Integrand size = 22, antiderivative size = 103 \[ \int \frac {x^5 \left (c+d x^2\right )^2}{a+b x^2} \, dx=-\frac {a (b c-a d)^2 x^2}{2 b^4}+\frac {(b c-a d)^2 x^4}{4 b^3}+\frac {d (2 b c-a d) x^6}{6 b^2}+\frac {d^2 x^8}{8 b}+\frac {a^2 (b c-a d)^2 \log \left (a+b x^2\right )}{2 b^5} \] Output:
-1/2*a*(-a*d+b*c)^2*x^2/b^4+1/4*(-a*d+b*c)^2*x^4/b^3+1/6*d*(-a*d+2*b*c)*x^ 6/b^2+1/8*d^2*x^8/b+1/2*a^2*(-a*d+b*c)^2*ln(b*x^2+a)/b^5
Time = 0.04 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.13 \[ \int \frac {x^5 \left (c+d x^2\right )^2}{a+b x^2} \, dx=-\frac {a (-b c+a d)^2 x^2}{2 b^4}+\frac {(b c-a d)^2 x^4}{4 b^3}+\frac {d (2 b c-a d) x^6}{6 b^2}+\frac {d^2 x^8}{8 b}+\frac {\left (a^2 b^2 c^2-2 a^3 b c d+a^4 d^2\right ) \log \left (a+b x^2\right )}{2 b^5} \] Input:
Integrate[(x^5*(c + d*x^2)^2)/(a + b*x^2),x]
Output:
-1/2*(a*(-(b*c) + a*d)^2*x^2)/b^4 + ((b*c - a*d)^2*x^4)/(4*b^3) + (d*(2*b* c - a*d)*x^6)/(6*b^2) + (d^2*x^8)/(8*b) + ((a^2*b^2*c^2 - 2*a^3*b*c*d + a^ 4*d^2)*Log[a + b*x^2])/(2*b^5)
Time = 0.28 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {354, 99, 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.
\(\displaystyle \int \frac {x^5 \left (c+d x^2\right )^2}{a+b x^2} \, dx\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {1}{2} \int \frac {x^4 \left (d x^2+c\right )^2}{b x^2+a}dx^2\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {1}{2} \int \left (\frac {d^2 x^6}{b}+\frac {d (2 b c-a d) x^4}{b^2}+\frac {(b c-a d)^2 x^2}{b^3}-\frac {a (a d-b c)^2}{b^4}+\frac {a^2 (a d-b c)^2}{b^4 \left (b x^2+a\right )}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {a^2 (b c-a d)^2 \log \left (a+b x^2\right )}{b^5}-\frac {a x^2 (b c-a d)^2}{b^4}+\frac {x^4 (b c-a d)^2}{2 b^3}+\frac {d x^6 (2 b c-a d)}{3 b^2}+\frac {d^2 x^8}{4 b}\right )\) |
Input:
Int[(x^5*(c + d*x^2)^2)/(a + b*x^2),x]
Output:
(-((a*(b*c - a*d)^2*x^2)/b^4) + ((b*c - a*d)^2*x^4)/(2*b^3) + (d*(2*b*c - a*d)*x^6)/(3*b^2) + (d^2*x^8)/(4*b) + (a^2*(b*c - a*d)^2*Log[a + b*x^2])/b ^5)/2
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Time = 0.60 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.22
method | result | size |
norman | \(\frac {d^{2} x^{8}}{8 b}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x^{4}}{4 b^{3}}-\frac {a \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x^{2}}{2 b^{4}}-\frac {d \left (a d -2 b c \right ) x^{6}}{6 b^{2}}+\frac {a^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (b \,x^{2}+a \right )}{2 b^{5}}\) | \(126\) |
default | \(-\frac {-\frac {d^{2} x^{8} b^{3}}{4}+\frac {\left (\left (a d -b c \right ) d \,b^{2}-d \,b^{3} c \right ) x^{6}}{3}+\frac {\left (\left (a d -b c \right ) b^{2} c -d b \left (d \,a^{2}-a b c \right )\right ) x^{4}}{2}+\left (a d -b c \right ) \left (d \,a^{2}-a b c \right ) x^{2}}{2 b^{4}}+\frac {a^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (b \,x^{2}+a \right )}{2 b^{5}}\) | \(139\) |
