Integrand size = 27, antiderivative size = 216 \[ \int \frac {(d+e x)^3 \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^2} \, dx=-\frac {3 e^2 (A c d-a (3 C d+B e)) x}{2 a c^2}-\frac {(A c-2 a C) e^3 x^2}{2 a c^2}-\frac {(a B-(A c-a C) x) (d+e x)^3}{2 a c \left (a+c x^2\right )}+\frac {\left (A c d \left (c d^2+3 a e^2\right )-a \left (3 a e^2 (3 C d+B e)-c d^2 (C d+3 B e)\right )\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} c^{5/2}}-\frac {e \left (2 a C e^2-c \left (3 C d^2+e (3 B d+A e)\right )\right ) \log \left (a+c x^2\right )}{2 c^3} \]
-3/2*e^2*(A*c*d-a*(B*e+3*C*d))*x/a/c^2-1/2*(A*c-2*C*a)*e^3*x^2/a/c^2-1/2*( a*B-(A*c-C*a)*x)*(e*x+d)^3/a/c/(c*x^2+a)+1/2*(A*c*d*(3*a*e^2+c*d^2)-a*(3*a *e^2*(B*e+3*C*d)-c*d^2*(3*B*e+C*d)))*arctan(x*c^(1/2)/a^(1/2))/a^(3/2)/c^( 5/2)-1/2*e*(2*a*C*e^2-c*(3*C*d^2+e*(A*e+3*B*d)))*ln(c*x^2+a)/c^3
Time = 0.13 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.08 \[ \int \frac {(d+e x)^3 \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^2} \, dx=\frac {2 c e^2 (3 C d+B e) x+c C e^3 x^2+\frac {-a^3 C e^3+A c^3 d^3 x-a c^2 d \left (C d^2 x+3 A e (d+e x)+B d (d+3 e x)\right )+a^2 c e (3 C d (d+e x)+e (3 B d+A e+B e x))}{a \left (a+c x^2\right )}+\frac {\sqrt {c} \left (A c d \left (c d^2+3 a e^2\right )+a \left (-3 a e^2 (3 C d+B e)+c d^2 (C d+3 B e)\right )\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{3/2}}+e \left (3 c C d^2-2 a C e^2+c e (3 B d+A e)\right ) \log \left (a+c x^2\right )}{2 c^3} \]
(2*c*e^2*(3*C*d + B*e)*x + c*C*e^3*x^2 + (-(a^3*C*e^3) + A*c^3*d^3*x - a*c ^2*d*(C*d^2*x + 3*A*e*(d + e*x) + B*d*(d + 3*e*x)) + a^2*c*e*(3*C*d*(d + e *x) + e*(3*B*d + A*e + B*e*x)))/(a*(a + c*x^2)) + (Sqrt[c]*(A*c*d*(c*d^2 + 3*a*e^2) + a*(-3*a*e^2*(3*C*d + B*e) + c*d^2*(C*d + 3*B*e)))*ArcTan[(Sqrt [c]*x)/Sqrt[a]])/a^(3/2) + e*(3*c*C*d^2 - 2*a*C*e^2 + c*e*(3*B*d + A*e))*L og[a + c*x^2])/(2*c^3)
Time = 0.52 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2176, 25, 657, 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.
