Integrand size = 26, antiderivative size = 180 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^3}{e+f x^2} \, dx=-\frac {\left (b (d e-c f)^3-a d f \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) x}{f^4}-\frac {d \left (a d f (d e-3 c f)-b \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) x^3}{3 f^3}-\frac {d^2 (b d e-3 b c f-a d f) x^5}{5 f^2}+\frac {b d^3 x^7}{7 f}+\frac {(b e-a f) (d e-c f)^3 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{9/2}} \] Output:
-(b*(-c*f+d*e)^3-a*d*f*(3*c^2*f^2-3*c*d*e*f+d^2*e^2))*x/f^4-1/3*d*(a*d*f*( -3*c*f+d*e)-b*(3*c^2*f^2-3*c*d*e*f+d^2*e^2))*x^3/f^3-1/5*d^2*(-a*d*f-3*b*c *f+b*d*e)*x^5/f^2+1/7*b*d^3*x^7/f+(-a*f+b*e)*(-c*f+d*e)^3*arctan(f^(1/2)*x /e^(1/2))/e^(1/2)/f^(9/2)
Time = 0.09 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^3}{e+f x^2} \, dx=\frac {\left (-b (d e-c f)^3+a d f \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) x}{f^4}+\frac {d \left (a d f (-d e+3 c f)+b \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) x^3}{3 f^3}+\frac {d^2 (-b d e+3 b c f+a d f) x^5}{5 f^2}+\frac {b d^3 x^7}{7 f}+\frac {(b e-a f) (d e-c f)^3 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{9/2}} \] Input:
Integrate[((a + b*x^2)*(c + d*x^2)^3)/(e + f*x^2),x]
Output:
((-(b*(d*e - c*f)^3) + a*d*f*(d^2*e^2 - 3*c*d*e*f + 3*c^2*f^2))*x)/f^4 + ( d*(a*d*f*(-(d*e) + 3*c*f) + b*(d^2*e^2 - 3*c*d*e*f + 3*c^2*f^2))*x^3)/(3*f ^3) + (d^2*(-(b*d*e) + 3*b*c*f + a*d*f)*x^5)/(5*f^2) + (b*d^3*x^7)/(7*f) + ((b*e - a*f)*(d*e - c*f)^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*f^(9/2))
Time = 0.52 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.38, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {403, 25, 403, 25, 403, 299, 218}
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.
| \(\displaystyle \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^3}{e+f x^2} \, dx\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\int -\frac {\left (d x^2+c\right )^2 \left ((7 b d e-6 b c f-7 a d f) x^2+c (b e-7 a f)\right )}{f x^2+e}dx}{7 f}+\frac {b x \left (c+d x^2\right )^3}{7 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b x \left (c+d x^2\right )^3}{7 f}-\frac {\int \frac {\left (d x^2+c\right )^2 \left ((7 b d e-6 b c f-7 a d f) x^2+c (b e-7 a f)\right )}{f x^2+e}dx}{7 