Integrand size = 26, antiderivative size = 292 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^2} \, dx=\frac {e \left (2 A c^3 d^2-5 b^3 B e^2-b c^2 d (B d+2 A e)+b^2 c e (8 B d+3 A e)\right ) \sqrt {d+e x}}{b^2 c^3}+\frac {e \left (6 A c^2 d+5 b^2 B e-3 b c (B d+A e)\right ) (d+e x)^{3/2}}{3 b^2 c^2}-\frac {(d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \left (b x+c x^2\right )}-\frac {d^{5/2} (2 b B d-4 A c d+7 A b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}+\frac {(c d-b e)^{5/2} \left (2 b B c d-4 A c^2 d+5 b^2 B e-3 A b c e\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{7/2}} \]
1/3*e*(6*A*c^2*d+5*b^2*B*e-3*b*c*(A*e+B*d))*(e*x+d)^(3/2)/b^2/c^2-(e*x+d)^ (5/2)*(A*b*c*d+(2*A*c^2*d+b^2*B*e-b*c*(A*e+B*d))*x)/b^2/c/(c*x^2+b*x)-d^(5 /2)*(7*A*b*e-4*A*c*d+2*B*b*d)*arctanh((e*x+d)^(1/2)/d^(1/2))/b^3+(-b*e+c*d )^(5/2)*(-3*A*b*c*e-4*A*c^2*d+5*B*b^2*e+2*B*b*c*d)*arctanh(c^(1/2)*(e*x+d) ^(1/2)/(-b*e+c*d)^(1/2))/b^3/c^(7/2)+e*(2*A*c^3*d^2-5*b^3*B*e^2-b*c^2*d*(2 *A*e+B*d)+b^2*c*e*(3*A*e+8*B*d))*(e*x+d)^(1/2)/b^2/c^3
Time = 0.74 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.90 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^2} \, dx=\frac {\frac {b \sqrt {d+e x} \left (-3 A c \left (2 c^3 d^3 x-3 b^3 e^3 x+b c^2 d^2 (d-3 e x)+b^2 c e^2 x (3 d-2 e x)\right )+b B x \left (3 c^3 d^3-15 b^3 e^3+b^2 c e^2 (29 d-10 e x)+b c^2 e \left (-9 d^2+20 d e x+2 e^2 x^2\right )\right )\right )}{c^3 x (b+c x)}-\frac {3 (-c d+b e)^{5/2} \left (-2 b B c d+4 A c^2 d-5 b^2 B e+3 A b c e\right ) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{c^{7/2}}-3 d^{5/2} (2 b B d-4 A c d+7 A b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{3 b^3} \]
((b*Sqrt[d + e*x]*(-3*A*c*(2*c^3*d^3*x - 3*b^3*e^3*x + b*c^2*d^2*(d - 3*e* x) + b^2*c*e^2*x*(3*d - 2*e*x)) + b*B*x*(3*c^3*d^3 - 15*b^3*e^3 + b^2*c*e^ 2*(29*d - 10*e*x) + b*c^2*e*(-9*d^2 + 20*d*e*x + 2*e^2*x^2))))/(c^3*x*(b + c*x)) - (3*(-(c*d) + b*e)^(5/2)*(-2*b*B*c*d + 4*A*c^2*d - 5*b^2*B*e + 3*A *b*c*e)*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/c^(7/2) - 3*d^ (5/2)*(2*b*B*d - 4*A*c*d + 7*A*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(3*b^3 )
Time = 0.95 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {1233, 27, 1196, 1196, 1197, 25, 27, 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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {\int \frac {(d+e x)^{3/2} \left (c d (2 b B d-4 A c d+7 A b e)+e \left (5 B e b^2-3 c (B d+A e) b+6 A c^2 d\right ) x\right )}{2 \left (c x^2+b x\right )}dx}{b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(d+e x)^{3/2} \left (c d (2 b B d-4 A c d+7 A b e)+e \left (5 B e b^2-3 c (B d+A e) b+6 A c^2 d\right ) x\right )}{c x^2+b x}dx}{2 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1196 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {d+e