Integrand size = 32, antiderivative size = 143 \[ \int \frac {(d+e x) \left (A+B x+C x^2\right )}{\sqrt {d^2-e^2 x^2}} \, dx=-\frac {\left (2 C d^2+3 e (B d+A e)\right ) \sqrt {d^2-e^2 x^2}}{3 e^3}-\frac {(C d+B e) x \sqrt {d^2-e^2 x^2}}{2 e^2}-\frac {C x^2 \sqrt {d^2-e^2 x^2}}{3 e}+\frac {d \left (C d^2+e (B d+2 A e)\right ) \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^3} \]
1/2*d*(C*d^2+e*(2*A*e+B*d))*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^3-1/3*(2*C* d^2+3*e*(A*e+B*d))*(-e^2*x^2+d^2)^(1/2)/e^3-1/2*(B*e+C*d)*x*(-e^2*x^2+d^2) ^(1/2)/e^2-1/3*C*x^2*(-e^2*x^2+d^2)^(1/2)/e
Time = 0.50 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.80 \[ \int \frac {(d+e x) \left (A+B x+C x^2\right )}{\sqrt {d^2-e^2 x^2}} \, dx=-\frac {\sqrt {d^2-e^2 x^2} \left (3 e (2 B d+2 A e+B e x)+C \left (4 d^2+3 d e x+2 e^2 x^2\right )\right )+6 d \left (C d^2+e (B d+2 A e)\right ) \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{6 e^3} \]
-1/6*(Sqrt[d^2 - e^2*x^2]*(3*e*(2*B*d + 2*A*e + B*e*x) + C*(4*d^2 + 3*d*e* x + 2*e^2*x^2)) + 6*d*(C*d^2 + e*(B*d + 2*A*e))*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])])/e^3
Time = 0.44 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2346, 25, 2346, 25, 27, 455, 224, 216}
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.
\(\displaystyle \int \frac {(d+e x) \left (A+B x+C x^2\right )}{\sqrt {d^2-e^2 x^2}} \, dx\) |
\(\Big \downarrow \) 2346 |
\(\displaystyle -\frac {\int -\frac {3 (C d+B e) x^2 e^2+3 A d e^2+\left (2 C d^2+3 e (B d+A e)\right ) x e}{\sqrt {d^2-e^2 x^2}}dx}{3 e^2}-\frac {C x^2 \sqrt {d^2-e^2 x^2}}{3 e}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {3 (C d+B e) x^2 e^2+3 A d e^2+\left (2 C d^2+3 e (B d+A e)\right ) x e}{\sqrt {d^2-e^2 x^2}}dx}{3 e^2}-\frac {C x^2 \sqrt {d^2-e^2 x^2}}{3 e}\) |
\(\Big \downarrow \) 2346 |
\(\displaystyle \frac {-\frac {\int -\frac {e^2 \left (3 d \left (C d^2+e (B d+2 A e)\right )+2 e \left (2 C d^2+3 e (B d+A e)\right ) x\right )}{\sqrt {d^2-e^2 x^2}}dx}{2 e^2}-\frac {3}{2} x \sqrt {d^2-e^2 x^2} (B e+C d)}{3 e^2}-\frac {C x^2 \sqrt {d^2-e^2 x^2}}{3 e}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {\int \frac {e^2 \left (3 d \left (C d^2+e (B d+2 A e)\right )+2 e \left (2 C d^2+3 e (B d+A e)\right ) x\right )}{\sqrt {d^2-e^2 x^2}}dx}{2 e^2}-\frac {3}{2} x \sqrt {d^2-e^2 x^2} (B e+C d)}{3 e^2}-\frac {C x^2 \sqrt {d^2-e^2 x^2}}{3 e}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {1}{2} \int \frac {3 d \left (C d^2+e (B d+2 A e)\right )+2 e \left (2 C d^2+3 e (B d+A e)\right ) x}{\sqrt {d^2-e^2 x^2}}dx-\frac {3}{2} x \sqrt {d^2-e^2 x^2} (B e+C d)}{3 e^2}-\frac {C x^2 \sqrt {d^2-e^2 x^2}}{3 e}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle \frac {\frac {1}{2} \left (3 d \left (e (2 A e+B d)+C d^2\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx-\frac {2 \sqrt {d^2-e^2 x^2} \left (3 e (A e+B d)+2 C d^2\right )}{e}\right )-\frac {3}{2} x \sqrt {d^2-e^2 x^2} (B e+C d)}{3 e^2}-\frac {C x^2 \sqrt {d^2-e^2 x^2}}{3 e}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {\frac {1}{2} \left (3 d \left (e (2 A e+B d)+C d^2\right ) \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}-\frac {2 \sqrt {d^2-e^2 x^2} \left (3 e (A e+B d)+2 C d^2\right )}{e}\right )-\frac {3}{2} x \sqrt {d^2-e^2 x^2} (B e+C d)}{3 e^2}-\frac {C x^2 \sqrt {d^2-e^2 x^2}}{3 e}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\frac {1}{2} \left (\frac {3 d \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right ) \left (e (2 A e+B d)+C d^2\right )}{e}-\frac {2 \sqrt {d^2-e^2 x^2} \left (3 e (A e+B d)+2 C d^2\right )}{e}\right )-\frac {3}{2} x \sqrt {d^2-e^2 x^2} (B e+C d)}{3 e^2}-\frac {C x^2 \sqrt {d^2-e^2 x^2}}{3 e}\) |
