Integrand size = 20, antiderivative size = 87 \[ \int \frac {x \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {d (b c-a d)^2 x^2}{2 b^3}+\frac {(b c-a d) \left (c+d x^2\right )^2}{4 b^2}+\frac {\left (c+d x^2\right )^3}{6 b}+\frac {(b c-a d)^3 \log \left (a+b x^2\right )}{2 b^4} \] Output:
1/2*d*(-a*d+b*c)^2*x^2/b^3+1/4*(-a*d+b*c)*(d*x^2+c)^2/b^2+1/6*(d*x^2+c)^3/ b+1/2*(-a*d+b*c)^3*ln(b*x^2+a)/b^4
Time = 0.02 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.94 \[ \int \frac {x \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {b d x^2 \left (6 a^2 d^2-3 a b d \left (6 c+d x^2\right )+b^2 \left (18 c^2+9 c d x^2+2 d^2 x^4\right )\right )+6 (b c-a d)^3 \log \left (a+b x^2\right )}{12 b^4} \] Input:
Integrate[(x*(c + d*x^2)^3)/(a + b*x^2),x]
Output:
(b*d*x^2*(6*a^2*d^2 - 3*a*b*d*(6*c + d*x^2) + b^2*(18*c^2 + 9*c*d*x^2 + 2* d^2*x^4)) + 6*(b*c - a*d)^3*Log[a + b*x^2])/(12*b^4)
Time = 0.22 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {353, 49, 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.
\(\displaystyle \int \frac {x \left (c+d x^2\right )^3}{a+b x^2} \, dx\) |
\(\Big \downarrow \) 353 |
\(\displaystyle \frac {1}{2} \int \frac {\left (d x^2+c\right )^3}{b x^2+a}dx^2\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {1}{2} \int \left (\frac {(b c-a d)^3}{b^3 \left (b x^2+a\right )}+\frac {d (b c-a d)^2}{b^3}+\frac {d \left (d x^2+c\right ) (b c-a d)}{b^2}+\frac {d \left (d x^2+c\right )^2}{b}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {(b c-a d)^3 \log \left (a+b x^2\right )}{b^4}+\frac {d x^2 (b c-a d)^2}{b^3}+\frac {\left (c+d x^2\right )^2 (b c-a d)}{2 b^2}+\frac {\left (c+d x^2\right )^3}{3 b}\right )\) |
Input:
Int[(x*(c + d*x^2)^3)/(a + b*x^2),x]
Output:
((d*(b*c - a*d)^2*x^2)/b^3 + ((b*c - a*d)*(c + d*x^2)^2)/(2*b^2) + (c + d* x^2)^3/(3*b) + ((b*c - a*d)^3*Log[a + b*x^2])/b^4)/2
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/2 Subst[Int[(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]
Time = 0.60 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.29
method | result | size |
norman | \(\frac {d^{3} x^{6}}{6 b}-\frac {d^{2} \left (a d -3 b c \right ) x^{4}}{4 b^{2}}+\frac {d \left (a^{2} d^{2}-3 a b c d +3 b^{2} c^{2}\right ) x^{2}}{2 b^{3}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b \,x^{2}+a \right )}{2 b^{4}}\) | \(112\) |
default | \(\frac {d \left (\frac {1}{3} b^{2} d^{2} x^{6}-\frac {1}{2} x^{4} a b \,d^{2}+\frac {3}{2} x^{4} b^{2} c d +a^{2} d^{2} x^{2}-3 a b c d \,x^{2}+3 b^{2} c^{2} x^{2}\right )}{2 b^{3}}+\frac {\left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \ln \left (b \,x^{2}+a \right )}{2 b^{4}}\) | \(119\) |
parallelrisch | \(-\frac {-2 b^{3} d^{3} x^{6}+3 a \,b^{2} d^{3} x^{4}-9 b^{3} c \,d^{2} x^{4}-6 x^{2} a^{2} b \,d^{3}+18 x^{2} a \,b^{2} c \,d^{2}-18 x^{2} b^{3} c^{2} d +6 \ln \left (b \,x^{2}+a \right ) a^{3} d^{3}-18 \ln \left (b \,x^{2}+a \right ) a^{2} b c \,d^{2}+18 \ln \left (b \,x^{2}+a \right ) a \,b^{2} c^{2} d -6 \ln \left (b \,x^{2}+a \right ) b^{3} c^{3}}{12 b^{4}}\) | \(147\) |
risch | \(\frac {d^{3} x^{6}}{6 b}-\frac {d^{3} x^{4} a}{4 b^{2}}+\frac {3 d^{2} x^{4} c}{4 b}+\frac {d^{3} a^{2} x^{2}}{2 b^{3}}-\frac {3 d^{2} a c \,x^{2}}{2 b^{2}}+\frac {3 d \,c^{2} x^{2}}{2 b}-\frac {\ln \left (b \,x^{2}+a \right ) a^{3} d^{3}}{2 b^{4}}+\frac {3 \ln \left (b \,x^{2}+a \right ) a^{2} c \,d^{2}}{2 b^{3}}-\frac {3 \ln \left (b \,x^{2}+a \right ) a \,c^{2} d}{2 b^{2}}+\frac {\ln \left (b \,x^{2}+a \right ) c^{3}}{2 b}\) | \(149\) |
Input:
int(x*(d*x^2+c)^3/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
1/6/b*d^3*x^6-1/4/b^2*d^2*(a*d-3*b*c)*x^4+1/2*d*(a^2*d^2-3*a*b*c*d+3*b^2*c ^2)/b^3*x^2-1/2/b^4*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*ln(b*x^2 +a)
