\(\int \frac {(d+e x)^{7/2}}{(b x+c x^2)^3} \, dx\) [123]

Optimal result

Integrand size = 21, antiderivative size = 305 \[ \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^3} \, dx=\frac {(c d-b e) \left (12 c^2 d^2-15 b c d e+2 b^2 e^2\right ) \sqrt {d+e x}}{4 b^3 c (b+c x)^2}+\frac {(2 c d-b e) \left (12 c^2 d^2-12 b c d e-b^2 e^2\right ) \sqrt {d+e x}}{4 b^4 c (b+c x)}+\frac {d (8 c d-9 b e) (d+e x)^{3/2}}{4 b^2 x (b+c x)^2}-\frac {d (d+e x)^{5/2}}{2 b x^2 (b+c x)^2}-\frac {d^{3/2} \left (48 c^2 d^2-84 b c d e+35 b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5}+\frac {(c d-b e)^{3/2} \left (48 c^2 d^2-12 b c d e-b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 c^{3/2}} \] Output:

1/4*(-b*e+c*d)*(2*b^2*e^2-15*b*c*d*e+12*c^2*d^2)*(e*x+d)^(1/2)/b^3/c/(c*x+ 
b)^2+1/4*(-b*e+2*c*d)*(-b^2*e^2-12*b*c*d*e+12*c^2*d^2)*(e*x+d)^(1/2)/b^4/c 
/(c*x+b)+1/4*d*(-9*b*e+8*c*d)*(e*x+d)^(3/2)/b^2/x/(c*x+b)^2-1/2*d*(e*x+d)^ 
(5/2)/b/x^2/(c*x+b)^2-1/4*d^(3/2)*(35*b^2*e^2-84*b*c*d*e+48*c^2*d^2)*arcta 
nh((e*x+d)^(1/2)/d^(1/2))/b^5+1/4*(-b*e+c*d)^(3/2)*(-b^2*e^2-12*b*c*d*e+48 
*c^2*d^2)*arctanh(c^(1/2)*(e*x+d)^(1/2)/(-b*e+c*d)^(1/2))/b^5/c^(3/2)
 

Mathematica [A] (verified)

Time = 1.54 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.82 \[ \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^3} \, dx=-\frac {-\frac {b \sqrt {d+e x} \left (-b^4 e^3 x^2+24 c^4 d^3 x^3+36 b c^3 d^2 x^2 (d-e x)+b^2 c^2 d x \left (8 d^2-55 d e x+10 e^2 x^2\right )+b^3 c \left (-2 d^3-13 d^2 e x+16 d e^2 x^2+e^3 x^3\right )\right )}{c x^2 (b+c x)^2}+\frac {(-c d+b e)^{3/2} \left (48 c^2 d^2-12 b c d e-b^2 e^2\right ) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{c^{3/2}}+d^{3/2} \left (48 c^2 d^2-84 b c d e+35 b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5} \] Input:

Integrate[(d + e*x)^(7/2)/(b*x + c*x^2)^3,x]
 

Output:

-1/4*(-((b*Sqrt[d + e*x]*(-(b^4*e^3*x^2) + 24*c^4*d^3*x^3 + 36*b*c^3*d^2*x 
^2*(d - e*x) + b^2*c^2*d*x*(8*d^2 - 55*d*e*x + 10*e^2*x^2) + b^3*c*(-2*d^3 
 - 13*d^2*e*x + 16*d*e^2*x^2 + e^3*x^3)))/(c*x^2*(b + c*x)^2)) + ((-(c*d) 
+ b*e)^(3/2)*(48*c^2*d^2 - 12*b*c*d*e - b^2*e^2)*ArcTan[(Sqrt[c]*Sqrt[d + 
e*x])/Sqrt[-(c*d) + b*e]])/c^(3/2) + d^(3/2)*(48*c^2*d^2 - 84*b*c*d*e + 35 
*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b^5
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.87, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {1164, 27, 1233, 27, 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.