| \(\displaystyle \int \frac {(d+e x)^3 \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 2176 |
| \(\displaystyle -\frac {\int -\frac {(d+e x)^2 (A c d+a C d+3 a B e-2 (A c-2 a C) e x)}{c x^2+a}dx}{2 a c}-\frac {(d+e x)^3 (a B-x (A c-a C))}{2 a c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {(d+e x)^2 (A c d+a C d+3 a B e-2 (A c-2 a C) e x)}{c x^2+a}dx}{2 a c}-\frac {(d+e x)^3 (a B-x (A c-a C))}{2 a c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 657 |
| \(\displaystyle \frac {\int \left (-\frac {2 (A c-2 a C) x e^3}{c}-\frac {3 (A c d-3 a C d-a B e) e^2}{c}+\frac {A c d \left (c d^2+3 a e^2\right )-a \left (3 a e^2 (3 C d+B e)-c d^2 (C d+3 B e)\right )-2 a e \left (2 a C e^2-c \left (3 C d^2+e (3 B d+A e)\right )\right ) x}{c \left (c x^2+a\right )}\right )dx}{2 a c}-\frac {(d+e x)^3 (a B-x (A c-a C))}{2 a c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (A c d \left (3 a e^2+c d^2\right )-a \left (3 a e^2 (B e+3 C d)-c d^2 (3 B e+C d)\right )\right )}{\sqrt {a} c^{3/2}}-\frac {a e \log \left (a+c x^2\right ) \left (2 a C e^2-c \left (e (A e+3 B d)+3 C d^2\right )\right )}{c^2}-\frac {3 e^2 x (-a B e-3 a C d+A c d)}{c}-\frac {e^3 x^2 (A c-2 a C)}{c}}{2 a c}-\frac {(d+e x)^3 (a B-x (A c-a C))}{2 a c \left (a+c x^2\right )}\) |
-1/2*((a*B - (A*c - a*C)*x)*(d + e*x)^3)/(a*c*(a + c*x^2)) + ((-3*e^2*(A*c *d - 3*a*C*d - a*B*e)*x)/c - ((A*c - 2*a*C)*e^3*x^2)/c + ((A*c*d*(c*d^2 + 3*a*e^2) - a*(3*a*e^2*(3*C*d + B*e) - c*d^2*(C*d + 3*B*e)))*ArcTan[(Sqrt[c ]*x)/Sqrt[a]])/(Sqrt[a]*c^(3/2)) - (a*e*(2*a*C*e^2 - c*(3*C*d^2 + e*(3*B*d + A*e)))*Log[a + c*x^2])/c^2)/(2*a*c)
3.1.50.3.1 Defintions of rubi rules used
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_) + (c_.)*( x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g*x)^n/(a + c*x^ 2)), x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IntegersQ[n]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{Qx = PolynomialQuotient[Pq, a + b*x^2, x], R = Coeff[PolynomialRema inder[Pq, a + b*x^2, x], x, 0], S = Coeff[PolynomialRemainder[Pq, a + b*x^2 , x], x, 1]}, Simp[(d + e*x)^m*(a + b*x^2)^(p + 1)*((a*S - b*R*x)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*b*(p + 1)) Int[(d + e*x)^(m - 1)*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*b*(p + 1)*(d + e*x)*Qx - a*e*S*m + b*d*R*(2*p + 3) + b*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e}, x] && PolyQ[Pq, x ] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && !(IGtQ[m, 0] && R ationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Time = 0.65 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.31
| method | result | size |
| default | \(\frac {e^{2} \left (\frac {1}{2} C \,x^{2} e +B e x +3 C d x \right )}{c^{2}}+\frac {\frac {-\frac {\left (3 A a c d \,e^{2}-A \,d^{3} c^{2}-a^{2} B \,e^{3}+3 B a c \,d^{2} e -3 C \,a^{2} d \,e^{2}+C a c \,d^{3}\right ) x}{2 a}+\frac {A a c \,e^{3}-3 A \,c^{2} d^{2} e +3 B a c d \,e^{2}-B \,c^{2} d^{3}-C \,a^{2} e^{3}+3 C a c \,d^{2} e}{2 c}}{c \,x^{2}+a}+\frac {\frac {\left (2 A a c \,e^{3}+6 B a c d \,e^{2}-4 C \,a^{2} e^{3}+6 C a c \,d^{2} e \right ) \ln \left (c \,x^{2}+a \right )}{2 c}+\frac {\left (3 A a c d \,e^{2}+A \,d^{3} c^{2}-3 a^{2} B \,e^{3}+3 B a c \,d^{2} e -9 C \,a^{2} d \,e^{2}+C a c \,d^{3}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{2 a}}{c^{2}}\) | \(284\) |
| risch | \(\text {Expression too large to display}\) | \(1628\) |
e^2/c^2*(1/2*C*x^2*e+B*e*x+3*C*d*x)+1/c^2*((-1/2*(3*A*a*c*d*e^2-A*c^2*d^3- B*a^2*e^3+3*B*a*c*d^2*e-3*C*a^2*d*e^2+C*a*c*d^3)/a*x+1/2*(A*a*c*e^3-3*A*c^ 2*d^2*e+3*B*a*c*d*e^2-B*c^2*d^3-C*a^2*e^3+3*C*a*c*d^2*e)/c)/(c*x^2+a)+1/2/ a*(1/2*(2*A*a*c*e^3+6*B*a*c*d*e^2-4*C*a^2*e^3+6*C*a*c*d^2*e)/c*ln(c*x^2+a) +(3*A*a*c*d*e^2+A*c^2*d^3-3*B*a^2*e^3+3*B*a*c*d^2*e-9*C*a^2*d*e^2+C*a*c*d^ 3)/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 454 vs. \(2 (196) = 392\).