parallelrisch | \(\frac {3 d^{2} x^{8} b^{4}-4 x^{6} a \,b^{3} d^{2}+8 x^{6} b^{4} c d +6 x^{4} a^{2} b^{2} d^{2}-12 x^{4} a \,b^{3} c d +6 x^{4} b^{4} c^{2}-12 a^{3} b \,d^{2} x^{2}+24 a^{2} b^{2} c d \,x^{2}-12 a \,b^{3} c^{2} x^{2}+12 \ln \left (b \,x^{2}+a \right ) a^{4} d^{2}-24 \ln \left (b \,x^{2}+a \right ) a^{3} b c d +12 \ln \left (b \,x^{2}+a \right ) a^{2} b^{2} c^{2}}{24 b^{5}}\) | \(164\) |
risch | \(-\frac {c^{4}}{24 b \,d^{2}}+\frac {7 d^{2} a^{4}}{24 b^{5}}-\frac {d^{2} a^{3} x^{2}}{2 b^{4}}-\frac {d a c \,x^{4}}{2 b^{2}}-\frac {d^{2} a \,x^{6}}{6 b^{2}}+\frac {d c \,x^{6}}{3 b}+\frac {d^{2} a^{2} x^{4}}{4 b^{3}}+\frac {c^{2} x^{4}}{4 b}-\frac {5 d \,a^{3} c}{6 b^{4}}+\frac {3 a^{2} c^{2}}{4 b^{3}}-\frac {a \,c^{3}}{6 b^{2} d}+\frac {d \,a^{2} c \,x^{2}}{b^{3}}-\frac {a \,c^{2} x^{2}}{2 b^{2}}+\frac {d^{2} x^{8}}{8 b}+\frac {a^{4} \ln \left (b \,x^{2}+a \right ) d^{2}}{2 b^{5}}-\frac {a^{3} \ln \left (b \,x^{2}+a \right ) c d}{b^{4}}+\frac {a^{2} \ln \left (b \,x^{2}+a \right ) c^{2}}{2 b^{3}}\) | \(220\) |
Input:
int(x^5*(d*x^2+c)^2/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
1/8*d^2*x^8/b+1/4*(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^3*x^4-1/2*a*(a^2*d^2-2*a*b *c*d+b^2*c^2)/b^4*x^2-1/6*d*(a*d-2*b*c)/b^2*x^6+1/2*a^2*(a^2*d^2-2*a*b*c*d +b^2*c^2)/b^5*ln(b*x^2+a)
Time = 0.13 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.34 \[ \int \frac {x^5 \left (c+d x^2\right )^2}{a+b x^2} \, dx=\frac {3 \, b^{4} d^{2} x^{8} + 4 \, {\left (2 \, b^{4} c d - a b^{3} d^{2}\right )} x^{6} + 6 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4} - 12 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2} + 12 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \log \left (b x^{2} + a\right )}{24 \, b^{5}} \] Input:
integrate(x^5*(d*x^2+c)^2/(b*x^2+a),x, algorithm="fricas")
Output:
1/24*(3*b^4*d^2*x^8 + 4*(2*b^4*c*d - a*b^3*d^2)*x^6 + 6*(b^4*c^2 - 2*a*b^3 *c*d + a^2*b^2*d^2)*x^4 - 12*(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*x^2 + 12*(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)*log(b*x^2 + a))/b^5
Time = 0.24 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.18 \[ \int \frac {x^5 \left (c+d x^2\right )^2}{a+b x^2} \, dx=\frac {a^{2} \left (a d - b c\right )^{2} \log {\left (a + b x^{2} \right )}}{2 b^{5}} + x^{6} \left (- \frac {a d^{2}}{6 b^{2}} + \frac {c d}{3 b}\right ) + x^{4} \left (\frac {a^{2} d^{2}}{4 b^{3}} - \frac {a c d}{2 b^{2}} + \frac {c^{2}}{4 b}\right ) + x^{2} \left (- \frac {a^{3} d^{2}}{2 b^{4}} + \frac {a^{2} c d}{b^{3}} - \frac {a c^{2}}{2 b^{2}}\right ) + \frac {d^{2} x^{8}}{8 b} \] Input:
integrate(x**5*(d*x**2+c)**2/(b*x**2+a),x)
Output:
a**2*(a*d - b*c)**2*log(a + b*x**2)/(2*b**5) + x**6*(-a*d**2/(6*b**2) + c* d/(3*b)) + x**4*(a**2*d**2/(4*b**3) - a*c*d/(2*b**2) + c**2/(4*b)) + x**2* (-a**3*d**2/(2*b**4) + a**2*c*d/b**3 - a*c**2/(2*b**2)) + d**2*x**8/(8*b)
Time = 0.06 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.33 \[ \int \frac {x^5 \left (c+d x^2\right )^2}{a+b x^2} \, dx=\frac {3 \, b^{3} d^{2} x^{8} + 4 \, {\left (2 \, b^{3} c d - a b^{2} d^{2}\right )} x^{6} + 6 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{4} - 12 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2}}{24 \, b^{4}} + \frac {{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, b^{5}} \] Input:
integrate(x^5*(d*x^2+c)^2/(b*x^2+a),x, algorithm="maxima")
Output:
1/24*(3*b^3*d^2*x^8 + 4*(2*b^3*c*d - a*b^2*d^2)*x^6 + 6*(b^3*c^2 - 2*a*b^2 *c*d + a^2*b*d^2)*x^4 - 12*(a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2)*x^2)/b^4 + 1/2*(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)*log(b*x^2 + a)/b^5
Time = 0.13 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.44 \[ \int \frac {x^5 \left (c+d x^2\right )^2}{a+b x^2} \, dx=\frac {3 \, b^{3} d^{2} x^{8} + 8 \, b^{3} c d x^{6} - 4 \, a b^{2} d^{2} x^{6} + 6 \, b^{3} c^{2} x^{4} - 12 \, a b^{2} c d x^{4} + 6 \, a^{2} b d^{2} x^{4} - 12 \, a b^{2} c^{2} x^{2} + 24 \, a^{2} b c d x^{2} - 12 \, a^{3} d^{2} x^{2}}{24 \, b^{4}} + \frac {{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{5}} \] Input:
integrate(x^5*(d*x^2+c)^2/(b*x^2+a),x, algorithm="giac")
Output:
1/24*(3*b^3*d^2*x^8 + 8*b^3*c*d*x^6 - 4*a*b^2*d^2*x^6 + 6*b^3*c^2*x^4 - 12 *a*b^2*c*d*x^4 + 6*a^2*b*d^2*x^4 - 12*a*b^2*c^2*x^2 + 24*a^2*b*c*d*x^2 - 1 2*a^3*d^2*x^2)/b^4 + 1/2*(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)*log(abs(b*x ^2 + a))/b^5
Time = 0.06 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.42 \[ \int \frac {x^5 \left (c+d x^2\right )^2}{a+b x^2} \, dx=x^4\,\left (\frac {c^2}{4\,b}+\frac {a\,\left (\frac {a\,d^2}{b^2}-\frac {2\,c\,d}{b}\right )}{4\,b}\right )-x^6\,\left (\frac {a\,d^2}{6\,b^2}-\frac {c\,d}{3\,b}\right )+\frac {\ln \left (b\,x^2+a\right )\,\left (a^4\,d^2-2\,a^3\,b\,c\,d+a^2\,b^2\,c^2\right )}{2\,b^5}+\frac {d^2\,x^8}{8\,b}-\frac {a\,x^2\,\left (\frac {c^2}{b}+\frac {a\,\left (\frac {a\,d^2}{b^2}-\frac {2\,c\,d}{b}\right )}{b}\right )}{2\,b} \] Input:
int((x^5*(c + d*x^2)^2)/(a + b*x^2),x)
Output:
x^4*(c^2/(4*b) + (a*((a*d^2)/b^2 - (2*c*d)/b))/(4*b)) - x^6*((a*d^2)/(6*b^ 2) - (c*d)/(3*b)) + (log(a + b*x^2)*(a^4*d^2 + a^2*b^2*c^2 - 2*a^3*b*c*d)) /(2*b^5) + (d^2*x^8)/(8*b) - (a*x^2*(c^2/b + (a*((a*d^2)/b^2 - (2*c*d)/b)) /b))/(2*b)
Time = 0.20 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.58 \[ \int \frac {x^5 \left (c+d x^2\right )^2}{a+b x^2} \, dx=\frac {12 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{4} d^{2}-24 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{3} b c d +12 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} b^{2} c^{2}-12 a^{3} b \,d^{2} x^{2}+24 a^{2} b^{2} c d \,x^{2}+6 a^{2} b^{2} d^{2} x^{4}-12 a \,b^{3} c^{2} x^{2}-12 a \,b^{3} c d \,x^{4}-4 a \,b^{3} d^{2} x^{6}+6 b^{4} c^{2} x^{4}+8 b^{4} c d \,x^{6}+3 b^{4} d^{2} x^{8}}{24 b^{5}} \] Input:
int(x^5*(d*x^2+c)^2/(b*x^2+a),x)
Output:
(12*log(a + b*x**2)*a**4*d**2 - 24*log(a + b*x**2)*a**3*b*c*d + 12*log(a + b*x**2)*a**2*b**2*c**2 - 12*a**3*b*d**2*x**2 + 24*a**2*b**2*c*d*x**2 + 6* a**2*b**2*d**2*x**4 - 12*a*b**3*c**2*x**2 - 12*a*b**3*c*d*x**4 - 4*a*b**3* d**2*x**6 + 6*b**4*c**2*x**4 + 8*b**4*c*d*x**6 + 3*b**4*d**2*x**8)/(24*b** 5)