Integrand size = 44, antiderivative size = 168 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^3} \, dx=-\frac {2 g \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (2 c d-b e) (d+e x)^3}-\frac {\sqrt {c} g \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{e^2} \]
-2/3*(-d*g+e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/e^2/(-b*e+2*c*d)/(e *x+d)^3-g*arctan(1/2*e*(2*c*x+b)/c^(1/2)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^ (1/2))*c^(1/2)/e^2-2*g*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/e^2/(e*x+d)
Time = 0.25 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.95 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^3} \, dx=\frac {2 \sqrt {(d+e x) (-b e+c (d-e x))} \left (\frac {b e (2 d g+e (f+3 g x))-c \left (5 d^2 g-e^2 f x+d e (f+7 g x)\right )}{(2 c d-b e) (d+e x)^2}+\frac {3 \sqrt {c} g \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {c} \sqrt {d+e x}}\right )}{\sqrt {d+e x} \sqrt {-b e+c (d-e x)}}\right )}{3 e^2} \]
(2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*((b*e*(2*d*g + e*(f + 3*g*x)) - c*(5*d^2*g - e^2*f*x + d*e*(f + 7*g*x)))/((2*c*d - b*e)*(d + e*x)^2) + (3* Sqrt[c]*g*ArcTan[Sqrt[c*d - b*e - c*e*x]/(Sqrt[c]*Sqrt[d + e*x])])/(Sqrt[d + e*x]*Sqrt[-(b*e) + c*(d - e*x)])))/(3*e^2)
Time = 0.43 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {1216, 1218, 1124, 1092, 217}
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.
| \(\displaystyle \int \frac {(f+g x) \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2}}{(d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 1216 |
| \(\displaystyle \int \frac {(f+g x) \left (\frac {c d^2-b d e}{d}-c e x\right )^3}{\left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 1218 |
| \(\displaystyle \frac {g \int \frac {(c d-b e-c e x)^2}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{e}-\frac {2 (e f-d g) (-b e+c d-c e x)^3}{3 e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1124 |
| \(\displaystyle \frac {g \left (-c \int \frac {1}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx-\frac {2 (-b e+c d-c e x)}{e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{e}-\frac {2 (e f-d g) (-b e+c d-c e x)^3}{3 e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {g \left (-2 c \int \frac {1}{-\frac {(b+2 c x)^2 e^4}{-c x^2 e^2-b x e^2+d (c d-b e)}-4 c e^2}d\left (-\frac {e^2 (b+2 c x)}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}\right )-\frac {2 (-b e+c d-c e x)}{e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{e}-\frac {2 (e f-d g) (-b e+c d-c e x)^3}{3 e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {g \left (-\frac {\sqrt {c} \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{e}-\frac {2 (-b e+c d-c e x)}{e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{e}-\frac {2 (e f-d g) (-b e+c d-c e x)^3}{3 e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\) |
(-2*(e*f - d*g)*(c*d - b*e - c*e*x)^3)/(3*e^2*(2*c*d - b*e)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) + (g*((-2*(c*d - b*e - c*e*x))/(e*Sqrt[d*(c *d - b*e) - b*e^2*x - c*e^2*x^2]) - (Sqrt[c]*ArcTan[(e*(b + 2*c*x))/(2*Sqr t[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/e))/e
3.22.77.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[((d_.) + (e_.)*(x_))^(m_.)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x _Symbol] :> Simp[-2*e*(2*c*d - b*e)^(m - 2)*((d + e*x)/(c^(m - 1)*Sqrt[a + b*x + c*x^2])), x] + Simp[e^2/c^(m - 1) Int[(1/Sqrt[a + b*x + c*x^2])*Exp andToSum[((2*c*d - b*e)^(m - 1) - c^(m - 1)*(d + e*x)^(m - 1))/(c*d - b*e - c*e*x), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e ^2, 0] && IGtQ[m, 0]
Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_.)*Sqrt[(a_.) + (b_.)* (x_) + (c_.)*(x_)^2], x_Symbol] :> Int[((f + g*x)^n*(a + b*x + c*x^2)^(m + 1/2))/(a/d + c*(x/e))^m, x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c* d^2 - b*d*e + a*e^2, 0] && ILtQ[m, 0] && IntegerQ[n]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + ( c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(g*(c*d - b*e) + c*e*f)*(d + e*x)^m*(( a + b*x + c*x^2)^(p + 1)/(c*(p + 1)*(2*c*d - b*e))), x] - Simp[e*((m*(g*(c* d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g))/(c*(p + 1)*(2*c*d - b*e))) I nt[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d , e, f, g}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0]
Time = 1.02 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.78
| method | result | size |
| default | \(\frac {g \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {2 c \,e^{2} \left (\sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}+\frac {\left (-b \,e^{2}+2 c d e \right ) \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {d}{e}-\frac {-b \,e^{2}+2 c d e}{2 c \,e^{2}}\right )}{\sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {c \,e^{2}}}\right )}{-b \,e^{2}+2 c d e}\right )}{e^{3}}-\frac {2 \left (-d g +e f \right ) \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3 e^{4} \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{3}}\) | \(299\) |
