\(\int \frac {x^4 (c+d x^2)^3}{(a+b x^2)^2} \, dx\) [674]

Optimal result

Integrand size = 22, antiderivative size = 150 \[ \int \frac {x^4 \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {(b c-4 a d) (b c-a d)^2 x}{b^5}+\frac {d (b c-a d)^2 x^3}{b^4}+\frac {d^2 (3 b c-2 a d) x^5}{5 b^3}+\frac {d^3 x^7}{7 b^2}+\frac {a (b c-a d)^3 x}{2 b^5 \left (a+b x^2\right )}-\frac {3 \sqrt {a} (b c-3 a d) (b c-a d)^2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{11/2}} \] Output:

(-4*a*d+b*c)*(-a*d+b*c)^2*x/b^5+d*(-a*d+b*c)^2*x^3/b^4+1/5*d^2*(-2*a*d+3*b 
*c)*x^5/b^3+1/7*d^3*x^7/b^2+1/2*a*(-a*d+b*c)^3*x/b^5/(b*x^2+a)-3/2*a^(1/2) 
*(-3*a*d+b*c)*(-a*d+b*c)^2*arctan(b^(1/2)*x/a^(1/2))/b^(11/2)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.01 \[ \int \frac {x^4 \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {(b c-4 a d) (b c-a d)^2 x}{b^5}+\frac {d (b c-a d)^2 x^3}{b^4}+\frac {d^2 (3 b c-2 a d) x^5}{5 b^3}+\frac {d^3 x^7}{7 b^2}+\frac {a (b c-a d)^3 x}{2 b^5 \left (a+b x^2\right )}+\frac {3 \sqrt {a} (b c-a d)^2 (-b c+3 a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{11/2}} \] Input:

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

Output:

((b*c - 4*a*d)*(b*c - a*d)^2*x)/b^5 + (d*(b*c - a*d)^2*x^3)/b^4 + (d^2*(3* 
b*c - 2*a*d)*x^5)/(5*b^3) + (d^3*x^7)/(7*b^2) + (a*(b*c - a*d)^3*x)/(2*b^5 
*(a + b*x^2)) + (3*Sqrt[a]*(b*c - a*d)^2*(-(b*c) + 3*a*d)*ArcTan[(Sqrt[b]* 
x)/Sqrt[a]])/(2*b^(11/2))
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.15, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {369, 27, 437, 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.

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

\(\Big \downarrow \) 369

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 437

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 \left (\frac {d x^3 \left (3 a^2 d^2-7 a b c d+5 b^2 c^2\right )}{3 b^3}-\frac {\sqrt {a} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-3 a d) (b c-a d)^2}{b^{9/2}}+\frac {x (b c-3 a d) (b c-a d)^2}{b^4}+\frac {d^2 x^5 (7 b c-3 a d)}{5 b^2}+\frac {3 d^3 x^7}{7 b}\right )}{2 b}-\frac {x^3 \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}\)

Input:

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

Output:

-1/2*(x^3*(c + d*x^2)^3)/(b*(a + b*x^2)) + (3*(((b*c - 3*a*d)*(b*c - a*d)^ 
2*x)/b^4 + (d*(5*b^2*c^2 - 7*a*b*c*d + 3*a^2*d^2)*x^3)/(3*b^3) + (d^2*(7*b 
*c - 3*a*d)*x^5)/(5*b^2) + (3*d^3*x^7)/(7*b) - (Sqrt[a]*(b*c - 3*a*d)*(b*c 
 - a*d)^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(9/2)))/(2*b)
 

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 369

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2* 
b*(p + 1))), x] - Simp[e^2/(2*b*(p + 1))   Int[(e*x)^(m - 2)*(a + b*x^2)^(p 
 + 1)*(c + d*x^2)^(q - 1)*Simp[c*(m - 1) + d*(m + 2*q - 1)*x^2, x], x], x] 
/; FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 0 
] && GtQ[m, 1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
 

rule 437

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q 
_.)*((e_) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*( 
a + b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^r, x], x] /; FreeQ[{a, b, c, d, e, f 
, g, m}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.51

