Integrand size = 26, antiderivative size = 301 \[ \int \frac {A+B x}{(d+e x)^{9/2} \left (b x+c x^2\right )} \, dx=\frac {2 (B d-A e)}{7 d (c d-b e) (d+e x)^{7/2}}+\frac {2 \left (B c d^2-A e (2 c d-b e)\right )}{5 d^2 (c d-b e)^2 (d+e x)^{5/2}}+\frac {2 \left (B c^2 d^3-A e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right )}{3 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac {2 \left (B c^3 d^4-A e \left (4 c^3 d^3-6 b c^2 d^2 e+4 b^2 c d e^2-b^3 e^3\right )\right )}{d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {2 A \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b d^{9/2}}-\frac {2 c^{7/2} (b B-A c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b (c d-b e)^{9/2}} \] Output:
2/7*(-A*e+B*d)/d/(-b*e+c*d)/(e*x+d)^(7/2)+2/5*(B*c*d^2-A*e*(-b*e+2*c*d))/d ^2/(-b*e+c*d)^2/(e*x+d)^(5/2)+2/3*(B*c^2*d^3-A*e*(b^2*e^2-3*b*c*d*e+3*c^2* d^2))/d^3/(-b*e+c*d)^3/(e*x+d)^(3/2)+2*(B*c^3*d^4-A*e*(-b^3*e^3+4*b^2*c*d* e^2-6*b*c^2*d^2*e+4*c^3*d^3))/d^4/(-b*e+c*d)^4/(e*x+d)^(1/2)-2*A*arctanh(( e*x+d)^(1/2)/d^(1/2))/b/d^(9/2)-2*c^(7/2)*(-A*c+B*b)*arctanh(c^(1/2)*(e*x+ d)^(1/2)/(-b*e+c*d)^(1/2))/b/(-b*e+c*d)^(9/2)
Time = 1.56 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.21 \[ \int \frac {A+B x}{(d+e x)^{9/2} \left (b x+c x^2\right )} \, dx=\frac {2 \left (B d^4 \left (-15 b^3 e^3+3 b^2 c e^2 (22 d+7 e x)-b c^2 e \left (122 d^2+112 d e x+35 e^2 x^2\right )+c^3 \left (176 d^3+406 d^2 e x+350 d e^2 x^2+105 e^3 x^3\right )\right )+A e \left (15 b c^2 d^2 e \left (66 d^3+161 d^2 e x+140 d e^2 x^2+42 e^3 x^3\right )+b^3 e^3 \left (176 d^3+406 d^2 e x+350 d e^2 x^2+105 e^3 x^3\right )-3 c^3 d^3 \left (194 d^3+504 d^2 e x+455 d e^2 x^2+140 e^3 x^3\right )-b^2 c d e^2 \left (689 d^3+1624 d^2 e x+1400 d e^2 x^2+420 e^3 x^3\right )\right )\right )}{105 d^4 (c d-b e)^4 (d+e x)^{7/2}}-\frac {2 c^{7/2} (-b B+A c) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{b (-c d+b e)^{9/2}}-\frac {2 A \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b d^{9/2}} \] Input:
Integrate[(A + B*x)/((d + e*x)^(9/2)*(b*x + c*x^2)),x]
Output:
(2*(B*d^4*(-15*b^3*e^3 + 3*b^2*c*e^2*(22*d + 7*e*x) - b*c^2*e*(122*d^2 + 1 12*d*e*x + 35*e^2*x^2) + c^3*(176*d^3 + 406*d^2*e*x + 350*d*e^2*x^2 + 105* e^3*x^3)) + A*e*(15*b*c^2*d^2*e*(66*d^3 + 161*d^2*e*x + 140*d*e^2*x^2 + 42 *e^3*x^3) + b^3*e^3*(176*d^3 + 406*d^2*e*x + 350*d*e^2*x^2 + 105*e^3*x^3) - 3*c^3*d^3*(194*d^3 + 504*d^2*e*x + 455*d*e^2*x^2 + 140*e^3*x^3) - b^2*c* d*e^2*(689*d^3 + 1624*d^2*e*x + 1400*d*e^2*x^2 + 420*e^3*x^3))))/(105*d^4* (c*d - b*e)^4*(d + e*x)^(7/2)) - (2*c^(7/2)*(-(b*B) + A*c)*ArcTan[(Sqrt[c] *Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/(b*(-(c*d) + b*e)^(9/2)) - (2*A*ArcTa nh[Sqrt[d + e*x]/Sqrt[d]])/(b*d^(9/2))
Time = 1.41 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.25, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {1198, 1198, 1198, 1198, 1197, 25, 1480, 221}
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.