f}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {b x \left (c+d x^2\right )^3}{7 f}-\frac {\frac {\int -\frac {\left (d x^2+c\right ) \left (c (b e (7 d e-11 c f)-7 a f (d e-5 c f))-\left (7 a d f (5 d e-9 c f)-b \left (35 d^2 e^2-63 c d f e+24 c^2 f^2\right )\right ) x^2\right )}{f x^2+e}dx}{5 f}+\frac {x \left (c+d x^2\right )^2 (-7 a d f-6 b c f+7 b d e)}{5 f}}{7 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b x \left (c+d x^2\right )^3}{7 f}-\frac {\frac {x \left (c+d x^2\right )^2 (-7 a d f-6 b c f+7 b d e)}{5 f}-\frac {\int \frac {\left (d x^2+c\right ) \left (c (b e (7 d e-11 c f)-7 a f (d e-5 c f))-\left (7 a d f (5 d e-9 c f)-b \left (35 d^2 e^2-63 c d f e+24 c^2 f^2\right )\right ) x^2\right )}{f x^2+e}dx}{5 f}}{7 f}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {b x \left (c+d x^2\right )^3}{7 f}-\frac {\frac {x \left (c+d x^2\right )^2 (-7 a d f-6 b c f+7 b d e)}{5 f}-\frac {\frac {\int \frac {\left (7 a d f \left (15 d^2 e^2-40 c d f e+33 c^2 f^2\right )-b \left (105 d^3 e^3-280 c d^2 f e^2+231 c^2 d f^2 e-48 c^3 f^3\right )\right ) x^2+c \left (7 a f \left (5 d^2 e^2-12 c d f e+15 c^2 f^2\right )-b e \left (35 d^2 e^2-84 c d f e+57 c^2 f^2\right )\right )}{f x^2+e}dx}{3 f}-\frac {x \left (c+d x^2\right ) \left (7 a d f (5 d e-9 c f)-b \left (24 c^2 f^2-63 c d e f+35 d^2 e^2\right )\right )}{3 f}}{5 f}}{7 f}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {b x \left (c+d x^2\right )^3}{7 f}-\frac {\frac {x \left (c+d x^2\right )^2 (-7 a d f-6 b c f+7 b d e)}{5 f}-\frac {\frac {\frac {105 (b e-a f) (d e-c f)^3 \int \frac {1}{f x^2+e}dx}{f}+\frac {x \left (7 a d f \left (33 c^2 f^2-40 c d e f+15 d^2 e^2\right )-b \left (-48 c^3 f^3+231 c^2 d e f^2-280 c d^2 e^2 f+105 d^3 e^3\right )\right )}{f}}{3 f}-\frac {x \left (c+d x^2\right ) \left (7 a d f (5 d e-9 c f)-b \left (24 c^2 f^2-63 c d e f+35 d^2 e^2\right )\right )}{3 f}}{5 f}}{7 f}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {b x \left (c+d x^2\right )^3}{7 f}-\frac {\frac {x \left (c+d x^2\right )^2 (-7 a d f-6 b c f+7 b d e)}{5 f}-\frac {\frac {\frac {105 (b e-a f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (d e-c f)^3}{\sqrt {e} f^{3/2}}+\frac {x \left (7 a d f \left (33 c^2 f^2-40 c d e f+15 d^2 e^2\right )-b \left (-48 c^3 f^3+231 c^2 d e f^2-280 c d^2 e^2 f+105 d^3 e^3\right )\right )}{f}}{3 f}-\frac {x \left (c+d x^2\right ) \left (7 a d f (5 d e-9 c f)-b \left (24 c^2 f^2-63 c d e f+35 d^2 e^2\right )\right )}{3 f}}{5 f}}{7 f}\) |
