\(\int \frac {(d+e x)^{11/2}}{(b x+c x^2)^3} \, dx\) [121]

Optimal result

Integrand size = 21, antiderivative size = 378 \[ \int \frac {(d+e x)^{11/2}}{\left (b x+c x^2\right )^3} \, dx=-\frac {3 e \left (c^2 d^2-b c d e-b^2 e^2\right ) \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right ) \sqrt {d+e x}}{4 b^4 c^3}+\frac {(c d-b e) (2 c d-b e) \left (12 c^2 d^2-12 b c d e-5 b^2 e^2\right ) (d+e x)^{3/2}}{4 b^4 c^2 (b+c x)}+\frac {(c d-b e) \left (12 c^2 d^2-19 b c d e+2 b^2 e^2\right ) (d+e x)^{5/2}}{4 b^3 c (b+c x)^2}+\frac {d (8 c d-13 b e) (d+e x)^{7/2}}{4 b^2 x (b+c x)^2}-\frac {d (d+e x)^{9/2}}{2 b x^2 (b+c x)^2}-\frac {3 d^{7/2} \left (16 c^2 d^2-44 b c d e+33 b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5}+\frac {3 (c d-b e)^{7/2} \left (16 c^2 d^2+12 b c d e+5 b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 c^{7/2}} \] Output:

-3/4*e*(-b^2*e^2-b*c*d*e+c^2*d^2)*(5*b^2*e^2-8*b*c*d*e+8*c^2*d^2)*(e*x+d)^ 
(1/2)/b^4/c^3+1/4*(-b*e+c*d)*(-b*e+2*c*d)*(-5*b^2*e^2-12*b*c*d*e+12*c^2*d^ 
2)*(e*x+d)^(3/2)/b^4/c^2/(c*x+b)+1/4*(-b*e+c*d)*(2*b^2*e^2-19*b*c*d*e+12*c 
^2*d^2)*(e*x+d)^(5/2)/b^3/c/(c*x+b)^2+1/4*d*(-13*b*e+8*c*d)*(e*x+d)^(7/2)/ 
b^2/x/(c*x+b)^2-1/2*d*(e*x+d)^(9/2)/b/x^2/(c*x+b)^2-3/4*d^(7/2)*(33*b^2*e^ 
2-44*b*c*d*e+16*c^2*d^2)*arctanh((e*x+d)^(1/2)/d^(1/2))/b^5+3/4*(-b*e+c*d) 
^(7/2)*(5*b^2*e^2+12*b*c*d*e+16*c^2*d^2)*arctanh(c^(1/2)*(e*x+d)^(1/2)/(-b 
*e+c*d)^(1/2))/b^5/c^(7/2)
 

Mathematica [A] (verified)

Time = 1.16 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.83 \[ \int \frac {(d+e x)^{11/2}}{\left (b x+c x^2\right )^3} \, dx=\frac {\frac {b \sqrt {d+e x} \left (15 b^6 e^5 x^2+24 c^6 d^5 x^3+12 b c^5 d^4 x^2 (3 d-5 e x)+b^5 c e^4 x^2 (-14 d+25 e x)+2 b^4 c^2 e^3 x^2 \left (-7 d^2-12 d e x+4 e^2 x^2\right )+b^2 c^4 d^3 x \left (8 d^2-91 d e x+36 e^2 x^2\right )+b^3 c^3 d^2 \left (-2 d^3-21 d^2 e x+56 d e^2 x^2+6 e^3 x^3\right )\right )}{c^3 x^2 (b+c x)^2}-\frac {3 (-c d+b e)^{7/2} \left (16 c^2 d^2+12 b c d e+5 b^2 e^2\right ) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{c^{7/2}}-3 d^{7/2} \left (16 c^2 d^2-44 b c d e+33 b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5} \] Input:

Integrate[(d + e*x)^(11/2)/(b*x + c*x^2)^3,x]
 

Output:

((b*Sqrt[d + e*x]*(15*b^6*e^5*x^2 + 24*c^6*d^5*x^3 + 12*b*c^5*d^4*x^2*(3*d 
 - 5*e*x) + b^5*c*e^4*x^2*(-14*d + 25*e*x) + 2*b^4*c^2*e^3*x^2*(-7*d^2 - 1 
2*d*e*x + 4*e^2*x^2) + b^2*c^4*d^3*x*(8*d^2 - 91*d*e*x + 36*e^2*x^2) + b^3 
*c^3*d^2*(-2*d^3 - 21*d^2*e*x + 56*d*e^2*x^2 + 6*e^3*x^3)))/(c^3*x^2*(b + 
c*x)^2) - (3*(-(c*d) + b*e)^(7/2)*(16*c^2*d^2 + 12*b*c*d*e + 5*b^2*e^2)*Ar 
cTan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/c^(7/2) - 3*d^(7/2)*(16* 
c^2*d^2 - 44*b*c*d*e + 33*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5)
 

Rubi [A] (verified)

Time = 1.50 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {1164, 27, 1233, 27, 1196, 1196, 1197, 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.

