Integrand size = 20, antiderivative size = 170 \[ \int \frac {x^3 \left (a+b x^2\right )^2}{c+d x} \, dx=\frac {c^2 \left (b c^2+a d^2\right )^2 x}{d^7}-\frac {c \left (b c^2+a d^2\right )^2 x^2}{2 d^6}+\frac {\left (b c^2+a d^2\right )^2 x^3}{3 d^5}-\frac {b c \left (b c^2+2 a d^2\right ) x^4}{4 d^4}+\frac {b \left (b c^2+2 a d^2\right ) x^5}{5 d^3}-\frac {b^2 c x^6}{6 d^2}+\frac {b^2 x^7}{7 d}-\frac {c^3 \left (b c^2+a d^2\right )^2 \log (c+d x)}{d^8} \] Output:
c^2*(a*d^2+b*c^2)^2*x/d^7-1/2*c*(a*d^2+b*c^2)^2*x^2/d^6+1/3*(a*d^2+b*c^2)^ 2*x^3/d^5-1/4*b*c*(2*a*d^2+b*c^2)*x^4/d^4+1/5*b*(2*a*d^2+b*c^2)*x^5/d^3-1/ 6*b^2*c*x^6/d^2+1/7*b^2*x^7/d-c^3*(a*d^2+b*c^2)^2*ln(d*x+c)/d^8
Time = 0.06 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.94 \[ \int \frac {x^3 \left (a+b x^2\right )^2}{c+d x} \, dx=\frac {420 d \left (b c^3+a c d^2\right )^2 x-210 c d^2 \left (b c^2+a d^2\right )^2 x^2+140 d^3 \left (b c^2+a d^2\right )^2 x^3-105 b c d^4 \left (b c^2+2 a d^2\right ) x^4+84 b d^5 \left (b c^2+2 a d^2\right ) x^5-70 b^2 c d^6 x^6+60 b^2 d^7 x^7-420 c^3 \left (b c^2+a d^2\right )^2 \log (c+d x)}{420 d^8} \] Input:
Integrate[(x^3*(a + b*x^2)^2)/(c + d*x),x]
Output:
(420*d*(b*c^3 + a*c*d^2)^2*x - 210*c*d^2*(b*c^2 + a*d^2)^2*x^2 + 140*d^3*( b*c^2 + a*d^2)^2*x^3 - 105*b*c*d^4*(b*c^2 + 2*a*d^2)*x^4 + 84*b*d^5*(b*c^2 + 2*a*d^2)*x^5 - 70*b^2*c*d^6*x^6 + 60*b^2*d^7*x^7 - 420*c^3*(b*c^2 + a*d ^2)^2*Log[c + d*x])/(420*d^8)
Time = 0.61 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {522, 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.
| \(\displaystyle \int \frac {x^3 \left (a+b x^2\right )^2}{c+d x} \, dx\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle \int \left (\frac {\left (a c d^2+b c^3\right )^2}{d^7}-\frac {c x \left (a d^2+b c^2\right )^2}{d^6}+\frac {x^2 \left (a d^2+b c^2\right )^2}{d^5}-\frac {b c x^3 \left (2 a d^2+b c^2\right )}{d^4}+\frac {b x^4 \left (2 a d^2+b c^2\right )}{d^3}-\frac {c^3 \left (a d^2+b c^2\right )^2}{d^7 (c+d x)}-\frac {b^2 c x^5}{d^2}+\frac {b^2 x^6}{d}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {c^2 x \left (a d^2+b c^2\right )^2}{d^7}-\frac {c x^2 \left (a d^2+b c^2\right )^2}{2 d^6}+\frac {x^3 \left (a d^2+b c^2\right )^2}{3 d^5}-\frac {b c x^4 \left (2 a d^2+b c^2\right )}{4 d^4}+\frac {b x^5 \left (2 a d^2+b c^2\right )}{5 d^3}-\frac {c^3 \left (a d^2+b c^2\right )^2 \log (c+d x)}{d^8}-\frac {b^2 c x^6}{6 d^2}+\frac {b^2 x^7}{7 d}\) |
Input:
Int[(x^3*(a + b*x^2)^2)/(c + d*x),x]
Output:
(c^2*(b*c^2 + a*d^2)^2*x)/d^7 - (c*(b*c^2 + a*d^2)^2*x^2)/(2*d^6) + ((b*c^ 2 + a*d^2)^2*x^3)/(3*d^5) - (b*c*(b*c^2 + 2*a*d^2)*x^4)/(4*d^4) + (b*(b*c^ 2 + 2*a*d^2)*x^5)/(5*d^3) - (b^2*c*x^6)/(6*d^2) + (b^2*x^7)/(7*d) - (c^3*( b*c^2 + a*d^2)^2*Log[c + d*x])/d^8
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Time = 0.26 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.22
| method | result | size |
| norman | \(\frac {c^{2} \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) x}{d^{7}}+\frac {b^{2} x^{7}}{7 d}+\frac {\left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) x^{3}}{3 d^{5}}+\frac {b \left (2 a \,d^{2}+b \,c^{2}\right ) x^{5}}{5 d^{3}}-\frac {b^{2} c \,x^{6}}{6 d^{2}}-\frac {c \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) x^{2}}{2 d^{6}}-\frac {b c \left (2 a \,d^{2}+b \,c^{2}\right ) x^{4}}{4 d^{4}}-\frac {c^{3} \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \ln \left (d x +c \right )}{d^{8}}\) | \(207\) |
| default | \(\frac {\frac {1}{7} b^{2} d^{6} x^{7}-\frac {1}{6} b^{2} c \,d^{5} x^{6}+\frac {2}{5} a b \,d^{6} x^{5}+\frac {1}{5} b^{2} c^{2} d^{4} x^{5}-\frac {1}{2} a b c \,d^{5} x^{4}-\frac {1}{4} b^{2} c^{3} d^{3} x^{4}+\frac {1}{3} a^{2} d^{6} x^{3}+\frac {2}{3} a b \,c^{2} d^{4} x^{3}+\frac {1}{3} b^{2} c^{4} d^{2} x^{3}-\frac {1}{2} a^{2} c \,d^{5} x^{2}-a b \,c^{3} d^{3} x^{2}-\frac {1}{2} b^{2} c^{5} d \,x^{2}+a^{2} c^{2} d^{4} x +2 a b \,c^{4} d^{2} x +b^{2} c^{6} x}{d^{7}}-\frac {c^{3} \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \ln \left (d x +c \right )}{d^{8}}\) | \(223\) |
| risch | \(\frac {b^{2} x^{7}}{7 d}-\frac {b^{2} c \,x^{6}}{6 d^{2}}+\frac {2 a b \,x^{5}}{5 d}+\frac {b^{2} c^{2} x^{5}}{5 d^{3}}-\frac {a b c \,x^{4}}{2 d^{2}}-\frac {b^{2} c^{3} x^{4}}{4 d^{4}}+\frac {a^{2} x^{3}}{3 d}+\frac {2 a b \,c^{2} x^{3}}{3 d^{3}}+\frac {b^{2} c^{4} x^{3}}{3 d^{5}}-\frac {a^{2} c \,x^{2}}{2 d^{2}}-\frac {a b \,c^{3} x^{2}}{d^{4}}-\frac {b^{2} c^{5} x^{2}}{2 d^{6}}+\frac {a^{2} c^{2} x}{d^{3}}+\frac {2 a b \,c^{4} x}{d^{5}}+\frac {b^{2} c^{6} x}{d^{7}}-\frac {c^{3} \ln \left (d x +c \right ) a^{2}}{d^{4}}-\frac {2 c^{5} \ln \left (d x +c \right ) b a}{d^{6}}-\frac {c^{7} \ln \left (d x +c \right ) b^{2}}{d^{8}}\) | \(234\) |
| parallelrisch | \(-\frac {-60 x^{7} b^{2} d^{7}+70 c \,b^{2} x^{6} d^{6}-168 x^{5} a b \,d^{7}-84 x^{5} b^{2} c^{2} d^{5}+210 x^{4} a b c \,d^{6}+105 x^{4} b^{2} c^{3} d^{4}-140 x^{3} a^{2} d^{7}-280 x^{3} a b \,c^{2} d^{5}-140 x^{3} b^{2} c^{4} d^{3}+210 x^{2} a^{2} c \,d^{6}+420 x^{2} a b \,c^{3} d^{4}+210 x^{2} b^{2} c^{5} d^{2}+420 \ln \left (d x +c \right ) a^{2} c^{3} d^{4}+840 \ln \left (d x +c \right ) a b \,c^{5} d^{2}+420 \ln \left (d x +c \right ) b^{2} c^{7}-420 x \,a^{2} c^{2} d^{5}-840 x a b \,c^{4} d^{3}-420 x \,b^{2} c^{6} d}{420 d^{8}}\) | \(236\) |