Input:
Int[((a + b*x^2)*(c + d*x^2)^3)/(e + f*x^2),x]
Output:
(b*x*(c + d*x^2)^3)/(7*f) - (((7*b*d*e - 6*b*c*f - 7*a*d*f)*x*(c + d*x^2)^ 2)/(5*f) - (-1/3*((7*a*d*f*(5*d*e - 9*c*f) - b*(35*d^2*e^2 - 63*c*d*e*f + 24*c^2*f^2))*x*(c + d*x^2))/f + (((7*a*d*f*(15*d^2*e^2 - 40*c*d*e*f + 33*c ^2*f^2) - b*(105*d^3*e^3 - 280*c*d^2*e^2*f + 231*c^2*d*e*f^2 - 48*c^3*f^3) )*x)/f + (105*(b*e - a*f)*(d*e - c*f)^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt [e]*f^(3/2)))/(3*f))/(5*f))/(7*f)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
Time = 0.54 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.67
| method | result | size |
| default | \(\frac {\frac {1}{7} b \,d^{3} x^{7} f^{3}+\frac {1}{5} a \,d^{3} f^{3} x^{5}+\frac {3}{5} b c \,d^{2} f^{3} x^{5}-\frac {1}{5} b \,d^{3} e \,f^{2} x^{5}+a c \,d^{2} f^{3} x^{3}-\frac {1}{3} a \,d^{3} e \,f^{2} x^{3}+b \,c^{2} d \,f^{3} x^{3}-b c \,d^{2} e \,f^{2} x^{3}+\frac {1}{3} b \,d^{3} e^{2} f \,x^{3}+3 a \,c^{2} d \,f^{3} x -3 a c \,d^{2} e \,f^{2} x +a \,d^{3} e^{2} f x +b \,c^{3} f^{3} x -3 b \,c^{2} d e \,f^{2} x +3 b c \,d^{2} e^{2} f x -b \,d^{3} e^{3} x}{f^{4}}+\frac {\left (c^{3} a \,f^{4}-3 a \,c^{2} d e \,f^{3}+3 a c \,d^{2} e^{2} f^{2}-a \,d^{3} e^{3} f -b \,c^{3} e \,f^{3}+3 b \,c^{2} d \,e^{2} f^{2}-3 b c \,d^{2} e^{3} f +e^{4} b \,d^{3}\right ) \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{f^{4} \sqrt {e f}}\) | \(300\) |
| risch | \(\frac {3 b c \,d^{2} x^{5}}{5 f}-\frac {b \,d^{3} e \,x^{5}}{5 f^{2}}+\frac {a \,d^{3} x^{5}}{5 f}+\frac {b \,c^{3} x}{f}-\frac {\ln \left (f x +\sqrt {-e f}\right ) c^{3} a}{2 \sqrt {-e f}}+\frac {\ln \left (-f x +\sqrt {-e f}\right ) c^{3} a}{2 \sqrt {-e f}}-\frac {3 \ln \left (-f x +\sqrt {-e f}\right ) b c \,d^{2} e^{3}}{2 f^{3} \sqrt {-e f}}-\frac {a \,d^{3} e \,x^{3}}{3 f^{2}}+\frac {b \,c^{2} d \,x^{3}}{f}+\frac {b \,d^{3} e^{2} x^{3}}{3 f^{3}}+\frac {3 a \,c^{2} d x}{f}+\frac {a \,d^{3} e^{2} x}{f^{3}}-\frac {b \,d^{3} e^{3} x}{f^{4}}+\frac {a c \,d^{2} x^{3}}{f}-\frac {b c \,d^{2} e \,x^{3}}{f^{2}}-\frac {3 a c \,d^{2} e x}{f^{2}}-\frac {3 b \,c^{2} d e