x} \left (c^2 (2 b B d-4 A c d+7 A b e) d^2+e \left (-5 B e^2 b^3+c e (8 B d+3 A e) b^2-c^2 d (B d+2 A e) b+2 A c^3 d^2\right ) x\right )}{c x^2+b x}dx}{c}+\frac {2 e (d+e x)^{3/2} \left (-3 b c (A e+B d)+6 A c^2 d+5 b^2 B e\right )}{3 c}}{2 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1196 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {c^3 d^3 (2 b B d-4 A c d+7 A b e)-e \left (-5 B e^3 b^4+c e^2 (13 B d+3 A e) b^3-c^2 d e (9 B d+5 A e) b^2-c^3 d^2 (B d+3 A e) b+2 A c^4 d^3\right ) x}{\sqrt {d+e x} \left (c x^2+b x\right )}dx}{c}+\frac {2 e \sqrt {d+e x} \left (b^2 c e (3 A e+8 B d)-b c^2 d (2 A e+B d)+2 A c^3 d^2-5 b^3 B e^2\right )}{c}}{c}+\frac {2 e (d+e x)^{3/2} \left (-3 b c (A e+B d)+6 A c^2 d+5 b^2 B e\right )}{3 c}}{2 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle \frac {\frac {\frac {2 \int -\frac {e \left (d (c d-b e) \left (-5 B e^2 b^3+c e (8 B d+3 A e) b^2-c^2 d (B d+2 A e) b+2 A c^3 d^2\right )+\left (-5 B e^3 b^4+c e^2 (13 B d+3 A e) b^3-c^2 d e (9 B d+5 A e) b^2-c^3 d^2 (B d+3 A e) b+2 A c^4 d^3\right ) (d+e x)\right )}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{c}+\frac {2 e \sqrt {d+e x} \left (b^2 c e (3 A e+8 B d)-b c^2 d (2 A e+B d)+2 A c^3 d^2-5 b^3 B e^2\right )}{c}}{c}+\frac {2 e (d+e x)^{3/2} \left (-3 b c (A e+B d)+6 A c^2 d+5 b^2 B e\right )}{3 c}}{2 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {2 e \sqrt {d+e x} \left (b^2 c e (3 A e+8 B d)-b c^2 d (2 A e+B d)+2 A c^3 d^2-5 b^3 B e^2\right )}{c}-\frac {2 \int \frac {e \left (d (c d-b e) \left (-5 B e^2 b^3+c e (8 B d+3 A e) b^2-c^2 d (B d+2 A e) b+2 A c^3 d^2\right )+\left (-5 B e^3 b^4+c e^2 (13 B d+3 A e) b^3-c^2 d e (9 B d+5 A e) b^2-c^3 d^2 (B d+3 A e) b+2 A c^4 d^3\right ) (d+e x)\right )}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{c}}{c}+\frac {2 e (d+e x)^{3/2} \left (-3 b c (A e+B d)+6 A c^2 d+5 b^2 B e\right )}{3 c}}{2 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 e \sqrt {d+e x} \left (b^2 c e (3 A e+8 B d)-b c^2 d (2 A e+B d)+2 A c^3 d^2-5 b^3 B e^2\right )}{c}-\frac {2 e \int \frac {d (c d-b e) \left (-5 B e^2 b^3+c e (8 B d+3 A e) b^2-c^2 d (B d+2 A e) b+2 A c^3 d^2\right )+\left (-5 B e^3 b^4+c e^2 (13 B d+3 A e) b^3-c^2 d e (9 B d+5 A e) b^2-c^3 d^2 (B d+3 A e) b+2 A c^4 d^3\right ) (d+e x)}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{c}}{c}+\frac {2 e (d+e x)^{3/2} \left (-3 b c (A e+B d)+6 A c^2 d+5 b^2 B e\right )}{3 c}}{2 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {\frac {2 e \sqrt {d+e x} \left (b^2 c e (3 A e+8 B d)-b c^2 d (2 A e+B d)+2 A c^3 d^2-5 b^3 B e^2\right )}{c}-\frac {2 e \left (-\frac {(c d-b e)^3 \left (-b c (2 B d-3 A e)+4 A c^2 d-5 b^2 B e\right ) \int \frac {1}{-c d+b e+c (d+e x)}d\sqrt {d+e x}}{b e}-\frac {c^4 d^3 (7 A b e-4 A c d+2 b B d) \int \frac {1}{c (d+e x)-c