Time = 0.08 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.38 \[ \int \frac {x \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {2 \, b^{3} d^{3} x^{6} + 3 \, {\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{4} + 6 \, {\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x^{2} + a\right )}{12 \, b^{4}} \] Input:
integrate(x*(d*x^2+c)^3/(b*x^2+a),x, algorithm="fricas")
Output:
1/12*(2*b^3*d^3*x^6 + 3*(3*b^3*c*d^2 - a*b^2*d^3)*x^4 + 6*(3*b^3*c^2*d - 3 *a*b^2*c*d^2 + a^2*b*d^3)*x^2 + 6*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*log(b*x^2 + a))/b^4
Time = 0.25 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.08 \[ \int \frac {x \left (c+d x^2\right )^3}{a+b x^2} \, dx=x^{4} \left (- \frac {a d^{3}}{4 b^{2}} + \frac {3 c d^{2}}{4 b}\right ) + x^{2} \left (\frac {a^{2} d^{3}}{2 b^{3}} - \frac {3 a c d^{2}}{2 b^{2}} + \frac {3 c^{2} d}{2 b}\right ) + \frac {d^{3} x^{6}}{6 b} - \frac {\left (a d - b c\right )^{3} \log {\left (a + b x^{2} \right )}}{2 b^{4}} \] Input:
integrate(x*(d*x**2+c)**3/(b*x**2+a),x)
Output:
x**4*(-a*d**3/(4*b**2) + 3*c*d**2/(4*b)) + x**2*(a**2*d**3/(2*b**3) - 3*a* c*d**2/(2*b**2) + 3*c**2*d/(2*b)) + d**3*x**6/(6*b) - (a*d - b*c)**3*log(a + b*x**2)/(2*b**4)
Time = 0.03 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.37 \[ \int \frac {x \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {2 \, b^{2} d^{3} x^{6} + 3 \, {\left (3 \, b^{2} c d^{2} - a b d^{3}\right )} x^{4} + 6 \, {\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}}{12 \, b^{3}} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, b^{4}} \] Input:
integrate(x*(d*x^2+c)^3/(b*x^2+a),x, algorithm="maxima")
Output:
1/12*(2*b^2*d^3*x^6 + 3*(3*b^2*c*d^2 - a*b*d^3)*x^4 + 6*(3*b^2*c^2*d - 3*a *b*c*d^2 + a^2*d^3)*x^2)/b^3 + 1/2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^ 2 - a^3*d^3)*log(b*x^2 + a)/b^4
Time = 0.12 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.43 \[ \int \frac {x \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {2 \, b^{2} d^{3} x^{6} + 9 \, b^{2} c d^{2} x^{4} - 3 \, a b d^{3} x^{4} + 18 \, b^{2} c^{2} d x^{2} - 18 \, a b c d^{2} x^{2} + 6 \, a^{2} d^{3} x^{2}}{12 \, b^{3}} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4}} \] Input:
integrate(x*(d*x^2+c)^3/(b*x^2+a),x, algorithm="giac")
Output:
1/12*(2*b^2*d^3*x^6 + 9*b^2*c*d^2*x^4 - 3*a*b*d^3*x^4 + 18*b^2*c^2*d*x^2 - 18*a*b*c*d^2*x^2 + 6*a^2*d^3*x^2)/b^3 + 1/2*(b^3*c^3 - 3*a*b^2*c^2*d + 3* a^2*b*c*d^2 - a^3*d^3)*log(abs(b*x^2 + a))/b^4
Time = 0.07 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.41 \[ \int \frac {x \left (c+d x^2\right )^3}{a+b x^2} \, dx=x^2\,\left (\frac {3\,c^2\,d}{2\,b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{2\,b}\right )-x^4\,\left (\frac {a\,d^3}{4\,b^2}-\frac {3\,c\,d^2}{4\,b}\right )-\frac {\ln \left (b\,x^2+a\right )\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{2\,b^4}+\frac {d^3\,x^6}{6\,b} \] Input:
int((x*(c + d*x^2)^3)/(a + b*x^2),x)
Output:
x^2*((3*c^2*d)/(2*b) + (a*((a*d^3)/b^2 - (3*c*d^2)/b))/(2*b)) - x^4*((a*d^ 3)/(4*b^2) - (3*c*d^2)/(4*b)) - (log(a + b*x^2)*(a^3*d^3 - b^3*c^3 + 3*a*b ^2*c^2*d - 3*a^2*b*c*d^2))/(2*b^4) + (d^3*x^6)/(6*b)
Time = 0.21 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.68 \[ \int \frac {x \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {-6 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{3} d^{3}+18 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} b c \,d^{2}-18 \,\mathrm {log}\left (b \,x^{2}+a \right ) a \,b^{2} c^{2} d +6 \,\mathrm {log}\left (b \,x^{2}+a \right ) b^{3} c^{3}+6 a^{2} b \,d^{3} x^{2}-18 a \,b^{2} c \,d^{2} x^{2}-3 a \,b^{2} d^{3} x^{4}+18 b^{3} c^{2} d \,x^{2}+9 b^{3} c \,d^{2} x^{4}+2 b^{3} d^{3} x^{6}}{12 b^{4}} \] Input:
int(x*(d*x^2+c)^3/(b*x^2+a),x)
Output:
( - 6*log(a + b*x**2)*a**3*d**3 + 18*log(a + b*x**2)*a**2*b*c*d**2 - 18*lo g(a + b*x**2)*a*b**2*c**2*d + 6*log(a + b*x**2)*b**3*c**3 + 6*a**2*b*d**3* x**2 - 18*a*b**2*c*d**2*x**2 - 3*a*b**2*d**3*x**4 + 18*b**3*c**2*d*x**2 + 9*b**3*c*d**2*x**4 + 2*b**3*d**3*x**6)/(12*b**4)