x}{f^{2}}+\frac {3 b c \,d^{2} e^{2} x}{f^{3}}+\frac {\ln \left (f x +\sqrt {-e f}\right ) a \,d^{3} e^{3}}{2 f^{3} \sqrt {-e f}}+\frac {\ln \left (f x +\sqrt {-e f}\right ) b \,c^{3} e}{2 f \sqrt {-e f}}-\frac {\ln \left (f x +\sqrt {-e f}\right ) e^{4} b \,d^{3}}{2 f^{4} \sqrt {-e f}}-\frac {3 \ln \left (-f x +\sqrt {-e f}\right ) a \,c^{2} d e}{2 f \sqrt {-e f}}+\frac {3 \ln \left (-f x +\sqrt {-e f}\right ) a c \,d^{2} e^{2}}{2 f^{2} \sqrt {-e f}}+\frac {3 \ln \left (-f x +\sqrt {-e f}\right ) b \,c^{2} d \,e^{2}}{2 f^{2} \sqrt {-e f}}-\frac {3 \ln \left (f x +\sqrt {-e f}\right ) a c \,d^{2} e^{2}}{2 f^{2} \sqrt {-e f}}-\frac {\ln \left (-f x +\sqrt {-e f}\right ) a \,d^{3} e^{3}}{2 f^{3} \sqrt {-e f}}-\frac {\ln \left (-f x +\sqrt {-e f}\right ) b \,c^{3} e}{2 f \sqrt {-e f}}+\frac {\ln \left (-f x +\sqrt {-e f}\right ) e^{4} b \,d^{3}}{2 f^{4} \sqrt {-e f}}-\frac {3 \ln \left (f x +\sqrt {-e f}\right ) b \,c^{2} d \,e^{2}}{2 f^{2} \sqrt {-e f}}+\frac {3 \ln \left (f x +\sqrt {-e f}\right ) b c \,d^{2} e^{3}}{2 f^{3} \sqrt {-e f}}+\frac {3 \ln \left (f x +\sqrt {-e f}\right ) a \,c^{2} d e}{2 f \sqrt {-e f}}+\frac {b \,d^{3} x^{7}}{7 f}\) | \(661\) |
Input:
int((b*x^2+a)*(d*x^2+c)^3/(f*x^2+e),x,method=_RETURNVERBOSE)
Output:
1/f^4*(1/7*b*d^3*x^7*f^3+1/5*a*d^3*f^3*x^5+3/5*b*c*d^2*f^3*x^5-1/5*b*d^3*e *f^2*x^5+a*c*d^2*f^3*x^3-1/3*a*d^3*e*f^2*x^3+b*c^2*d*f^3*x^3-b*c*d^2*e*f^2 *x^3+1/3*b*d^3*e^2*f*x^3+3*a*c^2*d*f^3*x-3*a*c*d^2*e*f^2*x+a*d^3*e^2*f*x+b *c^3*f^3*x-3*b*c^2*d*e*f^2*x+3*b*c*d^2*e^2*f*x-b*d^3*e^3*x)+(a*c^3*f^4-3*a *c^2*d*e*f^3+3*a*c*d^2*e^2*f^2-a*d^3*e^3*f-b*c^3*e*f^3+3*b*c^2*d*e^2*f^2-3 *b*c*d^2*e^3*f+b*d^3*e^4)/f^4/(e*f)^(1/2)*arctan(f*x/(e*f)^(1/2))
Time = 0.09 (sec) , antiderivative size = 586, normalized size of antiderivative = 3.26 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^3}{e+f x^2} \, dx=\left [\frac {30 \, b d^{3} e f^{4} x^{7} - 42 \, {\left (b d^{3} e^{2} f^{3} - {\left (3 \, b c d^{2} + a d^{3}\right )} e f^{4}\right )} x^{5} + 70 \, {\left (b d^{3} e^{3} f^{2} - {\left (3 \, b c d^{2} + a d^{3}\right )} e^{2} f^{3} + 3 \, {\left (b c^{2} d + a c d^{2}\right )} e f^{4}\right )} x^{3} - 105 \, {\left (b d^{3} e^{4} + a c^{3} f^{4} - {\left (3 \, b c d^{2} + a d^{3}\right )} e^{3} f + 3 \, {\left (b c^{2} d + a c d^{2}\right )} e^{2} f^{2} - {\left (b c^{3} + 3 \, a c^{2} d\right )} e f^{3}\right )} \sqrt {-e f} \log \left (\frac {f x^{2} - 2 \, \sqrt {-e f} x - e}{f x^{2} + e}\right ) - 210 \, {\left (b d^{3} e^{4} f - {\left (3 \, b c d^{2} + a d^{3}\right )} e^{3} f^{2} + 3 \, {\left (b c^{2} d + a c d^{2}\right )} e^{2} f^{3} - {\left (b c^{3} + 3 \, a c^{2} d\right )} e f^{4}\right )} x}{210 \, e f^{5}}, \frac {15 \, b d^{3} e f^{4} x^{7} - 21 \, {\left (b d^{3} e^{2} f^{3} - {\left (3 \, b c d^{2} + a d^{3}\right )} e f^{4}\right )} x^{5} + 35 \, {\left (b d^{3} e^{3} f^{2} - {\left (3 \, b c d^{2} + a d^{3}\right )} e^{2} f^{3} + 3 \, {\left (b c^{2} d + a c d^{2}\right )} e f^{4}\right )} x^{3} + 105 \, {\left (b d^{3} e^{4} + a c^{3} f^{4} - {\left (3 \, b c d^{2} + a d^{3}\right )} e^{3} f + 3 \, {\left (b c^{2} d + a c d^{2}\right )} e^{2} f^{2} - {\left (b c^{3} + 3 \, a c^{2} d\right )} e f^{3}\right )} \sqrt {e f} \arctan \left (\frac {\sqrt {e f} x}{e}\right ) - 105 \, {\left (b d^{3} e^{4} f - {\left (3 \, b c d^{2} + a d^{3}\right )} e^{3} f^{2} + 3 \, {\left (b c^{2} d + a c d^{2}\right )} e^{2} f^{3} - {\left (b c^{3} + 3 \, a c^{2} d\right )} e f^{4}\right )} x}{105 \, e f^{5}}\right ] \] Input:
integrate((b*x^2+a)*(d*x^2+c)^3/(f*x^2+e),x, algorithm="fricas")
Output:
[1/210*(30*b*d^3*e*f^4*x^7 - 42*(b*d^3*e^2*f^3 - (3*b*c*d^2 + a*d^3)*e*f^4 )*x^5 + 70*(b*d^3*e^3*f^2 - (3*b*c*d^2 + a*d^3)*e^2*f^3 + 3*(b*c^2*d + a*c *d^2)*e*f^4)*x^3 - 105*(b*d^3*e^4 + a*c^3*f^4 - (3*b*c*d^2 + a*d^3)*e^3*f + 3*(b*c^2*d + a*c*d^2)*e^2*f^2 - (b*c^3 + 3*a*c^2*d)*e*f^3)*sqrt(-e*f)*lo g((f*x^2 - 2*sqrt(-e*f)*x - e)/(f*x^2 + e)) - 210*(b*d^3*e^4*f - (3*b*c*d^ 2 + a*d^3)*e^3*f^2 + 3*(b*c^2*d + a*c*d^2)*e^2*f^3 - (b*c^3 + 3*a*c^2*d)*e *f^4)*x)/(e*f^5), 1/105*(15*b*d^3*e*f^4*x^7 - 21*(b*d^3*e^2*f^3 - (3*b*c*d ^2 + a*d^3)*e*f^4)*x^5 + 35*(b*d^3*e^3*f^2 - (3*b*c*d^2 + a*d^3)*e^2*f^3 + 3*(b*c^2*d + a*c*d^2)*e*f^4)*x^3 + 105*(b*d^3*e^4 + a*c^3*f^4 - (3*b*c*d^ 2 + a*d^3)*e^3*f + 3*(b*c^2*d + a*c*d^2)*e^2*f^2 - (b*c^3 + 3*a*c^2*d)*e*f ^3)*sqrt(e*f)*arctan(sqrt(e*f)*x/e) - 105*(b*d^3*e^4*f - (3*b*c*d^2 + a*d^ 3)*e^3*f^2 + 3*(b*c^2*d + a*c*d^2)*e^2*f^3 - (b*c^3 + 3*a*c^2*d)*e*f^4)*x) /(e*f^5)]
Leaf count of result is larger than twice the leaf count of optimal. 508 vs. \(2 (172) = 344\).