Time = 0.31 (sec) , antiderivative size = 931, normalized size of antiderivative = 4.31 \[ \int \frac {(d+e x)^3 \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^2} \, dx=\left [\frac {2 \, C a^{2} c^{2} e^{3} x^{4} + 2 \, C a^{3} c e^{3} x^{2} - 2 \, B a^{2} c^{2} d^{3} + 6 \, B a^{3} c d e^{2} + 6 \, {\left (C a^{3} c - A a^{2} c^{2}\right )} d^{2} e - 2 \, {\left (C a^{4} - A a^{3} c\right )} e^{3} + 4 \, {\left (3 \, C a^{2} c^{2} d e^{2} + B a^{2} c^{2} e^{3}\right )} x^{3} + {\left (3 \, B a^{2} c d^{2} e - 3 \, B a^{3} e^{3} + {\left (C a^{2} c + A a c^{2}\right )} d^{3} - 3 \, {\left (3 \, C a^{3} - A a^{2} c\right )} d e^{2} + {\left (3 \, B a c^{2} d^{2} e - 3 \, B a^{2} c e^{3} + {\left (C a c^{2} + A c^{3}\right )} d^{3} - 3 \, {\left (3 \, C a^{2} c - A a c^{2}\right )} d e^{2}\right )} x^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} + 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) - 2 \, {\left (3 \, B a^{2} c^{2} d^{2} e - 3 \, B a^{3} c e^{3} + {\left (C a^{2} c^{2} - A a c^{3}\right )} d^{3} - 3 \, {\left (3 \, C a^{3} c - A a^{2} c^{2}\right )} d e^{2}\right )} x + 2 \, {\left (3 \, C a^{3} c d^{2} e + 3 \, B a^{3} c d e^{2} - {\left (2 \, C a^{4} - A a^{3} c\right )} e^{3} + {\left (3 \, C a^{2} c^{2} d^{2} e + 3 \, B a^{2} c^{2} d e^{2} - {\left (2 \, C a^{3} c - A a^{2} c^{2}\right )} e^{3}\right )} x^{2}\right )} \log \left (c x^{2} + a\right )}{4 \, {\left (a^{2} c^{4} x^{2} + a^{3} c^{3}\right )}}, \frac {C a^{2} c^{2} e^{3} x^{4} + C a^{3} c e^{3} x^{2} - B a^{2} c^{2} d^{3} + 3 \, B a^{3} c d e^{2} + 3 \, {\left (C a^{3} c - A a^{2} c^{2}\right )} d^{2} e - {\left (C a^{4} - A a^{3} c\right )} e^{3} + 2 \, {\left (3 \, C a^{2} c^{2} d e^{2} + B a^{2} c^{2} e^{3}\right )} x^{3} + {\left (3 \, B a^{2} c d^{2} e - 3 \, B a^{3} e^{3} + {\left (C a^{2} c + A a c^{2}\right )} d^{3} - 3 \, {\left (3 \, C a^{3} - A a^{2} c\right )} d e^{2} + {\left (3 \, B a c^{2} d^{2} e - 3 \, B a^{2} c e^{3} + {\left (C a c^{2} + A c^{3}\right )} d^{3} - 3 \, {\left (3 \, C a^{2} c - A a c^{2}\right )} d