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

\(\Big \downarrow \) 1164

\(\displaystyle -\frac {\int \frac {(d+e x)^{3/2} (d (12 c d-11 b e)+e (2 c d-b e) x)}{2 \left (c x^2+b x\right )^2}dx}{2 b^2}-\frac {(d+e x)^{5/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {(d+e x)^{3/2} (d (12 c d-11 b e)+e (2 c d-b e) x)}{\left (c x^2+b x\right )^2}dx}{4 b^2}-\frac {(d+e x)^{5/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1233

\(\displaystyle -\frac {\frac {\int -\frac {c \left (48 c^2 d^2-84 b c e d+35 b^2 e^2\right ) d^2+e (2 c d-b e) \left (12 c^2 d^2-12 b c e d-b^2 e^2\right ) x}{2 \sqrt {d+e x} \left (c x^2+b x\right )}dx}{b^2 c}-\frac {\sqrt {d+e x} \left (x (2 c d-b e) \left (b^2 e^2-12 b c d e+12 c^2 d^2\right )+b c d^2 (12 c d-11 b e)\right )}{b^2 c \left (b x+c x^2\right )}}{4 b^2}-\frac {(d+e x)^{5/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {c \left (48 c^2 d^2-84 b c e d+35 b^2 e^2\right ) d^2+e (2 c d-b e) \left (12 c^2 d^2-12 b c e d-b^2 e^2\right ) x}{\sqrt {d+e x} \left (c x^2+b x\right )}dx}{2 b^2 c}-\frac {\sqrt {d+e x} \left (x (2 c d-b e) \left (b^2 e^2-12 b c d e+12 c^2 d^2\right )+b c d^2 (12 c d-11 b e)\right )}{b^2 c \left (b x+c x^2\right )}}{4 b^2}-\frac {(d+e x)^{5/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1197

\(\displaystyle -\frac {-\frac {\int \frac {e \left (d (c d-b e) \left (24 c^2 d^2-24 b c e d+b^2 e^2\right )+(2 c d-b e) \left (12 c^2 d^2-12 b c e d-b^2 e^2\right ) (d+e x)\right )}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{b^2 c}-\frac {\sqrt {d+e x} \left (x (2 c d-b e) \left (b^2 e^2-12 b c d e+12 c^2 d^2\right )+b c d^2 (12 c d-11 b e)\right )}{b^2 c \left (b x+c x^2\right )}}{4 b^2}-\frac {(d+e x)^{5/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {e \int \frac {d (c d-b e) \left (24 c^2 d^2-24 b c e d+b^2 e^2\right )+(2 c d-b e) \left (12 c^2 d^2-12 b c e d-b^2 e^2\right ) (d+e x)}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{b^2 c}-\frac {\sqrt {d+e x} \left (x (2 c d-b e) \left (b^2 e^2-12 b c d e+12 c^2 d^2\right )+b c d^2 (12 c d-11 b e)\right )}{b^2 c \left (b x+c x^2\right )}}{4 b^2}-\frac {(d+e x)^{5/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1480

\(\displaystyle -\frac {-\frac {e \left (\frac {c^2 d^2 \left (35 b^2 e^2-84 b c d e+48 c^2 d^2\right ) \int \frac {1}{c (d+e x)-c d}d\sqrt {d+e x}}{b e}-\frac {(c d-b e)^2 \left (-b^2 e^2-12 b c d e+48 c^2 d^2\right ) \int \frac {1}{-c d+b e+c (d+e x)}d\sqrt {d+e x}}{b e}\right )}{b^2 c}-\frac {\sqrt {d+e x} \left (x (2 c d-b e) \left (b^2 e^2-12 b c d e+12 c^2 d^2\right )+b c d^2 (12 c d-11 b e)\right )}{b^2 c \left (b x+c x^2\right )}}{4 b^2}-\frac {(d+e x)^{5/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {-\frac {e \left (\frac {(c d-b e)^{3/2} \left (-b^2 e^2-12 b c d e+48 c^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b \sqrt {c} e}-\frac {c d^{3/2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (35 b^2 e^2-84 b c d e+48 c^2 d^2\right )}{b e}\right )}{b^2 c}-\frac {\sqrt {d+e x} \left (x (2 c d-b e) \left (b^2 e^2-12 b c d e+12 c^2 d^2\right )+b c d^2 (12 c d-11 b e)\right )}{b^2 c \left (b x+c x^2\right )}}{4 b^2}-\frac {(d+e x)^{5/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}\)