Time = 0.73 (sec) , antiderivative size = 508, normalized size of antiderivative = 2.82 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^3}{e+f x^2} \, dx=\frac {b d^{3} x^{7}}{7 f} + x^{5} \left (\frac {a d^{3}}{5 f} + \frac {3 b c d^{2}}{5 f} - \frac {b d^{3} e}{5 f^{2}}\right ) + x^{3} \left (\frac {a c d^{2}}{f} - \frac {a d^{3} e}{3 f^{2}} + \frac {b c^{2} d}{f} - \frac {b c d^{2} e}{f^{2}} + \frac {b d^{3} e^{2}}{3 f^{3}}\right ) + x \left (\frac {3 a c^{2} d}{f} - \frac {3 a c d^{2} e}{f^{2}} + \frac {a d^{3} e^{2}}{f^{3}} + \frac {b c^{3}}{f} - \frac {3 b c^{2} d e}{f^{2}} + \frac {3 b c d^{2} e^{2}}{f^{3}} - \frac {b d^{3} e^{3}}{f^{4}}\right ) - \frac {\sqrt {- \frac {1}{e f^{9}}} \left (a f - b e\right ) \left (c f - d e\right )^{3} \log {\left (- \frac {e f^{4} \sqrt {- \frac {1}{e f^{9}}} \left (a f - b e\right ) \left (c f - d e\right )^{3}}{a c^{3} f^{4} - 3 a c^{2} d e f^{3} + 3 a c d^{2} e^{2} f^{2} - a d^{3} e^{3} f - b c^{3} e f^{3} + 3 b c^{2} d e^{2} f^{2} - 3 b c d^{2} e^{3} f + b d^{3} e^{4}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{e f^{9}}} \left (a f - b e\right ) \left (c f - d e\right )^{3} \log {\left (\frac {e f^{4} \sqrt {- \frac {1}{e f^{9}}} \left (a f - b e\right ) \left (c f - d e\right )^{3}}{a c^{3} f^{4} - 3 a c^{2} d e f^{3} + 3 a c d^{2} e^{2} f^{2} - a d^{3} e^{3} f - b c^{3} e f^{3} + 3 b c^{2} d e^{2} f^{2} - 3 b c d^{2} e^{3} f + b d^{3} e^{4}} + x \right )}}{2} \] Input:
integrate((b*x**2+a)*(d*x**2+c)**3/(f*x**2+e),x)
Output:
b*d**3*x**7/(7*f) + x**5*(a*d**3/(5*f) + 3*b*c*d**2/(5*f) - b*d**3*e/(5*f* *2)) + x**3*(a*c*d**2/f - a*d**3*e/(3*f**2) + b*c**2*d/f - b*c*d**2*e/f**2 + b*d**3*e**2/(3*f**3)) + x*(3*a*c**2*d/f - 3*a*c*d**2*e/f**2 + a*d**3*e* *2/f**3 + b*c**3/f - 3*b*c**2*d*e/f**2 + 3*b*c*d**2*e**2/f**3 - b*d**3*e** 3/f**4) - sqrt(-1/(e*f**9))*(a*f - b*e)*(c*f - d*e)**3*log(-e*f**4*sqrt(-1 /(e*f**9))*(a*f - b*e)*(c*f - d*e)**3/(a*c**3*f**4 - 3*a*c**2*d*e*f**3 + 3 *a*c*d**2*e**2*f**2 - a*d**3*e**3*f - b*c**3*e*f**3 + 3*b*c**2*d*e**2*f**2 - 3*b*c*d**2*e**3*f + b*d**3*e**4) + x)/2 + sqrt(-1/(e*f**9))*(a*f - b*e) *(c*f - d*e)**3*log(e*f**4*sqrt(-1/(e*f**9))*(a*f - b*e)*(c*f - d*e)**3/(a *c**3*f**4 - 3*a*c**2*d*e*f**3 + 3*a*c*d**2*e**2*f**2 - a*d**3*e**3*f - b* c**3*e*f**3 + 3*b*c**2*d*e**2*f**2 - 3*b*c*d**2*e**3*f + b*d**3*e**4) + x) /2