| \(\displaystyle \int \frac {A+B x}{\left (b x+c x^2\right ) (d+e x)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle \frac {\int \frac {A (c d-b e)+c (B d-A e) x}{(d+e x)^{7/2} \left (c x^2+b x\right )}dx}{d (c d-b e)}+\frac {2 (B d-A e)}{7 d (d+e x)^{7/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle \frac {\frac {\int \frac {A (c d-b e)^2+c \left (B c d^2-A e (2 c d-b e)\right ) x}{(d+e x)^{5/2} \left (c x^2+b x\right )}dx}{d (c d-b e)}+\frac {2 \left (B c d^2-A e (2 c d-b e)\right )}{5 d (d+e x)^{5/2} (c d-b e)}}{d (c d-b e)}+\frac {2 (B d-A e)}{7 d (d+e x)^{7/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {A (c d-b e)^3+c \left (B c^2 d^3-A e \left (3 c^2 d^2-3 b c e d+b^2 e^2\right )\right ) x}{(d+e x)^{3/2} \left (c x^2+b x\right )}dx}{d (c d-b e)}+\frac {2 \left (B c^2 d^3-A e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )\right )}{3 d (d+e x)^{3/2} (c d-b e)}}{d (c d-b e)}+\frac {2 \left (B c d^2-A e (2 c d-b e)\right )}{5 d (d+e x)^{5/2} (c d-b e)}}{d (c d-b e)}+\frac {2 (B d-A e)}{7 d (d+e x)^{7/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {A (c d-b e)^4+c \left (B c^3 d^4-A e \left (4 c^3 d^3-6 b c^2 e d^2+4 b^2 c e^2 d-b^3 e^3\right )\right ) x}{\sqrt {d+e x} \left (c x^2+b x\right )}dx}{d (c d-b e)}+\frac {2 \left (B c^3 d^4-A e \left (-b^3 e^3+4 b^2 c d e^2-6 b c^2 d^2 e+4 c^3 d^3\right )\right )}{d \sqrt {d+e x} (c d-b e)}}{d (c d-b e)}+\frac {2 \left (B c^2 d^3-A e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )\right )}{3 d (d+e x)^{3/2} (c d-b e)}}{d (c d-b e)}+\frac {2 \left (B c d^2-A e (2 c d-b e)\right )}{5 d (d+e x)^{5/2} (c d-b e)}}{d (c d-b e)}+\frac {2 (B d-A e)}{7 d (d+e x)^{7/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \int -\frac {B c^4 d^5-A e \left (5 c^4 d^4-10 b c^3 e d^3+10 b^2 c^2 e^2 d^2-5 b^3 c e^3 d+b^4 e^4\right )-c \left (B c^3 d^4-A e \left (4 c^3 d^3-6 b c^2 e d^2+4 b^2 c e^2 d-b^3 e^3\right )\right ) (d+e x)}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{d (c d-b e)}+\frac {2 \left (B c^3 d^4-A e \left (-b^3 e^3+4 b^2 c d e^2-6 b c^2 d^2 e+4 c^3 d^3\right )\right )}{d \sqrt {d+e x} (c d-b e)}}{d (c d-b e)}+\frac {2 \left (B c^2 d^3-A e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )\right )}{3 d (d+e x)^{3/2} (c d-b e)}}{d (c d-b e)}+\frac {2 \left (B c d^2-A e (2 c d-b e)\right )}{5 d (d+e x)^{5/2} (c d-b e)}}{d (c d-b e)}+\frac {2 (B d-A e)}{7 d (d+e x)^{7/2} (c d-b e)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \left (B c^3 d^4-A e \left (-b^3 e^3+4 b^2 c d e^2-6 b c^2 d^2 e+4 c^3 d^3\right )\right )}{d \sqrt {d+e x} (c d-b e)}-\frac {2 \int \frac {B c^4 d^5-A e \left (5 c^4 d^4-10 b c^3 e d^3+10 b^2 c^2 e^2 d^2-5 b^3 c e^3 d+b^4 e^4\right )-c \left (B c^3 d^4-A e \left (4 c^3 d^3-6 b c^2 e d^2+4 b^2 c e^2 d-b^3 e^3\right )\right ) (d+e x)}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{d (c d-b e)}}{d (c d-b e)}+\frac {2 \left (B c^2 d^3-A