method	result	size
default	\(-\frac {-\frac {1}{7} b^{3} d^{3} x^{7}+\frac {2}{5} a \,b^{2} d^{3} x^{5}-\frac {3}{5} b^{3} c \,d^{2} x^{5}-a^{2} b \,d^{3} x^{3}+2 a \,b^{2} c \,d^{2} x^{3}-b^{3} c^{2} d \,x^{3}+4 a^{3} d^{3} x -9 a^{2} b c \,d^{2} x +6 a \,b^{2} c^{2} d x -b^{3} c^{3} x}{b^{5}}+\frac {a \left (\frac {\left (-\frac {1}{2} a^{3} d^{3}+\frac {3}{2} a^{2} b c \,d^{2}-\frac {3}{2} a \,b^{2} c^{2} d +\frac {1}{2} b^{3} c^{3}\right ) x}{b \,x^{2}+a}+\frac {3 \left (3 a^{3} d^{3}-7 a^{2} b c \,d^{2}+5 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{5}}\)	\(227\)
risch	\(\frac {d^{3} x^{7}}{7 b^{2}}-\frac {2 a \,d^{3} x^{5}}{5 b^{3}}+\frac {3 c \,d^{2} x^{5}}{5 b^{2}}+\frac {a^{2} d^{3} x^{3}}{b^{4}}-\frac {2 a c \,d^{2} x^{3}}{b^{3}}+\frac {c^{2} d \,x^{3}}{b^{2}}-\frac {4 a^{3} d^{3} x}{b^{5}}+\frac {9 a^{2} c \,d^{2} x}{b^{4}}-\frac {6 a \,c^{2} d x}{b^{3}}+\frac {c^{3} x}{b^{2}}+\frac {\left (-\frac {1}{2} a^{4} d^{3}+\frac {3}{2} a^{3} b c \,d^{2}-\frac {3}{2} a^{2} b^{2} c^{2} d +\frac {1}{2} a \,b^{3} c^{3}\right ) x}{b^{5} \left (b \,x^{2}+a \right )}+\frac {9 \sqrt {-a b}\, \ln \left (-\sqrt {-a b}\, x +a \right ) a^{3} d^{3}}{4 b^{6}}-\frac {21 \sqrt {-a b}\, \ln \left (-\sqrt {-a b}\, x +a \right ) a^{2} c \,d^{2}}{4 b^{5}}+\frac {15 \sqrt {-a b}\, \ln \left (-\sqrt {-a b}\, x +a \right ) a \,c^{2} d}{4 b^{4}}-\frac {3 \sqrt {-a b}\, \ln \left (-\sqrt {-a b}\, x +a \right ) c^{3}}{4 b^{3}}-\frac {9 \sqrt {-a b}\, \ln \left (\sqrt {-a b}\, x +a \right ) a^{3} d^{3}}{4 b^{6}}+\frac {21 \sqrt {-a b}\, \ln \left (\sqrt {-a b}\, x +a \right ) a^{2} c \,d^{2}}{4 b^{5}}-\frac {15 \sqrt {-a b}\, \ln \left (\sqrt {-a b}\, x +a \right ) a \,c^{2} d}{4 b^{4}}+\frac {3 \sqrt {-a b}\, \ln \left (\sqrt {-a b}\, x +a \right ) c^{3}}{4 b^{3}}\)	\(394\)

Input:

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

Output:

-1/b^5*(-1/7*b^3*d^3*x^7+2/5*a*b^2*d^3*x^5-3/5*b^3*c*d^2*x^5-a^2*b*d^3*x^3 
+2*a*b^2*c*d^2*x^3-b^3*c^2*d*x^3+4*a^3*d^3*x-9*a^2*b*c*d^2*x+6*a*b^2*c^2*d 
*x-b^3*c^3*x)+a/b^5*((-1/2*a^3*d^3+3/2*a^2*b*c*d^2-3/2*a*b^2*c^2*d+1/2*b^3 
*c^3)*x/(b*x^2+a)+3/2*(3*a^3*d^3-7*a^2*b*c*d^2+5*a*b^2*c^2*d-b^3*c^3)/(a*b 
)^(1/2)*arctan(b*x/(a*b)^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (134) = 268\).