Exception generated. \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^3}{e+f x^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x^2+a)*(d*x^2+c)^3/(f*x^2+e),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Time = 0.13 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.74 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^3}{e+f x^2} \, dx=\frac {{\left (b d^{3} e^{4} - 3 \, b c d^{2} e^{3} f - a d^{3} e^{3} f + 3 \, b c^{2} d e^{2} f^{2} + 3 \, a c d^{2} e^{2} f^{2} - b c^{3} e f^{3} - 3 \, a c^{2} d e f^{3} + a c^{3} f^{4}\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{\sqrt {e f} f^{4}} + \frac {15 \, b d^{3} f^{6} x^{7} - 21 \, b d^{3} e f^{5} x^{5} + 63 \, b c d^{2} f^{6} x^{5} + 21 \, a d^{3} f^{6} x^{5} + 35 \, b d^{3} e^{2} f^{4} x^{3} - 105 \, b c d^{2} e f^{5} x^{3} - 35 \, a d^{3} e f^{5} x^{3} + 105 \, b c^{2} d f^{6} x^{3} + 105 \, a c d^{2} f^{6} x^{3} - 105 \, b d^{3} e^{3} f^{3} x + 315 \, b c d^{2} e^{2} f^{4} x + 105 \, a d^{3} e^{2} f^{4} x - 315 \, b c^{2} d e f^{5} x - 315 \, a c d^{2} e f^{5} x + 105 \, b c^{3} f^{6} x + 315 \, a c^{2} d f^{6} x}{105 \, f^{7}} \] Input:
integrate((b*x^2+a)*(d*x^2+c)^3/(f*x^2+e),x, algorithm="giac")
Output:
(b*d^3*e^4 - 3*b*c*d^2*e^3*f - a*d^3*e^3*f + 3*b*c^2*d*e^2*f^2 + 3*a*c*d^2 *e^2*f^2 - b*c^3*e*f^3 - 3*a*c^2*d*e*f^3 + a*c^3*f^4)*arctan(f*x/sqrt(e*f) )/(sqrt(e*f)*f^4) + 1/105*(15*b*d^3*f^6*x^7 - 21*b*d^3*e*f^5*x^5 + 63*b*c* d^2*f^6*x^5 + 21*a*d^3*f^6*x^5 + 35*b*d^3*e^2*f^4*x^3 - 105*b*c*d^2*e*f^5* x^3 - 35*a*d^3*e*f^5*x^3 + 105*b*c^2*d*f^6*x^3 + 105*a*c*d^2*f^6*x^3 - 105 *b*d^3*e^3*f^3*x + 315*b*c*d^2*e^2*f^4*x + 105*a*d^3*e^2*f^4*x - 315*b*c^2 *d*e*f^5*x - 315*a*c*d^2*e*f^5*x + 105*b*c^3*f^6*x + 315*a*c^2*d*f^6*x)/f^ 7
Time = 1.71 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.73 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^3}{e+f x^2} \, dx=x\,\left (\frac {b\,c^3+3\,a\,d\,c^2}{f}+\frac {e\,\left (\frac {e\,\left (\frac {a\,d^3+3\,b\,c\,d^2}{f}-\frac {b\,d^3\,e}{f^2}\right )}{f}-\frac {3\,c\,d\,\left (a\,d+b\,c\right )}{f}\right )}{f}\right )+x^5\,\left (\frac {a\,d^3+3\,b\,c\,d^2}{5\,f}-\frac {b\,d^3\,e}{5\,f^2}\right )-x^3\,\left (\frac {e\,\left (\frac {a\,d^3+3\,b\,c\,d^2}{f}-\frac {b\,d^3\,e}{f^2}\right )}{3\,f}-\frac {c\,d\,\left (a\,d+b\,c\right )}{f}\right )+\frac {b\,d^3\,x^7}{7\,f}+\frac {\mathrm {atan}\left (\frac {\sqrt {f}\,x\,\left (a\,f-b\,e\right )\,{\left (c\,f-d\,e\right )}^3}{\sqrt {e}\,\left (-b\,c^3\,e\,f^3+a\,c^3\,f^4+3\,b\,c^2\,d\,e^2\,f^2-3\,a\,c^2\,d\,e\,f^3-3\,b\,c\,d^2\,e^3\,f+3\,a\,c\,d^2\,e^2\,f^2+b\,d^3\,e^4-a\,d^3\,e^3\,f\right )}\right )\,\left (a\,f-b\,e\right )\,{\left (c\,f-d\,e\right )}^3}{\sqrt {e}\,f^{9/2}} \] Input:
int(((a + b*x^2)*(c + d*x^2)^3)/(e + f*x^2),x)
Output:
x*((b*c^3 + 3*a*c^2*d)/f + (e*((e*((a*d^3 + 3*b*c*d^2)/f - (b*d^3*e)/f^2)) /f - (3*c*d*(a*d + b*c))/f))/f) + x^5*((a*d^3 + 3*b*c*d^2)/(5*f) - (b*d^3* e)/(5*f^2)) - x^3*((e*((a*d^3 + 3*b*c*d^2)/f - (b*d^3*e)/f^2))/(3*f) - (c* d*(a*d + b*c))/f) + (b*d^3*x^7)/(7*f) + (atan((f^(1/2)*x*(a*f - b*e)*(c*f - d*e)^3)/(e^(1/2)*(a*c^3*f^4 + b*d^3*e^4 - a*d^3*e^3*f - b*c^3*e*f^3 - 3* a*c^2*d*e*f^3 - 3*b*c*d^2*e^3*f + 3*a*c*d^2*e^2*f^2 + 3*b*c^2*d*e^2*f^2))) *(a*f - b*e)*(c*f - d*e)^3)/(e^(1/2)*f^(9/2))