e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )\right )}{3 d (d+e x)^{3/2} (c d-b e)}}{d (c d-b e)}+\frac {2 \left (B c d^2-A e (2 c d-b e)\right )}{5 d (d+e x)^{5/2} (c d-b e)}}{d (c d-b e)}+\frac {2 (B d-A e)}{7 d (d+e x)^{7/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \left (\frac {c^4 d^4 (b B-A c) \int \frac {1}{-c d+b e+c (d+e x)}d\sqrt {d+e x}}{b}+\frac {A c (c d-b e)^4 \int \frac {1}{c (d+e x)-c d}d\sqrt {d+e x}}{b}\right )}{d (c d-b e)}+\frac {2 \left (B c^3 d^4-A e \left (-b^3 e^3+4 b^2 c d e^2-6 b c^2 d^2 e+4 c^3 d^3\right )\right )}{d \sqrt {d+e x} (c d-b e)}}{d (c d-b e)}+\frac {2 \left (B c^2 d^3-A e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )\right )}{3 d (d+e x)^{3/2} (c d-b e)}}{d (c d-b e)}+\frac {2 \left (B c d^2-A e (2 c d-b e)\right )}{5 d (d+e x)^{5/2} (c d-b e)}}{d (c d-b e)}+\frac {2 (B d-A e)}{7 d (d+e x)^{7/2} (c d-b e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \left (-\frac {c^{7/2} d^4 (b B-A c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b \sqrt {c d-b e}}-\frac {A \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (c d-b e)^4}{b \sqrt {d}}\right )}{d (c d-b e)}+\frac {2 \left (B c^3 d^4-A e \left (-b^3 e^3+4 b^2 c d e^2-6 b c^2 d^2 e+4 c^3 d^3\right )\right )}{d \sqrt {d+e x} (c d-b e)}}{d (c d-b e)}+\frac {2 \left (B c^2 d^3-A e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )\right )}{3 d (d+e x)^{3/2} (c d-b e)}}{d (c d-b e)}+\frac {2 \left (B c d^2-A e (2 c d-b e)\right )}{5 d (d+e x)^{5/2} (c d-b e)}}{d (c d-b e)}+\frac {2 (B d-A e)}{7 d (d+e x)^{7/2} (c d-b e)}\) |
Input:
Int[(A + B*x)/((d + e*x)^(9/2)*(b*x + c*x^2)),x]
Output:
(2*(B*d - A*e))/(7*d*(c*d - b*e)*(d + e*x)^(7/2)) + ((2*(B*c*d^2 - A*e*(2* c*d - b*e)))/(5*d*(c*d - b*e)*(d + e*x)^(5/2)) + ((2*(B*c^2*d^3 - A*e*(3*c ^2*d^2 - 3*b*c*d*e + b^2*e^2)))/(3*d*(c*d - b*e)*(d + e*x)^(3/2)) + ((2*(B *c^3*d^4 - A*e*(4*c^3*d^3 - 6*b*c^2*d^2*e + 4*b^2*c*d*e^2 - b^3*e^3)))/(d* (c*d - b*e)*Sqrt[d + e*x]) + (2*(-((A*(c*d - b*e)^4*ArcTanh[Sqrt[d + e*x]/ Sqrt[d]])/(b*Sqrt[d])) - (c^(7/2)*(b*B - A*c)*d^4*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*Sqrt[c*d - b*e])))/(d*(c*d - b*e)))/(d*(c*d - b*e)))/(d*(c*d - b*e)))/(d*(c*d - b*e))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr eeQ[{a, b, c, d, e, f, g}, x]
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*f - d*g)*((d + e*x)^(m + 1)/((m + 1)*(c *d^2 - b*d*e + a*e^2))), x] + Simp[1/(c*d^2 - b*d*e + a*e^2) Int[(d + e*x )^(m + 1)*(Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x]/(a + b*x + c*x^ 2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && FractionQ[m] && LtQ[m, -1 ]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Time = 1.74 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.95
| method | result | size |