Input:
int(x^3*(b*x^2+a)^2/(d*x+c),x,method=_RETURNVERBOSE)
Output:
c^2*(a^2*d^4+2*a*b*c^2*d^2+b^2*c^4)/d^7*x+1/7*b^2*x^7/d+1/3/d^5*(a^2*d^4+2 *a*b*c^2*d^2+b^2*c^4)*x^3+1/5*b*(2*a*d^2+b*c^2)*x^5/d^3-1/6*b^2*c*x^6/d^2- 1/2*c/d^6*(a^2*d^4+2*a*b*c^2*d^2+b^2*c^4)*x^2-1/4*b*c*(2*a*d^2+b*c^2)*x^4/ d^4-c^3*(a^2*d^4+2*a*b*c^2*d^2+b^2*c^4)/d^8*ln(d*x+c)
Time = 0.08 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.24 \[ \int \frac {x^3 \left (a+b x^2\right )^2}{c+d x} \, dx=\frac {60 \, b^{2} d^{7} x^{7} - 70 \, b^{2} c d^{6} x^{6} + 84 \, {\left (b^{2} c^{2} d^{5} + 2 \, a b d^{7}\right )} x^{5} - 105 \, {\left (b^{2} c^{3} d^{4} + 2 \, a b c d^{6}\right )} x^{4} + 140 \, {\left (b^{2} c^{4} d^{3} + 2 \, a b c^{2} d^{5} + a^{2} d^{7}\right )} x^{3} - 210 \, {\left (b^{2} c^{5} d^{2} + 2 \, a b c^{3} d^{4} + a^{2} c d^{6}\right )} x^{2} + 420 \, {\left (b^{2} c^{6} d + 2 \, a b c^{4} d^{3} + a^{2} c^{2} d^{5}\right )} x - 420 \, {\left (b^{2} c^{7} + 2 \, a b c^{5} d^{2} + a^{2} c^{3} d^{4}\right )} \log \left (d x + c\right )}{420 \, d^{8}} \] Input:
integrate(x^3*(b*x^2+a)^2/(d*x+c),x, algorithm="fricas")
Output:
1/420*(60*b^2*d^7*x^7 - 70*b^2*c*d^6*x^6 + 84*(b^2*c^2*d^5 + 2*a*b*d^7)*x^ 5 - 105*(b^2*c^3*d^4 + 2*a*b*c*d^6)*x^4 + 140*(b^2*c^4*d^3 + 2*a*b*c^2*d^5 + a^2*d^7)*x^3 - 210*(b^2*c^5*d^2 + 2*a*b*c^3*d^4 + a^2*c*d^6)*x^2 + 420* (b^2*c^6*d + 2*a*b*c^4*d^3 + a^2*c^2*d^5)*x - 420*(b^2*c^7 + 2*a*b*c^5*d^2 + a^2*c^3*d^4)*log(d*x + c))/d^8
Time = 0.21 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.21 \[ \int \frac {x^3 \left (a+b x^2\right )^2}{c+d x} \, dx=- \frac {b^{2} c x^{6}}{6 d^{2}} + \frac {b^{2} x^{7}}{7 d} - \frac {c^{3} \left (a d^{2} + b c^{2}\right )^{2} \log {\left (c + d x \right )}}{d^{8}} + x^{5} \cdot \left (\frac {2 a b}{5 d} + \frac {b^{2} c^{2}}{5 d^{3}}\right ) + x^{4} \left (- \frac {a b c}{2 d^{2}} - \frac {b^{2} c^{3}}{4 d^{4}}\right ) + x^{3} \left (\frac {a^{2}}{3 d} + \frac {2 a b c^{2}}{3 d^{3}} + \frac {b^{2} c^{4}}{3 d^{5}}\right ) + x^{2} \left (- \frac {a^{2} c}{2 d^{2}} - \frac {a b c^{3}}{d^{4}} - \frac {b^{2} c^{5}}{2 d^{6}}\right ) + x \left (\frac {a^{2} c^{2}}{d^{3}} + \frac {2 a b c^{4}}{d^{5}} + \frac {b^{2} c^{6}}{d^{7}}\right ) \] Input:
integrate(x**3*(b*x**2+a)**2/(d*x+c),x)
Output:
-b**2*c*x**6/(6*d**2) + b**2*x**7/(7*d) - c**3*(a*d**2 + b*c**2)**2*log(c + d*x)/d**8 + x**5*(2*a*b/(5*d) + b**2*c**2/(5*d**3)) + x**4*(-a*b*c/(2*d* *2) - b**2*c**3/(4*d**4)) + x**3*(a**2/(3*d) + 2*a*b*c**2/(3*d**3) + b**2* c**4/(3*d**5)) + x**2*(-a**2*c/(2*d**2) - a*b*c**3/d**4 - b**2*c**5/(2*d** 6)) + x*(a**2*c**2/d**3 + 2*a*b*c**4/d**5 + b**2*c**6/d**7)
Time = 0.03 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.25 \[ \int \frac {x^3 \left (a+b x^2\right )^2}{c+d x} \, dx=\frac {60 \, b^{2} d^{6} x^{7} - 70 \, b^{2} c d^{5} x^{6} + 84 \, {\left (b^{2} c^{2} d^{4} + 2 \, a b d^{6}\right )} x^{5} - 105 \, {\left (b^{2} c^{3} d^{3} + 2 \, a b c d^{5}\right )} x^{4} + 140 \, {\left (b^{2} c^{4} d^{2} + 2 \, a b c^{2} d^{4} + a^{2} d^{6}\right )} x^{3} - 210 \, {\left (b^{2} c^{5} d + 2 \, a b c^{3} d^{3} + a^{2} c d^{5}\right )} x^{2} + 420 \, {\left (b^{2} c^{6} + 2 \, a b c^{4} d^{2} + a^{2} c^{2} d^{4}\right )} x}{420 \, d^{7}} - \frac {{\left (b^{2} c^{7} + 2 \, a b c^{5} d^{2} + a^{2} c^{3} d^{4}\right )} \log \left (d x + c\right )}{d^{8}} \] Input:
integrate(x^3*(b*x^2+a)^2/(d*x+c),x, algorithm="maxima")
Output:
1/420*(60*b^2*d^6*x^7 - 70*b^2*c*d^5*x^6 + 84*(b^2*c^2*d^4 + 2*a*b*d^6)*x^ 5 - 105*(b^2*c^3*d^3 + 2*a*b*c*d^5)*x^4 + 140*(b^2*c^4*d^2 + 2*a*b*c^2*d^4 + a^2*d^6)*x^3 - 210*(b^2*c^5*d + 2*a*b*c^3*d^3 + a^2*c*d^5)*x^2 + 420*(b ^2*c^6 + 2*a*b*c^4*d^2 + a^2*c^2*d^4)*x)/d^7 - (b^2*c^7 + 2*a*b*c^5*d^2 + a^2*c^3*d^4)*log(d*x + c)/d^8
Time = 0.13 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.33 \[ \int \frac {x^3 \left (a+b x^2\right )^2}{c+d x} \, dx=\frac {60 \, b^{2} d^{6} x^{7} - 70 \, b^{2} c d^{5} x^{6} + 84 \, b^{2} c^{2} d^{4} x^{5} + 168 \, a b d^{6} x^{5} - 105 \, b^{2} c^{3} d^{3} x^{4} - 210 \, a b c d^{5} x^{4} + 140 \, b^{2} c^{4} d^{2} x^{3} + 280 \, a b c^{2} d^{4} x^{3} + 140 \, a^{2} d^{6} x^{3} - 210 \, b^{2} c^{5} d x^{2} - 420 \, a b c^{3} d^{3} x^{2} - 210 \, a^{2} c d^{5} x^{2} + 420 \, b^{2} c^{6} x + 840 \, a b c^{4} d^{2} x + 420 \, a^{2} c^{2} d^{4} x}{420 \, d^{7}} - \frac {{\left (b^{2} c^{7} + 2 \, a b c^{5} d^{2} + a^{2} c^{3} d^{4}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{8}} \] Input:
integrate(x^3*(b*x^2+a)^2/(d*x+c),x, algorithm="giac")
Output:
1/420*(60*b^2*d^6*x^7 - 70*b^2*c*d^5*x^6 + 84*b^2*c^2*d^4*x^5 + 168*a*b*d^ 6*x^5 - 105*b^2*c^3*d^3*x^4 - 210*a*b*c*d^5*x^4 + 140*b^2*c^4*d^2*x^3 + 28 0*a*b*c^2*d^4*x^3 + 140*a^2*d^6*x^3 - 210*b^2*c^5*d*x^2 - 420*a*b*c^3*d^3* x^2 - 210*a^2*c*d^5*x^2 + 420*b^2*c^6*x + 840*a*b*c^4*d^2*x + 420*a^2*c^2* d^4*x)/d^7 - (b^2*c^7 + 2*a*b*c^5*d^2 + a^2*c^3*d^4)*log(abs(d*x + c))/d^8