Time = 0.16 (sec) , antiderivative size = 441, normalized size of antiderivative = 2.45 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^3}{e+f x^2} \, dx=\frac {105 \sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) a \,c^{3} f^{4}-315 \sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) a \,c^{2} d e \,f^{3}+315 \sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) a c \,d^{2} e^{2} f^{2}-105 \sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) a \,d^{3} e^{3} f -105 \sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) b \,c^{3} e \,f^{3}+315 \sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) b \,c^{2} d \,e^{2} f^{2}-315 \sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) b c \,d^{2} e^{3} f +105 \sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) b \,d^{3} e^{4}+315 a \,c^{2} d e \,f^{4} x -315 a c \,d^{2} e^{2} f^{3} x +105 a c \,d^{2} e \,f^{4} x^{3}+105 a \,d^{3} e^{3} f^{2} x -35 a \,d^{3} e^{2} f^{3} x^{3}+21 a \,d^{3} e \,f^{4} x^{5}+105 b \,c^{3} e \,f^{4} x -315 b \,c^{2} d \,e^{2} f^{3} x +105 b \,c^{2} d e \,f^{4} x^{3}+315 b c \,d^{2} e^{3} f^{2} x -105 b c \,d^{2} e^{2} f^{3} x^{3}+63 b c \,d^{2} e \,f^{4} x^{5}-105 b \,d^{3} e^{4} f x +35 b \,d^{3} e^{3} f^{2} x^{3}-21 b \,d^{3} e^{2} f^{3} x^{5}+15 b \,d^{3} e \,f^{4} x^{7}}{105 e \,f^{5}} \] Input:
int((b*x^2+a)*(d*x^2+c)^3/(f*x^2+e),x)
Output:
(105*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*c**3*f**4 - 315*sqrt( f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*c**2*d*e*f**3 + 315*sqrt(f)*sqr t(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*c*d**2*e**2*f**2 - 105*sqrt(f)*sqrt(e )*atan((f*x)/(sqrt(f)*sqrt(e)))*a*d**3*e**3*f - 105*sqrt(f)*sqrt(e)*atan(( f*x)/(sqrt(f)*sqrt(e)))*b*c**3*e*f**3 + 315*sqrt(f)*sqrt(e)*atan((f*x)/(sq rt(f)*sqrt(e)))*b*c**2*d*e**2*f**2 - 315*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt( f)*sqrt(e)))*b*c*d**2*e**3*f + 105*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqr t(e)))*b*d**3*e**4 + 315*a*c**2*d*e*f**4*x - 315*a*c*d**2*e**2*f**3*x + 10 5*a*c*d**2*e*f**4*x**3 + 105*a*d**3*e**3*f**2*x - 35*a*d**3*e**2*f**3*x**3 + 21*a*d**3*e*f**4*x**5 + 105*b*c**3*e*f**4*x - 315*b*c**2*d*e**2*f**3*x + 105*b*c**2*d*e*f**4*x**3 + 315*b*c*d**2*e**3*f**2*x - 105*b*c*d**2*e**2* f**3*x**3 + 63*b*c*d**2*e*f**4*x**5 - 105*b*d**3*e**4*f*x + 35*b*d**3*e**3 *f**2*x**3 - 21*b*d**3*e**2*f**3*x**5 + 15*b*d**3*e*f**4*x**7)/(105*e*f**5 )