e^{2}\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - {\left (3 \, B a^{2} c^{2} d^{2} e - 3 \, B a^{3} c e^{3} + {\left (C a^{2} c^{2} - A a c^{3}\right )} d^{3} - 3 \, {\left (3 \, C a^{3} c - A a^{2} c^{2}\right )} d e^{2}\right )} x + {\left (3 \, C a^{3} c d^{2} e + 3 \, B a^{3} c d e^{2} - {\left (2 \, C a^{4} - A a^{3} c\right )} e^{3} + {\left (3 \, C a^{2} c^{2} d^{2} e + 3 \, B a^{2} c^{2} d e^{2} - {\left (2 \, C a^{3} c - A a^{2} c^{2}\right )} e^{3}\right )} x^{2}\right )} \log \left (c x^{2} + a\right )}{2 \, {\left (a^{2} c^{4} x^{2} + a^{3} c^{3}\right )}}\right ] \]
[1/4*(2*C*a^2*c^2*e^3*x^4 + 2*C*a^3*c*e^3*x^2 - 2*B*a^2*c^2*d^3 + 6*B*a^3* c*d*e^2 + 6*(C*a^3*c - A*a^2*c^2)*d^2*e - 2*(C*a^4 - A*a^3*c)*e^3 + 4*(3*C *a^2*c^2*d*e^2 + B*a^2*c^2*e^3)*x^3 + (3*B*a^2*c*d^2*e - 3*B*a^3*e^3 + (C* a^2*c + A*a*c^2)*d^3 - 3*(3*C*a^3 - A*a^2*c)*d*e^2 + (3*B*a*c^2*d^2*e - 3* B*a^2*c*e^3 + (C*a*c^2 + A*c^3)*d^3 - 3*(3*C*a^2*c - A*a*c^2)*d*e^2)*x^2)* sqrt(-a*c)*log((c*x^2 + 2*sqrt(-a*c)*x - a)/(c*x^2 + a)) - 2*(3*B*a^2*c^2* d^2*e - 3*B*a^3*c*e^3 + (C*a^2*c^2 - A*a*c^3)*d^3 - 3*(3*C*a^3*c - A*a^2*c ^2)*d*e^2)*x + 2*(3*C*a^3*c*d^2*e + 3*B*a^3*c*d*e^2 - (2*C*a^4 - A*a^3*c)* e^3 + (3*C*a^2*c^2*d^2*e + 3*B*a^2*c^2*d*e^2 - (2*C*a^3*c - A*a^2*c^2)*e^3 )*x^2)*log(c*x^2 + a))/(a^2*c^4*x^2 + a^3*c^3), 1/2*(C*a^2*c^2*e^3*x^4 + C *a^3*c*e^3*x^2 - B*a^2*c^2*d^3 + 3*B*a^3*c*d*e^2 + 3*(C*a^3*c - A*a^2*c^2) *d^2*e - (C*a^4 - A*a^3*c)*e^3 + 2*(3*C*a^2*c^2*d*e^2 + B*a^2*c^2*e^3)*x^3 + (3*B*a^2*c*d^2*e - 3*B*a^3*e^3 + (C*a^2*c + A*a*c^2)*d^3 - 3*(3*C*a^3 - A*a^2*c)*d*e^2 + (3*B*a*c^2*d^2*e - 3*B*a^2*c*e^3 + (C*a*c^2 + A*c^3)*d^3 - 3*(3*C*a^2*c - A*a*c^2)*d*e^2)*x^2)*sqrt(a*c)*arctan(sqrt(a*c)*x/a) - ( 3*B*a^2*c^2*d^2*e - 3*B*a^3*c*e^3 + (C*a^2*c^2 - A*a*c^3)*d^3 - 3*(3*C*a^3 *c - A*a^2*c^2)*d*e^2)*x + (3*C*a^3*c*d^2*e + 3*B*a^3*c*d*e^2 - (2*C*a^4 - A*a^3*c)*e^3 + (3*C*a^2*c^2*d^2*e + 3*B*a^2*c^2*d*e^2 - (2*C*a^3*c - A*a^ 2*c^2)*e^3)*x^2)*log(c*x^2 + a))/(a^2*c^4*x^2 + a^3*c^3)]
Leaf count of result is larger than twice the leaf count of optimal. 952 vs. \(2 (197) = 394\).
