\(\int \frac {1}{(d+e x)^2 (b x+c x^2)^{5/2}} \, dx\) [177]

Optimal result

Integrand size = 21, antiderivative size = 348 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 c d-5 b e}{3 b d^2 (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {\left (\frac {4 c^2}{b^2}-\frac {5 e^2}{d^2}\right ) x}{d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {c \left (16 c^3 d^3-16 b c^2 d^2 e-10 b^2 c d e^2+15 b^3 e^3\right ) x^2}{3 b^3 d^3 (c d-b e)^2 \left (b x+c x^2\right )^{3/2}}-\frac {e}{d (c d-b e) (d+e x) \left (b x+c x^2\right )^{3/2}}+\frac {c \left (32 c^4 d^4-64 b c^3 d^3 e+12 b^2 c^2 d^2 e^2+20 b^3 c d e^3-15 b^4 e^4\right ) x}{3 b^4 d^3 (c d-b e)^3 \sqrt {b x+c x^2}}+\frac {5 e^4 (2 c d-b e) \text {arctanh}\left (\frac {\sqrt {c d-b e} x}{\sqrt {d} \sqrt {b x+c x^2}}\right )}{d^{7/2} (c d-b e)^{7/2}} \] Output:

-1/3*(-5*b*e+2*c*d)/b/d^2/(-b*e+c*d)/(c*x^2+b*x)^(3/2)+(4*c^2/b^2-5*e^2/d^ 
2)*x/d/(-b*e+c*d)/(c*x^2+b*x)^(3/2)+1/3*c*(15*b^3*e^3-10*b^2*c*d*e^2-16*b* 
c^2*d^2*e+16*c^3*d^3)*x^2/b^3/d^3/(-b*e+c*d)^2/(c*x^2+b*x)^(3/2)-e/d/(-b*e 
+c*d)/(e*x+d)/(c*x^2+b*x)^(3/2)+1/3*c*(-15*b^4*e^4+20*b^3*c*d*e^3+12*b^2*c 
^2*d^2*e^2-64*b*c^3*d^3*e+32*c^4*d^4)*x/b^4/d^3/(-b*e+c*d)^3/(c*x^2+b*x)^( 
1/2)+5*e^4*(-b*e+2*c*d)*arctanh((-b*e+c*d)^(1/2)*x/d^(1/2)/(c*x^2+b*x)^(1/ 
2))/d^(7/2)/(-b*e+c*d)^(7/2)
 

Mathematica [A] (verified)

Time = 2.04 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{5/2}} \, dx=\frac {x \left (\frac {\sqrt {d} (b+c x) \left (-32 c^6 d^4 x^3 (d+e x)+16 b c^5 d^3 x^2 \left (-3 d^2+d e x+4 e^2 x^2\right )+b^6 e^3 \left (-2 d^2+10 d e x+15 e^2 x^2\right )-12 b^2 c^4 d^2 x \left (d^3-7 d^2 e x-7 d e^2 x^2+e^3 x^3\right )+6 b^5 c e^2 \left (d^3-3 d^2 e x+5 e^3 x^3\right )+2 b^3 c^3 d \left (d^4+13 d^3 e x+3 d^2 e^2 x^2-19 d e^3 x^3-10 e^4 x^4\right )-3 b^4 c^2 e \left (2 d^4+2 d^3 e x+14 d^2 e^2 x^2+10 d e^3 x^3-5 e^4 x^4\right )\right )}{b^4 (-c d+b e)^3 (d+e x)}+\frac {15 e^4 (2 c d-b e) x^{3/2} (b+c x)^{5/2} \arctan \left (\frac {-e \sqrt {x} \sqrt {b+c x}+\sqrt {c} (d+e x)}{\sqrt {d} \sqrt {-c d+b e}}\right )}{(-c d+b e)^{7/2}}\right )}{3 d^{7/2} (x (b+c x))^{5/2}} \] Input:

Integrate[1/((d + e*x)^2*(b*x + c*x^2)^(5/2)),x]
 

Output:

(x*((Sqrt[d]*(b + c*x)*(-32*c^6*d^4*x^3*(d + e*x) + 16*b*c^5*d^3*x^2*(-3*d 
^2 + d*e*x + 4*e^2*x^2) + b^6*e^3*(-2*d^2 + 10*d*e*x + 15*e^2*x^2) - 12*b^ 
2*c^4*d^2*x*(d^3 - 7*d^2*e*x - 7*d*e^2*x^2 + e^3*x^3) + 6*b^5*c*e^2*(d^3 - 
 3*d^2*e*x + 5*e^3*x^3) + 2*b^3*c^3*d*(d^4 + 13*d^3*e*x + 3*d^2*e^2*x^2 - 
19*d*e^3*x^3 - 10*e^4*x^4) - 3*b^4*c^2*e*(2*d^4 + 2*d^3*e*x + 14*d^2*e^2*x 
^2 + 10*d*e^3*x^3 - 5*e^4*x^4)))/(b^4*(-(c*d) + b*e)^3*(d + e*x)) + (15*e^ 
4*(2*c*d - b*e)*x^(3/2)*(b + c*x)^(5/2)*ArcTan[(-(e*Sqrt[x]*Sqrt[b + c*x]) 
 + Sqrt[c]*(d + e*x))/(Sqrt[d]*Sqrt[-(c*d) + b*e])])/(-(c*d) + b*e)^(7/2)) 
)/(3*d^(7/2)*(x*(b + c*x))^(5/2))
 

Rubi [A] (verified)

Time = 1.11 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1165, 27, 1235, 27, 1228, 1154, 219}

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.

\(\displaystyle \int \frac {1}{\left (b x+c x^2\right )^{5/2} (d+e x)^2} \, dx\)

\(\Big \downarrow \) 1165

\(\displaystyle -\frac {2 \int \frac {8 c^2 d^2-2 b c e d-5 b^2 e^2+6 c e (2 c d-b e) x}{2 (d+e x)^2 \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (d+e x) (c d-b e)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {8 c^2 d^2-2 b c e d-5 b^2 e^2+6 c e (2 c d-b e) x}{(d+e x)^2 \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (d+e x) (c d-b e)}\)

\(\Big \downarrow \) 1235

\(\displaystyle -\frac {-\frac {2 \int \frac {e \left (b \left (16 c^3 d^3-16 b c^2 e d^2-10 b^2 c e^2 d+15 b^3 e^3\right )+2 c (2 c d-b e) \left (8 c^2 d^2-8 b c e d-5 b^2 e^2\right ) x\right )}{2 (d+e x)^2 \sqrt {c x^2+b x}}dx}{b^2 d (c d-b e)}-\frac {2 \left (c x (2 c d-b e) \left (-5 b^2 e^2-8 b c d e+8 c^2 d^2\right )+b (c d-b e) \left (-5 b^2 e^2-2 b c d e+8 c^2 d^2\right )\right )}{b^2 d \sqrt {b x+c x^2} (d+e x) (c d-b e)}}{3 b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (d+e x) (c d-b e)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {e \int \frac {b \left (16 c^3 d^3-16 b c^2 e d^2-10 b^2 c e^2 d+15 b^3 e^3\right )+2 c (2 c d-b e) \left (8 c^2 d^2-8 b c e d-5 b^2 e^2\right ) x}{(d+e x)^2 \sqrt {c x^2+b x}}dx}{b^2 d (c d-b e)}-\frac {2 \left (c x (2 c d-b e) \left (-5 b^2 e^2-8 b c d e+8 c^2 d^2\right )+b (c d-b e) \left (-5 b^2 e^2-2 b c d e+8 c^2 d^2\right )\right )}{b^2 d \sqrt {b x+c x^2} (d+e x) (c d-b e)}}{3 b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (d+e x) (c d-b e)}\)

\(\Big \downarrow \) 1228

\(\displaystyle -\frac {-\frac {e \left (\frac {15 b^4 e^3 (2 c d-b e) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{2 d (c d-b e)}+\frac {\sqrt {b x+c x^2} \left (-15 b^4 e^4+20 b^3 c d e^3+12 b^2 c^2 d^2 e^2-64 b c^3 d^3 e+32 c^4 d^4\right )}{d (d+e x) (c d-b e)}\right )}{b^2 d (c d-b e)}-\frac {2 \left (c x (2 c d-b e) \left (-5 b^2 e^2-8 b c d e+8 c^2 d^2\right )+b (c d-b e) \left (-5 b^2 e^2-2 b c d e+8 c^2 d^2\right )\right )}{b^2 d \sqrt {b x+c x^2} (d+e x) (c d-b e)}}{3 b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (d+e x) (c d-b e)}\)