| pseudoelliptic | \(\frac {\frac {2 A e}{7}-\frac {2 B d}{7}}{d \left (b e -c d \right ) \left (e x +d \right )^{\frac {7}{2}}}+\frac {\frac {2}{5} A b \,e^{2}-\frac {4}{5} A c d e +\frac {2}{5} B c \,d^{2}}{d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {5}{2}}}+\frac {\frac {2}{3} A \,b^{2} e^{3}-2 A b c d \,e^{2}+2 A \,c^{2} d^{2} e -\frac {2}{3} B \,c^{2} d^{3}}{d^{3} \left (b e -c d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 \left (e^{4} b^{3}-4 b^{2} c d \,e^{3}+6 d^{2} e^{2} b \,c^{2}-4 d^{3} e \,c^{3}\right ) A +2 B \,c^{3} d^{4}}{\sqrt {e x +d}\, d^{4} \left (b e -c d \right )^{4}}-\frac {2 A \,\operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b \,d^{\frac {9}{2}}}-\frac {2 c^{4} \left (A c -B b \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{\left (b e -c d \right )^{4} b \sqrt {c \left (b e -c d \right )}}\) | \(285\) |
| derivativedivides | \(-\frac {2 \left (-A e +B d \right )}{7 d \left (b e -c d \right ) \left (e x +d \right )^{\frac {7}{2}}}-\frac {2 \left (-A b \,e^{2}+2 A c d e -B c \,d^{2}\right )}{5 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 \left (-A \,b^{2} e^{3}+3 A b c d \,e^{2}-3 A \,c^{2} d^{2} e +B \,c^{2} d^{3}\right )}{3 d^{3} \left (b e -c d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (-A \,b^{3} e^{4}+4 A \,b^{2} c d \,e^{3}-6 A b \,c^{2} d^{2} e^{2}+4 A \,c^{3} d^{3} e -B \,c^{3} d^{4}\right )}{d^{4} \left (b e -c d \right )^{4} \sqrt {e x +d}}-\frac {2 A \,\operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b \,d^{\frac {9}{2}}}-\frac {2 c^{4} \left (A c -B b \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{\left (b e -c d \right )^{4} b \sqrt {c \left (b e -c d \right )}}\) | \(290\) |
| default | \(-\frac {2 \left (-A e +B d \right )}{7 d \left (b e -c d \right ) \left (e x +d \right )^{\frac {7}{2}}}-\frac {2 \left (-A b \,e^{2}+2 A c d e -B c \,d^{2}\right )}{5 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 \left (-A \,b^{2} e^{3}+3 A b c d \,e^{2}-3 A \,c^{2} d^{2} e +B \,c^{2} d^{3}\right )}{3 d^{3} \left (b e -c d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (-A \,b^{3} e^{4}+4 A \,b^{2} c d \,e^{3}-6 A b \,c^{2} d^{2} e^{2}+4 A \,c^{3} d^{3} e -B \,c^{3} d^{4}\right )}{d^{4} \left (b e -c d \right )^{4} \sqrt {e x +d}}-\frac {2 A \,\operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b \,d^{\frac {9}{2}}}-\frac {2 c^{4} \left (A c -B b \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{\left (b e -c d \right )^{4} b \sqrt {c \left (b e -c d \right )}}\) | \(290\) |
Input:
int((B*x+A)/(e*x+d)^(9/2)/(c*x^2+b*x),x,method=_RETURNVERBOSE)
Output:
2/7*(A*e-B*d)/d/(b*e-c*d)/(e*x+d)^(7/2)+2/5*(A*b*e^2-2*A*c*d*e+B*c*d^2)/d^ 2/(b*e-c*d)^2/(e*x+d)^(5/2)+2/3*(A*b^2*e^3-3*A*b*c*d*e^2+3*A*c^2*d^2*e-B*c ^2*d^3)/d^3/(b*e-c*d)^3/(e*x+d)^(3/2)+2*((b^3*e^4-4*b^2*c*d*e^3+6*b*c^2*d^ 2*e^2-4*c^3*d^3*e)*A+B*c^3*d^4)/(e*x+d)^(1/2)/d^4/(b*e-c*d)^4-2*A*arctanh( (e*x+d)^(1/2)/d^(1/2))/b/d^(9/2)-2/(b*e-c*d)^4*c^4*(A*c-B*b)/b/(c*(b*e-c*d ))^(1/2)*arctan(c*(e*x+d)^(1/2)/(c*(b*e-c*d))^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 1159 vs. \(2 (271) = 542\).
