\(\int \frac {(d+e x)^{9/2}}{(b x+c x^2)^3} \, dx\) [122]

Optimal result

Integrand size = 21, antiderivative size = 312 \[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^3} \, dx=\frac {3 (c d-b e) (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) \sqrt {d+e x}}{4 b^4 c^2 (b+c x)}+\frac {(c d-b e) \left (12 c^2 d^2-17 b c d e+2 b^2 e^2\right ) (d+e x)^{3/2}}{4 b^3 c (b+c x)^2}+\frac {d (8 c d-11 b e) (d+e x)^{5/2}}{4 b^2 x (b+c x)^2}-\frac {d (d+e x)^{7/2}}{2 b x^2 (b+c x)^2}-\frac {3 d^{5/2} \left (16 c^2 d^2-36 b c d e+21 b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5}+\frac {3 (c d-b e)^{5/2} \left (16 c^2 d^2+4 b c d e+b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 c^{5/2}} \] Output:

3/4*(-b*e+c*d)*(-b*e+2*c*d)*(-b^2*e^2-4*b*c*d*e+4*c^2*d^2)*(e*x+d)^(1/2)/b 
^4/c^2/(c*x+b)+1/4*(-b*e+c*d)*(2*b^2*e^2-17*b*c*d*e+12*c^2*d^2)*(e*x+d)^(3 
/2)/b^3/c/(c*x+b)^2+1/4*d*(-11*b*e+8*c*d)*(e*x+d)^(5/2)/b^2/x/(c*x+b)^2-1/ 
2*d*(e*x+d)^(7/2)/b/x^2/(c*x+b)^2-3/4*d^(5/2)*(21*b^2*e^2-36*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)^(5/2)*(b^2*e^2+4* 
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^ 
(5/2)
 

Mathematica [A] (verified)

Time = 1.28 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.88 \[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^3} \, dx=\frac {\frac {b \sqrt {d+e x} \left (-3 b^5 e^4 x^2+24 c^5 d^4 x^3+12 b c^4 d^3 x^2 (3 d-4 e x)-5 b^4 c e^3 x^2 (d+e x)+b^2 c^3 d^2 x \left (8 d^2-73 d e x+21 e^2 x^2\right )+b^3 c^2 d \left (-2 d^3-17 d^2 e x+33 d e^2 x^2+3 e^3 x^3\right )\right )}{c^2 x^2 (b+c x)^2}+\frac {3 (-c d+b e)^{5/2} \left (16 c^2 d^2+4 b c d e+b^2 e^2\right ) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{c^{5/2}}-3 d^{5/2} \left (16 c^2 d^2-36 b c d e+21 b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5} \] Input:

Integrate[(d + e*x)^(9/2)/(b*x + c*x^2)^3,x]
 

Output:

((b*Sqrt[d + e*x]*(-3*b^5*e^4*x^2 + 24*c^5*d^4*x^3 + 12*b*c^4*d^3*x^2*(3*d 
 - 4*e*x) - 5*b^4*c*e^3*x^2*(d + e*x) + b^2*c^3*d^2*x*(8*d^2 - 73*d*e*x + 
21*e^2*x^2) + b^3*c^2*d*(-2*d^3 - 17*d^2*e*x + 33*d*e^2*x^2 + 3*e^3*x^3))) 
/(c^2*x^2*(b + c*x)^2) + (3*(-(c*d) + b*e)^(5/2)*(16*c^2*d^2 + 4*b*c*d*e + 
 b^2*e^2)*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/c^(5/2) - 3* 
d^(5/2)*(16*c^2*d^2 - 36*b*c*d*e + 21*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[ 
d]])/(4*b^5)
 

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1164, 27, 1233, 27, 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.