Input:

Int[(d + e*x)^(7/2)/(b*x + c*x^2)^3,x]
 

Output:

-1/2*((d + e*x)^(5/2)*(b*d + (2*c*d - b*e)*x))/(b^2*(b*x + c*x^2)^2) - (-( 
(Sqrt[d + e*x]*(b*c*d^2*(12*c*d - 11*b*e) + (2*c*d - b*e)*(12*c^2*d^2 - 12 
*b*c*d*e + b^2*e^2)*x))/(b^2*c*(b*x + c*x^2))) - (e*(-((c*d^(3/2)*(48*c^2* 
d^2 - 84*b*c*d*e + 35*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*e)) + (( 
c*d - b*e)^(3/2)*(48*c^2*d^2 - 12*b*c*d*e - b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt 
[d + e*x])/Sqrt[c*d - b*e]])/(b*Sqrt[c]*e)))/(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 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 = 0.76 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.90

method	result	size
derivativedivides	\(2 e^{5} \left (-\frac {d^{2} \left (\frac {\left (\frac {13}{8} b^{2} e^{2}-\frac {3}{2} b c d e \right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {11}{8} d \,e^{2} b^{2}+\frac {3}{2} d^{2} e b c \right ) \sqrt {e x +d}}{e^{2} x^{2}}+\frac {\left (35 b^{2} e^{2}-84 b c d e +48 c^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 \sqrt {d}}\right )}{b^{5} e^{5}}+\frac {\left (b e -c d \right )^{2} \left (\frac {\left (\frac {1}{8} b^{2} e^{2}+\frac {3}{2} b c d e \right ) \left (e x +d \right )^{\frac {3}{2}}-\frac {b e \left (b^{2} e^{2}-13 b c d e +12 c^{2} d^{2}\right ) \sqrt {e x +d}}{8 c}}{\left (\left (e x +d \right ) c +b e -c d \right )^{2}}+\frac {\left (b^{2} e^{2}+12 b c d e -48 c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{8 c \sqrt {c \left (b e -c d \right )}}\right )}{b^{5} e^{5}}\right )\)	\(273\)
default	\(2 e^{5} \left (-\frac {d^{2} \left (\frac {\left (\frac {13}{8} b^{2} e^{2}-\frac {3}{2} b c d e \right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {11}{8} d \,e^{2} b^{2}+\frac {3}{2} d^{2} e b c \right ) \sqrt {e x +d}}{e^{2} x^{2}}+\frac {\left (35 b^{2} e^{2}-84 b c d e +48 c^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 \sqrt {d}}\right )}{b^{5} e^{5}}+\frac {\left (b e -c d \right )^{2} \left (\frac {\left (\frac {1}{8} b^{2} e^{2}+\frac {3}{2} b c d e \right ) \left (e x +d \right )^{\frac {3}{2}}-\frac {b e \left (b^{2} e^{2}-13 b c d e +12 c^{2} d^{2}\right ) \sqrt {e x +d}}{8 c}}{\left (\left (e x +d \right ) c +b e -c d \right )^{2}}+\frac {\left (b^{2} e^{2}+12 b c d e -48 c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{8 c \sqrt {c \left (b e -c d \right )}}\right )}{b^{5} e^{5}}\right )\)	\(273\)
pseudoelliptic	\(-\frac {-\frac {\left (b^{2} e^{2}+12 b c d e -48 c^{2} d^{2}\right ) x^{2} \left (c x +b \right )^{2} \sqrt {d}\, \left (b e -c d \right )^{2} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{2}+\left (\frac {x^{2} \left (c x +b \right )^{2} c \left (35 b^{2} e^{2}-84 b c d e +48 c^{2} d^{2}\right ) d^{2} \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2}+\sqrt {d}\, \left (-12 d^{3} c^{4} x^{3}-18 b \,d^{2} x^{2} \left (-e x +d \right ) c^{3}-4 x \left (\frac {5}{4} e^{2} x^{2}-\frac {55}{8} d e x +d^{2}\right ) b^{2} d \,c^{2}+b^{3} \left (\frac {13}{2} d^{2} e x -8 d \,e^{2} x^{2}-\frac {1}{2} e^{3} x^{3}+d^{3}\right ) c +\frac {b^{4} e^{3} x^{2}}{2}\right ) \sqrt {e x +d}\, b \right ) \sqrt {c \left (b e -c d \right )}}{2 \sqrt {d}\, \sqrt {c \left (b e -c d \right )}\, b^{5} x^{2} \left (c x +b \right )^{2} c}\)	\(282\)
risch	\(-\frac {d^{2} \sqrt {e x +d}\, \left (13 b e x -12 c d x +2 b d \right )}{4 b^{4} x^{2}}-\frac {e \left (\frac {d^{\frac {3}{2}} \left (35 b^{2} e^{2}-84 b c d e +48 c^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{e b}+\frac {\frac {8 \left (\left (-\frac {1}{8} b^{4} e^{4}-\frac {5}{4} d \,e^{3} b^{3} c +\frac {23}{8} d^{2} e^{2} b^{2} c^{2}-\frac {3}{2} d^{3} e b \,c^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}+\frac {b e \left (b^{4} e^{4}-15 d \,e^{3} b^{3} c +39 d^{2} e^{2} b^{2} c^{2}-37 d^{3} e b \,c^{3}+12 d^{4} c^{4}\right ) \sqrt {e x +d}}{8 c}\right )}{\left (\left (e x +d \right ) c +b e -c d \right )^{2}}-\frac {\left (b^{4} e^{4}+10 d \,e^{3} b^{3} c -71 d^{2} e^{2} b^{2} c^{2}+108 d^{3} e b \,c^{3}-48 d^{4} c^{4}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{c \sqrt {c \left (b e -c d \right )}}}{b e}\right )}{4 b^{4}}\)	\(319\)