d}d\sqrt {d+e x}}{b e}\right )}{c}}{c}+\frac {2 e (d+e x)^{3/2} \left (-3 b c (A e+B d)+6 A c^2 d+5 b^2 B e\right )}{3 c}}{2 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {2 e \sqrt {d+e x} \left (b^2 c e (3 A e+8 B d)-b c^2 d (2 A e+B d)+2 A c^3 d^2-5 b^3 B e^2\right )}{c}-\frac {2 e \left (\frac {(c d-b e)^{5/2} \left (-b c (2 B d-3 A e)+4 A c^2 d-5 b^2 B e\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b \sqrt {c} e}+\frac {c^3 d^{5/2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (7 A b e-4 A c d+2 b B d)}{b e}\right )}{c}}{c}+\frac {2 e (d+e x)^{3/2} \left (-3 b c (A e+B d)+6 A c^2 d+5 b^2 B e\right )}{3 c}}{2 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}\) |
-(((d + e*x)^(5/2)*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/ (b^2*c*(b*x + c*x^2))) + ((2*e*(6*A*c^2*d + 5*b^2*B*e - 3*b*c*(B*d + A*e)) *(d + e*x)^(3/2))/(3*c) + ((2*e*(2*A*c^3*d^2 - 5*b^3*B*e^2 - b*c^2*d*(B*d + 2*A*e) + b^2*c*e*(8*B*d + 3*A*e))*Sqrt[d + e*x])/c - (2*e*((c^3*d^(5/2)* (2*b*B*d - 4*A*c*d + 7*A*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*e) + ((c* d - b*e)^(5/2)*(4*A*c^2*d - 5*b^2*B*e - b*c*(2*B*d - 3*A*e))*ArcTanh[(Sqrt [c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*Sqrt[c]*e)))/c)/c)/(2*b^2*c)
3.13.39.3.1 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[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[g*((d + e*x)^m/(c*m)), x] + Simp[1/c Int [(d + e*x)^(m - 1)*(Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && FractionQ[m] & & GtQ[m, 0]
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)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2) ^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c *(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f *(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | | !ILtQ[m + 2*p + 3, 0])
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 = 0.56 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.94
| method | result | size |
| pseudoelliptic | \(-\frac {-4 x \left (c x +b \right ) \sqrt {d}\, \left (-b e +c d \right )^{3} \left (A \,c^{2} d +\frac {3 \left (A e -\frac {2 B d}{3}\right ) b c}{4}-\frac {5 b^{2} B e}{4}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )+\sqrt {\left (b e -c d \right ) c}\, \left (7 c^{3} x \left (c x +b \right ) \left (-\frac {4 A c d}{7}+b \left (A e +\frac {2 B d}{7}\right )\right ) d^{3} \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )+\sqrt {e x +d}\, \sqrt {d}\, \left (2 A \,c^{4} d^{3} x +d^{2} \left (\left (-B x +A \right ) d -3 A e x \right ) b \,c^{3}+3 \left (B \,d^{2}+e \left (-\frac {20 B x}{9}+A \right ) d -\frac {2 \left (\frac {B x}{3}+A \right ) x \,e^{2}}{3}\right ) x e \,b^{2} c^{2}-3 x \left (\frac {29 B d}{9}+e \left (-\frac {10 B x}{9}+A \right )\right ) e^{2} b^{3} c +5 B \,b^{4} e^{3} x \right ) b \right )}{\sqrt {\left (b e -c d \right ) c}\, \sqrt {d}\, c^{3} \left (c x +b \right ) b^{3} x}\) | \(275\) |