Time = 0.11 (sec) , antiderivative size = 580, normalized size of antiderivative = 3.87 \[ \int \frac {x^4 \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\left [\frac {20 \, b^{4} d^{3} x^{9} + 12 \, {\left (7 \, b^{4} c d^{2} - 3 \, a b^{3} d^{3}\right )} x^{7} + 28 \, {\left (5 \, b^{4} c^{2} d - 7 \, a b^{3} c d^{2} + 3 \, a^{2} b^{2} d^{3}\right )} x^{5} + 140 \, {\left (b^{4} c^{3} - 5 \, a b^{3} c^{2} d + 7 \, a^{2} b^{2} c d^{2} - 3 \, a^{3} b d^{3}\right )} x^{3} - 105 \, {\left (a b^{3} c^{3} - 5 \, a^{2} b^{2} c^{2} d + 7 \, a^{3} b c d^{2} - 3 \, a^{4} d^{3} + {\left (b^{4} c^{3} - 5 \, a b^{3} c^{2} d + 7 \, a^{2} b^{2} c d^{2} - 3 \, a^{3} b d^{3}\right )} x^{2}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 210 \, {\left (a b^{3} c^{3} - 5 \, a^{2} b^{2} c^{2} d + 7 \, a^{3} b c d^{2} - 3 \, a^{4} d^{3}\right )} x}{140 \, {\left (b^{6} x^{2} + a b^{5}\right )}}, \frac {10 \, b^{4} d^{3} x^{9} + 6 \, {\left (7 \, b^{4} c d^{2} - 3 \, a b^{3} d^{3}\right )} x^{7} + 14 \, {\left (5 \, b^{4} c^{2} d - 7 \, a b^{3} c d^{2} + 3 \, a^{2} b^{2} d^{3}\right )} x^{5} + 70 \, {\left (b^{4} c^{3} - 5 \, a b^{3} c^{2} d + 7 \, a^{2} b^{2} c d^{2} - 3 \, a^{3} b d^{3}\right )} x^{3} - 105 \, {\left (a b^{3} c^{3} - 5 \, a^{2} b^{2} c^{2} d + 7 \, a^{3} b c d^{2} - 3 \, a^{4} d^{3} + {\left (b^{4} c^{3} - 5 \, a b^{3} c^{2} d + 7 \, a^{2} b^{2} c d^{2} - 3 \, a^{3} b d^{3}\right )} x^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 105 \, {\left (a b^{3} c^{3} - 5 \, a^{2} b^{2} c^{2} d + 7 \, a^{3} b c d^{2} - 3 \, a^{4} d^{3}\right )} x}{70 \, {\left (b^{6} x^{2} + a b^{5}\right )}}\right ] \] Input:

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

Output:

[1/140*(20*b^4*d^3*x^9 + 12*(7*b^4*c*d^2 - 3*a*b^3*d^3)*x^7 + 28*(5*b^4*c^ 
2*d - 7*a*b^3*c*d^2 + 3*a^2*b^2*d^3)*x^5 + 140*(b^4*c^3 - 5*a*b^3*c^2*d + 
7*a^2*b^2*c*d^2 - 3*a^3*b*d^3)*x^3 - 105*(a*b^3*c^3 - 5*a^2*b^2*c^2*d + 7* 
a^3*b*c*d^2 - 3*a^4*d^3 + (b^4*c^3 - 5*a*b^3*c^2*d + 7*a^2*b^2*c*d^2 - 3*a 
^3*b*d^3)*x^2)*sqrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) 
+ 210*(a*b^3*c^3 - 5*a^2*b^2*c^2*d + 7*a^3*b*c*d^2 - 3*a^4*d^3)*x)/(b^6*x^ 
2 + a*b^5), 1/70*(10*b^4*d^3*x^9 + 6*(7*b^4*c*d^2 - 3*a*b^3*d^3)*x^7 + 14* 
(5*b^4*c^2*d - 7*a*b^3*c*d^2 + 3*a^2*b^2*d^3)*x^5 + 70*(b^4*c^3 - 5*a*b^3* 
c^2*d + 7*a^2*b^2*c*d^2 - 3*a^3*b*d^3)*x^3 - 105*(a*b^3*c^3 - 5*a^2*b^2*c^ 
2*d + 7*a^3*b*c*d^2 - 3*a^4*d^3 + (b^4*c^3 - 5*a*b^3*c^2*d + 7*a^2*b^2*c*d 
^2 - 3*a^3*b*d^3)*x^2)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) + 105*(a*b^3*c^3 
- 5*a^2*b^2*c^2*d + 7*a^3*b*c*d^2 - 3*a^4*d^3)*x)/(b^6*x^2 + a*b^5)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (139) = 278\).

