\(\int \frac {(d+e x)^{7/2}}{(b x+c x^2)^2} \, dx\) [114]

Optimal result

Integrand size = 21, antiderivative size = 200 \[ \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^2} \, dx=\frac {e \left (2 c^2 d^2-2 b c d e+3 b^2 e^2\right ) \sqrt {d+e x}}{b^2 c^2}-\frac {(c d-b e) (2 c d-b e) (d+e x)^{3/2}}{b^2 c (b+c x)}-\frac {d (d+e x)^{5/2}}{b x (b+c x)}+\frac {d^{5/2} (4 c d-7 b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}-\frac {(c d-b e)^{5/2} (4 c d+3 b e) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{5/2}} \] Output:

e*(3*b^2*e^2-2*b*c*d*e+2*c^2*d^2)*(e*x+d)^(1/2)/b^2/c^2-(-b*e+c*d)*(-b*e+2 
*c*d)*(e*x+d)^(3/2)/b^2/c/(c*x+b)-d*(e*x+d)^(5/2)/b/x/(c*x+b)+d^(5/2)*(-7* 
b*e+4*c*d)*arctanh((e*x+d)^(1/2)/d^(1/2))/b^3-(-b*e+c*d)^(5/2)*(3*b*e+4*c* 
d)*arctanh(c^(1/2)*(e*x+d)^(1/2)/(-b*e+c*d)^(1/2))/b^3/c^(5/2)
 

Mathematica [A] (verified)

Time = 0.74 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.84 \[ \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^2} \, dx=\frac {-\frac {b \sqrt {d+e x} \left (2 c^3 d^3 x-3 b^3 e^3 x+b c^2 d^2 (d-3 e x)+b^2 c e^2 x (3 d-2 e x)\right )}{c^2 x (b+c x)}-\frac {(-c d+b e)^{5/2} (4 c d+3 b e) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{c^{5/2}}+d^{5/2} (4 c d-7 b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3} \] Input:

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

Output:

(-((b*Sqrt[d + e*x]*(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)))/(c^2*x*(b + c*x))) - ((-(c*d) + b*e)^(5/2)*(4*c 
*d + 3*b*e)*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/c^(5/2) + 
d^(5/2)*(4*c*d - 7*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b^3
 

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {1164, 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.

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

\(\Big \downarrow \) 1164

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1196

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

\(\Big \downarrow \) 1196

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

\(\Big \downarrow \) 1197

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1480

\(\displaystyle -\frac {\frac {\frac {2 e \left (\frac {c^3 d^3 (4 c d-7 b e) \int \frac {1}{c (d+e x)-c d}d\sqrt {d+e x}}{b e}-\frac {(c d-b e)^3 (3 b e+4 c d) \int \frac {1}{-c d+b e+c (d+e x)}d\sqrt {d+e x}}{b e}\right )}{c}-\frac {2 e \sqrt {d+e x} \left (3 b^2 e^2-2 b c d e+2 c^2 d^2\right )}{c}}{c}-\frac {2 e (d+e x)^{3/2} (2 c d-b e)}{c}}{2 b^2}-\frac {(d+e x)^{5/2} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )}\)

\(\Big \downarrow \) 221

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

Input:

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

Output:

-(((d + e*x)^(5/2)*(b*d + (2*c*d - b*e)*x))/(b^2*(b*x + c*x^2))) - ((-2*e* 
(2*c*d - b*e)*(d + e*x)^(3/2))/c + ((-2*e*(2*c^2*d^2 - 2*b*c*d*e + 3*b^2*e 
^2)*Sqrt[d + e*x])/c + (2*e*(-((c^2*d^(5/2)*(4*c*d - 7*b*e)*ArcTanh[Sqrt[d 
 + e*x]/Sqrt[d]])/(b*e)) + ((c*d - b*e)^(5/2)*(4*c*d + 3*b*e)*ArcTanh[(Sqr 
t[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*Sqrt[c]*e)))/c)/c)/(2*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 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 = 0.64 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.98