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

\(\Big \downarrow \) 1164

\(\displaystyle -\frac {\int \frac {(d+e x)^{5/2} (d (12 c d-13 b e)-e (2 c d-b e) x)}{2 \left (c x^2+b x\right )^2}dx}{2 b^2}-\frac {(d+e x)^{7/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {(d+e x)^{5/2} (d (12 c d-13 b e)-e (2 c d-b e) x)}{\left (c x^2+b x\right )^2}dx}{4 b^2}-\frac {(d+e x)^{7/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1233

\(\displaystyle -\frac {\frac {\int -\frac {3 \sqrt {d+e x} \left (c d^2 \left (16 c^2 d^2-36 b c e d+21 b^2 e^2\right )-e (2 c d-b e) \left (4 c^2 d^2-4 b c e d-b^2 e^2\right ) x\right )}{2 \left (c x^2+b x\right )}dx}{b^2 c}-\frac {(d+e x)^{3/2} \left (x (2 c d-b e) \left (-b^2 e^2-12 b c d e+12 c^2 d^2\right )+b c d^2 (12 c d-13 b e)\right )}{b^2 c \left (b x+c x^2\right )}}{4 b^2}-\frac {(d+e x)^{7/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {3 \int \frac {\sqrt {d+e x} \left (c d^2 \left (16 c^2 d^2-36 b c e d+21 b^2 e^2\right )-e (2 c d-b e) \left (4 c^2 d^2-4 b c e d-b^2 e^2\right ) x\right )}{c x^2+b x}dx}{2 b^2 c}-\frac {(d+e x)^{3/2} \left (x (2 c d-b e) \left (-b^2 e^2-12 b c d e+12 c^2 d^2\right )+b c d^2 (12 c d-13 b e)\right )}{b^2 c \left (b x+c x^2\right )}}{4 b^2}-\frac {(d+e x)^{7/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1196

\(\displaystyle -\frac {-\frac {3 \left (\frac {\int \frac {c^2 \left (16 c^2 d^2-36 b c e d+21 b^2 e^2\right ) d^3+e \left (8 c^4 d^4-16 b c^3 e d^3+7 b^2 c^2 e^2 d^2+b^3 c e^3 d+b^4 e^4\right ) x}{\sqrt {d+e x} \left (c x^2+b x\right )}dx}{c}-\frac {2 e \sqrt {d+e x} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )}{c}\right )}{2 b^2 c}-\frac {(d+e x)^{3/2} \left (x (2 c d-b e) \left (-b^2 e^2-12 b c d e+12 c^2 d^2\right )+b c d^2 (12 c d-13 b e)\right )}{b^2 c \left (b x+c x^2\right )}}{4 b^2}-\frac {(d+e x)^{7/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1197

\(\displaystyle -\frac {-\frac {3 \left (\frac {2 \int \frac {e \left (d (c d-b e) (2 c d-b e) \left (4 c^2 d^2-4 b c e d-b^2 e^2\right )+\left (8 c^4 d^4-16 b c^3 e d^3+7 b^2 c^2 e^2 d^2+b^3 c e^3 d+b^4 e^4\right ) (d+e x)\right )}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{c}-\frac {2 e \sqrt {d+e x} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )}{c}\right )}{2 b^2 c}-\frac {(d+e x)^{3/2} \left (x (2 c d-b e) \left (-b^2 e^2-12 b c d e+12 c^2 d^2\right )+b c d^2 (12 c d-13 b e)\right )}{b^2 c \left (b x+c x^2\right )}}{4 b^2}-\frac {(d+e x)^{7/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {3 \left (\frac {2 e \int \frac {d (c d-b e) (2 c d-b e) \left (4 c^2 d^2-4 b c e d-b^2 e^2\right )+\left (8 c^4 d^4-16 b c^3 e d^3+7 b^2 c^2 e^2 d^2+b^3 c e^3 d+b^4 e^4\right ) (d+e x)}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{c}-\frac {2 e \sqrt {d+e x} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )}{c}\right )}{2 b^2 c}-\frac {(d+e x)^{3/2} \left (x (2 c d-b e) \left (-b^2 e^2-12 b c d e+12 c^2 d^2\right )+b c d^2 (12 c d-13 b e)\right )}{b^2 c \left (b x+c x^2\right )}}{4 b^2}-\frac {(d+e x)^{7/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1480