Input:

int((e*x+d)^(7/2)/(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)
 

Output:

2*e^5*(-d^2/b^5/e^5*(((13/8*b^2*e^2-3/2*b*c*d*e)*(e*x+d)^(3/2)+(-11/8*d*e^ 
2*b^2+3/2*d^2*e*b*c)*(e*x+d)^(1/2))/e^2/x^2+1/8*(35*b^2*e^2-84*b*c*d*e+48* 
c^2*d^2)/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2)))+(b*e-c*d)^2/b^5/e^5*(((1/ 
8*b^2*e^2+3/2*b*c*d*e)*(e*x+d)^(3/2)-1/8*b*e*(b^2*e^2-13*b*c*d*e+12*c^2*d^ 
2)/c*(e*x+d)^(1/2))/((e*x+d)*c+b*e-c*d)^2+1/8*(b^2*e^2+12*b*c*d*e-48*c^2*d 
^2)/c/(c*(b*e-c*d))^(1/2)*arctan(c*(e*x+d)^(1/2)/(c*(b*e-c*d))^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 2021, normalized size of antiderivative = 6.63 \[ \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^(7/2)/(c*x^2+b*x)^3,x, algorithm="fricas")
 

Output:

[1/8*(((48*c^5*d^3 - 60*b*c^4*d^2*e + 11*b^2*c^3*d*e^2 + b^3*c^2*e^3)*x^4 
+ 2*(48*b*c^4*d^3 - 60*b^2*c^3*d^2*e + 11*b^3*c^2*d*e^2 + b^4*c*e^3)*x^3 + 
 (48*b^2*c^3*d^3 - 60*b^3*c^2*d^2*e + 11*b^4*c*d*e^2 + b^5*e^3)*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)) + ((48*c^5*d^3 - 84*b*c^4*d^2*e + 35*b^2*c^3*d*e^2)*x^4 
+ 2*(48*b*c^4*d^3 - 84*b^2*c^3*d^2*e + 35*b^3*c^2*d*e^2)*x^3 + (48*b^2*c^3 
*d^3 - 84*b^3*c^2*d^2*e + 35*b^4*c*d*e^2)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e 
*x + d)*sqrt(d) + 2*d)/x) - 2*(2*b^4*c*d^3 - (24*b*c^4*d^3 - 36*b^2*c^3*d^ 
2*e + 10*b^3*c^2*d*e^2 + b^4*c*e^3)*x^3 - (36*b^2*c^3*d^3 - 55*b^3*c^2*d^2 
*e + 16*b^4*c*d*e^2 - b^5*e^3)*x^2 - (8*b^3*c^2*d^3 - 13*b^4*c*d^2*e)*x)*s 
qrt(e*x + d))/(b^5*c^3*x^4 + 2*b^6*c^2*x^3 + b^7*c*x^2), 1/8*(2*((48*c^5*d 
^3 - 60*b*c^4*d^2*e + 11*b^2*c^3*d*e^2 + b^3*c^2*e^3)*x^4 + 2*(48*b*c^4*d^ 
3 - 60*b^2*c^3*d^2*e + 11*b^3*c^2*d*e^2 + b^4*c*e^3)*x^3 + (48*b^2*c^3*d^3 
 - 60*b^3*c^2*d^2*e + 11*b^4*c*d*e^2 + b^5*e^3)*x^2)*sqrt(-(c*d - b*e)/c)* 
arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + ((48*c^5*d^3 - 
 84*b*c^4*d^2*e + 35*b^2*c^3*d*e^2)*x^4 + 2*(48*b*c^4*d^3 - 84*b^2*c^3*d^2 
*e + 35*b^3*c^2*d*e^2)*x^3 + (48*b^2*c^3*d^3 - 84*b^3*c^2*d^2*e + 35*b^4*c 
*d*e^2)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*b 
^4*c*d^3 - (24*b*c^4*d^3 - 36*b^2*c^3*d^2*e + 10*b^3*c^2*d*e^2 + b^4*c*e^3 