| derivativedivides | \(2 e^{2} \left (\frac {\frac {B c \left (e x +d \right )^{\frac {3}{2}}}{3}+A c e \sqrt {e x +d}-2 b e B \sqrt {e x +d}+3 B c d \sqrt {e x +d}}{c^{3}}-\frac {\frac {\left (-\frac {1}{2} A \,b^{4} c \,e^{4}+\frac {3}{2} A \,b^{3} c^{2} d \,e^{3}-\frac {3}{2} A \,b^{2} c^{3} d^{2} e^{2}+\frac {1}{2} A b \,c^{4} d^{3} e +\frac {1}{2} b^{5} B \,e^{4}-\frac {3}{2} B \,b^{4} c d \,e^{3}+\frac {3}{2} B \,b^{3} c^{2} d^{2} e^{2}-\frac {1}{2} B \,b^{2} c^{3} d^{3} e \right ) \sqrt {e x +d}}{c \left (e x +d \right )+b e -c d}+\frac {\left (3 A \,b^{4} c \,e^{4}-5 A \,b^{3} c^{2} d \,e^{3}-3 A \,b^{2} c^{3} d^{2} e^{2}+9 A b \,c^{4} d^{3} e -4 d^{4} A \,c^{5}-5 b^{5} B \,e^{4}+13 B \,b^{4} c d \,e^{3}-9 B \,b^{3} c^{2} d^{2} e^{2}-B \,b^{2} c^{3} d^{3} e +2 B b \,c^{4} d^{4}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}}{b^{3} e^{2} c^{3}}-\frac {d^{3} \left (\frac {A b \sqrt {e x +d}}{2 x}+\frac {\left (7 A b e -4 A c d +2 B b d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}\right )}{e^{2} b^{3}}\right )\) | \(405\) |
| default | \(2 e^{2} \left (\frac {\frac {B c \left (e x +d \right )^{\frac {3}{2}}}{3}+A c e \sqrt {e x +d}-2 b e B \sqrt {e x +d}+3 B c d \sqrt {e x +d}}{c^{3}}-\frac {\frac {\left (-\frac {1}{2} A \,b^{4} c \,e^{4}+\frac {3}{2} A \,b^{3} c^{2} d \,e^{3}-\frac {3}{2} A \,b^{2} c^{3} d^{2} e^{2}+\frac {1}{2} A b \,c^{4} d^{3} e +\frac {1}{2} b^{5} B \,e^{4}-\frac {3}{2} B \,b^{4} c d \,e^{3}+\frac {3}{2} B \,b^{3} c^{2} d^{2} e^{2}-\frac {1}{2} B \,b^{2} c^{3} d^{3} e \right ) \sqrt {e x +d}}{c \left (e x +d \right )+b e -c d}+\frac {\left (3 A \,b^{4} c \,e^{4}-5 A \,b^{3} c^{2} d \,e^{3}-3 A \,b^{2} c^{3} d^{2} e^{2}+9 A b \,c^{4} d^{3} e -4 d^{4} A \,c^{5}-5 b^{5} B \,e^{4}+13 B \,b^{4} c d \,e^{3}-9 B \,b^{3} c^{2} d^{2} e^{2}-B \,b^{2} c^{3} d^{3} e +2 B b \,c^{4} d^{4}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}}{b^{3} e^{2} c^{3}}-\frac {d^{3} \left (\frac {A b \sqrt {e x +d}}{2 x}+\frac {\left (7 A b e -4 A c d +2 B b d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}\right )}{e^{2} b^{3}}\right )\) | \(405\) |
| risch | \(-\frac {d^{3} A \sqrt {e x +d}}{b^{2} x}+\frac {e \left (\frac {2 b^{2} e \left (\frac {B c \left (e x +d \right )^{\frac {3}{2}}}{3}+A c e \sqrt {e x +d}-2 b e B \sqrt {e x +d}+3 B c d \sqrt {e x +d}\right )}{c^{3}}-\frac {2 \left (\frac {\left (-\frac {1}{2} A \,b^{4} c \,e^{4}+\frac {3}{2} A \,b^{3} c^{2} d \,e^{3}-\frac {3}{2} A \,b^{2} c^{3} d^{2} e^{2}+\frac {1}{2} A b \,c^{4} d^{3} e +\frac {1}{2} b^{5} B \,e^{4}-\frac {3}{2} B \,b^{4} c d \,e^{3}+\frac {3}{2} B \,b^{3} c^{2} d^{2} e^{2}-\frac {1}{2} B \,b^{2} c^{3} d^{3} e \right ) \sqrt {e x +d}}{c \left (e x +d \right )+b e -c d}+\frac {\left (3 A \,b^{4} c \,e^{4}-5 A \,b^{3} c^{2} d \,e^{3}-3 A \,b^{2} c^{3} d^{2} e^{2}+9 A b \,c^{4} d^{3} e -4 d^{4} A \,c^{5}-5 b^{5} B \,e^{4}+13 B \,b^{4} c d \,e^{3}-9 B \,b^{3} c^{2} d^{2} e^{2}-B \,b^{2} c^{3} d^{3} e +2 B b \,c^{4} d^{4}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}\right )}{e b \,c^{3}}-\frac {d^{\frac {5}{2}} \left (7 A b e -4 A c d +2 B b d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b e}\right )}{b^{2}}\) | \(410\) |
-1/((b*e-c*d)*c)^(1/2)*(-4*x*(c*x+b)*d^(1/2)*(-b*e+c*d)^3*(A*c^2*d+3/4*(A* e-2/3*B*d)*b*c-5/4*b^2*B*e)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))+(( b*e-c*d)*c)^(1/2)*(7*c^3*x*(c*x+b)*(-4/7*A*c*d+b*(A*e+2/7*B*d))*d^3*arctan h((e*x+d)^(1/2)/d^(1/2))+(e*x+d)^(1/2)*d^(1/2)*(2*A*c^4*d^3*x+d^2*((-B*x+A )*d-3*A*e*x)*b*c^3+3*(B*d^2+e*(-20/9*B*x+A)*d-2/3*(1/3*B*x+A)*x*e^2)*x*e*b ^2*c^2-3*x*(29/9*B*d+e*(-10/9*B*x+A))*e^2*b^3*c+5*B*b^4*e^3*x)*b))/d^(1/2) /c^3/(c*x+b)/b^3/x
Time = 43.77 (sec) , antiderivative size = 2104, normalized size of antiderivative = 7.21 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
[-1/6*(3*((2*(B*b*c^4 - 2*A*c^5)*d^3 + (B*b^2*c^3 + 5*A*b*c^4)*d^2*e - 2*( 4*B*b^3*c^2 - A*b^2*c^3)*d*e^2 + (5*B*b^4*c - 3*A*b^3*c^2)*e^3)*x^2 + (2*( B*b^2*c^3 - 2*A*b*c^4)*d^3 + (B*b^3*c^2 + 5*A*b^2*c^3)*d^2*e - 2*(4*B*b^4* c - A*b^3*c^2)*d*e^2 + (5*B*b^5 - 3*A*b^4*c)*e^3)*x)*sqrt((c*d - b*e)/c)*l og((c*e*x + 2*c*d - b*e - 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b) ) - 3*((7*A*b*c^4*d^2*e + 2*(B*b*c^4 - 2*A*c^5)*d^3)*x^2 + (7*A*b^2*c^3*d^ 2*e + 2*(B*b^2*c^3 - 2*A*b*c^4)*d^3)*x)*sqrt(d)*log((e*x - 2*sqrt(e*x + d) *sqrt(d) + 2*d)/x) - 2*(2*B*b^3*c^2*e^3*x^3 - 3*A*b^2*c^3*d^3 + 2*(10*B*b^ 3*c^2*d*e^2 - (5*B*b^4*c - 3*A*b^3*c^2)*e^3)*x^2 + (3*(B*b^2*c^3 - 2*A*b*c ^4)*d^3 - 9*(B*b^3*c^2 - A*b^2*c^3)*d^2*e + (29*B*b^4*c - 9*A*b^3*c^2)*d*e ^2 - 3*(5*B*b^5 - 3*A*b^4*c)*e^3)*x)*sqrt(e*x + d))/(b^3*c^4*x^2 + b^4*c^3 *x), 1/6*(6*((2*(B*b*c^4 - 2*A*c^5)*d^3 + (B*b^2*c^3 + 5*A*b*c^4)*d^2*e - 2*(4*B*b^3*c^2 - A*b^2*c^3)*d*e^2 + (5*B*b^4*c - 3*A*b^3*c^2)*e^3)*x^2 + ( 2*(B*b^2*c^3 - 2*A*b*c^4)*d^3 + (B*b^3*c^2 + 5*A*b^2*c^3)*d^2*e - 2*(4*B*b ^4*c - A*b^3*c^2)*d*e^2 + (5*B*b^5 - 3*A*b^4*c)*e^3)*x)*sqrt(-(c*d - b*e)/ c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + 3*((7*A*b*c ^4*d^2*e + 2*(B*b*c^4 - 2*A*c^5)*d^3)*x^2 + (7*A*b^2*c^3*d^2*e + 2*(B*b^2* c^3 - 2*A*b*c^4)*d^3)*x)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d) /x) + 2*(2*B*b^3*c^2*e^3*x^3 - 3*A*b^2*c^3*d^3 + 2*(10*B*b^3*c^2*d*e^2 - ( 5*B*b^4*c - 3*A*b^3*c^2)*e^3)*x^2 + (3*(B*b^2*c^3 - 2*A*b*c^4)*d^3 - 9*...
Timed out. \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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-c*d>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 625 vs. \(2 (268) = 536\).