\(\Big \downarrow \) 1154

\(\displaystyle -\frac {-\frac {e \left (\frac {\sqrt {b x+c x^2} \left (-15 b^4 e^4+20 b^3 c d e^3+12 b^2 c^2 d^2 e^2-64 b c^3 d^3 e+32 c^4 d^4\right )}{d (d+e x) (c d-b e)}-\frac {15 b^4 e^3 (2 c d-b e) \int \frac {1}{4 d (c d-b e)-\frac {(b d+(2 c d-b e) x)^2}{c x^2+b x}}d\left (-\frac {b d+(2 c d-b e) x}{\sqrt {c x^2+b x}}\right )}{d (c d-b e)}\right )}{b^2 d (c d-b e)}-\frac {2 \left (c x (2 c d-b e) \left (-5 b^2 e^2-8 b c d e+8 c^2 d^2\right )+b (c d-b e) \left (-5 b^2 e^2-2 b c d e+8 c^2 d^2\right )\right )}{b^2 d \sqrt {b x+c x^2} (d+e x) (c d-b e)}}{3 b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (d+e x) (c d-b e)}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {-\frac {e \left (\frac {15 b^4 e^3 (2 c d-b e) \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{2 d^{3/2} (c d-b e)^{3/2}}+\frac {\sqrt {b x+c x^2} \left (-15 b^4 e^4+20 b^3 c d e^3+12 b^2 c^2 d^2 e^2-64 b c^3 d^3 e+32 c^4 d^4\right )}{d (d+e x) (c d-b e)}\right )}{b^2 d (c d-b e)}-\frac {2 \left (c x (2 c d-b e) \left (-5 b^2 e^2-8 b c d e+8 c^2 d^2\right )+b (c d-b e) \left (-5 b^2 e^2-2 b c d e+8 c^2 d^2\right )\right )}{b^2 d \sqrt {b x+c x^2} (d+e x) (c d-b e)}}{3 b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (d+e x) (c d-b e)}\)

Input:

Int[1/((d + e*x)^2*(b*x + c*x^2)^(5/2)),x]
 

Output:

(-2*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(3*b^2*d*(c*d - b*e)*(d + e*x)*(b 
*x + c*x^2)^(3/2)) - ((-2*(b*(c*d - b*e)*(8*c^2*d^2 - 2*b*c*d*e - 5*b^2*e^ 
2) + c*(2*c*d - b*e)*(8*c^2*d^2 - 8*b*c*d*e - 5*b^2*e^2)*x))/(b^2*d*(c*d - 
 b*e)*(d + e*x)*Sqrt[b*x + c*x^2]) - (e*(((32*c^4*d^4 - 64*b*c^3*d^3*e + 1 
2*b^2*c^2*d^2*e^2 + 20*b^3*c*d*e^3 - 15*b^4*e^4)*Sqrt[b*x + c*x^2])/(d*(c* 
d - b*e)*(d + e*x)) + (15*b^4*e^3*(2*c*d - b*e)*ArcTanh[(b*d + (2*c*d - b* 
e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(2*d^(3/2)*(c*d - b* 
e)^(3/2))))/(b^2*d*(c*d - b*e)))/(3*b^2*d*(c*d - b*e))
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1165

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) 
*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 
2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d 
+ e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p 
+ 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + 
 b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] 
 && IntQuadraticQ[a, b, c, d, e, m, p, x]
 

rule 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + 
 b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e 
*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^ 
(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 1235

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 
*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a 
+ b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] 
 + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^m 
*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 
 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* 
m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - 
f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] 
)
 

Maple [A] (verified)