Time = 0.68 (sec) , antiderivative size = 389, normalized size of antiderivative = 2.59 \[ \int \frac {x^4 \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=x^{5} \left (- \frac {2 a d^{3}}{5 b^{3}} + \frac {3 c d^{2}}{5 b^{2}}\right ) + x^{3} \left (\frac {a^{2} d^{3}}{b^{4}} - \frac {2 a c d^{2}}{b^{3}} + \frac {c^{2} d}{b^{2}}\right ) + x \left (- \frac {4 a^{3} d^{3}}{b^{5}} + \frac {9 a^{2} c d^{2}}{b^{4}} - \frac {6 a c^{2} d}{b^{3}} + \frac {c^{3}}{b^{2}}\right ) + \frac {x \left (- a^{4} d^{3} + 3 a^{3} b c d^{2} - 3 a^{2} b^{2} c^{2} d + a b^{3} c^{3}\right )}{2 a b^{5} + 2 b^{6} x^{2}} - \frac {3 \sqrt {- \frac {a}{b^{11}}} \left (a d - b c\right )^{2} \cdot \left (3 a d - b c\right ) \log {\left (- \frac {3 b^{5} \sqrt {- \frac {a}{b^{11}}} \left (a d - b c\right )^{2} \cdot \left (3 a d - b c\right )}{9 a^{3} d^{3} - 21 a^{2} b c d^{2} + 15 a b^{2} c^{2} d - 3 b^{3} c^{3}} + x \right )}}{4} + \frac {3 \sqrt {- \frac {a}{b^{11}}} \left (a d - b c\right )^{2} \cdot \left (3 a d - b c\right ) \log {\left (\frac {3 b^{5} \sqrt {- \frac {a}{b^{11}}} \left (a d - b c\right )^{2} \cdot \left (3 a d - b c\right )}{9 a^{3} d^{3} - 21 a^{2} b c d^{2} + 15 a b^{2} c^{2} d - 3 b^{3} c^{3}} + x \right )}}{4} + \frac {d^{3} x^{7}}{7 b^{2}} \] Input:

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

Output:

x**5*(-2*a*d**3/(5*b**3) + 3*c*d**2/(5*b**2)) + x**3*(a**2*d**3/b**4 - 2*a 
*c*d**2/b**3 + c**2*d/b**2) + x*(-4*a**3*d**3/b**5 + 9*a**2*c*d**2/b**4 - 
6*a*c**2*d/b**3 + c**3/b**2) + x*(-a**4*d**3 + 3*a**3*b*c*d**2 - 3*a**2*b* 
*2*c**2*d + a*b**3*c**3)/(2*a*b**5 + 2*b**6*x**2) - 3*sqrt(-a/b**11)*(a*d 
- b*c)**2*(3*a*d - b*c)*log(-3*b**5*sqrt(-a/b**11)*(a*d - b*c)**2*(3*a*d - 
 b*c)/(9*a**3*d**3 - 21*a**2*b*c*d**2 + 15*a*b**2*c**2*d - 3*b**3*c**3) + 
x)/4 + 3*sqrt(-a/b**11)*(a*d - b*c)**2*(3*a*d - b*c)*log(3*b**5*sqrt(-a/b* 
*11)*(a*d - b*c)**2*(3*a*d - b*c)/(9*a**3*d**3 - 21*a**2*b*c*d**2 + 15*a*b 
**2*c**2*d - 3*b**3*c**3) + x)/4 + d**3*x**7/(7*b**2)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.52 \[ \int \frac {x^4 \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x}{2 \, {\left (b^{6} x^{2} + a b^{5}\right )}} - \frac {3 \, {\left (a b^{3} c^{3} - 5 \, a^{2} b^{2} c^{2} d + 7 \, a^{3} b c d^{2} - 3 \, a^{4} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{5}} + \frac {5 \, b^{3} d^{3} x^{7} + 7 \, {\left (3 \, b^{3} c d^{2} - 2 \, a b^{2} d^{3}\right )} x^{5} + 35 \, {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{3} + 35 \, {\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3}\right )} x}{35 \, b^{5}} \] Input:

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

Output:

1/2*(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*x/(b^6*x^2 + a 
*b^5) - 3/2*(a*b^3*c^3 - 5*a^2*b^2*c^2*d + 7*a^3*b*c*d^2 - 3*a^4*d^3)*arct 
an(b*x/sqrt(a*b))/(sqrt(a*b)*b^5) + 1/35*(5*b^3*d^3*x^7 + 7*(3*b^3*c*d^2 - 
 2*a*b^2*d^3)*x^5 + 35*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^3 + 35*(b 
^3*c^3 - 6*a*b^2*c^2*d + 9*a^2*b*c*d^2 - 4*a^3*d^3)*x)/b^5
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.61 \[ \int \frac {x^4 \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=-\frac {3 \, {\left (a b^{3} c^{3} - 5 \, a^{2} b^{2} c^{2} d + 7 \, a^{3} b c d^{2} - 3 \, a^{4} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{5}} + \frac {a b^{3} c^{3} x - 3 \, a^{2} b^{2} c^{2} d x + 3 \, a^{3} b c d^{2} x - a^{4} d^{3} x}{2 \, {\left (b x^{2} + a\right )} b^{5}} + \frac {5 \, b^{12} d^{3} x^{7} + 21 \, b^{12} c d^{2} x^{5} - 14 \, a b^{11} d^{3} x^{5} + 35 \, b^{12} c^{2} d x^{3} - 70 \, a b^{11} c d^{2} x^{3} + 35 \, a^{2} b^{10} d^{3} x^{3} + 35 \, b^{12} c^{3} x - 210 \, a b^{11} c^{2} d x + 315 \, a^{2} b^{10} c d^{2} x - 140 \, a^{3} b^{9} d^{3} x}{35 \, b^{14}} \] Input:

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

Output:

-3/2*(a*b^3*c^3 - 5*a^2*b^2*c^2*d + 7*a^3*b*c*d^2 - 3*a^4*d^3)*arctan(b*x/ 
sqrt(a*b))/(sqrt(a*b)*b^5) + 1/2*(a*b^3*c^3*x - 3*a^2*b^2*c^2*d*x + 3*a^3* 
b*c*d^2*x - a^4*d^3*x)/((b*x^2 + a)*b^5) + 1/35*(5*b^12*d^3*x^7 + 21*b^12* 
c*d^2*x^5 - 14*a*b^11*d^3*x^5 + 35*b^12*c^2*d*x^3 - 70*a*b^11*c*d^2*x^3 + 
35*a^2*b^10*d^3*x^3 + 35*b^12*c^3*x - 210*a*b^11*c^2*d*x + 315*a^2*b^10*c* 
d^2*x - 140*a^3*b^9*d^3*x)/b^14
 

Mupad [B] (verification not implemented)

Time = 0.59 (sec) , antiderivative size = 328, normalized size of antiderivative = 2.19 \[ \int \frac {x^4 \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=x\,\left (\frac {c^3}{b^2}-\frac {2\,a\,\left (\frac {3\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{b^4}\right )}{b}+\frac {a^2\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b^2}\right )-x^5\,\left (\frac {2\,a\,d^3}{5\,b^3}-\frac {3\,c\,d^2}{5\,b^2}\right )+x^3\,\left (\frac {c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{3\,b}-\frac {a^2\,d^3}{3\,b^4}\right )-\frac {x\,\left (\frac {a^4\,d^3}{2}-\frac {3\,a^3\,b\,c\,d^2}{2}+\frac {3\,a^2\,b^2\,c^2\,d}{2}-\frac {a\,b^3\,c^3}{2}\right )}{b^6\,x^2+a\,b^5}+\frac {d^3\,x^7}{7\,b^2}+\frac {3\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d-b\,c\right )}{3\,a^4\,d^3-7\,a^3\,b\,c\,d^2+5\,a^2\,b^2\,c^2\,d-a\,b^3\,c^3}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d-b\,c\right )}{2\,b^{11/2}} \] Input:

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

Output:

x*(c^3/b^2 - (2*a*((3*c^2*d)/b^2 + (2*a*((2*a*d^3)/b^3 - (3*c*d^2)/b^2))/b 
 - (a^2*d^3)/b^4))/b + (a^2*((2*a*d^3)/b^3 - (3*c*d^2)/b^2))/b^2) - x^5*(( 
2*a*d^3)/(5*b^3) - (3*c*d^2)/(5*b^2)) + x^3*((c^2*d)/b^2 + (2*a*((2*a*d^3) 
/b^3 - (3*c*d^2)/b^2))/(3*b) - (a^2*d^3)/(3*b^4)) - (x*((a^4*d^3)/2 - (a*b 
^3*c^3)/2 + (3*a^2*b^2*c^2*d)/2 - (3*a^3*b*c*d^2)/2))/(a*b^5 + b^6*x^2) + 
(d^3*x^7)/(7*b^2) + (3*a^(1/2)*atan((a^(1/2)*b^(1/2)*x*(a*d - b*c)^2*(3*a* 
d - b*c))/(3*a^4*d^3 - a*b^3*c^3 + 5*a^2*b^2*c^2*d - 7*a^3*b*c*d^2))*(a*d 
- b*c)^2*(3*a*d - b*c))/(2*b^(11/2))
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 406, normalized size of antiderivative = 2.71 \[ \int \frac {x^4 \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {315 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{4} d^{3}-735 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b c \,d^{2}+315 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b \,d^{3} x^{2}+525 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{2} c^{2} d -735 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{2} c \,d^{2} x^{2}-105 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{3} c^{3}+525 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{3} c^{2} d \,x^{2}-105 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{4} c^{3} x^{2}-315 a^{4} b \,d^{3} x +735 a^{3} b^{2} c \,d^{2} x -210 a^{3} b^{2} d^{3} x^{3}-525 a^{2} b^{3} c^{2} d x +490 a^{2} b^{3} c \,d^{2} x^{3}+42 a^{2} b^{3} d^{3} x^{5}+105 a \,b^{4} c^{3} x -350 a \,b^{4} c^{2} d \,x^{3}-98 a \,b^{4} c \,d^{2} x^{5}-18 a \,b^{4} d^{3} x^{7}+70 b^{5} c^{3} x^{3}+70 b^{5} c^{2} d \,x^{5}+42 b^{5} c \,d^{2} x^{7}+10 b^{5} d^{3} x^{9}}{70 b^{6} \left (b \,x^{2}+a \right )} \] Input:

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

Output:

(315*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*d**3 - 735*sqrt(b) 
*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b*c*d**2 + 315*sqrt(b)*sqrt(a) 
*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b*d**3*x**2 + 525*sqrt(b)*sqrt(a)*atan 
((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**2*c**2*d - 735*sqrt(b)*sqrt(a)*atan((b*x 
)/(sqrt(b)*sqrt(a)))*a**2*b**2*c*d**2*x**2 - 105*sqrt(b)*sqrt(a)*atan((b*x 
)/(sqrt(b)*sqrt(a)))*a*b**3*c**3 + 525*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b) 
*sqrt(a)))*a*b**3*c**2*d*x**2 - 105*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sq 
rt(a)))*b**4*c**3*x**2 - 315*a**4*b*d**3*x + 735*a**3*b**2*c*d**2*x - 210* 
a**3*b**2*d**3*x**3 - 525*a**2*b**3*c**2*d*x + 490*a**2*b**3*c*d**2*x**3 + 
 42*a**2*b**3*d**3*x**5 + 105*a*b**4*c**3*x - 350*a*b**4*c**2*d*x**3 - 98* 
a*b**4*c*d**2*x**5 - 18*a*b**4*d**3*x**7 + 70*b**5*c**3*x**3 + 70*b**5*c** 
2*d*x**5 + 42*b**5*c*d**2*x**7 + 10*b**5*d**3*x**9)/(70*b**6*(a + b*x**2))