(a)*atan((b*x) 
/(sqrt(b)*sqrt(a)))*a**3*b**3*c**5*d**5*x**4 + 84*sqrt(b)*sqrt(a)*atan(...