Time = 25.72 (sec) , antiderivative size = 952, normalized size of antiderivative = 4.41 \[ \int \frac {(d+e x)^3 \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^2} \, dx=\frac {C e^{3} x^{2}}{2 c^{2}} + x \left (\frac {B e^{3}}{c^{2}} + \frac {3 C d e^{2}}{c^{2}}\right ) + \left (- \frac {e \left (- A c e^{2} - 3 B c d e + 2 C a e^{2} - 3 C c d^{2}\right )}{2 c^{3}} - \frac {\sqrt {- a^{3} c^{7}} \left (- 3 A a c d e^{2} - A c^{2} d^{3} + 3 B a^{2} e^{3} - 3 B a c d^{2} e + 9 C a^{2} d e^{2} - C a c d^{3}\right )}{4 a^{3} c^{6}}\right ) \log {\left (x + \frac {2 A a^{2} c e^{3} + 6 B a^{2} c d e^{2} - 4 C a^{3} e^{3} + 6 C a^{2} c d^{2} e - 4 a^{2} c^{3} \left (- \frac {e \left (- A c e^{2} - 3 B c d e + 2 C a e^{2} - 3 C c d^{2}\right )}{2 c^{3}} - \frac {\sqrt {- a^{3} c^{7}} \left (- 3 A a c d e^{2} - A c^{2} d^{3} + 3 B a^{2} e^{3} - 3 B a c d^{2} e + 9 C a^{2} d e^{2} - C a c d^{3}\right )}{4 a^{3} c^{6}}\right )}{- 3 A a c^{2} d e^{2} - A c^{3} d^{3} + 3 B a^{2} c e^{3} - 3 B a c^{2} d^{2} e + 9 C a^{2} c d e^{2} - C a c^{2} d^{3}} \right )} + \left (- \frac {e \left (- A c e^{2} - 3 B c d e + 2 C a e^{2} - 3 C c d^{2}\right )}{2 c^{3}} + \frac {\sqrt {- a^{3} c^{7}} \left (- 3 A a c d e^{2} - A c^{2} d^{3} + 3 B a^{2} e^{3} - 3 B a c d^{2} e + 9 C a^{2} d e^{2} - C a c d^{3}\right )}{4 a^{3} c^{6}}\right ) \log {\left (x + \frac {2 A a^{2} c e^{3} + 6 B a^{2} c d e^{2} - 4 C a^{3} e^{3} + 6 C a^{2} c d^{2} e - 4 a^{2} c^{3} \left (- \frac {e \left (- A c e^{2} - 3 B c d e + 2 C a e^{2} - 3 C c d^{2}\right )}{2 c^{3}} + \frac {\sqrt {- a^{3} c^{7}} \left (- 3 A a c d e^{2} - A c^{2} d^{3} + 3 B a^{2} e^{3} - 3 B a c d^{2} e + 9 C a^{2} d e^{2} - C a c d^{3}\right )}{4 a^{3} c^{6}}\right )}{- 3 A a c^{2} d e^{2} - A c^{3} d^{3} + 3 B a^{2} c e^{3} - 3 B a c^{2} d^{2} e + 9 C a^{2} c d e^{2} - C a c^{2} d^{3}} \right )} + \frac {A a^{2} c e^{3} - 3 A a c^{2} d^{2} e + 3 B a^{2} c d e^{2} - B a c^{2} d^{3} - C a^{3} e^{3} + 3 C a^{2} c d^{2} e + x \left (- 3 A a c^{2} d e^{2} + A c^{3} d^{3} + B a^{2} c e^{3} - 3 B a c^{2} d^{2} e + 3 C a^{2} c d e^{2} - C a c^{2} d^{3}\right )}{2 a^{2} c^{3} + 2 a c^{4} x^{2}} \]
C*e**3*x**2/(2*c**2) + x*(B*e**3/c**2 + 3*C*d*e**2/c**2) + (-e*(-A*c*e**2 - 3*B*c*d*e + 2*C*a*e**2 - 3*C*c*d**2)/(2*c**3) - sqrt(-a**3*c**7)*(-3*A*a *c*d*e**2 - A*c**2*d**3 + 3*B*a**2*e**3 - 3*B*a*c*d**2*e + 9*C*a**2*d*e**2 - C*a*c*d**3)/(4*a**3*c**6))*log(x + (2*A*a**2*c*e**3 + 6*B*a**2*c*d*e**2 - 4*C*a**3*e**3 + 6*C*a**2*c*d**2*e - 4*a**2*c**3*(-e*(-A*c*e**2 - 3*B*c* d*e + 2*C*a*e**2 - 3*C*c*d**2)/(2*c**3) - sqrt(-a**3*c**7)*(-3*A*a*c*d*e** 2 - A*c**2*d**3 + 3*B*a**2*e**3 - 3*B*a*c*d**2*e + 9*C*a**2*d*e**2 - C*a*c *d**3)/(4*a**3*c**6)))/(-3*A*a*c**2*d*e**2 - A*c**3*d**3 + 3*B*a**2*c*e**3 - 3*B*a*c**2*d**2*e + 9*C*a**2*c*d*e**2 - C*a*c**2*d**3)) + (-e*(-A*c*e** 2 - 3*B*c*d*e + 2*C*a*e**2 - 3*C*c*d**2)/(2*c**3) + sqrt(-a**3*c**7)*(-3*A *a*c*d*e**2 - A*c**2*d**3 + 3*B*a**2*e**3 - 3*B*a*c*d**2*e + 9*C*a**2*d*e* *2 - C*a*c*d**3)/(4*a**3*c**6))*log(x + (2*A*a**2*c*e**3 + 6*B*a**2*c*d*e* *2 - 4*C*a**3*e**3 + 6*C*a**2*c*d**2*e - 4*a**2*c**3*(-e*(-A*c*e**2 - 3*B* c*d*e + 2*C*a*e**2 - 3*C*c*d**2)/(2*c**3) + sqrt(-a**3*c**7)*(-3*A*a*c*d*e **2 - A*c**2*d**3 + 3*B*a**2*e**3 - 3*B*a*c*d**2*e + 9*C*a**2*d*e**2 - C*a *c*d**3)/(4*a**3*c**6)))/(-3*A*a*c**2*d*e**2 - A*c**3*d**3 + 3*B*a**2*c*e* *3 - 3*B*a*c**2*d**2*e + 9*C*a**2*c*d*e**2 - C*a*c**2*d**3)) + (A*a**2*c*e **3 - 3*A*a*c**2*d**2*e + 3*B*a**2*c*d*e**2 - B*a*c**2*d**3 - C*a**3*e**3 + 3*C*a**2*c*d**2*e + x*(-3*A*a*c**2*d*e**2 + A*c**3*d**3 + B*a**2*c*e**3 - 3*B*a*c**2*d**2*e + 3*C*a**2*c*d*e**2 - C*a*c**2*d**3))/(2*a**2*c**3 ...
Time = 0.28 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.33 \[ \int \frac {(d+e x)^3 \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^2} \, dx=-\frac {B a c^{2} d^{3} - 3 \, B a^{2} c d e^{2} - 3 \, {\left (C a^{2} c - A a c^{2}\right )} d^{2} e + {\left (C a^{3} - A a^{2} c\right )} e^{3} + {\left (3 \, B a c^{2} d^{2} e - B a^{2} c e^{3} + {\left (C a c^{2} - A c^{3}\right )} d^{3} - 3 \, {\left (C a^{2} c - A a c^{2}\right )} d e^{2}\right )} x}{2 \, {\left (a c^{4} x^{2} + a^{2} c^{3}\right )}} + \frac {C e^{3} x^{2} + 2 \, {\left (3 \, C d e^{2} + B e^{3}\right )} x}{2 \, c^{2}} + \frac {{\left (3 \, C c d^{2} e + 3 \, B c d e^{2} - {\left (2 \, C a - A c\right )} e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, c^{3}} + \frac {{\left (3 \, B a c d^{2} e - 3 \, B a^{2} e^{3} + {\left (C a c + A c^{2}\right )} d^{3} - 3 \, {\left (3 \, C a^{2} - A a c\right )} d e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a c^{2}} \]
-1/2*(B*a*c^2*d^3 - 3*B*a^2*c*d*e^2 - 3*(C*a^2*c - A*a*c^2)*d^2*e + (C*a^3 - A*a^2*c)*e^3 + (3*B*a*c^2*d^2*e - B*a^2*c*e^3 + (C*a*c^2 - A*c^3)*d^3 - 3*(C*a^2*c - A*a*c^2)*d*e^2)*x)/(a*c^4*x^2 + a^2*c^3) + 1/2*(C*e^3*x^2 + 2*(3*C*d*e^2 + B*e^3)*x)/c^2 + 1/2*(3*C*c*d^2*e + 3*B*c*d*e^2 - (2*C*a - A *c)*e^3)*log(c*x^2 + a)/c^3 + 1/2*(3*B*a*c*d^2*e - 3*B*a^2*e^3 + (C*a*c + A*c^2)*d^3 - 3*(3*C*a^2 - A*a*c)*d*e^2)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a *c^2)
