\(\int \frac {(d+e x)^{5/2}}{b x+c x^2} \, dx\) [106]

Optimal result

Integrand size = 21, antiderivative size = 118 \[ \int \frac {(d+e x)^{5/2}}{b x+c x^2} \, dx=\frac {2 e (2 c d-b e) \sqrt {d+e x}}{c^2}+\frac {2 e (d+e x)^{3/2}}{3 c}-\frac {2 d^{5/2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b}+\frac {2 (c d-b e)^{5/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b c^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.91 \[ \int \frac {(d+e x)^{5/2}}{b x+c x^2} \, dx=\frac {2 e \sqrt {d+e x} (7 c d-3 b e+c e x)}{3 c^2}+\frac {2 (-c d+b e)^{5/2} \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{b c^{5/2}}-\frac {2 d^{5/2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b} \] Input:

Integrate[(d + e*x)^(5/2)/(b*x + c*x^2),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.17, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1146, 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.

\(\displaystyle \int \frac {(d+e x)^{5/2}}{b x+c x^2} \, dx\)

\(\Big \downarrow \) 1146

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

\(\Big \downarrow \) 1196

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

\(\Big \downarrow \) 1197

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 221

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

Input:

Int[(d + e*x)^(5/2)/(b*x + c*x^2),x]
 

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 1146

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

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 = 0.54 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.97

method	result	size
pseudoelliptic	\(-\frac {2 \left (-\left (b e -c d \right )^{3} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )+\left (\operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) d^{\frac {5}{2}} c^{2}+e \sqrt {e x +d}\, \left (\frac {\left (-e x -7 d \right ) c}{3}+b e \right ) b \right ) \sqrt {c \left (b e -c d \right )}\right )}{\sqrt {c \left (b e -c d \right )}\, c^{2} b}\)	\(114\)
derivativedivides	\(2 e \left (-\frac {-\frac {c \left (e x +d \right )^{\frac {3}{2}}}{3}+b e \sqrt {e x +d}-2 c d \sqrt {e x +d}}{c^{2}}+\frac {\left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{c^{2} b e \sqrt {c \left (b e -c d \right )}}-\frac {d^{\frac {5}{2}} \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b e}\right )\)	\(145\)
default	\(2 e \left (-\frac {-\frac {c \left (e x +d \right )^{\frac {3}{2}}}{3}+b e \sqrt {e x +d}-2 c d \sqrt {e x +d}}{c^{2}}+\frac {\left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{c^{2} b e \sqrt {c \left (b e -c d \right )}}-\frac {d^{\frac {5}{2}} \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b e}\right )\)	\(145\)

Input:

int((e*x+d)^(5/2)/(c*x^2+b*x),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 592, normalized size of antiderivative = 5.02 \[ \int \frac {(d+e x)^{5/2}}{b x+c x^2} \, dx=\left [\frac {3 \, c^{2} d^{\frac {5}{2}} \log \left (\frac {e x - 2 \, \sqrt {e x + d} \sqrt {d} + 2 \, d}{x}\right ) + 3 \, {\left (c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} \sqrt {\frac {c d - b e}{c}} \log \left (\frac {c e x + 2 \, c d - b e + 2 \, \sqrt {e x + d} c \sqrt {\frac {c d - b e}{c}}}{c x + b}\right ) + 2 \, {\left (b c e^{2} x + 7 \, b c d e - 3 \, b^{2} e^{2}\right )} \sqrt {e x + d}}{3 \, b c^{2}}, \frac {3 \, c^{2} d^{\frac {5}{2}} \log \left (\frac {e x - 2 \, \sqrt {e x + d} \sqrt {d} + 2 \, d}{x}\right ) + 6 \, {\left (c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} \sqrt {-\frac {c d - b e}{c}} \arctan \left (-\frac {\sqrt {e x + d} c \sqrt {-\frac {c d - b e}{c}}}{c d - b e}\right ) + 2 \, {\left (b c e^{2} x + 7 \, b c d e - 3 \, b^{2} e^{2}\right )} \sqrt {e x + d}}{3 \, b c^{2}}, \frac {6 \, c^{2} \sqrt {-d} d^{2} \arctan \left (\frac {\sqrt {-d}}{\sqrt {e x + d}}\right ) + 3 \, {\left (c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} \sqrt {\frac {c d - b e}{c}} \log \left (\frac {c e x + 2 \, c d - b e + 2 \, \sqrt {e x + d} c \sqrt {\frac {c d - b e}{c}}}{c x + b}\right ) + 2 \, {\left (b c e^{2} x + 7 \, b c d e - 3 \, b^{2} e^{2}\right )} \sqrt {e x + d}}{3 \, b c^{2}}, \frac {2 \, {\left (3 \, c^{2} \sqrt {-d} d^{2} \arctan \left (\frac {\sqrt {-d}}{\sqrt {e x + d}}\right ) + 3 \, {\left (c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} \sqrt {-\frac {c d - b e}{c}} \arctan \left (-\frac {\sqrt {e x + d} c \sqrt {-\frac {c d - b e}{c}}}{c d - b e}\right ) + {\left (b c e^{2} x + 7 \, b c d e - 3 \, b^{2} e^{2}\right )} \sqrt {e x + d}\right )}}{3 \, b c^{2}}\right ] \] Input:

integrate((e*x+d)^(5/2)/(c*x^2+b*x),x, algorithm="fricas")
 

Output:

[1/3*(3*c^2*d^(5/2)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 3*(c^2* 
d^2 - 2*b*c*d*e + b^2*e^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)) + 2*(b*c*e^2*x + 7*b*c*d 
*e - 3*b^2*e^2)*sqrt(e*x + d))/(b*c^2), 1/3*(3*c^2*d^(5/2)*log((e*x - 2*sq 
rt(e*x + d)*sqrt(d) + 2*d)/x) + 6*(c^2*d^2 - 2*b*c*d*e + b^2*e^2)*sqrt(-(c 
*d - b*e)/c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + 2 
*(b*c*e^2*x + 7*b*c*d*e - 3*b^2*e^2)*sqrt(e*x + d))/(b*c^2), 1/3*(6*c^2*sq 
rt(-d)*d^2*arctan(sqrt(-d)/sqrt(e*x + d)) + 3*(c^2*d^2 - 2*b*c*d*e + b^2*e 
^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)) + 2*(b*c*e^2*x + 7*b*c*d*e - 3*b^2*e^2)*sqrt(e* 
x + d))/(b*c^2), 2/3*(3*c^2*sqrt(-d)*d^2*arctan(sqrt(-d)/sqrt(e*x + d)) + 
3*(c^2*d^2 - 2*b*c*d*e + b^2*e^2)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x + 
d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + (b*c*e^2*x + 7*b*c*d*e - 3*b^2*e^ 
2)*sqrt(e*x + d))/(b*c^2)]
 

Sympy [A] (verification not implemented)

Time = 2.54 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.67 \[ \int \frac {(d+e x)^{5/2}}{b x+c x^2} \, dx=\begin {cases} \frac {2 \left (\frac {e^{2} \left (d + e x\right )^{\frac {3}{2}}}{3 c} + \frac {\sqrt {d + e x} \left (- b e^{3} + 2 c d e^{2}\right )}{c^{2}} + \frac {d^{3} e \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d}} \right )}}{b \sqrt {- d}} + \frac {e \left (b e - c d\right )^{3} \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {b e - c d}{c}}} \right )}}{b c^{3} \sqrt {\frac {b e - c d}{c}}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (- \frac {2 c \left (\begin {cases} \frac {\frac {b}{2 c} + x}{b} & \text {for}\: c = 0 \\- \frac {\log {\left (b - 2 c \left (\frac {b}{2 c} + x\right ) \right )}}{2 c} & \text {otherwise} \end {cases}\right )}{b} - \frac {2 c \left (\begin {cases} \frac {\frac {b}{2 c} + x}{b} & \text {for}\: c = 0 \\\frac {\log {\left (b + 2 c \left (\frac {b}{2 c} + x\right ) \right )}}{2 c} & \text {otherwise} \end {cases}\right )}{b}\right ) & \text {otherwise} \end {cases} \] Input:

integrate((e*x+d)**(5/2)/(c*x**2+b*x),x)
 