Time = 0.29 (sec) , antiderivative size = 625, normalized size of antiderivative = 2.14 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^2} \, dx=\frac {{\left (2 \, B b d^{4} - 4 \, A c d^{4} + 7 \, A b d^{3} e\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d}} - \frac {{\left (2 \, B b c^{4} d^{4} - 4 \, A c^{5} d^{4} - B b^{2} c^{3} d^{3} e + 9 \, A b c^{4} d^{3} e - 9 \, B b^{3} c^{2} d^{2} e^{2} - 3 \, A b^{2} c^{3} d^{2} e^{2} + 13 \, B b^{4} c d e^{3} - 5 \, A b^{3} c^{2} d e^{3} - 5 \, B b^{5} e^{4} + 3 \, A b^{4} c e^{4}\right )} \arctan \left (\frac {\sqrt {e x + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{\sqrt {-c^{2} d + b c e} b^{3} c^{3}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} B c^{4} e^{2} + 9 \, \sqrt {e x + d} B c^{4} d e^{2} - 6 \, \sqrt {e x + d} B b c^{3} e^{3} + 3 \, \sqrt {e x + d} A c^{4} e^{3}\right )}}{3 \, c^{6}} + \frac {{\left (e x + d\right )}^{\frac {3}{2}} B b c^{3} d^{3} e - 2 \, {\left (e x + d\right )}^{\frac {3}{2}} A c^{4} d^{3} e - \sqrt {e x + d} B b c^{3} d^{4} e + 2 \, \sqrt {e x + d} A c^{4} d^{4} e - 3 \, {\left (e x + d\right )}^{\frac {3}{2}} B b^{2} c^{2} d^{2} e^{2} + 3 \, {\left (e x + d\right )}^{\frac {3}{2}} A b c^{3} d^{2} e^{2} + 3 \, \sqrt {e x + d} B b^{2} c^{2} d^{3} e^{2} - 4 \, \sqrt {e x + d} A b c^{3} d^{3} e^{2} + 3 \, {\left (e x + d\right )}^{\frac {3}{2}} B b^{3} c d e^{3} - 3 \, {\left (e x + d\right )}^{\frac {3}{2}} A b^{2} c^{2} d e^{3} - 3 \, \sqrt {e x + d} B b^{3} c d^{2} e^{3} + 3 \, \sqrt {e x + d} A b^{2} c^{2} d^{2} e^{3} - {\left (e x + d\right )}^{\frac {3}{2}} B b^{4} e^{4} + {\left (e x + d\right )}^{\frac {3}{2}} A b^{3} c e^{4} + \sqrt {e x + d} B b^{4} d e^{4} - \sqrt {e x + d} A b^{3} c d e^{4}}{{\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} + {\left (e x + d\right )} b e - b d e\right )} b^{2} c^{3}} \]
(2*B*b*d^4 - 4*A*c*d^4 + 7*A*b*d^3*e)*arctan(sqrt(e*x + d)/sqrt(-d))/(b^3* sqrt(-d)) - (2*B*b*c^4*d^4 - 4*A*c^5*d^4 - B*b^2*c^3*d^3*e + 9*A*b*c^4*d^3 *e - 9*B*b^3*c^2*d^2*e^2 - 3*A*b^2*c^3*d^2*e^2 + 13*B*b^4*c*d*e^3 - 5*A*b^ 3*c^2*d*e^3 - 5*B*b^5*e^4 + 3*A*b^4*c*e^4)*arctan(sqrt(e*x + d)*c/sqrt(-c^ 2*d + b*c*e))/(sqrt(-c^2*d + b*c*e)*b^3*c^3) + 2/3*((e*x + d)^(3/2)*B*c^4* e^2 + 9*sqrt(e*x + d)*B*c^4*d*e^2 - 6*sqrt(e*x + d)*B*b*c^3*e^3 + 3*sqrt(e *x + d)*A*c^4*e^3)/c^6 + ((e*x + d)^(3/2)*B*b*c^3*d^3*e - 2*(e*x + d)^(3/2 )*A*c^4*d^3*e - sqrt(e*x + d)*B*b*c^3*d^4*e + 2*sqrt(e*x + d)*A*c^4*d^4*e - 3*(e*x + d)^(3/2)*B*b^2*c^2*d^2*e^2 + 3*(e*x + d)^(3/2)*A*b*c^3*d^2*e^2 + 3*sqrt(e*x + d)*B*b^2*c^2*d^3*e^2 - 4*sqrt(e*x + d)*A*b*c^3*d^3*e^2 + 3* (e*x + d)^(3/2)*B*b^3*c*d*e^3 - 3*(e*x + d)^(3/2)*A*b^2*c^2*d*e^3 - 3*sqrt (e*x + d)*B*b^3*c*d^2*e^3 + 3*sqrt(e*x + d)*A*b^2*c^2*d^2*e^3 - (e*x + d)^ (3/2)*B*b^4*e^4 + (e*x + d)^(3/2)*A*b^3*c*e^4 + sqrt(e*x + d)*B*b^4*d*e^4 - sqrt(e*x + d)*A*b^3*c*d*e^4)/(((e*x + d)^2*c - 2*(e*x + d)*c*d + c*d^2 + (e*x + d)*b*e - b*d*e)*b^2*c^3)
Time = 12.44 (sec) , antiderivative size = 7328, normalized size of antiderivative = 25.10 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
(((d + e*x)^(3/2)*(B*b^4*e^4 - A*b^3*c*e^4 + 2*A*c^4*d^3*e - 3*A*b*c^3*d^2 *e^2 + 3*A*b^2*c^2*d*e^3 + 3*B*b^2*c^2*d^2*e^2 - B*b*c^3*d^3*e - 3*B*b^3*c *d*e^3))/b^2 - ((d + e*x)^(1/2)*(2*A*c^4*d^4*e + B*b^4*d*e^4 - 4*A*b*c^3*d ^3*e^2 - 3*B*b^3*c*d^2*e^3 + 3*A*b^2*c^2*d^2*e^3 + 3*B*b^2*c^2*d^3*e^2 - A *b^3*c*d*e^4 - B*b*c^3*d^4*e))/b^2)/((2*c^4*d - b*c^3*e)*(d + e*x) - c^4*( d + e*x)^2 - c^4*d^2 + b*c^3*d*e) + ((2*A*e^3 - 2*B*d*e^2)/c^2 + (2*B*e^2* (4*c^2*d - 2*b*c*e))/c^4)*(d + e*x)^(1/2) + (atan(((((((12*A*b^9*c^5*d*e^6 - 20*B*b^10*c^4*d*e^6 - 8*A*b^6*c^8*d^4*e^3 + 16*A*b^7*c^7*d^3*e^4 - 20*A *b^8*c^6*d^2*e^5 + 4*B*b^7*c^7*d^4*e^3 - 36*B*b^8*c^6*d^3*e^4 + 52*B*b^9*c ^5*d^2*e^5)/(b^6*c^5) + ((4*b^7*c^7*e^3 - 8*b^6*c^8*d*e^2)*(d^5)^(1/2)*(d + e*x)^(1/2)*(7*A*b*e - 4*A*c*d + 2*B*b*d))/(b^7*c^5))*(d^5)^(1/2)*(7*A*b* e - 4*A*c*d + 2*B*b*d))/(2*b^3) + (2*(d + e*x)^(1/2)*(25*B^2*b^10*e^10 + 9 *A^2*b^8*c^2*e^10 + 32*A^2*c^10*d^8*e^2 + 154*A^2*b^2*c^8*d^6*e^4 - 14*A^2 *b^3*c^7*d^5*e^5 - 105*A^2*b^4*c^6*d^4*e^6 + 84*A^2*b^5*c^5*d^3*e^7 + 7*A^ 2*b^6*c^4*d^2*e^8 + 8*B^2*b^2*c^8*d^8*e^2 - 4*B^2*b^3*c^7*d^7*e^3 - 35*B^2 *b^4*c^6*d^6*e^4 + 70*B^2*b^5*c^5*d^5*e^5 + 35*B^2*b^6*c^4*d^4*e^6 - 224*B ^2*b^7*c^3*d^3*e^7 + 259*B^2*b^8*c^2*d^2*e^8 - 130*B^2*b^9*c*d*e^9 - 128*A ^2*b*c^9*d^7*e^3 - 30*A^2*b^7*c^3*d*e^9 - 30*A*B*b^9*c*e^10 - 32*A*B*b*c^9 *d^8*e^2 + 128*A*B*b^8*c^2*d*e^9 + 72*A*B*b^2*c^8*d^7*e^3 + 42*A*B*b^3*c^7 *d^6*e^4 - 280*A*B*b^4*c^6*d^5*e^5 + 350*A*B*b^5*c^5*d^4*e^6 - 84*A*B*b...