\(\int \frac {(d+e x)^{3/2}}{(b x+c x^2)^2} \, dx\) [116]

Optimal result

Integrand size = 21, antiderivative size = 149 \[ \int \frac {(d+e x)^{3/2}}{\left (b x+c x^2\right )^2} \, dx=-\frac {(2 c d-b e) \sqrt {d+e x}}{b^2 (b+c x)}-\frac {d \sqrt {d+e x}}{b x (b+c x)}+\frac {\sqrt {d} (4 c d-3 b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}-\frac {\sqrt {c d-b e} (4 c d-b e) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 \sqrt {c}} \] Output:

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

Mathematica [A] (verified)

Time = 0.78 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.93 \[ \int \frac {(d+e x)^{3/2}}{\left (b x+c x^2\right )^2} \, dx=\frac {\frac {b \sqrt {d+e x} (-b d-2 c d x+b e x)}{x (b+c x)}+\frac {\left (4 c^2 d^2-5 b c d e+b^2 e^2\right ) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{\sqrt {c} \sqrt {-c d+b e}}+\sqrt {d} (4 c d-3 b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1164, 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.

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

\(\Big \downarrow \) 1164

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1197

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-((Sqrt[d + e*x]*(b*d + (2*c*d - b*e)*x))/(b^2*(b*x + c*x^2))) - (e*(-((Sq 
rt[d]*(4*c*d - 3*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*e)) + (Sqrt[c*d - 
 b*e]*(4*c*d - b*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*S 
qrt[c]*e)))/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 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 = 0.58 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.97