\(\displaystyle -\frac {-\frac {3 \left (\frac {2 e \left (\frac {c^3 d^3 \left (21 b^2 e^2-36 b c d e+16 c^2 d^2\right ) \int \frac {1}{c (d+e x)-c d}d\sqrt {d+e x}}{b e}-\frac {(c d-b e)^3 \left (b^2 e^2+4 b c d e+16 c^2 d^2\right ) \int \frac {1}{-c d+b e+c (d+e x)}d\sqrt {d+e x}}{b e}\right )}{c}-\frac {2 e \sqrt {d+e x} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )}{c}\right )}{2 b^2 c}-\frac {(d+e x)^{3/2} \left (x (2 c d-b e) \left (-b^2 e^2-12 b c d e+12 c^2 d^2\right )+b c d^2 (12 c d-13 b e)\right )}{b^2 c \left (b x+c x^2\right )}}{4 b^2}-\frac {(d+e x)^{7/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {-\frac {3 \left (\frac {2 e \left (\frac {(c d-b e)^{5/2} \left (b^2 e^2+4 b c d e+16 c^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b \sqrt {c} e}-\frac {c^2 d^{5/2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (21 b^2 e^2-36 b c d e+16 c^2 d^2\right )}{b e}\right )}{c}-\frac {2 e \sqrt {d+e x} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )}{c}\right )}{2 b^2 c}-\frac {(d+e x)^{3/2} \left (x (2 c d-b e) \left (-b^2 e^2-12 b c d e+12 c^2 d^2\right )+b c d^2 (12 c d-13 b e)\right )}{b^2 c \left (b x+c x^2\right )}}{4 b^2}-\frac {(d+e x)^{7/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}\)

Input:

Int[(d + e*x)^(9/2)/(b*x + c*x^2)^3,x]
 

Output:

-1/2*((d + e*x)^(7/2)*(b*d + (2*c*d - b*e)*x))/(b^2*(b*x + c*x^2)^2) - (-( 
((d + e*x)^(3/2)*(b*c*d^2*(12*c*d - 13*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))) - (3*((-2*e*(2*c*d - b*e) 
*(4*c^2*d^2 - 4*b*c*d*e - b^2*e^2)*Sqrt[d + e*x])/c + (2*e*(-((c^2*d^(5/2) 
*(16*c^2*d^2 - 36*b*c*d*e + 21*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b 
*e)) + ((c*d - b*e)^(5/2)*(16*c^2*d^2 + 4*b*c*d*e + b^2*e^2)*ArcTanh[(Sqrt 
[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*Sqrt[c]*e)))/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 = 0.75 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(2 e^{5} \left (\frac {\left (b e -c d \right )^{3} \left (\frac {-\frac {b e \left (5 b e +12 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 c}-\frac {3 b e \left (b^{2} e^{2}+3 b c d e -4 c^{2} d^{2}\right ) \sqrt {e x +d}}{8 c^{2}}}{\left (\left (e x +d \right ) c +b e -c d \right )^{2}}+\frac {3 \left (b^{2} e^{2}+4 b c d e +16 c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{8 c^{2} \sqrt {c \left (b e -c d \right )}}\right )}{b^{5} e^{5}}-\frac {d^{3} \left (\frac {\left (\frac {17}{8} b^{2} e^{2}-\frac {3}{2} b c d e \right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {15}{8} d \,e^{2} b^{2}+\frac {3}{2} d^{2} e b c \right ) \sqrt {e x +d}}{e^{2} x^{2}}+\frac {3 \left (21 b^{2} e^{2}-36 b c d e +16 c^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 \sqrt {d}}\right )}{b^{5} e^{5}}\right )\)	\(273\)
default	\(2 e^{5} \left (\frac {\left (b e -c d \right )^{3} \left (\frac {-\frac {b e \left (5 b e +12 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 c}-\frac {3 b e \left (b^{2} e^{2}+3 b c d e -4 c^{2} d^{2}\right ) \sqrt {e x +d}}{8 c^{2}}}{\left (\left (e x +d \right ) c +b e -c d \right )^{2}}+\frac {3 \left (b^{2} e^{2}+4 b c d e +16 c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{8 c^{2} \sqrt {c \left (b e -c d \right )}}\right )}{b^{5} e^{5}}-\frac {d^{3} \left (\frac {\left (\frac {17}{8} b^{2} e^{2}-\frac {3}{2} b c d e \right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {15}{8} d \,e^{2} b^{2}+\frac {3}{2} d^{2} e b c \right ) \sqrt {e x +d}}{e^{2} x^{2}}+\frac {3 \left (21 b^{2} e^{2}-36 b c d e +16 c^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 \sqrt {d}}\right )}{b^{5} e^{5}}\right )\)	\(273\)
pseudoelliptic	\(-\frac {3 \left (-\frac {\left (b e -c d \right )^{3} x^{2} \left (c x +b \right )^{2} \sqrt {d}\, \left (b^{2} e^{2}+4 b c d e +16 c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{2}+\frac {\sqrt {c \left (b e -c d \right )}\, \left (\frac {3 x^{2} \left (c x +b \right )^{2} c^{2} \left (21 b^{2} e^{2}-36 b c d e +16 c^{2} d^{2}\right ) d^{3} \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2}+\left (-12 c^{5} d^{4} x^{3}-18 x^{2} \left (-\frac {4 e x}{3}+d \right ) b \,d^{3} c^{4}-4 x \left (\frac {21}{8} e^{2} x^{2}-\frac {73}{8} d e x +d^{2}\right ) b^{2} d^{2} c^{3}+b^{3} d \left (d^{3}+\frac {17}{2} d^{2} e x -\frac {33}{2} d \,e^{2} x^{2}-\frac {3}{2} e^{3} x^{3}\right ) c^{2}+\frac {5 b^{4} e^{3} x^{2} \left (e x +d \right ) c}{2}+\frac {3 b^{5} e^{4} x^{2}}{2}\right ) \sqrt {d}\, \sqrt {e x +d}\, b \right )}{3}\right )}{2 \sqrt {d}\, \sqrt {c \left (b e -c d \right )}\, b^{5} x^{2} \left (c x +b \right )^{2} c^{2}}\)	\(307\)
risch	\(-\frac {d^{3} \sqrt {e x +d}\, \left (17 b e x -12 c d x +2 b d \right )}{4 b^{4} x^{2}}+\frac {e \left (-\frac {3 d^{\frac {5}{2}} \left (21 b^{2} e^{2}-36 b c d e +16 c^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b e}+\frac {\frac {8 \left (-\frac {b e \left (5 b^{4} e^{4}-3 d \,e^{3} b^{3} c -21 d^{2} e^{2} b^{2} c^{2}+31 d^{3} e b \,c^{3}-12 d^{4} c^{4}\right ) \left (e x +d \right )^{\frac {3}{2}}}{8 c}-\frac {3 b e \left (b^{5} e^{5}-10 b^{3} d^{2} e^{3} c^{2}+20 b^{2} c^{3} d^{3} e^{2}-15 b \,c^{4} d^{4} e +4 d^{5} c^{5}\right ) \sqrt {e x +d}}{8 c^{2}}\right )}{\left (\left (e x +d \right ) c +b e -c d \right )^{2}}+\frac {3 \left (b^{5} e^{5}+b^{4} c d \,e^{4}+7 b^{3} d^{2} e^{3} c^{2}-37 b^{2} c^{3} d^{3} e^{2}+44 b \,c^{4} d^{4} e -16 d^{5} c^{5}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{c^{2} \sqrt {c \left (b e -c d \right )}}}{b e}\right )}{4 b^{4}}\)	\(351\)

Input:

int((e*x+d)^(9/2)/(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)
 

Output:

2*e^5*((b*e-c*d)^3/b^5/e^5*((-1/8*b*e*(5*b*e+12*c*d)/c*(e*x+d)^(3/2)-3/8*b 
/c^2*e*(b^2*e^2+3*b*c*d*e-4*c^2*d^2)*(e*x+d)^(1/2))/((e*x+d)*c+b*e-c*d)^2+ 
3/8*(b^2*e^2+4*b*c*d*e+16*c^2*d^2)/c^2/(c*(b*e-c*d))^(1/2)*arctan(c*(e*x+d 
)^(1/2)/(c*(b*e-c*d))^(1/2)))-d^3/b^5/e^5*(((17/8*b^2*e^2-3/2*b*c*d*e)*(e* 
x+d)^(3/2)+(-15/8*d*e^2*b^2+3/2*d^2*e*b*c)*(e*x+d)^(1/2))/e^2/x^2+3/8*(21* 
b^2*e^2-36*b*c*d*e+16*c^2*d^2)/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 583 vs. \(2 (276) = 552\).