\(\displaystyle \int \frac {(d+e x)^{11/2}}{\left (b x+c x^2\right )^3} \, dx\)

\(\Big \downarrow \) 1164

\(\displaystyle -\frac {\int \frac {3 (d+e x)^{7/2} (d (4 c d-5 b e)-e (2 c d-b e) x)}{2 \left (c x^2+b x\right )^2}dx}{2 b^2}-\frac {(d+e x)^{9/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3 \int \frac {(d+e x)^{7/2} (d (4 c d-5 b e)-e (2 c d-b e) x)}{\left (c x^2+b x\right )^2}dx}{4 b^2}-\frac {(d+e x)^{9/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1233

\(\displaystyle -\frac {3 \left (\frac {\int -\frac {(d+e x)^{3/2} \left (c d^2 \left (16 c^2 d^2-44 b c e d+33 b^2 e^2\right )-e (2 c d-b e) \left (12 c^2 d^2-12 b c e d-5 b^2 e^2\right ) x\right )}{2 \left (c x^2+b x\right )}dx}{b^2 c}-\frac {(d+e x)^{5/2} \left (x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b c d^2 (4 c d-5 b e)\right )}{b^2 c \left (b x+c x^2\right )}\right )}{4 b^2}-\frac {(d+e x)^{9/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3 \left (-\frac {\int \frac {(d+e x)^{3/2} \left (c d^2 \left (16 c^2 d^2-44 b c e d+33 b^2 e^2\right )-e (2 c d-b e) \left (12 c^2 d^2-12 b c e d-5 b^2 e^2\right ) x\right )}{c x^2+b x}dx}{2 b^2 c}-\frac {(d+e x)^{5/2} \left (x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b c d^2 (4 c d-5 b e)\right )}{b^2 c \left (b x+c x^2\right )}\right )}{4 b^2}-\frac {(d+e x)^{9/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1196

\(\displaystyle -\frac {3 \left (-\frac {\frac {\int \frac {\sqrt {d+e x} \left (c^2 d^3 \left (16 c^2 d^2-44 b c e d+33 b^2 e^2\right )-e \left (c^2 d^2-b c e d-b^2 e^2\right ) \left (8 c^2 d^2-8 b c e d+5 b^2 e^2\right ) x\right )}{c x^2+b x}dx}{c}-\frac {2 e (d+e x)^{3/2} (2 c d-b e) \left (-5 b^2 e^2-12 b c d e+12 c^2 d^2\right )}{3 c}}{2 b^2 c}-\frac {(d+e x)^{5/2} \left (x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b c d^2 (4 c d-5 b e)\right )}{b^2 c \left (b x+c x^2\right )}\right )}{4 b^2}-\frac {(d+e x)^{9/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1196

\(\displaystyle -\frac {3 \left (-\frac {\frac {\frac {\int \frac {c^3 \left (16 c^2 d^2-44 b c e d+33 b^2 e^2\right ) d^4+e (2 c d-b e) \left (4 c^4 d^4-8 b c^3 e d^3+2 b^2 c^2 e^2 d^2+2 b^3 c e^3 d+5 b^4 e^4\right ) x}{\sqrt {d+e x} \left (c x^2+b x\right )}dx}{c}-\frac {2 e \sqrt {d+e x} \left (-b^2 e^2-b c d e+c^2 d^2\right ) \left (5 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{c}}{c}-\frac {2 e (d+e x)^{3/2} (2 c d-b e) \left (-5 b^2 e^2-12 b c d e+12 c^2 d^2\right )}{3 c}}{2 b^2 c}-\frac {(d+e x)^{5/2} \left (x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b c d^2 (4 c d-5 b e)\right )}{b^2 c \left (b x+c x^2\right )}\right )}{4 b^2}-\frac {(d+e x)^{9/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1197