Time = 12.78 (sec) , antiderivative size = 4712, normalized size of antiderivative = 15.65 \[ \int \frac {A+B x}{(d+e x)^{9/2} \left (b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate((B*x+A)/(e*x+d)^(9/2)/(c*x^2+b*x),x, algorithm="fricas")
Output:
Too large to include
Time = 11.99 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.36 \[ \int \frac {A+B x}{(d+e x)^{9/2} \left (b x+c x^2\right )} \, dx=\begin {cases} \frac {2 \left (\frac {A e \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d}} \right )}}{b d^{4} \sqrt {- d}} - \frac {e \left (- A e + B d\right )}{7 d \left (d + e x\right )^{\frac {7}{2}} \left (b e - c d\right )} + \frac {e \left (A b e^{2} - 2 A c d e + B c d^{2}\right )}{5 d^{2} \left (d + e x\right )^{\frac {5}{2}} \left (b e - c d\right )^{2}} + \frac {e \left (A b^{2} e^{3} - 3 A b c d e^{2} + 3 A c^{2} d^{2} e - B c^{2} d^{3}\right )}{3 d^{3} \left (d + e x\right )^{\frac {3}{2}} \left (b e - c d\right )^{3}} + \frac {e \left (A b^{3} e^{4} - 4 A b^{2} c d e^{3} + 6 A b c^{2} d^{2} e^{2} - 4 A c^{3} d^{3} e + B c^{3} d^{4}\right )}{d^{4} \sqrt {d + e x} \left (b e - c d\right )^{4}} + \frac {c^{3} e \left (- A c + B b\right ) \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {b e - c d}{c}}} \right )}}{b \sqrt {\frac {b e - c d}{c}} \left (b e - c d\right )^{4}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {\frac {B \log {\left (b x + c x^{2} \right )}}{2 c} + \left (A - \frac {B b}{2 c}\right ) \left (- \frac {2 c \left (\begin {cases} \frac {\frac {b}{2 c} + x}{b} & \text {for}\: c = 0 \\- \frac {\log {\left (b - 2 c \left (\frac {b}{2 c} + x\right ) \right )}}{2 c} & \text {otherwise} \end {cases}\right )}{b} - \frac {2 c \left (\begin {cases} \frac {\frac {b}{2 c} + x}{b} & \text {for}\: c = 0 \\\frac {\log {\left (b + 2 c \left (\frac {b}{2 c} + x\right ) \right )}}{2 c} & \text {otherwise} \end {cases}\right )}{b}\right )}{d^{\frac {9}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate((B*x+A)/(e*x+d)**(9/2)/(c*x**2+b*x),x)
Output:
Piecewise((2*(A*e*atan(sqrt(d + e*x)/sqrt(-d))/(b*d**4*sqrt(-d)) - e*(-A*e + B*d)/(7*d*(d + e*x)**(7/2)*(b*e - c*d)) + e*(A*b*e**2 - 2*A*c*d*e + B*c *d**2)/(5*d**2*(d + e*x)**(5/2)*(b*e - c*d)**2) + e*(A*b**2*e**3 - 3*A*b*c *d*e**2 + 3*A*c**2*d**2*e - B*c**2*d**3)/(3*d**3*(d + e*x)**(3/2)*(b*e - c *d)**3) + e*(A*b**3*e**4 - 4*A*b**2*c*d*e**3 + 6*A*b*c**2*d**2*e**2 - 4*A* c**3*d**3*e + B*c**3*d**4)/(d**4*sqrt(d + e*x)*(b*e - c*d)**4) + c**3*e*(- A*c + B*b)*atan(sqrt(d + e*x)/sqrt((b*e - c*d)/c))/(b*sqrt((b*e - c*d)/c)* (b*e - c*d)**4))/e, Ne(e, 0)), ((B*log(b*x + c*x**2)/(2*c) + (A - B*b/(2*c ))*(-2*c*Piecewise(((b/(2*c) + x)/b, Eq(c, 0)), (-log(b - 2*c*(b/(2*c) + x ))/(2*c), True))/b - 2*c*Piecewise(((b/(2*c) + x)/b, Eq(c, 0)), (log(b + 2 *c*(b/(2*c) + x))/(2*c), True))/b))/d**(9/2), True))
Exception generated. \[ \int \frac {A+B x}{(d+e x)^{9/2} \left (b x+c x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((B*x+A)/(e*x+d)^(9/2)/(c*x^2+b*x),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(b*e-c*d>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 604 vs. \(2 (271) = 542\).