Time = 0.96 (sec) , antiderivative size = 2367, normalized size of antiderivative = 7.59 \[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^(9/2)/(c*x^2+b*x)^3,x, algorithm="fricas")
 

Output:

[1/8*(3*((16*c^6*d^4 - 28*b*c^5*d^3*e + 9*b^2*c^4*d^2*e^2 + 2*b^3*c^3*d*e^ 
3 + b^4*c^2*e^4)*x^4 + 2*(16*b*c^5*d^4 - 28*b^2*c^4*d^3*e + 9*b^3*c^3*d^2* 
e^2 + 2*b^4*c^2*d*e^3 + b^5*c*e^4)*x^3 + (16*b^2*c^4*d^4 - 28*b^3*c^3*d^3* 
e + 9*b^4*c^2*d^2*e^2 + 2*b^5*c*d*e^3 + b^6*e^4)*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^6*d^4 - 36*b*c^5*d^3*e + 21*b^2*c^4*d^2*e^2)*x^4 + 2*(16*b*c 
^5*d^4 - 36*b^2*c^4*d^3*e + 21*b^3*c^3*d^2*e^2)*x^3 + (16*b^2*c^4*d^4 - 36 
*b^3*c^3*d^3*e + 21*b^4*c^2*d^2*e^2)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + 
d)*sqrt(d) + 2*d)/x) - 2*(2*b^4*c^2*d^4 - (24*b*c^5*d^4 - 48*b^2*c^4*d^3*e 
 + 21*b^3*c^3*d^2*e^2 + 3*b^4*c^2*d*e^3 - 5*b^5*c*e^4)*x^3 - (36*b^2*c^4*d 
^4 - 73*b^3*c^3*d^3*e + 33*b^4*c^2*d^2*e^2 - 5*b^5*c*d*e^3 - 3*b^6*e^4)*x^ 
2 - (8*b^3*c^3*d^4 - 17*b^4*c^2*d^3*e)*x)*sqrt(e*x + d))/(b^5*c^4*x^4 + 2* 
b^6*c^3*x^3 + b^7*c^2*x^2), 1/8*(6*((16*c^6*d^4 - 28*b*c^5*d^3*e + 9*b^2*c 
^4*d^2*e^2 + 2*b^3*c^3*d*e^3 + b^4*c^2*e^4)*x^4 + 2*(16*b*c^5*d^4 - 28*b^2 
*c^4*d^3*e + 9*b^3*c^3*d^2*e^2 + 2*b^4*c^2*d*e^3 + b^5*c*e^4)*x^3 + (16*b^ 
2*c^4*d^4 - 28*b^3*c^3*d^3*e + 9*b^4*c^2*d^2*e^2 + 2*b^5*c*d*e^3 + b^6*e^4 
)*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*((16*c^6*d^4 - 36*b*c^5*d^3*e + 21*b^2*c^4*d^2*e^2)*x^4 + 
2*(16*b*c^5*d^4 - 36*b^2*c^4*d^3*e + 21*b^3*c^3*d^2*e^2)*x^3 + (16*b^2*c^4 
*d^4 - 36*b^3*c^3*d^3*e + 21*b^4*c^2*d^2*e^2)*x^2)*sqrt(d)*log((e*x - 2...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate((e*x+d)**(9/2)/(c*x**2+b*x)**3,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x+d)^(9/2)/(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. 617 vs. \(2 (276) = 552\).