\(\displaystyle -\frac {3 \left (-\frac {\frac {\frac {2 \int \frac {e \left (d (c d-b e) \left (c^2 d^2-b c e d-b^2 e^2\right ) \left (8 c^2 d^2-8 b c e d+5 b^2 e^2\right )+(2 c d-b e) \left (4 c^4 d^4-8 b c^3 e d^3+2 b^2 c^2 e^2 d^2+2 b^3 c e^3 d+5 b^4 e^4\right ) (d+e x)\right )}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{c}-\frac {2 e \sqrt {d+e x} \left (-b^2 e^2-b c d e+c^2 d^2\right ) \left (5 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{c}}{c}-\frac {2 e (d+e x)^{3/2} (2 c d-b e) \left (-5 b^2 e^2-12 b c d e+12 c^2 d^2\right )}{3 c}}{2 b^2 c}-\frac {(d+e x)^{5/2} \left (x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b c d^2 (4 c d-5 b e)\right )}{b^2 c \left (b x+c x^2\right )}\right )}{4 b^2}-\frac {(d+e x)^{9/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3 \left (-\frac {\frac {\frac {2 e \int \frac {d (c d-b e) \left (c^2 d^2-b c e d-b^2 e^2\right ) \left (8 c^2 d^2-8 b c e d+5 b^2 e^2\right )+(2 c d-b e) \left (4 c^4 d^4-8 b c^3 e d^3+2 b^2 c^2 e^2 d^2+2 b^3 c e^3 d+5 b^4 e^4\right ) (d+e x)}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{c}-\frac {2 e \sqrt {d+e x} \left (-b^2 e^2-b c d e+c^2 d^2\right ) \left (5 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{c}}{c}-\frac {2 e (d+e x)^{3/2} (2 c d-b e) \left (-5 b^2 e^2-12 b c d e+12 c^2 d^2\right )}{3 c}}{2 b^2 c}-\frac {(d+e x)^{5/2} \left (x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b c d^2 (4 c d-5 b e)\right )}{b^2 c \left (b x+c x^2\right )}\right )}{4 b^2}-\frac {(d+e x)^{9/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1480

\(\displaystyle -\frac {3 \left (-\frac {\frac {\frac {2 e \left (\frac {c^4 d^4 \left (33 b^2 e^2-44 b c d e+16 c^2 d^2\right ) \int \frac {1}{c (d+e x)-c d}d\sqrt {d+e x}}{b e}-\frac {(c d-b e)^4 \left (5 b^2 e^2+12 b c d e+16 c^2 d^2\right ) \int \frac {1}{-c d+b e+c (d+e x)}d\sqrt {d+e x}}{b e}\right )}{c}-\frac {2 e \sqrt {d+e x} \left (-b^2 e^2-b c d e+c^2 d^2\right ) \left (5 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{c}}{c}-\frac {2 e (d+e x)^{3/2} (2 c d-b e) \left (-5 b^2 e^2-12 b c d e+12 c^2 d^2\right )}{3 c}}{2 b^2 c}-\frac {(d+e x)^{5/2} \left (x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b c d^2 (4 c d-5 b e)\right )}{b^2 c \left (b x+c x^2\right )}\right )}{4 b^2}-\frac {(d+e x)^{9/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {3 \left (-\frac {\frac {\frac {2 e \left (\frac {(c d-b e)^{7/2} \left (5 b^2 e^2+12 b c d e+16 c^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b \sqrt {c} e}-\frac {c^3 d^{7/2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (33 b^2 e^2-44 b c d e+16 c^2 d^2\right )}{b e}\right )}{c}-\frac {2 e \sqrt {d+e x} \left (-b^2 e^2-b c d e+c^2 d^2\right ) \left (5 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{c}}{c}-\frac {2 e (d+e x)^{3/2} (2 c d-b e) \left (-5 b^2 e^2-12 b c d e+12 c^2 d^2\right )}{3 c}}{2 b^2 c}-\frac {(d+e x)^{5/2} \left (x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b c d^2 (4 c d-5 b e)\right )}{b^2 c \left (b x+c x^2\right )}\right )}{4 b^2}-\frac {(d+e x)^{9/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}\)

Input:

Int[(d + e*x)^(11/2)/(b*x + c*x^2)^3,x]
 

Output:

-1/2*((d + e*x)^(9/2)*(b*d + (2*c*d - b*e)*x))/(b^2*(b*x + c*x^2)^2) - (3* 
(-(((d + e*x)^(5/2)*(b*c*d^2*(4*c*d - 5*b*e) + (2*c*d - b*e)*(4*c^2*d^2 - 
4*b*c*d*e - b^2*e^2)*x))/(b^2*c*(b*x + c*x^2))) - ((-2*e*(2*c*d - b*e)*(12 
*c^2*d^2 - 12*b*c*d*e - 5*b^2*e^2)*(d + e*x)^(3/2))/(3*c) + ((-2*e*(c^2*d^ 
2 - b*c*d*e - b^2*e^2)*(8*c^2*d^2 - 8*b*c*d*e + 5*b^2*e^2)*Sqrt[d + e*x])/ 
c + (2*e*(-((c^3*d^(7/2)*(16*c^2*d^2 - 44*b*c*d*e + 33*b^2*e^2)*ArcTanh[Sq 
rt[d + e*x]/Sqrt[d]])/(b*e)) + ((c*d - b*e)^(7/2)*(16*c^2*d^2 + 12*b*c*d*e 
 + 5*b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*Sqrt[c] 
*e)))/c)/c)/(2*b^2*c)))/(4*b^2)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 221

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]
 

rule 1164

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x 
+ c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a* 
c))   Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2* 
c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p 
+ 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1] && Int 
QuadraticQ[a, b, c, d, e, m, p, x]
 

rule 1196

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]
 

rule 1197

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]
 

rule 1233

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])
 

rule 1480

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]
 

Maple [A] (verified)

Time = 0.81 (sec) , antiderivative size = 336, normalized size of antiderivative = 0.89