g/e^3*(-2/(-b*e^2+2*c*d*e)/(x+d/e)^2*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x +d/e))^(3/2)-2*c*e^2/(-b*e^2+2*c*d*e)*((-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)* (x+d/e))^(1/2)+1/2*(-b*e^2+2*c*d*e)/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+ d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e) )^(1/2))))-2/3*(-d*g+e*f)/e^4/(-b*e^2+2*c*d*e)/(x+d/e)^3*(-c*e^2*(x+d/e)^2 +(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)
Time = 1.82 (sec) , antiderivative size = 579, normalized size of antiderivative = 3.45 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^3} \, dx=\left [\frac {3 \, {\left ({\left (2 \, c d e^{2} - b e^{3}\right )} g x^{2} + 2 \, {\left (2 \, c d^{2} e - b d e^{2}\right )} g x + {\left (2 \, c d^{3} - b d^{2} e\right )} g\right )} \sqrt {-c} \log \left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2} - 4 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {-c}\right ) - 4 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left ({\left (c d e - b e^{2}\right )} f + {\left (5 \, c d^{2} - 2 \, b d e\right )} g - {\left (c e^{2} f - {\left (7 \, c d e - 3 \, b e^{2}\right )} g\right )} x\right )}}{6 \, {\left (2 \, c d^{3} e^{2} - b d^{2} e^{3} + {\left (2 \, c d e^{4} - b e^{5}\right )} x^{2} + 2 \, {\left (2 \, c d^{2} e^{3} - b d e^{4}\right )} x\right )}}, \frac {3 \, {\left ({\left (2 \, c d e^{2} - b e^{3}\right )} g x^{2} + 2 \, {\left (2 \, c d^{2} e - b d e^{2}\right )} g x + {\left (2 \, c d^{3} - b d^{2} e\right )} g\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {c}}{2 \, {\left (c^{2} e^{2} x^{2} + b c e^{2} x - c^{2} d^{2} + b c d e\right )}}\right ) - 2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left ({\left (c d e - b e^{2}\right )} f + {\left (5 \, c d^{2} - 2 \, b d e\right )} g - {\left (c e^{2} f - {\left (7 \, c d e - 3 \, b e^{2}\right )} g\right )} x\right )}}{3 \, {\left (2 \, c d^{3} e^{2} - b d^{2} e^{3} + {\left (2 \, c d e^{4} - b e^{5}\right )} x^{2} + 2 \, {\left (2 \, c d^{2} e^{3} - b d e^{4}\right )} x\right )}}\right ] \]
[1/6*(3*((2*c*d*e^2 - b*e^3)*g*x^2 + 2*(2*c*d^2*e - b*d*e^2)*g*x + (2*c*d^ 3 - b*d^2*e)*g)*sqrt(-c)*log(8*c^2*e^2*x^2 + 8*b*c*e^2*x - 4*c^2*d^2 + 4*b *c*d*e + b^2*e^2 - 4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(-c)) - 4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*((c*d*e - b *e^2)*f + (5*c*d^2 - 2*b*d*e)*g - (c*e^2*f - (7*c*d*e - 3*b*e^2)*g)*x))/(2 *c*d^3*e^2 - b*d^2*e^3 + (2*c*d*e^4 - b*e^5)*x^2 + 2*(2*c*d^2*e^3 - b*d*e^ 4)*x), 1/3*(3*((2*c*d*e^2 - b*e^3)*g*x^2 + 2*(2*c*d^2*e - b*d*e^2)*g*x + ( 2*c*d^3 - b*d^2*e)*g)*sqrt(c)*arctan(1/2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(c)/(c^2*e^2*x^2 + b*c*e^2*x - c^2*d^2 + b*c *d*e)) - 2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*((c*d*e - b*e^2)*f + (5*c*d^2 - 2*b*d*e)*g - (c*e^2*f - (7*c*d*e - 3*b*e^2)*g)*x))/(2*c*d^3*e^ 2 - b*d^2*e^3 + (2*c*d*e^4 - b*e^5)*x^2 + 2*(2*c*d^2*e^3 - b*d*e^4)*x)]
\[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^3} \, dx=\int \frac {\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (f + g x\right )}{\left (d + e x\right )^{3}}\, dx \]
Exception generated. \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-2*c*d>0)', see `assume?` for more deta
Leaf count of result is larger than twice the leaf count of optimal. 517 vs. \(2 (154) = 308\).
Time = 1.29 (sec) , antiderivative size = 517, normalized size of antiderivative = 3.08 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^3} \, dx=\frac {c g \log \left ({\left | -b c^{2} d^{4} e^{2} + 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} \sqrt {-c} c^{2} d^{4} {\left | e \right |} + 4 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} b \sqrt {-c} c d^{3} e {\left | e \right |} + 8 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{2} c^{2} d^{3} e + 6 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{2} b c d^{2} e^{2} - 12 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{3} \sqrt {-c} c d^{2} {\left | e \right |} - 4 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{3} b \sqrt {-c} d e {\left | e \right |} - 8 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{4} c d e - {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{4} b e^{2} + 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{5} \sqrt {-c} {\left | e \right |} \right |}\right )}{5 \, \sqrt {-c} e {\left | e \right |}} \]
1/5*c*g*log(abs(-b*c^2*d^4*e^2 + 2*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e ^2*x + c*d^2 - b*d*e))*sqrt(-c)*c^2*d^4*abs(e) + 4*(sqrt(-c*e^2)*x - sqrt( -c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))*b*sqrt(-c)*c*d^3*e*abs(e) + 8*(sqrt (-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^2*c^2*d^3*e + 6*( sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^2*b*c*d^2*e^2 - 12*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^3*sqrt (-c)*c*d^2*abs(e) - 4*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^3*b*sqrt(-c)*d*e*abs(e) - 8*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^4*c*d*e - (sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b* e^2*x + c*d^2 - b*d*e))^4*b*e^2 + 2*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b* e^2*x + c*d^2 - b*d*e))^5*sqrt(-c)*abs(e)))/(sqrt(-c)*e*abs(e))
Timed out. \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^3} \, dx=\int \frac {\left (f+g\,x\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^3} \,d x \]