method	result	size
pseudoelliptic	\(-\frac {-4 \left (c d +\frac {3 b e}{4}\right ) x \left (c x +b \right ) \sqrt {d}\, \left (-b e +c d \right )^{3} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )+\sqrt {c \left (b e -c d \right )}\, \left (7 \left (b e -\frac {4 c d}{7}\right ) x \left (c x +b \right ) c^{2} d^{3} \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )+\left (2 d^{3} c^{3} x +d^{2} b \left (-3 e x +d \right ) c^{2}+3 e^{2} x \,b^{2} \left (-\frac {2 e x}{3}+d \right ) c -3 b^{3} e^{3} x \right ) \sqrt {e x +d}\, b \sqrt {d}\right )}{\sqrt {d}\, \sqrt {c \left (b e -c d \right )}\, c^{2} b^{3} x \left (c x +b \right )}\)	\(195\)
derivativedivides	\(2 e^{3} \left (\frac {\sqrt {e x +d}}{c^{2}}-\frac {d^{3} \left (\frac {b \sqrt {e x +d}}{2 x}+\frac {\left (7 b e -4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}\right )}{b^{3} e^{3}}-\frac {\frac {\left (-\frac {1}{2} b^{4} e^{4}+\frac {3}{2} d \,e^{3} b^{3} c -\frac {3}{2} d^{2} e^{2} b^{2} c^{2}+\frac {1}{2} d^{3} e b \,c^{3}\right ) \sqrt {e x +d}}{\left (e x +d \right ) c +b e -c d}+\frac {\left (3 b^{4} e^{4}-5 d \,e^{3} b^{3} c -3 d^{2} e^{2} b^{2} c^{2}+9 d^{3} e b \,c^{3}-4 d^{4} c^{4}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{2 \sqrt {c \left (b e -c d \right )}}}{c^{2} b^{3} e^{3}}\right )\)	\(236\)
default	\(2 e^{3} \left (\frac {\sqrt {e x +d}}{c^{2}}-\frac {d^{3} \left (\frac {b \sqrt {e x +d}}{2 x}+\frac {\left (7 b e -4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}\right )}{b^{3} e^{3}}-\frac {\frac {\left (-\frac {1}{2} b^{4} e^{4}+\frac {3}{2} d \,e^{3} b^{3} c -\frac {3}{2} d^{2} e^{2} b^{2} c^{2}+\frac {1}{2} d^{3} e b \,c^{3}\right ) \sqrt {e x +d}}{\left (e x +d \right ) c +b e -c d}+\frac {\left (3 b^{4} e^{4}-5 d \,e^{3} b^{3} c -3 d^{2} e^{2} b^{2} c^{2}+9 d^{3} e b \,c^{3}-4 d^{4} c^{4}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{2 \sqrt {c \left (b e -c d \right )}}}{c^{2} b^{3} e^{3}}\right )\)	\(236\)
risch	\(-\frac {d^{3} \sqrt {e x +d}}{b^{2} x}+\frac {e \left (\frac {2 b^{2} e^{2} \sqrt {e x +d}}{c^{2}}-\frac {d^{\frac {5}{2}} \left (7 b e -4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{e b}-\frac {2 \left (\frac {\left (-\frac {1}{2} b^{4} e^{4}+\frac {3}{2} d \,e^{3} b^{3} c -\frac {3}{2} d^{2} e^{2} b^{2} c^{2}+\frac {1}{2} d^{3} e b \,c^{3}\right ) \sqrt {e x +d}}{\left (e x +d \right ) c +b e -c d}+\frac {\left (3 b^{4} e^{4}-5 d \,e^{3} b^{3} c -3 d^{2} e^{2} b^{2} c^{2}+9 d^{3} e b \,c^{3}-4 d^{4} c^{4}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{2 \sqrt {c \left (b e -c d \right )}}\right )}{c^{2} b e}\right )}{b^{2}}\)	\(243\)

Input:

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

Output:

-1/d^(1/2)/(c*(b*e-c*d))^(1/2)*(-4*(c*d+3/4*b*e)*x*(c*x+b)*d^(1/2)*(-b*e+c 
*d)^3*arctan(c*(e*x+d)^(1/2)/(c*(b*e-c*d))^(1/2))+(c*(b*e-c*d))^(1/2)*(7*( 
b*e-4/7*c*d)*x*(c*x+b)*c^2*d^3*arctanh((e*x+d)^(1/2)/d^(1/2))+(2*d^3*c^3*x 
+d^2*b*(-3*e*x+d)*c^2+3*e^2*x*b^2*(-2/3*e*x+d)*c-3*b^3*e^3*x)*(e*x+d)^(1/2 
)*b*d^(1/2)))/c^2/b^3/x/(c*x+b)
 

Fricas [A] (verification not implemented)

Time = 0.50 (sec) , antiderivative size = 1267, normalized size of antiderivative = 6.34 \[ \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/2*(((4*c^4*d^3 - 5*b*c^3*d^2*e - 2*b^2*c^2*d*e^2 + 3*b^3*c*e^3)*x^2 + ( 
4*b*c^3*d^3 - 5*b^2*c^2*d^2*e - 2*b^3*c*d*e^2 + 3*b^4*e^3)*x)*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)) - ((4*c^4*d^3 - 7*b*c^3*d^2*e)*x^2 + (4*b*c^3*d^3 - 7*b^2*c^2*d 
^2*e)*x)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(2*b^3*c 
*e^3*x^2 - b^2*c^2*d^3 - (2*b*c^3*d^3 - 3*b^2*c^2*d^2*e + 3*b^3*c*d*e^2 - 
3*b^4*e^3)*x)*sqrt(e*x + d))/(b^3*c^3*x^2 + b^4*c^2*x), -1/2*(2*((4*c^4*d^ 
3 - 5*b*c^3*d^2*e - 2*b^2*c^2*d*e^2 + 3*b^3*c*e^3)*x^2 + (4*b*c^3*d^3 - 5* 
b^2*c^2*d^2*e - 2*b^3*c*d*e^2 + 3*b^4*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)) + ((4*c^4*d^3 - 7*b*c^3 
*d^2*e)*x^2 + (4*b*c^3*d^3 - 7*b^2*c^2*d^2*e)*x)*sqrt(d)*log((e*x - 2*sqrt 
(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*b^3*c*e^3*x^2 - b^2*c^2*d^3 - (2*b*c^3* 
d^3 - 3*b^2*c^2*d^2*e + 3*b^3*c*d*e^2 - 3*b^4*e^3)*x)*sqrt(e*x + d))/(b^3* 
c^3*x^2 + b^4*c^2*x), -1/2*(2*((4*c^4*d^3 - 7*b*c^3*d^2*e)*x^2 + (4*b*c^3* 
d^3 - 7*b^2*c^2*d^2*e)*x)*sqrt(-d)*arctan(sqrt(-d)/sqrt(e*x + d)) - ((4*c^ 
4*d^3 - 5*b*c^3*d^2*e - 2*b^2*c^2*d*e^2 + 3*b^3*c*e^3)*x^2 + (4*b*c^3*d^3 
- 5*b^2*c^2*d^2*e - 2*b^3*c*d*e^2 + 3*b^4*e^3)*x)*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)) - 
 2*(2*b^3*c*e^3*x^2 - b^2*c^2*d^3 - (2*b*c^3*d^3 - 3*b^2*c^2*d^2*e + 3*b^3 
*c*d*e^2 - 3*b^4*e^3)*x)*sqrt(e*x + d))/(b^3*c^3*x^2 + b^4*c^2*x), -(((...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((e*x+d)^(7/2)/(c*x^2+b*x)^2,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 = 0.19 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.66 \[ \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^2} \, dx=\frac {2 \, \sqrt {e x + d} e^{3}}{c^{2}} - \frac {{\left (4 \, c d^{4} - 7 \, b d^{3} e\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d}} + \frac {{\left (4 \, c^{4} d^{4} - 9 \, b c^{3} d^{3} e + 3 \, b^{2} c^{2} d^{2} e^{2} + 5 \, b^{3} c d e^{3} - 3 \, b^{4} e^{4}\right )} \arctan \left (\frac {\sqrt {e x + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{\sqrt {-c^{2} d + b c e} b^{3} c^{2}} - \frac {2 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{3} d^{3} e - 2 \, \sqrt {e x + d} c^{3} d^{4} e - 3 \, {\left (e x + d\right )}^{\frac {3}{2}} b c^{2} d^{2} e^{2} + 4 \, \sqrt {e x + d} b c^{2} d^{3} e^{2} + 3 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{2} c d e^{3} - 3 \, \sqrt {e x + d} b^{2} c d^{2} e^{3} - {\left (e x + d\right )}^{\frac {3}{2}} b^{3} e^{4} + \sqrt {e x + d} b^{3} d e^{4}}{{\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} + {\left (e x + d\right )} b e - b d e\right )} b^{2} c^{2}} \] Input:

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

Output:

2*sqrt(e*x + d)*e^3/c^2 - (4*c*d^4 - 7*b*d^3*e)*arctan(sqrt(e*x + d)/sqrt( 
-d))/(b^3*sqrt(-d)) + (4*c^4*d^4 - 9*b*c^3*d^3*e + 3*b^2*c^2*d^2*e^2 + 5*b 
^3*c*d*e^3 - 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^3*c^2) - (2*(e*x + d)^(3/2)*c^3*d^3*e - 2*sqrt(e*x + d) 
*c^3*d^4*e - 3*(e*x + d)^(3/2)*b*c^2*d^2*e^2 + 4*sqrt(e*x + d)*b*c^2*d^3*e 
^2 + 3*(e*x + d)^(3/2)*b^2*c*d*e^3 - 3*sqrt(e*x + d)*b^2*c*d^2*e^3 - (e*x 
+ d)^(3/2)*b^3*e^4 + sqrt(e*x + d)*b^3*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^2)
 

Mupad [B] (verification not implemented)

Time = 5.44 (sec) , antiderivative size = 2913, normalized size of antiderivative = 14.56 \[ \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(((d + e*x)^(1/2)*(b^3*d*e^4 - 2*c^3*d^4*e + 4*b*c^2*d^3*e^2 - 3*b^2*c*d^2 
*e^3))/b^2 - ((d + e*x)^(3/2)*(b^3*e^4 - 2*c^3*d^3*e + 3*b*c^2*d^2*e^2 - 3 
*b^2*c*d*e^3))/b^2)/((2*c^3*d - b*c^2*e)*(d + e*x) - c^3*(d + e*x)^2 - c^3 
*d^2 + b*c^2*d*e) + (2*e^3*(d + e*x)^(1/2))/c^2 + (atan(((((2*(d + e*x)^(1 
/2)*(9*b^8*e^10 + 32*c^8*d^8*e^2 - 128*b*c^7*d^7*e^3 + 154*b^2*c^6*d^6*e^4 
 - 14*b^3*c^5*d^5*e^5 - 105*b^4*c^4*d^4*e^6 + 84*b^5*c^3*d^3*e^7 + 7*b^6*c 
^2*d^2*e^8 - 30*b^7*c*d*e^9))/(b^4*c^3) + (((12*b^9*c^3*d*e^6 - 8*b^6*c^6* 
d^4*e^3 + 16*b^7*c^5*d^3*e^4 - 20*b^8*c^4*d^2*e^5)/(b^6*c^3) + ((4*b^7*c^5 
*e^3 - 8*b^6*c^6*d*e^2)*(7*b*e - 4*c*d)*(d^5)^(1/2)*(d + e*x)^(1/2))/(b^7* 
c^3))*(7*b*e - 4*c*d)*(d^5)^(1/2))/(2*b^3))*(7*b*e - 4*c*d)*(d^5)^(1/2)*1i 
)/(2*b^3) + (((2*(d + e*x)^(1/2)*(9*b^8*e^10 + 32*c^8*d^8*e^2 - 128*b*c^7* 
d^7*e^3 + 154*b^2*c^6*d^6*e^4 - 14*b^3*c^5*d^5*e^5 - 105*b^4*c^4*d^4*e^6 + 
 84*b^5*c^3*d^3*e^7 + 7*b^6*c^2*d^2*e^8 - 30*b^7*c*d*e^9))/(b^4*c^3) - ((( 
12*b^9*c^3*d*e^6 - 8*b^6*c^6*d^4*e^3 + 16*b^7*c^5*d^3*e^4 - 20*b^8*c^4*d^2 
*e^5)/(b^6*c^3) - ((4*b^7*c^5*e^3 - 8*b^6*c^6*d*e^2)*(7*b*e - 4*c*d)*(d^5) 
^(1/2)*(d + e*x)^(1/2))/(b^7*c^3))*(7*b*e - 4*c*d)*(d^5)^(1/2))/(2*b^3))*( 