Output:

Piecewise((2*(e**2*(d + e*x)**(3/2)/(3*c) + sqrt(d + e*x)*(-b*e**3 + 2*c*d 
*e**2)/c**2 + d**3*e*atan(sqrt(d + e*x)/sqrt(-d))/(b*sqrt(-d)) + e*(b*e - 
c*d)**3*atan(sqrt(d + e*x)/sqrt((b*e - c*d)/c))/(b*c**3*sqrt((b*e - c*d)/c 
)))/e, Ne(e, 0)), (d**(5/2)*(-2*c*Piecewise(((b/(2*c) + x)/b, Eq(c, 0)), ( 
-log(b - 2*c*(b/(2*c) + x))/(2*c), True))/b - 2*c*Piecewise(((b/(2*c) + x) 
/b, Eq(c, 0)), (log(b + 2*c*(b/(2*c) + x))/(2*c), True))/b), True))
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^{5/2}}{b x+c x^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x+d)^(5/2)/(c*x^2+b*x),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 = 0.16 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.31 \[ \int \frac {(d+e x)^{5/2}}{b x+c x^2} \, dx=\frac {2 \, d^{3} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-d}}\right )}{b \sqrt {-d}} - \frac {2 \, {\left (c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \arctan \left (\frac {\sqrt {e x + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{\sqrt {-c^{2} d + b c e} b c^{2}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} c^{2} e + 6 \, \sqrt {e x + d} c^{2} d e - 3 \, \sqrt {e x + d} b c e^{2}\right )}}{3 \, c^{3}} \] Input:

integrate((e*x+d)^(5/2)/(c*x^2+b*x),x, algorithm="giac")
 

Output:

2*d^3*arctan(sqrt(e*x + d)/sqrt(-d))/(b*sqrt(-d)) - 2*(c^3*d^3 - 3*b*c^2*d 
^2*e + 3*b^2*c*d*e^2 - b^3*e^3)*arctan(sqrt(e*x + d)*c/sqrt(-c^2*d + b*c*e 
))/(sqrt(-c^2*d + b*c*e)*b*c^2) + 2/3*((e*x + d)^(3/2)*c^2*e + 6*sqrt(e*x 
+ d)*c^2*d*e - 3*sqrt(e*x + d)*b*c*e^2)/c^3
 

Mupad [B] (verification not implemented)

Time = 5.67 (sec) , antiderivative size = 2048, normalized size of antiderivative = 17.36 \[ \int \frac {(d+e x)^{5/2}}{b x+c x^2} \, dx=\text {Too large to display} \] Input:

int((d + e*x)^(5/2)/(b*x + c*x^2),x)
                                                                                    
                                                                                    
 

Output:

(atan(((((8*(d + e*x)^(1/2)*(b^6*e^8 + 2*c^6*d^6*e^2 - 6*b*c^5*d^5*e^3 + 1 
5*b^2*c^4*d^4*e^4 - 20*b^3*c^3*d^3*e^5 + 15*b^4*c^2*d^2*e^6 - 6*b^5*c*d*e^ 
7))/c^3 + (((8*(b^4*c^3*d*e^5 + 2*b^2*c^5*d^3*e^3 - 3*b^3*c^4*d^2*e^4))/c^ 
3 + (8*(b^3*c^5*e^3 - 2*b^2*c^6*d*e^2)*(d^5)^(1/2)*(d + e*x)^(1/2))/(b*c^3 
))*(d^5)^(1/2))/b)*(d^5)^(1/2)*1i)/b + (((8*(d + e*x)^(1/2)*(b^6*e^8 + 2*c 
^6*d^6*e^2 - 6*b*c^5*d^5*e^3 + 15*b^2*c^4*d^4*e^4 - 20*b^3*c^3*d^3*e^5 + 1 
5*b^4*c^2*d^2*e^6 - 6*b^5*c*d*e^7))/c^3 - (((8*(b^4*c^3*d*e^5 + 2*b^2*c^5* 
d^3*e^3 - 3*b^3*c^4*d^2*e^4))/c^3 - (8*(b^3*c^5*e^3 - 2*b^2*c^6*d*e^2)*(d^ 
5)^(1/2)*(d + e*x)^(1/2))/(b*c^3))*(d^5)^(1/2))/b)*(d^5)^(1/2)*1i)/b)/((16 
*(b^5*d^3*e^8 - 3*c^5*d^8*e^3 + 12*b*c^4*d^7*e^4 - 6*b^4*c*d^4*e^7 - 19*b^ 
2*c^3*d^6*e^5 + 15*b^3*c^2*d^5*e^6))/c^3 - (((8*(d + e*x)^(1/2)*(b^6*e^8 + 
 2*c^6*d^6*e^2 - 6*b*c^5*d^5*e^3 + 15*b^2*c^4*d^4*e^4 - 20*b^3*c^3*d^3*e^5 
 + 15*b^4*c^2*d^2*e^6 - 6*b^5*c*d*e^7))/c^3 + (((8*(b^4*c^3*d*e^5 + 2*b^2* 
c^5*d^3*e^3 - 3*b^3*c^4*d^2*e^4))/c^3 + (8*(b^3*c^5*e^3 - 2*b^2*c^6*d*e^2) 
*(d^5)^(1/2)*(d + e*x)^(1/2))/(b*c^3))*(d^5)^(1/2))/b)*(d^5)^(1/2))/b + (( 
(8*(d + e*x)^(1/2)*(b^6*e^8 + 2*c^6*d^6*e^2 - 6*b*c^5*d^5*e^3 + 15*b^2*c^4 
*d^4*e^4 - 20*b^3*c^3*d^3*e^5 + 15*b^4*c^2*d^2*e^6 - 6*b^5*c*d*e^7))/c^3 - 
 (((8*(b^4*c^3*d*e^5 + 2*b^2*c^5*d^3*e^3 - 3*b^3*c^4*d^2*e^4))/c^3 - (8*(b 
^3*c^5*e^3 - 2*b^2*c^6*d*e^2)*(d^5)^(1/2)*(d + e*x)^(1/2))/(b*c^3))*(d^5)^ 
(1/2))/b)*(d^5)^(1/2))/b))*(d^5)^(1/2)*2i)/b + (2*e*(d + e*x)^(3/2))/(3...
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.89 \[ \int \frac {(d+e x)^{5/2}}{b x+c x^2} \, dx=\frac {6 \sqrt {c}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c}{\sqrt {c}\, \sqrt {b e -c d}}\right ) b^{2} e^{2}-12 \sqrt {c}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c}{\sqrt {c}\, \sqrt {b e -c d}}\right ) b c d e +6 \sqrt {c}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c}{\sqrt {c}\, \sqrt {b e -c d}}\right ) c^{2} d^{2}-6 \sqrt {e x +d}\, b^{2} c \,e^{2}+14 \sqrt {e x +d}\, b \,c^{2} d e +2 \sqrt {e x +d}\, b \,c^{2} e^{2} x +3 \sqrt {d}\, \mathrm {log}\left (\sqrt {e x +d}-\sqrt {d}\right ) c^{3} d^{2}-3 \sqrt {d}\, \mathrm {log}\left (\sqrt {e x +d}+\sqrt {d}\right ) c^{3} d^{2}}{3 b \,c^{3}} \] Input:

int((e*x+d)^(5/2)/(c*x^2+b*x),x)
 

Output:

(6*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d) 
))*b**2*e**2 - 12*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)* 
sqrt(b*e - c*d)))*b*c*d*e + 6*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)* 
c)/(sqrt(c)*sqrt(b*e - c*d)))*c**2*d**2 - 6*sqrt(d + e*x)*b**2*c*e**2 + 14 
*sqrt(d + e*x)*b*c**2*d*e + 2*sqrt(d + e*x)*b*c**2*e**2*x + 3*sqrt(d)*log( 
sqrt(d + e*x) - sqrt(d))*c**3*d**2 - 3*sqrt(d)*log(sqrt(d + e*x) + sqrt(d) 
)*c**3*d**2)/(3*b*c**3)