)*x^3 - (36*b^2*c^3*d^3 - 55*b^3*c^2*d^2*e + 16*b^4*c*d*e^2 - b^5*e^3)*...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate((e*x+d)**(7/2)/(c*x**2+b*x)**3,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x+d)^(7/2)/(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 [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.76 \[ \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^3} \, dx=\frac {{\left (48 \, c^{2} d^{4} - 84 \, b c d^{3} e + 35 \, b^{2} d^{2} e^{2}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-d}}\right )}{4 \, b^{5} \sqrt {-d}} - \frac {{\left (48 \, c^{4} d^{4} - 108 \, b c^{3} d^{3} e + 71 \, b^{2} c^{2} d^{2} e^{2} - 10 \, b^{3} c d e^{3} - b^{4} e^{4}\right )} \arctan \left (\frac {\sqrt {e x + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{4 \, \sqrt {-c^{2} d + b c e} b^{5} c} + \frac {24 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{4} d^{3} e - 72 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{4} d^{4} e + 72 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{4} d^{5} e - 24 \, \sqrt {e x + d} c^{4} d^{6} e - 36 \, {\left (e x + d\right )}^{\frac {7}{2}} b c^{3} d^{2} e^{2} + 144 \, {\left (e x + d\right )}^{\frac {5}{2}} b c^{3} d^{3} e^{2} - 180 \, {\left (e x + d\right )}^{\frac {3}{2}} b c^{3} d^{4} e^{2} + 72 \, \sqrt {e x + d} b c^{3} d^{5} e^{2} + 10 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{2} c^{2} d e^{3} - 85 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{2} c^{2} d^{2} e^{3} + 148 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{2} c^{2} d^{3} e^{3} - 73 \, \sqrt {e x + d} b^{2} c^{2} d^{4} e^{3} + {\left (e x + d\right )}^{\frac {7}{2}} b^{3} c e^{4} + 13 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{3} c d e^{4} - 42 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{3} c d^{2} e^{4} + 26 \, \sqrt {e x + d} b^{3} c d^{3} e^{4} - {\left (e x + d\right )}^{\frac {5}{2}} b^{4} e^{5} + 2 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{4} d e^{5} - \sqrt {e x + d} b^{4} d^{2} e^{5}}{4 \, {\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} + {\left (e x + d\right )} b e - b d e\right )}^{2} b^{4} c} \] Input:

integrate((e*x+d)^(7/2)/(c*x^2+b*x)^3,x, algorithm="giac")
 