method	result	size
pseudoelliptic	\(-\frac {\frac {3 x^{2} \left (c x +b \right )^{2} \sqrt {d}\, \left (5 b^{2} e^{2}+12 b c d e +16 c^{2} d^{2}\right ) \left (b e -c d \right )^{4} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{2}+\left (\frac {3 \left (33 b^{2} e^{2}-44 b c d e +16 c^{2} d^{2}\right ) x^{2} \left (c x +b \right )^{2} c^{3} d^{4} \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2}+\left (-12 c^{6} d^{5} x^{3}-18 x^{2} \left (-\frac {5 e x}{3}+d \right ) b \,d^{4} c^{5}-4 x \,b^{2} d^{3} \left (\frac {9}{2} e^{2} x^{2}-\frac {91}{8} d e x +d^{2}\right ) c^{4}+d^{2} b^{3} \left (-28 d \,e^{2} x^{2}+\frac {21}{2} d^{2} e x +d^{3}-3 e^{3} x^{3}\right ) c^{3}+7 e^{3} x^{2} \left (-\frac {2 e x}{7}+d \right ) \left (2 e x +d \right ) b^{4} c^{2}+7 e^{4} \left (-\frac {25 e x}{14}+d \right ) x^{2} b^{5} c -\frac {15 b^{6} x^{2} e^{5}}{2}\right ) \sqrt {d}\, \sqrt {e x +d}\, b \right ) \sqrt {c \left (b e -c d \right )}}{2 \sqrt {d}\, \sqrt {c \left (b e -c d \right )}\, c^{3} b^{5} x^{2} \left (c x +b \right )^{2}}\)	\(336\)
risch	\(-\frac {d^{4} \sqrt {e x +d}\, \left (21 b e x -12 c d x +2 b d \right )}{4 b^{4} x^{2}}+\frac {e \left (\frac {8 b^{4} e^{4} \sqrt {e x +d}}{c^{3}}-\frac {3 d^{\frac {7}{2}} \left (33 b^{2} e^{2}-44 b c d e +16 c^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b e}-\frac {8 \left (\frac {\left (-\frac {9}{8} b^{6} c \,e^{6}+3 b^{5} d \,e^{5} c^{2}-\frac {3}{4} b^{4} c^{3} d^{2} e^{4}-\frac {9}{2} b^{3} c^{4} d^{3} e^{3}+\frac {39}{8} b^{2} d^{4} e^{2} c^{5}-\frac {3}{2} b \,d^{5} e \,c^{6}\right ) \left (e x +d \right )^{\frac {3}{2}}-\frac {b e \left (7 b^{6} e^{6}-23 b^{5} d \,e^{5} c +10 b^{4} c^{2} d^{2} e^{4}+50 b^{3} d^{3} e^{3} c^{3}-85 b^{2} c^{4} d^{4} e^{2}+53 b \,c^{5} d^{5} e -12 d^{6} c^{6}\right ) \sqrt {e x +d}}{8}}{\left (\left (e x +d \right ) c +b e -c d \right )^{2}}+\frac {3 \left (5 b^{6} e^{6}-8 b^{5} d \,e^{5} c -2 b^{4} c^{2} d^{2} e^{4}-12 b^{3} d^{3} e^{3} c^{3}+53 b^{2} c^{4} d^{4} e^{2}-52 b \,c^{5} d^{5} e +16 d^{6} c^{6}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{8 \sqrt {c \left (b e -c d \right )}}\right )}{c^{3} b e}\right )}{4 b^{4}}\)	\(424\)
derivativedivides	\(2 e^{5} \left (\frac {\sqrt {e x +d}}{c^{3}}-\frac {\frac {\left (-\frac {9}{8} b^{6} c \,e^{6}+3 b^{5} d \,e^{5} c^{2}-\frac {3}{4} b^{4} c^{3} d^{2} e^{4}-\frac {9}{2} b^{3} c^{4} d^{3} e^{3}+\frac {39}{8} b^{2} d^{4} e^{2} c^{5}-\frac {3}{2} b \,d^{5} e \,c^{6}\right ) \left (e x +d \right )^{\frac {3}{2}}-\frac {b e \left (7 b^{6} e^{6}-23 b^{5} d \,e^{5} c +10 b^{4} c^{2} d^{2} e^{4}+50 b^{3} d^{3} e^{3} c^{3}-85 b^{2} c^{4} d^{4} e^{2}+53 b \,c^{5} d^{5} e -12 d^{6} c^{6}\right ) \sqrt {e x +d}}{8}}{\left (\left (e x +d \right ) c +b e -c d \right )^{2}}+\frac {3 \left (5 b^{6} e^{6}-8 b^{5} d \,e^{5} c -2 b^{4} c^{2} d^{2} e^{4}-12 b^{3} d^{3} e^{3} c^{3}+53 b^{2} c^{4} d^{4} e^{2}-52 b \,c^{5} d^{5} e +16 d^{6} c^{6}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{8 \sqrt {c \left (b e -c d \right )}}}{c^{3} b^{5} e^{5}}-\frac {d^{4} \left (\frac {\left (\frac {21}{8} b^{2} e^{2}-\frac {3}{2} b c d e \right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {19}{8} d \,e^{2} b^{2}+\frac {3}{2} d^{2} e b c \right ) \sqrt {e x +d}}{e^{2} x^{2}}+\frac {3 \left (33 b^{2} e^{2}-44 b c d e +16 c^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 \sqrt {d}}\right )}{b^{5} e^{5}}\right )\)	\(445\)
default	\(2 e^{5} \left (\frac {\sqrt {e x +d}}{c^{3}}-\frac {\frac {\left (-\frac {9}{8} b^{6} c \,e^{6}+3 b^{5} d \,e^{5} c^{2}-\frac {3}{4} b^{4} c^{3} d^{2} e^{4}-\frac {9}{2} b^{3} c^{4} d^{3} e^{3}+\frac {39}{8} b^{2} d^{4} e^{2} c^{5}-\frac {3}{2} b \,d^{5} e \,c^{6}\right ) \left (e x +d \right )^{\frac {3}{2}}-\frac {b e \left (7 b^{6} e^{6}-23 b^{5} d \,e^{5} c +10 b^{4} c^{2} d^{2} e^{4}+50 b^{3} d^{3} e^{3} c^{3}-85 b^{2} c^{4} d^{4} e^{2}+53 b \,c^{5} d^{5} e -12 d^{6} c^{6}\right ) \sqrt {e x +d}}{8}}{\left (\left (e x +d \right ) c +b e -c d \right )^{2}}+\frac {3 \left (5 b^{6} e^{6}-8 b^{5} d \,e^{5} c -2 b^{4} c^{2} d^{2} e^{4}-12 b^{3} d^{3} e^{3} c^{3}+53 b^{2} c^{4} d^{4} e^{2}-52 b \,c^{5} d^{5} e +16 d^{6} c^{6}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{8 \sqrt {c \left (b e -c d \right )}}}{c^{3} b^{5} e^{5}}-\frac {d^{4} \left (\frac {\left (\frac {21}{8} b^{2} e^{2}-\frac {3}{2} b c d e \right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {19}{8} d \,e^{2} b^{2}+\frac {3}{2} d^{2} e b c \right ) \sqrt {e x +d}}{e^{2} x^{2}}+\frac {3 \left (33 b^{2} e^{2}-44 b c d e +16 c^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 \sqrt {d}}\right )}{b^{5} e^{5}}\right )\)	\(445\)

Input:

int((e*x+d)^(11/2)/(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)
 

Output:

-1/2/d^(1/2)/(c*(b*e-c*d))^(1/2)*(3/2*x^2*(c*x+b)^2*d^(1/2)*(5*b^2*e^2+12* 
b*c*d*e+16*c^2*d^2)*(b*e-c*d)^4*arctan(c*(e*x+d)^(1/2)/(c*(b*e-c*d))^(1/2) 
)+(3/2*(33*b^2*e^2-44*b*c*d*e+16*c^2*d^2)*x^2*(c*x+b)^2*c^3*d^4*arctanh((e 
*x+d)^(1/2)/d^(1/2))+(-12*c^6*d^5*x^3-18*x^2*(-5/3*e*x+d)*b*d^4*c^5-4*x*b^ 
2*d^3*(9/2*e^2*x^2-91/8*d*e*x+d^2)*c^4+d^2*b^3*(-28*d*e^2*x^2+21/2*d^2*e*x 
+d^3-3*e^3*x^3)*c^3+7*e^3*x^2*(-2/7*e*x+d)*(2*e*x+d)*b^4*c^2+7*e^4*(-25/14 
*e*x+d)*x^2*b^5*c-15/2*b^6*x^2*e^5)*d^(1/2)*(e*x+d)^(1/2)*b)*(c*(b*e-c*d)) 
^(1/2))/c^3/b^5/x^2/(c*x+b)^2
 

Fricas [A] (verification not implemented)

Time = 2.90 (sec) , antiderivative size = 2702, normalized size of antiderivative = 7.15 \[ \int \frac {(d+e x)^{11/2}}{\left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^(11/2)/(c*x^2+b*x)^3,x, algorithm="fricas")
 

Output:

[-1/8*(3*((16*c^7*d^5 - 36*b*c^6*d^4*e + 17*b^2*c^5*d^3*e^2 + 5*b^3*c^4*d^ 
2*e^3 + 3*b^4*c^3*d*e^4 - 5*b^5*c^2*e^5)*x^4 + 2*(16*b*c^6*d^5 - 36*b^2*c^ 
5*d^4*e + 17*b^3*c^4*d^3*e^2 + 5*b^4*c^3*d^2*e^3 + 3*b^5*c^2*d*e^4 - 5*b^6 
*c*e^5)*x^3 + (16*b^2*c^5*d^5 - 36*b^3*c^4*d^4*e + 17*b^4*c^3*d^3*e^2 + 5* 
b^5*c^2*d^2*e^3 + 3*b^6*c*d*e^4 - 5*b^7*e^5)*x^2)*sqrt((c*d - b*e)/c)*log( 
(c*e*x + 2*c*d - b*e - 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) - 
 3*((16*c^7*d^5 - 44*b*c^6*d^4*e + 33*b^2*c^5*d^3*e^2)*x^4 + 2*(16*b*c^6*d 
^5 - 44*b^2*c^5*d^4*e + 33*b^3*c^4*d^3*e^2)*x^3 + (16*b^2*c^5*d^5 - 44*b^3 
*c^4*d^4*e + 33*b^4*c^3*d^3*e^2)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*s 
qrt(d) + 2*d)/x) - 2*(8*b^5*c^2*e^5*x^4 - 2*b^4*c^3*d^5 + (24*b*c^6*d^5 - 
60*b^2*c^5*d^4*e + 36*b^3*c^4*d^3*e^2 + 6*b^4*c^3*d^2*e^3 - 24*b^5*c^2*d*e 
^4 + 25*b^6*c*e^5)*x^3 + (36*b^2*c^5*d^5 - 91*b^3*c^4*d^4*e + 56*b^4*c^3*d 
^3*e^2 - 14*b^5*c^2*d^2*e^3 - 14*b^6*c*d*e^4 + 15*b^7*e^5)*x^2 + (8*b^3*c^ 
4*d^5 - 21*b^4*c^3*d^4*e)*x)*sqrt(e*x + d))/(b^5*c^5*x^4 + 2*b^6*c^4*x^3 + 
 b^7*c^3*x^2), 1/8*(6*((16*c^7*d^5 - 36*b*c^6*d^4*e + 17*b^2*c^5*d^3*e^2 + 
 5*b^3*c^4*d^2*e^3 + 3*b^4*c^3*d*e^4 - 5*b^5*c^2*e^5)*x^4 + 2*(16*b*c^6*d^ 
5 - 36*b^2*c^5*d^4*e + 17*b^3*c^4*d^3*e^2 + 5*b^4*c^3*d^2*e^3 + 3*b^5*c^2* 
d*e^4 - 5*b^6*c*e^5)*x^3 + (16*b^2*c^5*d^5 - 36*b^3*c^4*d^4*e + 17*b^4*c^3 
*d^3*e^2 + 5*b^5*c^2*d^2*e^3 + 3*b^6*c*d*e^4 - 5*b^7*e^5)*x^2)*sqrt(-(c*d 
- b*e)/c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + 3...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{11/2}}{\left (b x+c x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate((e*x+d)**(11/2)/(c*x**2+b*x)**3,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^{11/2}}{\left (b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x+d)^(11/2)/(c*x^2+b*x)^3,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
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 749 vs. \(2 (338) = 676\).