-1/3*(C*x^2*Sqrt[d^2 - e^2*x^2])/e + ((-3*(C*d + B*e)*x*Sqrt[d^2 - e^2*x^2 ])/2 + ((-2*(2*C*d^2 + 3*e*(B*d + A*e))*Sqrt[d^2 - e^2*x^2])/e + (3*d*(C*d ^2 + e*(B*d + 2*A*e))*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e)/2)/(3*e^2)
3.1.12.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1)) Int[(a + b*x^2)^p*ExpandToS um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && !LeQ[p, -1]
Time = 0.53 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.77
method | result | size |
risch | \(-\frac {\left (2 C \,e^{2} x^{2}+3 x B \,e^{2}+3 C d e x +6 A \,e^{2}+6 B d e +4 C \,d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{6 e^{3}}+\frac {d \left (2 A \,e^{2}+B d e +C \,d^{2}\right ) \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e^{2} \sqrt {e^{2}}}\) | \(110\) |
default | \(\frac {d A \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}+e C \left (-\frac {x^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3 e^{2}}-\frac {2 d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3 e^{4}}\right )+\left (B e +C d \right ) \left (-\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e^{2} \sqrt {e^{2}}}\right )-\frac {\left (A e +B d \right ) \sqrt {-e^{2} x^{2}+d^{2}}}{e^{2}}\) | \(170\) |
-1/6*(2*C*e^2*x^2+3*B*e^2*x+3*C*d*e*x+6*A*e^2+6*B*d*e+4*C*d^2)/e^3*(-e^2*x ^2+d^2)^(1/2)+1/2*d/e^2*(2*A*e^2+B*d*e+C*d^2)/(e^2)^(1/2)*arctan((e^2)^(1/ 2)*x/(-e^2*x^2+d^2)^(1/2))
Time = 0.31 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.76 \[ \int \frac {(d+e x) \left (A+B x+C x^2\right )}{\sqrt {d^2-e^2 x^2}} \, dx=-\frac {6 \, {\left (C d^{3} + B d^{2} e + 2 \, A d e^{2}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (2 \, C e^{2} x^{2} + 4 \, C d^{2} + 6 \, B d e + 6 \, A e^{2} + 3 \, {\left (C d e + B e^{2}\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{6 \, e^{3}} \]
-1/6*(6*(C*d^3 + B*d^2*e + 2*A*d*e^2)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/( e*x)) + (2*C*e^2*x^2 + 4*C*d^2 + 6*B*d*e + 6*A*e^2 + 3*(C*d*e + B*e^2)*x)* sqrt(-e^2*x^2 + d^2))/e^3
Time = 0.54 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.26 \[ \int \frac {(d+e x) \left (A+B x+C x^2\right )}{\sqrt {d^2-e^2 x^2}} \, dx=\begin {cases} \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {C x^{2}}{3 e} - \frac {x \left (B e + C d\right )}{2 e^{2}} - \frac {A e + B d + \frac {2 C d^{2}}{3 e}}{e^{2}}\right ) + \left (A d + \frac {d^{2} \left (B e + C d\right )}{2 e^{2}}\right ) \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {A d x + \frac {C e x^{4}}{4} + \frac {x^{3} \left (B e + C d\right )}{3} + \frac {x^{2} \left (A e + B d\right )}{2}}{\sqrt {d^{2}}} & \text {otherwise} \end {cases} \]
Piecewise((sqrt(d**2 - e**2*x**2)*(-C*x**2/(3*e) - x*(B*e + C*d)/(2*e**2) - (A*e + B*d + 2*C*d**2/(3*e))/e**2) + (A*d + d**2*(B*e + C*d)/(2*e**2))*P iecewise((log(-2*e**2*x + 2*sqrt(-e**2)*sqrt(d**2 - e**2*x**2))/sqrt(-e**2 ), Ne(d**2, 0)), (x*log(x)/sqrt(-e**2*x**2), True)), Ne(e**2, 0)), ((A*d*x + C*e*x**4/4 + x**3*(B*e + C*d)/3 + x**2*(A*e + B*d)/2)/sqrt(d**2), True) )