Output:

1/4*(48*c^2*d^4 - 84*b*c*d^3*e + 35*b^2*d^2*e^2)*arctan(sqrt(e*x + d)/sqrt 
(-d))/(b^5*sqrt(-d)) - 1/4*(48*c^4*d^4 - 108*b*c^3*d^3*e + 71*b^2*c^2*d^2* 
e^2 - 10*b^3*c*d*e^3 - b^4*e^4)*arctan(sqrt(e*x + d)*c/sqrt(-c^2*d + b*c*e 
))/(sqrt(-c^2*d + b*c*e)*b^5*c) + 1/4*(24*(e*x + d)^(7/2)*c^4*d^3*e - 72*( 
e*x + d)^(5/2)*c^4*d^4*e + 72*(e*x + d)^(3/2)*c^4*d^5*e - 24*sqrt(e*x + d) 
*c^4*d^6*e - 36*(e*x + d)^(7/2)*b*c^3*d^2*e^2 + 144*(e*x + d)^(5/2)*b*c^3* 
d^3*e^2 - 180*(e*x + d)^(3/2)*b*c^3*d^4*e^2 + 72*sqrt(e*x + d)*b*c^3*d^5*e 
^2 + 10*(e*x + d)^(7/2)*b^2*c^2*d*e^3 - 85*(e*x + d)^(5/2)*b^2*c^2*d^2*e^3 
 + 148*(e*x + d)^(3/2)*b^2*c^2*d^3*e^3 - 73*sqrt(e*x + d)*b^2*c^2*d^4*e^3 
+ (e*x + d)^(7/2)*b^3*c*e^4 + 13*(e*x + d)^(5/2)*b^3*c*d*e^4 - 42*(e*x + d 
)^(3/2)*b^3*c*d^2*e^4 + 26*sqrt(e*x + d)*b^3*c*d^3*e^4 - (e*x + d)^(5/2)*b 
^4*e^5 + 2*(e*x + d)^(3/2)*b^4*d*e^5 - sqrt(e*x + d)*b^4*d^2*e^5)/(((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)
 

Mupad [B] (verification not implemented)

Time = 5.57 (sec) , antiderivative size = 1792, normalized size of antiderivative = 5.88 \[ \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

int((d + e*x)^(7/2)/(b*x + c*x^2)^3,x)
 

Output:

- (((d + e*x)^(1/2)*(24*c^4*d^6*e + b^4*d^2*e^5 - 72*b*c^3*d^5*e^2 - 26*b^ 
3*c*d^3*e^4 + 73*b^2*c^2*d^4*e^3))/(4*b^4*c) - (e*(d + e*x)^(7/2)*(b^3*e^3 
 + 24*c^3*d^3 - 36*b*c^2*d^2*e + 10*b^2*c*d*e^2))/(4*b^4) - ((d + e*x)^(3/ 
2)*(b^4*d*e^5 + 36*c^4*d^5*e - 90*b*c^3*d^4*e^2 - 21*b^3*c*d^2*e^4 + 74*b^ 
2*c^2*d^3*e^3))/(2*b^4*c) + (e*(d + e*x)^(5/2)*(b^4*e^4 + 72*c^4*d^4 + 85* 
b^2*c^2*d^2*e^2 - 144*b*c^3*d^3*e - 13*b^3*c*d*e^3))/(4*b^4*c))/(c^2*(d + 
e*x)^4 - (d + e*x)*(4*c^2*d^3 + 2*b^2*d*e^2 - 6*b*c*d^2*e) - (4*c^2*d - 2* 
b*c*e)*(d + e*x)^3 + (d + e*x)^2*(b^2*e^2 + 6*c^2*d^2 - 6*b*c*d*e) + c^2*d 
^4 + b^2*d^2*e^2 - 2*b*c*d^3*e) - (atanh((35*e^12*(d^3)^(1/2)*(d + e*x)^(1 
/2))/(32*((35*d^2*e^12)/32 + (77*c*d^3*e^11)/(4*b) - (1551*c^2*d^4*e^10)/( 
16*b^2) + (5223*c^3*d^5*e^9)/(32*b^3) - (945*c^4*d^6*e^8)/(8*b^4) + (63*c^ 
5*d^7*e^7)/(2*b^5))) + (77*d*e^11*(d^3)^(1/2)*(d + e*x)^(1/2))/(4*((77*d^3 
*e^11)/4 + (35*b*d^2*e^12)/(32*c) - (1551*c*d^4*e^10)/(16*b) + (5223*c^2*d 
^5*e^9)/(32*b^2) - (945*c^3*d^6*e^8)/(8*b^3) + (63*c^4*d^7*e^7)/(2*b^4))) 
+ (5223*c^2*d^3*e^9*(d^3)^(1/2)*(d + e*x)^(1/2))/(32*((77*b^2*d^3*e^11)/4 
+ (5223*c^2*d^5*e^9)/32 - (945*c^3*d^6*e^8)/(8*b) + (35*b^3*d^2*e^12)/(32* 
c) + (63*c^4*d^7*e^7)/(2*b^2) - (1551*b*c*d^4*e^10)/16)) - (945*c^3*d^4*e^ 
8*(d^3)^(1/2)*(d + e*x)^(1/2))/(8*((77*b^3*d^3*e^11)/4 - (945*c^3*d^6*e^8) 
/8 + (5223*b*c^2*d^5*e^9)/32 - (1551*b^2*c*d^4*e^10)/16 + (63*c^4*d^7*e^7) 
/(2*b) + (35*b^4*d^2*e^12)/(32*c))) + (63*c^4*d^5*e^7*(d^3)^(1/2)*(d + ...
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 1308, normalized size of antiderivative = 4.29 \[ \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int((e*x+d)^(7/2)/(c*x^2+b*x)^3,x)
 

Output:

(2*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d) 
))*b**5*e**3*x**2 + 22*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqr 
t(c)*sqrt(b*e - c*d)))*b**4*c*d*e**2*x**2 + 4*sqrt(c)*sqrt(b*e - c*d)*atan 
((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**4*c*e**3*x**3 - 120*sqrt( 
c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**3* 
c**2*d**2*e*x**2 + 44*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt 
(c)*sqrt(b*e - c*d)))*b**3*c**2*d*e**2*x**3 + 2*sqrt(c)*sqrt(b*e - c*d)*at 
an((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**3*c**2*e**3*x**4 + 96*s 
qrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b 
**2*c**3*d**3*x**2 - 240*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(s 
qrt(c)*sqrt(b*e - c*d)))*b**2*c**3*d**2*e*x**3 + 22*sqrt(c)*sqrt(b*e - c*d 
)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**2*c**3*d*e**2*x**4 
+ 192*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c 
*d)))*b*c**4*d**3*x**3 - 120*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c 
)/(sqrt(c)*sqrt(b*e - c*d)))*b*c**4*d**2*e*x**4 + 96*sqrt(c)*sqrt(b*e - c* 
d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*c**5*d**3*x**4 - 2*sq 
rt(d + e*x)*b**5*c*e**3*x**2 - 4*sqrt(d + e*x)*b**4*c**2*d**3 - 26*sqrt(d 
+ e*x)*b**4*c**2*d**2*e*x + 32*sqrt(d + e*x)*b**4*c**2*d*e**2*x**2 + 2*sqr 
t(d + e*x)*b**4*c**2*e**3*x**3 + 16*sqrt(d + e*x)*b**3*c**3*d**3*x - 110*s 
qrt(d + e*x)*b**3*c**3*d**2*e*x**2 + 20*sqrt(d + e*x)*b**3*c**3*d*e**2*...