Time = 0.27 (sec) , antiderivative size = 604, normalized size of antiderivative = 2.01 \[ \int \frac {A+B x}{(d+e x)^{9/2} \left (b x+c x^2\right )} \, dx=\frac {2 \, {\left (B b c^{4} - A c^{5}\right )} \arctan \left (\frac {\sqrt {e x + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{{\left (b c^{4} d^{4} - 4 \, b^{2} c^{3} d^{3} e + 6 \, b^{3} c^{2} d^{2} e^{2} - 4 \, b^{4} c d e^{3} + b^{5} e^{4}\right )} \sqrt {-c^{2} d + b c e}} + \frac {2 \, {\left (105 \, {\left (e x + d\right )}^{3} B c^{3} d^{4} + 35 \, {\left (e x + d\right )}^{2} B c^{3} d^{5} + 21 \, {\left (e x + d\right )} B c^{3} d^{6} + 15 \, B c^{3} d^{7} - 420 \, {\left (e x + d\right )}^{3} A c^{3} d^{3} e - 35 \, {\left (e x + d\right )}^{2} B b c^{2} d^{4} e - 105 \, {\left (e x + d\right )}^{2} A c^{3} d^{4} e - 42 \, {\left (e x + d\right )} B b c^{2} d^{5} e - 42 \, {\left (e x + d\right )} A c^{3} d^{5} e - 45 \, B b c^{2} d^{6} e - 15 \, A c^{3} d^{6} e + 630 \, {\left (e x + d\right )}^{3} A b c^{2} d^{2} e^{2} + 210 \, {\left (e x + d\right )}^{2} A b c^{2} d^{3} e^{2} + 21 \, {\left (e x + d\right )} B b^{2} c d^{4} e^{2} + 105 \, {\left (e x + d\right )} A b c^{2} d^{4} e^{2} + 45 \, B b^{2} c d^{5} e^{2} + 45 \, A b c^{2} d^{5} e^{2} - 420 \, {\left (e x + d\right )}^{3} A b^{2} c d e^{3} - 140 \, {\left (e x + d\right )}^{2} A b^{2} c d^{2} e^{3} - 84 \, {\left (e x + d\right )} A b^{2} c d^{3} e^{3} - 15 \, B b^{3} d^{4} e^{3} - 45 \, A b^{2} c d^{4} e^{3} + 105 \, {\left (e x + d\right )}^{3} A b^{3} e^{4} + 35 \, {\left (e x + d\right )}^{2} A b^{3} d e^{4} + 21 \, {\left (e x + d\right )} A b^{3} d^{2} e^{4} + 15 \, A b^{3} d^{3} e^{4}\right )}}{105 \, {\left (c^{4} d^{8} - 4 \, b c^{3} d^{7} e + 6 \, b^{2} c^{2} d^{6} e^{2} - 4 \, b^{3} c d^{5} e^{3} + b^{4} d^{4} e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}}} + \frac {2 \, A \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-d}}\right )}{b \sqrt {-d} d^{4}} \] Input:
integrate((B*x+A)/(e*x+d)^(9/2)/(c*x^2+b*x),x, algorithm="giac")
Output:
2*(B*b*c^4 - A*c^5)*arctan(sqrt(e*x + d)*c/sqrt(-c^2*d + b*c*e))/((b*c^4*d ^4 - 4*b^2*c^3*d^3*e + 6*b^3*c^2*d^2*e^2 - 4*b^4*c*d*e^3 + b^5*e^4)*sqrt(- c^2*d + b*c*e)) + 2/105*(105*(e*x + d)^3*B*c^3*d^4 + 35*(e*x + d)^2*B*c^3* d^5 + 21*(e*x + d)*B*c^3*d^6 + 15*B*c^3*d^7 - 420*(e*x + d)^3*A*c^3*d^3*e - 35*(e*x + d)^2*B*b*c^2*d^4*e - 105*(e*x + d)^2*A*c^3*d^4*e - 42*(e*x + d )*B*b*c^2*d^5*e - 42*(e*x + d)*A*c^3*d^5*e - 45*B*b*c^2*d^6*e - 15*A*c^3*d ^6*e + 630*(e*x + d)^3*A*b*c^2*d^2*e^2 + 210*(e*x + d)^2*A*b*c^2*d^3*e^2 + 21*(e*x + d)*B*b^2*c*d^4*e^2 + 105*(e*x + d)*A*b*c^2*d^4*e^2 + 45*B*b^2*c *d^5*e^2 + 45*A*b*c^2*d^5*e^2 - 420*(e*x + d)^3*A*b^2*c*d*e^3 - 140*(e*x + d)^2*A*b^2*c*d^2*e^3 - 84*(e*x + d)*A*b^2*c*d^3*e^3 - 15*B*b^3*d^4*e^3 - 45*A*b^2*c*d^4*e^3 + 105*(e*x + d)^3*A*b^3*e^4 + 35*(e*x + d)^2*A*b^3*d*e^ 4 + 21*(e*x + d)*A*b^3*d^2*e^4 + 15*A*b^3*d^3*e^4)/((c^4*d^8 - 4*b*c^3*d^7 *e + 6*b^2*c^2*d^6*e^2 - 4*b^3*c*d^5*e^3 + b^4*d^4*e^4)*(e*x + d)^(7/2)) + 2*A*arctan(sqrt(e*x + d)/sqrt(-d))/(b*sqrt(-d)*d^4)
Time = 15.21 (sec) , antiderivative size = 11601, normalized size of antiderivative = 38.54 \[ \int \frac {A+B x}{(d+e x)^{9/2} \left (b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:
int((A + B*x)/((b*x + c*x^2)*(d + e*x)^(9/2)),x)
Output:
(A*atan((B^2*b^2*c^19*d^41*(d + e*x)^(1/2)*1i + A^2*b^21*d^20*e^21*(d + e* x)^(1/2)*1i - A^2*b^20*c*d^21*e^20*(d + e*x)^(1/2)*21i - B^2*b^3*c^18*d^40 *e*(d + e*x)^(1/2)*12i - A*B*b*c^20*d^41*(d + e*x)^(1/2)*2i - A^2*b^2*c^19 *d^39*e^2*(d + e*x)^(1/2)*144i + A^2*b^3*c^18*d^38*e^3*(d + e*x)^(1/2)*111 0i - A^2*b^4*c^17*d^37*e^4*(d + e*x)^(1/2)*5490i + A^2*b^5*c^16*d^36*e^5*( d + e*x)^(1/2)*19557i - A^2*b^6*c^15*d^35*e^6*(d + e*x)^(1/2)*53340i + A^2 *b^7*c^14*d^34*e^7*(d + e*x)^(1/2)*115488i - A^2*b^8*c^13*d^33*e^8*(d + e* x)^(1/2)*202995i + A^2*b^9*c^12*d^32*e^9*(d + e*x)^(1/2)*293710i - A^2*b^1 0*c^11*d^31*e^10*(d + e*x)^(1/2)*352650i + A^2*b^11*c^10*d^30*e^11*(d + e* x)^(1/2)*352704i - A^2*b^12*c^9*d^29*e^12*(d + e*x)^(1/2)*293929i + A^2*b^ 13*c^8*d^28*e^13*(d + e*x)^(1/2)*203490i - A^2*b^14*c^7*d^27*e^14*(d + e*x )^(1/2)*116280i + A^2*b^15*c^6*d^26*e^15*(d + e*x)^(1/2)*54264i - A^2*b^16 *c^5*d^25*e^16*(d + e*x)^(1/2)*20349i + A^2*b^17*c^4*d^24*e^17*(d + e*x)^( 1/2)*5985i - A^2*b^18*c^3*d^23*e^18*(d + e*x)^(1/2)*1330i + A^2*b^19*c^2*d ^22*e^19*(d + e*x)^(1/2)*210i + B^2*b^4*c^17*d^39*e^2*(d + e*x)^(1/2)*66i - B^2*b^5*c^16*d^38*e^3*(d + e*x)^(1/2)*220i + B^2*b^6*c^15*d^37*e^4*(d + e*x)^(1/2)*495i - B^2*b^7*c^14*d^36*e^5*(d + e*x)^(1/2)*792i + B^2*b^8*c^1 3*d^35*e^6*(d + e*x)^(1/2)*924i - B^2*b^9*c^12*d^34*e^7*(d + e*x)^(1/2)*79 2i + B^2*b^10*c^11*d^33*e^8*(d + e*x)^(1/2)*495i - B^2*b^11*c^10*d^32*e^9* (d + e*x)^(1/2)*220i + B^2*b^12*c^9*d^31*e^10*(d + e*x)^(1/2)*66i - B^2...
Time = 0.27 (sec) , antiderivative size = 2890, normalized size of antiderivative = 9.60 \[ \int \frac {A+B x}{(d+e x)^{9/2} \left (b x+c x^2\right )} \, dx =\text {Too large to display} \] Input:
int((B*x+A)/(e*x+d)^(9/2)/(c*x^2+b*x),x)
Output:
( - 210*sqrt(c)*sqrt(d + e*x)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt (c)*sqrt(b*e - c*d)))*a*c**4*d**8 - 630*sqrt(c)*sqrt(d + e*x)*sqrt(b*e - c *d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*a*c**4*d**7*e*x - 63 0*sqrt(c)*sqrt(d + e*x)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sq rt(b*e - c*d)))*a*c**4*d**6*e**2*x**2 - 210*sqrt(c)*sqrt(d + e*x)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*a*c**4*d**5*e**3 *x**3 + 210*sqrt(c)*sqrt(d + e*x)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/( sqrt(c)*sqrt(b*e - c*d)))*b**2*c**3*d**8 + 630*sqrt(c)*sqrt(d + e*x)*sqrt( b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**2*c**3*d** 7*e*x + 630*sqrt(c)*sqrt(d + e*x)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/( sqrt(c)*sqrt(b*e - c*d)))*b**2*c**3*d**6*e**2*x**2 + 210*sqrt(c)*sqrt(d + e*x)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b** 2*c**3*d**5*e**3*x**3 + 105*sqrt(d)*sqrt(d + e*x)*log(sqrt(d + e*x) - sqrt (d))*a*b**5*d**3*e**5 + 315*sqrt(d)*sqrt(d + e*x)*log(sqrt(d + e*x) - sqrt (d))*a*b**5*d**2*e**6*x + 315*sqrt(d)*sqrt(d + e*x)*log(sqrt(d + e*x) - sq rt(d))*a*b**5*d*e**7*x**2 + 105*sqrt(d)*sqrt(d + e*x)*log(sqrt(d + e*x) - sqrt(d))*a*b**5*e**8*x**3 - 525*sqrt(d)*sqrt(d + e*x)*log(sqrt(d + e*x) - sqrt(d))*a*b**4*c*d**4*e**4 - 1575*sqrt(d)*sqrt(d + e*x)*log(sqrt(d + e*x) - sqrt(d))*a*b**4*c*d**3*e**5*x - 1575*sqrt(d)*sqrt(d + e*x)*log(sqrt(d + e*x) - sqrt(d))*a*b**4*c*d**2*e**6*x**2 - 525*sqrt(d)*sqrt(d + e*x)*lo...