method	result	size
derivativedivides	\(2 e^{3} \left (\frac {\left (b e -c d \right ) \left (\frac {b e \sqrt {e x +d}}{2 \left (e x +d \right ) c +2 b e -2 c d}+\frac {\left (b e -4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{2 \sqrt {c \left (b e -c d \right )}}\right )}{b^{3} e^{3}}-\frac {d \left (\frac {b \sqrt {e x +d}}{2 x}+\frac {\left (3 b e -4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}\right )}{b^{3} e^{3}}\right )\)	\(144\)
default	\(2 e^{3} \left (\frac {\left (b e -c d \right ) \left (\frac {b e \sqrt {e x +d}}{2 \left (e x +d \right ) c +2 b e -2 c d}+\frac {\left (b e -4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{2 \sqrt {c \left (b e -c d \right )}}\right )}{b^{3} e^{3}}-\frac {d \left (\frac {b \sqrt {e x +d}}{2 x}+\frac {\left (3 b e -4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}\right )}{b^{3} e^{3}}\right )\)	\(144\)
pseudoelliptic	\(-\frac {-4 x \left (c x +b \right ) \left (c d -\frac {b e}{4}\right ) \left (-b e +c d \right ) \sqrt {d}\, \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )+\sqrt {c \left (b e -c d \right )}\, \left (3 x \left (c x +b \right ) d \left (b e -\frac {4 c d}{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )+\left (2 c d x +b \left (-e x +d \right )\right ) \sqrt {e x +d}\, b \sqrt {d}\right )}{\sqrt {d}\, \sqrt {c \left (b e -c d \right )}\, b^{3} x \left (c x +b \right )}\)	\(150\)
risch	\(-\frac {d \sqrt {e x +d}}{b^{2} x}-\frac {e \left (\frac {\sqrt {d}\, \left (3 b e -4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b e}+\frac {\frac {2 \left (-\frac {1}{2} b^{2} e^{2}+\frac {1}{2} b c d e \right ) \sqrt {e x +d}}{\left (e x +d \right ) c +b e -c d}-\frac {\left (b^{2} e^{2}-5 b c d e +4 c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{\sqrt {c \left (b e -c d \right )}}}{b e}\right )}{b^{2}}\)	\(163\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 764, normalized size of antiderivative = 5.13 \[ \int \frac {(d+e x)^{3/2}}{\left (b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((e*x+d)^(3/2)/(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 = 0.13 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.32 \[ \int \frac {(d+e x)^{3/2}}{\left (b x+c x^2\right )^2} \, dx=\frac {{\left (4 \, c^{2} d^{2} - 5 \, b c d e + b^{2} e^{2}\right )} \arctan \left (\frac {\sqrt {e x + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{\sqrt {-c^{2} d + b c e} b^{3}} - \frac {{\left (4 \, c d^{2} - 3 \, b d e\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d}} - \frac {2 \, {\left (e x + d\right )}^{\frac {3}{2}} c d e - 2 \, \sqrt {e x + d} c d^{2} e - {\left (e x + d\right )}^{\frac {3}{2}} b e^{2} + 2 \, \sqrt {e x + d} b d e^{2}}{{\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} + {\left (e x + d\right )} b e - b d e\right )} b^{2}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 5.56 (sec) , antiderivative size = 429, normalized size of antiderivative = 2.88 \[ \int \frac {(d+e x)^{3/2}}{\left (b x+c x^2\right )^2} \, dx=-\frac {\frac {2\,\left (b\,d\,e^2-c\,d^2\,e\right )\,\sqrt {d+e\,x}}{b^2}-\frac {e\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{3/2}}{b^2}}{\left (b\,e-2\,c\,d\right )\,\left (d+e\,x\right )+c\,{\left (d+e\,x\right )}^2+c\,d^2-b\,d\,e}-\frac {\sqrt {d}\,\mathrm {atanh}\left (\frac {6\,c\,\sqrt {d}\,e^7\,\sqrt {d+e\,x}}{6\,c\,d\,e^7-\frac {14\,c^2\,d^2\,e^6}{b}+\frac {8\,c^3\,d^3\,e^5}{b^2}}-\frac {14\,c^2\,d^{3/2}\,e^6\,\sqrt {d+e\,x}}{6\,b\,c\,d\,e^7-14\,c^2\,d^2\,e^6+\frac {8\,c^3\,d^3\,e^5}{b}}+\frac {8\,c^3\,d^{5/2}\,e^5\,\sqrt {d+e\,x}}{6\,b^2\,c\,d\,e^7-14\,b\,c^2\,d^2\,e^6+8\,c^3\,d^3\,e^5}\right )\,\left (3\,b\,e-4\,c\,d\right )}{b^3}-\frac {\mathrm {atanh}\left (\frac {2\,c\,d\,e^6\,\sqrt {c^2\,d-b\,c\,e}\,\sqrt {d+e\,x}}{2\,b\,c\,d\,e^7-10\,c^2\,d^2\,e^6+\frac {8\,c^3\,d^3\,e^5}{b}}-\frac {8\,c^2\,d^2\,e^5\,\sqrt {c^2\,d-b\,c\,e}\,\sqrt {d+e\,x}}{2\,b^2\,c\,d\,e^7-10\,b\,c^2\,d^2\,e^6+8\,c^3\,d^3\,e^5}\right )\,\sqrt {-c\,\left (b\,e-c\,d\right )}\,\left (b\,e-4\,c\,d\right )}{b^3\,c} \] Input:

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

Output:

- ((2*(b*d*e^2 - c*d^2*e)*(d + e*x)^(1/2))/b^2 - (e*(b*e - 2*c*d)*(d + e*x 
)^(3/2))/b^2)/((b*e - 2*c*d)*(d + e*x) + c*(d + e*x)^2 + c*d^2 - b*d*e) - 
(d^(1/2)*atanh((6*c*d^(1/2)*e^7*(d + e*x)^(1/2))/(6*c*d*e^7 - (14*c^2*d^2* 
e^6)/b + (8*c^3*d^3*e^5)/b^2) - (14*c^2*d^(3/2)*e^6*(d + e*x)^(1/2))/(6*b* 
c*d*e^7 - 14*c^2*d^2*e^6 + (8*c^3*d^3*e^5)/b) + (8*c^3*d^(5/2)*e^5*(d + e* 
x)^(1/2))/(8*c^3*d^3*e^5 - 14*b*c^2*d^2*e^6 + 6*b^2*c*d*e^7))*(3*b*e - 4*c 
*d))/b^3 - (atanh((2*c*d*e^6*(c^2*d - b*c*e)^(1/2)*(d + e*x)^(1/2))/(2*b*c 
*d*e^7 - 10*c^2*d^2*e^6 + (8*c^3*d^3*e^5)/b) - (8*c^2*d^2*e^5*(c^2*d - b*c 
*e)^(1/2)*(d + e*x)^(1/2))/(8*c^3*d^3*e^5 - 10*b*c^2*d^2*e^6 + 2*b^2*c*d*e 
^7))*(-c*(b*e - c*d))^(1/2)*(b*e - 4*c*d))/(b^3*c)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 404, normalized size of antiderivative = 2.71 \[ \int \frac {(d+e x)^{3/2}}{\left (b x+c x^2\right )^2} \, dx=\frac {2 \sqrt {c}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c}{\sqrt {c}\, \sqrt {b e -c d}}\right ) b^{2} e x -8 \sqrt {c}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c}{\sqrt {c}\, \sqrt {b e -c d}}\right ) b c d x +2 \sqrt {c}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c}{\sqrt {c}\, \sqrt {b e -c d}}\right ) b c e \,x^{2}-8 \sqrt {c}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c}{\sqrt {c}\, \sqrt {b e -c d}}\right ) c^{2} d \,x^{2}-2 \sqrt {e x +d}\, b^{2} c d +2 \sqrt {e x +d}\, b^{2} c e x -4 \sqrt {e x +d}\, b \,c^{2} d x +3 \sqrt {d}\, \mathrm {log}\left (\sqrt {e x +d}-\sqrt {d}\right ) b^{2} c e x -4 \sqrt {d}\, \mathrm {log}\left (\sqrt {e x +d}-\sqrt {d}\right ) b \,c^{2} d x +3 \sqrt {d}\, \mathrm {log}\left (\sqrt {e x +d}-\sqrt {d}\right ) b \,c^{2} e \,x^{2}-4 \sqrt {d}\, \mathrm {log}\left (\sqrt {e x +d}-\sqrt {d}\right ) c^{3} d \,x^{2}-3 \sqrt {d}\, \mathrm {log}\left (\sqrt {e x +d}+\sqrt {d}\right ) b^{2} c e x +4 \sqrt {d}\, \mathrm {log}\left (\sqrt {e x +d}+\sqrt {d}\right ) b \,c^{2} d x -3 \sqrt {d}\, \mathrm {log}\left (\sqrt {e x +d}+\sqrt {d}\right ) b \,c^{2} e \,x^{2}+4 \sqrt {d}\, \mathrm {log}\left (\sqrt {e x +d}+\sqrt {d}\right ) c^{3} d \,x^{2}}{2 b^{3} c x \left (c x +b \right )} \] Input:

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

Output:

(2*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d) 
))*b**2*e*x - 8*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sq 
rt(b*e - c*d)))*b*c*d*x + 2*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c) 
/(sqrt(c)*sqrt(b*e - c*d)))*b*c*e*x**2 - 8*sqrt(c)*sqrt(b*e - c*d)*atan((s 
qrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*c**2*d*x**2 - 2*sqrt(d + e*x)*b 
**2*c*d + 2*sqrt(d + e*x)*b**2*c*e*x - 4*sqrt(d + e*x)*b*c**2*d*x + 3*sqrt 
(d)*log(sqrt(d + e*x) - sqrt(d))*b**2*c*e*x - 4*sqrt(d)*log(sqrt(d + e*x) 
- sqrt(d))*b*c**2*d*x + 3*sqrt(d)*log(sqrt(d + e*x) - sqrt(d))*b*c**2*e*x* 
*2 - 4*sqrt(d)*log(sqrt(d + e*x) - sqrt(d))*c**3*d*x**2 - 3*sqrt(d)*log(sq 
rt(d + e*x) + sqrt(d))*b**2*c*e*x + 4*sqrt(d)*log(sqrt(d + e*x) + sqrt(d)) 
*b*c**2*d*x - 3*sqrt(d)*log(sqrt(d + e*x) + sqrt(d))*b*c**2*e*x**2 + 4*sqr 
t(d)*log(sqrt(d + e*x) + sqrt(d))*c**3*d*x**2)/(2*b**3*c*x*(b + c*x))