Time = 0.06 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.38 \[ \int \frac {x^3 \left (a+b x^2\right )^2}{c+d x} \, dx=x^5\,\left (\frac {b^2\,c^2}{5\,d^3}+\frac {2\,a\,b}{5\,d}\right )+x^3\,\left (\frac {a^2}{3\,d}+\frac {c^2\,\left (\frac {b^2\,c^2}{d^3}+\frac {2\,a\,b}{d}\right )}{3\,d^2}\right )-\frac {\ln \left (c+d\,x\right )\,\left (a^2\,c^3\,d^4+2\,a\,b\,c^5\,d^2+b^2\,c^7\right )}{d^8}+\frac {b^2\,x^7}{7\,d}-\frac {b^2\,c\,x^6}{6\,d^2}-\frac {c\,x^4\,\left (\frac {b^2\,c^2}{d^3}+\frac {2\,a\,b}{d}\right )}{4\,d}-\frac {c\,x^2\,\left (\frac {a^2}{d}+\frac {c^2\,\left (\frac {b^2\,c^2}{d^3}+\frac {2\,a\,b}{d}\right )}{d^2}\right )}{2\,d}+\frac {c^2\,x\,\left (\frac {a^2}{d}+\frac {c^2\,\left (\frac {b^2\,c^2}{d^3}+\frac {2\,a\,b}{d}\right )}{d^2}\right )}{d^2} \] Input:
int((x^3*(a + b*x^2)^2)/(c + d*x),x)
Output:
x^5*((b^2*c^2)/(5*d^3) + (2*a*b)/(5*d)) + x^3*(a^2/(3*d) + (c^2*((b^2*c^2) /d^3 + (2*a*b)/d))/(3*d^2)) - (log(c + d*x)*(b^2*c^7 + a^2*c^3*d^4 + 2*a*b *c^5*d^2))/d^8 + (b^2*x^7)/(7*d) - (b^2*c*x^6)/(6*d^2) - (c*x^4*((b^2*c^2) /d^3 + (2*a*b)/d))/(4*d) - (c*x^2*(a^2/d + (c^2*((b^2*c^2)/d^3 + (2*a*b)/d ))/d^2))/(2*d) + (c^2*x*(a^2/d + (c^2*((b^2*c^2)/d^3 + (2*a*b)/d))/d^2))/d ^2
Time = 0.21 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.38 \[ \int \frac {x^3 \left (a+b x^2\right )^2}{c+d x} \, dx=\frac {-420 \,\mathrm {log}\left (d x +c \right ) a^{2} c^{3} d^{4}-840 \,\mathrm {log}\left (d x +c \right ) a b \,c^{5} d^{2}-420 \,\mathrm {log}\left (d x +c \right ) b^{2} c^{7}+420 a^{2} c^{2} d^{5} x -210 a^{2} c \,d^{6} x^{2}+140 a^{2} d^{7} x^{3}+840 a b \,c^{4} d^{3} x -420 a b \,c^{3} d^{4} x^{2}+280 a b \,c^{2} d^{5} x^{3}-210 a b c \,d^{6} x^{4}+168 a b \,d^{7} x^{5}+420 b^{2} c^{6} d x -210 b^{2} c^{5} d^{2} x^{2}+140 b^{2} c^{4} d^{3} x^{3}-105 b^{2} c^{3} d^{4} x^{4}+84 b^{2} c^{2} d^{5} x^{5}-70 b^{2} c \,d^{6} x^{6}+60 b^{2} d^{7} x^{7}}{420 d^{8}} \] Input:
int(x^3*(b*x^2+a)^2/(d*x+c),x)
Output:
( - 420*log(c + d*x)*a**2*c**3*d**4 - 840*log(c + d*x)*a*b*c**5*d**2 - 420 *log(c + d*x)*b**2*c**7 + 420*a**2*c**2*d**5*x - 210*a**2*c*d**6*x**2 + 14 0*a**2*d**7*x**3 + 840*a*b*c**4*d**3*x - 420*a*b*c**3*d**4*x**2 + 280*a*b* c**2*d**5*x**3 - 210*a*b*c*d**6*x**4 + 168*a*b*d**7*x**5 + 420*b**2*c**6*d *x - 210*b**2*c**5*d**2*x**2 + 140*b**2*c**4*d**3*x**3 - 105*b**2*c**3*d** 4*x**4 + 84*b**2*c**2*d**5*x**5 - 70*b**2*c*d**6*x**6 + 60*b**2*d**7*x**7) /(420*d**8)