Time = 0.19 (sec) , antiderivative size = 749, normalized size of antiderivative = 1.98 \[ \int \frac {(d+e x)^{11/2}}{\left (b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)^(11/2)/(c*x^2+b*x)^3,x, algorithm="giac")
 

Output:

2*sqrt(e*x + d)*e^5/c^3 + 3/4*(16*c^2*d^6 - 44*b*c*d^5*e + 33*b^2*d^4*e^2) 
*arctan(sqrt(e*x + d)/sqrt(-d))/(b^5*sqrt(-d)) - 3/4*(16*c^6*d^6 - 52*b*c^ 
5*d^5*e + 53*b^2*c^4*d^4*e^2 - 12*b^3*c^3*d^3*e^3 - 2*b^4*c^2*d^2*e^4 - 8* 
b^5*c*d*e^5 + 5*b^6*e^6)*arctan(sqrt(e*x + d)*c/sqrt(-c^2*d + b*c*e))/(sqr 
t(-c^2*d + b*c*e)*b^5*c^3) + 1/4*(24*(e*x + d)^(7/2)*c^6*d^5*e - 72*(e*x + 
 d)^(5/2)*c^6*d^6*e + 72*(e*x + d)^(3/2)*c^6*d^7*e - 24*sqrt(e*x + d)*c^6* 
d^8*e - 60*(e*x + d)^(7/2)*b*c^5*d^4*e^2 + 216*(e*x + d)^(5/2)*b*c^5*d^5*e 
^2 - 252*(e*x + d)^(3/2)*b*c^5*d^6*e^2 + 96*sqrt(e*x + d)*b*c^5*d^7*e^2 + 
36*(e*x + d)^(7/2)*b^2*c^4*d^3*e^3 - 199*(e*x + d)^(5/2)*b^2*c^4*d^4*e^3 + 
 298*(e*x + d)^(3/2)*b^2*c^4*d^5*e^3 - 135*sqrt(e*x + d)*b^2*c^4*d^6*e^3 + 
 6*(e*x + d)^(7/2)*b^3*c^3*d^2*e^4 + 38*(e*x + d)^(5/2)*b^3*c^3*d^3*e^4 - 
115*(e*x + d)^(3/2)*b^3*c^3*d^4*e^4 + 69*sqrt(e*x + d)*b^3*c^3*d^5*e^4 - 2 
4*(e*x + d)^(7/2)*b^4*c^2*d*e^5 + 58*(e*x + d)^(5/2)*b^4*c^2*d^2*e^5 - 44* 
(e*x + d)^(3/2)*b^4*c^2*d^3*e^5 + 10*sqrt(e*x + d)*b^4*c^2*d^4*e^5 + 9*(e* 
x + d)^(7/2)*b^5*c*e^6 - 41*(e*x + d)^(5/2)*b^5*c*d*e^6 + 55*(e*x + d)^(3/ 
2)*b^5*c*d^2*e^6 - 23*sqrt(e*x + d)*b^5*c*d^3*e^6 + 7*(e*x + d)^(5/2)*b^6* 
e^7 - 14*(e*x + d)^(3/2)*b^6*d*e^7 + 7*sqrt(e*x + d)*b^6*d^2*e^7)/(((e*x + 
 d)^2*c - 2*(e*x + d)*c*d + c*d^2 + (e*x + d)*b*e - b*d*e)^2*b^4*c^3)
 

Mupad [B] (verification not implemented)

Time = 6.34 (sec) , antiderivative size = 4450, normalized size of antiderivative = 11.77 \[ \int \frac {(d+e x)^{11/2}}{\left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

int((d + e*x)^(11/2)/(b*x + c*x^2)^3,x)
 

Output:

(((d + e*x)^(5/2)*(7*b^6*e^7 - 72*c^6*d^6*e + 216*b*c^5*d^5*e^2 - 199*b^2* 
c^4*d^4*e^3 + 38*b^3*c^3*d^3*e^4 + 58*b^4*c^2*d^2*e^5 - 41*b^5*c*d*e^6))/( 
4*b^4) + ((d + e*x)^(1/2)*(7*b^6*d^2*e^7 - 24*c^6*d^8*e + 96*b*c^5*d^7*e^2 
 - 23*b^5*c*d^3*e^6 - 135*b^2*c^4*d^6*e^3 + 69*b^3*c^3*d^5*e^4 + 10*b^4*c^ 
2*d^4*e^5))/(4*b^4) + (3*(d + e*x)^(7/2)*(3*b^5*c*e^6 + 8*c^6*d^5*e - 20*b 
*c^5*d^4*e^2 - 8*b^4*c^2*d*e^5 + 12*b^2*c^4*d^3*e^3 + 2*b^3*c^3*d^2*e^4))/ 
(4*b^4) - ((d + e*x)^(3/2)*(14*b^6*d*e^7 - 72*c^6*d^7*e + 252*b*c^5*d^6*e^ 
2 - 55*b^5*c*d^2*e^6 - 298*b^2*c^4*d^5*e^3 + 115*b^3*c^3*d^4*e^4 + 44*b^4* 
c^2*d^3*e^5))/(4*b^4))/(c^5*(d + e*x)^4 - (d + e*x)*(4*c^5*d^3 + 2*b^2*c^3 
*d*e^2 - 6*b*c^4*d^2*e) - (4*c^5*d - 2*b*c^4*e)*(d + e*x)^3 + c^5*d^4 + (d 
 + e*x)^2*(6*c^5*d^2 + b^2*c^3*e^2 - 6*b*c^4*d*e) + b^2*c^3*d^2*e^2 - 2*b* 
c^4*d^3*e) + (2*e^5*(d + e*x)^(1/2))/c^3 + (atan((((((d + e*x)^(1/2)*(225* 
b^12*e^14 + 4608*c^12*d^12*e^2 - 27648*b*c^11*d^11*e^3 + 66528*b^2*c^10*d^ 
10*e^4 - 79200*b^3*c^9*d^9*e^5 + 45738*b^4*c^8*d^8*e^6 - 11880*b^5*c^7*d^7 
*e^7 + 8316*b^6*c^6*d^6*e^8 - 11880*b^7*c^5*d^5*e^9 + 6534*b^8*c^4*d^4*e^1 
0 - 792*b^9*c^3*d^3*e^11 + 396*b^10*c^2*d^2*e^12 - 720*b^11*c*d*e^13))/(8* 
b^8*c^5) - (3*((15*b^15*c^4*d*e^8 + 24*b^10*c^9*d^6*e^3 - 72*b^11*c^8*d^5* 
e^4 + 63*b^12*c^7*d^4*e^5 - 6*b^13*c^6*d^3*e^6 - 24*b^14*c^5*d^2*e^7)/(b^1 
2*c^5) - (3*(64*b^11*c^7*e^3 - 128*b^10*c^8*d*e^2)*(d^7)^(1/2)*(d + e*x)^( 
1/2)*(33*b^2*e^2 + 16*c^2*d^2 - 44*b*c*d*e))/(64*b^13*c^5))*(d^7)^(1/2)...
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 1744, normalized size of antiderivative = 4.61 \[ \int \frac {(d+e x)^{11/2}}{\left (b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int((e*x+d)^(11/2)/(c*x^2+b*x)^3,x)
 

Output:

( - 30*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - 
c*d)))*b**7*e**5*x**2 + 18*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/ 
(sqrt(c)*sqrt(b*e - c*d)))*b**6*c*d*e**4*x**2 - 60*sqrt(c)*sqrt(b*e - c*d) 
*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**6*c*e**5*x**3 + 30*s 
qrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b 
**5*c**2*d**2*e**3*x**2 + 36*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c 
)/(sqrt(c)*sqrt(b*e - c*d)))*b**5*c**2*d*e**4*x**3 - 30*sqrt(c)*sqrt(b*e - 
 c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**5*c**2*e**5*x** 
4 + 102*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - 
 c*d)))*b**4*c**3*d**3*e**2*x**2 + 60*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d 
 + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**4*c**3*d**2*e**3*x**3 + 18*sqrt(c 
)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**4*c 
**3*d*e**4*x**4 - 216*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt 
(c)*sqrt(b*e - c*d)))*b**3*c**4*d**4*e*x**2 + 204*sqrt(c)*sqrt(b*e - c*d)* 
atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**3*c**4*d**3*e**2*x**3 
 + 30*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c 
*d)))*b**3*c**4*d**2*e**3*x**4 + 96*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + 
 e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**2*c**5*d**5*x**2 - 432*sqrt(c)*sqrt 
(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**2*c**5*d* 
*4*e*x**3 + 102*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)...