Time = 0.27 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.38 \[ \int \frac {(d+e x)^3 \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^2} \, dx=\frac {{\left (3 \, C c d^{2} e + 3 \, B c d e^{2} - 2 \, C a e^{3} + A c e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, c^{3}} + \frac {{\left (C a c d^{3} + A c^{2} d^{3} + 3 \, B a c d^{2} e - 9 \, C a^{2} d e^{2} + 3 \, A a c d e^{2} - 3 \, B a^{2} e^{3}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a c^{2}} + \frac {C c^{2} e^{3} x^{2} + 6 \, C c^{2} d e^{2} x + 2 \, B c^{2} e^{3} x}{2 \, c^{4}} - \frac {B a c^{2} d^{3} - 3 \, C a^{2} c d^{2} e + 3 \, A a c^{2} d^{2} e - 3 \, B a^{2} c d e^{2} + C a^{3} e^{3} - A a^{2} c e^{3} + {\left (C a c^{2} d^{3} - A c^{3} d^{3} + 3 \, B a c^{2} d^{2} e - 3 \, C a^{2} c d e^{2} + 3 \, A a c^{2} d e^{2} - B a^{2} c e^{3}\right )} x}{2 \, {\left (c x^{2} + a\right )} a c^{3}} \]
1/2*(3*C*c*d^2*e + 3*B*c*d*e^2 - 2*C*a*e^3 + A*c*e^3)*log(c*x^2 + a)/c^3 + 1/2*(C*a*c*d^3 + A*c^2*d^3 + 3*B*a*c*d^2*e - 9*C*a^2*d*e^2 + 3*A*a*c*d*e^ 2 - 3*B*a^2*e^3)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a*c^2) + 1/2*(C*c^2*e^3* x^2 + 6*C*c^2*d*e^2*x + 2*B*c^2*e^3*x)/c^4 - 1/2*(B*a*c^2*d^3 - 3*C*a^2*c* d^2*e + 3*A*a*c^2*d^2*e - 3*B*a^2*c*d*e^2 + C*a^3*e^3 - A*a^2*c*e^3 + (C*a *c^2*d^3 - A*c^3*d^3 + 3*B*a*c^2*d^2*e - 3*C*a^2*c*d*e^2 + 3*A*a*c^2*d*e^2 - B*a^2*c*e^3)*x)/((c*x^2 + a)*a*c^3)
Time = 13.00 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.40 \[ \int \frac {(d+e x)^3 \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^2} \, dx=\frac {x\,\left (B\,e^3+3\,C\,d\,e^2\right )}{c^2}-\frac {\frac {C\,a^2\,e^3-3\,C\,a\,c\,d^2\,e-3\,B\,a\,c\,d\,e^2-A\,a\,c\,e^3+B\,c^2\,d^3+3\,A\,c^2\,d^2\,e}{2\,c}-\frac {x\,\left (3\,C\,a^2\,d\,e^2+B\,a^2\,e^3-C\,a\,c\,d^3-3\,B\,a\,c\,d^2\,e-3\,A\,a\,c\,d\,e^2+A\,c^2\,d^3\right )}{2\,a}}{c^3\,x^2+a\,c^2}+\frac {C\,e^3\,x^2}{2\,c^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (-9\,C\,a^2\,d\,e^2-3\,B\,a^2\,e^3+C\,a\,c\,d^3+3\,B\,a\,c\,d^2\,e+3\,A\,a\,c\,d\,e^2+A\,c^2\,d^3\right )}{2\,a^{3/2}\,c^{5/2}}+\frac {\ln \left (c\,x^2+a\right )\,\left (-32\,C\,a^4\,c^3\,e^3+48\,C\,a^3\,c^4\,d^2\,e+48\,B\,a^3\,c^4\,d\,e^2+16\,A\,a^3\,c^4\,e^3\right )}{32\,a^3\,c^6} \]
(x*(B*e^3 + 3*C*d*e^2))/c^2 - ((B*c^2*d^3 + C*a^2*e^3 - A*a*c*e^3 + 3*A*c^ 2*d^2*e - 3*B*a*c*d*e^2 - 3*C*a*c*d^2*e)/(2*c) - (x*(A*c^2*d^3 + B*a^2*e^3 - C*a*c*d^3 + 3*C*a^2*d*e^2 - 3*A*a*c*d*e^2 - 3*B*a*c*d^2*e))/(2*a))/(a*c ^2 + c^3*x^2) + (C*e^3*x^2)/(2*c^2) + (atan((c^(1/2)*x)/a^(1/2))*(A*c^2*d^ 3 - 3*B*a^2*e^3 + C*a*c*d^3 - 9*C*a^2*d*e^2 + 3*A*a*c*d*e^2 + 3*B*a*c*d^2* e))/(2*a^(3/2)*c^(5/2)) + (log(a + c*x^2)*(16*A*a^3*c^4*e^3 - 32*C*a^4*c^3 *e^3 + 48*B*a^3*c^4*d*e^2 + 48*C*a^3*c^4*d^2*e))/(32*a^3*c^6)