Time = 0.15 (sec) , antiderivative size = 617, normalized size of antiderivative = 1.98 \[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^3} \, dx=\frac {3 \, {\left (16 \, c^{2} d^{5} - 36 \, b c d^{4} e + 21 \, b^{2} d^{3} e^{2}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-d}}\right )}{4 \, b^{5} \sqrt {-d}} - \frac {3 \, {\left (16 \, c^{5} d^{5} - 44 \, b c^{4} d^{4} e + 37 \, b^{2} c^{3} d^{3} e^{2} - 7 \, b^{3} c^{2} d^{2} e^{3} - b^{4} c d e^{4} - b^{5} e^{5}\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^{2}} + \frac {24 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{5} d^{4} e - 72 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{5} d^{5} e + 72 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{5} d^{6} e - 24 \, \sqrt {e x + d} c^{5} d^{7} e - 48 \, {\left (e x + d\right )}^{\frac {7}{2}} b c^{4} d^{3} e^{2} + 180 \, {\left (e x + d\right )}^{\frac {5}{2}} b c^{4} d^{4} e^{2} - 216 \, {\left (e x + d\right )}^{\frac {3}{2}} b c^{4} d^{5} e^{2} + 84 \, \sqrt {e x + d} b c^{4} d^{6} e^{2} + 21 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{2} c^{3} d^{2} e^{3} - 136 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{2} c^{3} d^{3} e^{3} + 217 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{2} c^{3} d^{4} e^{3} - 102 \, \sqrt {e x + d} b^{2} c^{3} d^{5} e^{3} + 3 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{3} c^{2} d e^{4} + 24 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{3} c^{2} d^{2} e^{4} - 74 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{3} c^{2} d^{3} e^{4} + 45 \, \sqrt {e x + d} b^{3} c^{2} d^{4} e^{4} - 5 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{4} c e^{5} + 10 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{4} c d e^{5} - 5 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{4} c d^{2} e^{5} - 3 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{5} e^{6} + 6 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{5} d e^{6} - 3 \, \sqrt {e x + d} b^{5} d^{2} e^{6}}{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^{2}} \] Input:

integrate((e*x+d)^(9/2)/(c*x^2+b*x)^3,x, algorithm="giac")
 

Output:

3/4*(16*c^2*d^5 - 36*b*c*d^4*e + 21*b^2*d^3*e^2)*arctan(sqrt(e*x + d)/sqrt 
(-d))/(b^5*sqrt(-d)) - 3/4*(16*c^5*d^5 - 44*b*c^4*d^4*e + 37*b^2*c^3*d^3*e 
^2 - 7*b^3*c^2*d^2*e^3 - b^4*c*d*e^4 - b^5*e^5)*arctan(sqrt(e*x + d)*c/sqr 
t(-c^2*d + b*c*e))/(sqrt(-c^2*d + b*c*e)*b^5*c^2) + 1/4*(24*(e*x + d)^(7/2 
)*c^5*d^4*e - 72*(e*x + d)^(5/2)*c^5*d^5*e + 72*(e*x + d)^(3/2)*c^5*d^6*e 
- 24*sqrt(e*x + d)*c^5*d^7*e - 48*(e*x + d)^(7/2)*b*c^4*d^3*e^2 + 180*(e*x 
 + d)^(5/2)*b*c^4*d^4*e^2 - 216*(e*x + d)^(3/2)*b*c^4*d^5*e^2 + 84*sqrt(e* 
x + d)*b*c^4*d^6*e^2 + 21*(e*x + d)^(7/2)*b^2*c^3*d^2*e^3 - 136*(e*x + d)^ 
(5/2)*b^2*c^3*d^3*e^3 + 217*(e*x + d)^(3/2)*b^2*c^3*d^4*e^3 - 102*sqrt(e*x 
 + d)*b^2*c^3*d^5*e^3 + 3*(e*x + d)^(7/2)*b^3*c^2*d*e^4 + 24*(e*x + d)^(5/ 
2)*b^3*c^2*d^2*e^4 - 74*(e*x + d)^(3/2)*b^3*c^2*d^3*e^4 + 45*sqrt(e*x + d) 
*b^3*c^2*d^4*e^4 - 5*(e*x + d)^(7/2)*b^4*c*e^5 + 10*(e*x + d)^(5/2)*b^4*c* 
d*e^5 - 5*(e*x + d)^(3/2)*b^4*c*d^2*e^5 - 3*(e*x + d)^(5/2)*b^5*e^6 + 6*(e 
*x + d)^(3/2)*b^5*d*e^6 - 3*sqrt(e*x + d)*b^5*d^2*e^6)/(((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^2)
 

Mupad [B] (verification not implemented)

Time = 6.51 (sec) , antiderivative size = 3946, normalized size of antiderivative = 12.65 \[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

int((d + e*x)^(9/2)/(b*x + c*x^2)^3,x)
 

Output:

(((d + e*x)^(3/2)*(6*b^5*d*e^6 + 72*c^5*d^6*e - 216*b*c^4*d^5*e^2 - 5*b^4* 
c*d^2*e^5 + 217*b^2*c^3*d^4*e^3 - 74*b^3*c^2*d^3*e^4))/(4*b^4*c^2) - (3*(d 
 + e*x)^(1/2)*(8*c^5*d^7*e + b^5*d^2*e^6 - 28*b*c^4*d^6*e^2 + 34*b^2*c^3*d 
^5*e^3 - 15*b^3*c^2*d^4*e^4))/(4*b^4*c^2) + (e*(d + e*x)^(7/2)*(24*c^4*d^4 
 - 5*b^4*e^4 + 21*b^2*c^2*d^2*e^2 - 48*b*c^3*d^3*e + 3*b^3*c*d*e^3))/(4*b^ 
4*c) + ((b*e - 2*c*d)*(d + e*x)^(5/2)*(36*c^4*d^4*e - 3*b^4*e^5 - 72*b*c^3 
*d^3*e^2 + 32*b^2*c^2*d^2*e^3 + 4*b^3*c*d*e^4))/(4*b^4*c^2))/(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) + (atan(((((3*(d^5)^(1/2)*((3*b^14*c^3*d*e^7 
- 24*b^10*c^7*d^5*e^3 + 60*b^11*c^6*d^4*e^4 - 42*b^12*c^5*d^3*e^5 + 3*b^13 
*c^4*d^2*e^6)/(b^12*c^3) - (3*(64*b^11*c^5*e^3 - 128*b^10*c^6*d*e^2)*(d^5) 
^(1/2)*(d + e*x)^(1/2)*(21*b^2*e^2 + 16*c^2*d^2 - 36*b*c*d*e))/(64*b^13*c^ 
3))*(21*b^2*e^2 + 16*c^2*d^2 - 36*b*c*d*e))/(8*b^5) - ((d + e*x)^(1/2)*(9* 
b^10*e^12 + 4608*c^10*d^10*e^2 - 23040*b*c^9*d^9*e^3 + 45792*b^2*c^8*d^8*e 
^4 - 44928*b^3*c^7*d^7*e^5 + 21546*b^4*c^6*d^6*e^6 - 4158*b^5*c^5*d^5*e^7 
+ 567*b^6*c^4*d^4*e^8 - 540*b^7*c^3*d^3*e^9 + 135*b^8*c^2*d^2*e^10 + 18*b^ 
9*c*d*e^11))/(8*b^8*c^3))*(d^5)^(1/2)*(21*b^2*e^2 + 16*c^2*d^2 - 36*b*c*d* 
e)*3i)/(8*b^5) - (((3*(d^5)^(1/2)*((3*b^14*c^3*d*e^7 - 24*b^10*c^7*d^5*e^3 
 + 60*b^11*c^6*d^4*e^4 - 42*b^12*c^5*d^3*e^5 + 3*b^13*c^4*d^2*e^6)/(b^1...
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 1522, normalized size of antiderivative = 4.88 \[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int((e*x+d)^(9/2)/(c*x^2+b*x)^3,x)
 

Output:

(6*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d) 
))*b**6*e**4*x**2 + 12*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqr 
t(c)*sqrt(b*e - c*d)))*b**5*c*d*e**3*x**2 + 12*sqrt(c)*sqrt(b*e - c*d)*ata 
n((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**5*c*e**4*x**3 + 54*sqrt( 
c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**4* 
c**2*d**2*e**2*x**2 + 24*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(s 
qrt(c)*sqrt(b*e - c*d)))*b**4*c**2*d*e**3*x**3 + 6*sqrt(c)*sqrt(b*e - c*d) 
*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**4*c**2*e**4*x**4 - 1 
68*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d) 
))*b**3*c**3*d**3*e*x**2 + 108*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x) 
*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**3*c**3*d**2*e**2*x**3 + 12*sqrt(c)*sqrt( 
b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**3*c**3*d*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**4*d**4*x**2 - 336*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt 
(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**2*c**4*d**3*e*x**3 + 54*sqrt(c) 
*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**2*c* 
*4*d**2*e**2*x**4 + 192*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sq 
rt(c)*sqrt(b*e - c*d)))*b*c**5*d**4*x**3 - 168*sqrt(c)*sqrt(b*e - c*d)*ata 
n((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b*c**5*d**3*e*x**4 + 96*sqr 
t(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*...