Time = 0.27 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.20 \[ \int \frac {(d+e x) \left (A+B x+C x^2\right )}{\sqrt {d^2-e^2 x^2}} \, dx=-\frac {\sqrt {-e^{2} x^{2} + d^{2}} C x^{2}}{3 \, e} + \frac {A d \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{\sqrt {e^{2}}} + \frac {{\left (C d + B e\right )} d^{2} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{2 \, \sqrt {e^{2}} e^{2}} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} C d^{2}}{3 \, e^{3}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} B d}{e^{2}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} A}{e} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} {\left (C d + B e\right )} x}{2 \, e^{2}} \]
-1/3*sqrt(-e^2*x^2 + d^2)*C*x^2/e + A*d*arcsin(e^2*x/(d*sqrt(e^2)))/sqrt(e ^2) + 1/2*(C*d + B*e)*d^2*arcsin(e^2*x/(d*sqrt(e^2)))/(sqrt(e^2)*e^2) - 2/ 3*sqrt(-e^2*x^2 + d^2)*C*d^2/e^3 - sqrt(-e^2*x^2 + d^2)*B*d/e^2 - sqrt(-e^ 2*x^2 + d^2)*A/e - 1/2*sqrt(-e^2*x^2 + d^2)*(C*d + B*e)*x/e^2
Time = 0.30 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.78 \[ \int \frac {(d+e x) \left (A+B x+C x^2\right )}{\sqrt {d^2-e^2 x^2}} \, dx=-\frac {1}{6} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left ({\left (\frac {2 \, C x}{e} + \frac {3 \, {\left (C d e^{3} + B e^{4}\right )}}{e^{5}}\right )} x + \frac {2 \, {\left (2 \, C d^{2} e^{2} + 3 \, B d e^{3} + 3 \, A e^{4}\right )}}{e^{5}}\right )} + \frac {{\left (C d^{3} + B d^{2} e + 2 \, A d e^{2}\right )} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{2 \, e^{2} {\left | e \right |}} \]
-1/6*sqrt(-e^2*x^2 + d^2)*((2*C*x/e + 3*(C*d*e^3 + B*e^4)/e^5)*x + 2*(2*C* d^2*e^2 + 3*B*d*e^3 + 3*A*e^4)/e^5) + 1/2*(C*d^3 + B*d^2*e + 2*A*d*e^2)*ar csin(e*x/d)*sgn(d)*sgn(e)/(e^2*abs(e))
Time = 13.76 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.89 \[ \int \frac {(d+e x) \left (A+B x+C x^2\right )}{\sqrt {d^2-e^2 x^2}} \, dx=\left \{\begin {array}{cl} \frac {2\,C\,d\,x^3+3\,B\,d\,x^2+6\,A\,d\,x}{6\,\sqrt {d^2}} & \text {\ if\ \ }e=0\\ \frac {A\,d\,\ln \left (x\,\sqrt {-e^2}+\sqrt {d^2-e^2\,x^2}\right )}{\sqrt {-e^2}}-\frac {A\,\sqrt {d^2-e^2\,x^2}}{e}-\frac {B\,d\,\sqrt {d^2-e^2\,x^2}}{e^2}-\frac {B\,x\,\sqrt {d^2-e^2\,x^2}}{2\,e}-\frac {C\,\sqrt {d^2-e^2\,x^2}\,\left (2\,d^2+e^2\,x^2\right )}{3\,e^3}-\frac {C\,d^3\,\ln \left (2\,x\,\sqrt {-e^2}+2\,\sqrt {d^2-e^2\,x^2}\right )}{2\,{\left (-e^2\right )}^{3/2}}-\frac {B\,d^2\,e\,\ln \left (2\,x\,\sqrt {-e^2}+2\,\sqrt {d^2-e^2\,x^2}\right )}{2\,{\left (-e^2\right )}^{3/2}}-\frac {C\,d\,x\,\sqrt {d^2-e^2\,x^2}}{2\,e^2} & \text {\ if\ \ }e\neq 0 \end {array}\right . \]
piecewise(e == 0, (6*A*d*x + 3*B*d*x^2 + 2*C*d*x^3)/(6*(d^2)^(1/2)), e ~= 0, - (A*(d^2 - e^2*x^2)^(1/2))/e + (A*d*log(x*(-e^2)^(1/2) + (d^2 - e^2*x^ 2)^(1/2)))/(-e^2)^(1/2) - (B*d*(d^2 - e^2*x^2)^(1/2))/e^2 - (B*x*(d^2 - e^ 2*x^2)^(1/2))/(2*e) - (C*(d^2 - e^2*x^2)^(1/2)*(2*d^2 + e^2*x^2))/(3*e^3) - (C*d^3*log(2*x*(-e^2)^(1/2) + 2*(d^2 - e^2*x^2)^(1/2)))/(2*(-e^2)^(3/2)) - (B*d^2*e*log(2*x*(-e^2)^(1/2) + 2*(d^2 - e^2*x^2)^(1/2)))/(2*(-e^2)^(3/ 2)) - (C*d*x*(d^2 - e^2*x^2)^(1/2))/(2*e^2))