7*b*e - 4*c*d)*(d^5)^(1/2)*1i)/(2*b^3))/((2*(63*b^8*d^3*e^11 + 32*c^8*d^11 
*e^3 - 176*b*c^7*d^10*e^4 - 246*b^7*c*d^4*e^10 + 262*b^2*c^6*d^9*e^5 + 141 
*b^3*c^5*d^8*e^6 - 658*b^4*c^4*d^7*e^7 + 413*b^5*c^3*d^6*e^8 + 169*b^6*c^2 
*d^5*e^9))/(b^6*c^3) + (((2*(d + e*x)^(1/2)*(9*b^8*e^10 + 32*c^8*d^8*e^...
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 698, normalized size of antiderivative = 3.49 \[ \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 6*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c 
*d)))*b**4*e**3*x + 4*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt 
(c)*sqrt(b*e - c*d)))*b**3*c*d*e**2*x - 6*sqrt(c)*sqrt(b*e - c*d)*atan((sq 
rt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**3*c*e**3*x**2 + 10*sqrt(c)*sq 
rt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**2*c**2* 
d**2*e*x + 4*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt( 
b*e - c*d)))*b**2*c**2*d*e**2*x**2 - 8*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt( 
d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b*c**3*d**3*x + 10*sqrt(c)*sqrt(b*e 
 - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b*c**3*d**2*e*x* 
*2 - 8*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - 
c*d)))*c**4*d**3*x**2 + 6*sqrt(d + e*x)*b**4*c*e**3*x - 6*sqrt(d + e*x)*b* 
*3*c**2*d*e**2*x + 4*sqrt(d + e*x)*b**3*c**2*e**3*x**2 - 2*sqrt(d + e*x)*b 
**2*c**3*d**3 + 6*sqrt(d + e*x)*b**2*c**3*d**2*e*x - 4*sqrt(d + e*x)*b*c** 
4*d**3*x + 7*sqrt(d)*log(sqrt(d + e*x) - sqrt(d))*b**2*c**3*d**2*e*x - 4*s 
qrt(d)*log(sqrt(d + e*x) - sqrt(d))*b*c**4*d**3*x + 7*sqrt(d)*log(sqrt(d + 
 e*x) - sqrt(d))*b*c**4*d**2*e*x**2 - 4*sqrt(d)*log(sqrt(d + e*x) - sqrt(d 
))*c**5*d**3*x**2 - 7*sqrt(d)*log(sqrt(d + e*x) + sqrt(d))*b**2*c**3*d**2* 
e*x + 4*sqrt(d)*log(sqrt(d + e*x) + sqrt(d))*b*c**4*d**3*x - 7*sqrt(d)*log 
(sqrt(d + e*x) + sqrt(d))*b*c**4*d**2*e*x**2 + 4*sqrt(d)*log(sqrt(d + e*x) 
 + sqrt(d))*c**5*d**3*x**2)/(2*b**3*c**3*x*(b + c*x))