Time = 1.01 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.96

method	result	size
pseudoelliptic	\(-\frac {2 \left (\frac {15 b^{4} e^{4} x \sqrt {x \left (c x +b \right )}\, \left (e x +d \right ) \left (c x +b \right ) \left (b e -2 c d \right ) \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )}{2}+\left (\left (16 c^{6} x^{3}+24 b \,c^{5} x^{2}+6 b^{2} c^{4} x -b^{3} c^{3}\right ) d^{5}+3 e \left (\frac {16}{3} c^{4} x^{4}-\frac {8}{3} b \,c^{3} x^{3}-14 b^{2} c^{2} x^{2}-\frac {13}{3} b^{3} c x +b^{4}\right ) c^{2} d^{4}-3 \left (\frac {32}{3} c^{4} x^{4}+14 b \,c^{3} x^{3}+b^{2} c^{2} x^{2}-b^{3} c x +b^{4}\right ) e^{2} c b \,d^{3}+b^{2} e^{3} \left (6 c x +b \right ) \left (c x +b \right )^{3} d^{2}-5 b^{3} e^{4} x \left (-2 c x +b \right ) \left (c x +b \right )^{2} d -\frac {15 b^{4} e^{5} x^{2} \left (c x +b \right )^{2}}{2}\right ) \sqrt {d \left (b e -c d \right )}\right )}{3 \sqrt {x \left (c x +b \right )}\, \sqrt {d \left (b e -c d \right )}\, x \,d^{3} \left (b e -c d \right )^{3} \left (c x +b \right ) \left (e x +d \right ) b^{4}}\)	\(334\)
risch	\(-\frac {2 \left (c x +b \right ) \left (-6 b e x -8 c d x +b d \right )}{3 b^{4} d^{3} \sqrt {x \left (c x +b \right )}\, x}+\frac {\frac {d \,e^{2} b^{3} \left (\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )}-\frac {\left (b e -2 c d \right ) e \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{\left (b e -c d \right )^{2}}+\frac {d^{3} b \,c^{2} \left (\frac {2 \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}}{3 b \left (\frac {b}{c}+x \right )^{2}}+\frac {4 c \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}}{3 b^{2} \left (\frac {b}{c}+x \right )}\right )}{\left (b e -c d \right )^{2}}-\frac {2 b^{3} e^{3} \left (b e -2 c d \right ) \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (b e -c d \right )^{3} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {4 c^{3} d^{3} \left (2 b e -c d \right ) \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}}{\left (b e -c d \right )^{3} b \left (\frac {b}{c}+x \right )}}{b^{3} d^{3}}\)	\(602\)
default	\(\text {Expression too large to display}\)	\(989\)

Input:

int(1/(e*x+d)^2/(c*x^2+b*x)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-2/3/(x*(c*x+b))^(1/2)/(d*(b*e-c*d))^(1/2)*(15/2*b^4*e^4*x*(x*(c*x+b))^(1/ 
2)*(e*x+d)*(c*x+b)*(b*e-2*c*d)*arctan((x*(c*x+b))^(1/2)/x*d/(d*(b*e-c*d))^ 
(1/2))+((16*c^6*x^3+24*b*c^5*x^2+6*b^2*c^4*x-b^3*c^3)*d^5+3*e*(16/3*c^4*x^ 
4-8/3*b*c^3*x^3-14*b^2*c^2*x^2-13/3*b^3*c*x+b^4)*c^2*d^4-3*(32/3*c^4*x^4+1 
4*b*c^3*x^3+b^2*c^2*x^2-b^3*c*x+b^4)*e^2*c*b*d^3+b^2*e^3*(6*c*x+b)*(c*x+b) 
^3*d^2-5*b^3*e^4*x*(-2*c*x+b)*(c*x+b)^2*d-15/2*b^4*e^5*x^2*(c*x+b)^2)*(d*( 
b*e-c*d))^(1/2))/x/d^3/(b*e-c*d)^3/(c*x+b)/(e*x+d)/b^4
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 884 vs. \(2 (322) = 644\).

Time = 0.25 (sec) , antiderivative size = 1784, normalized size of antiderivative = 5.13 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x+d)^2/(c*x^2+b*x)^(5/2),x, algorithm="fricas")
 

Output:

[1/6*(15*((2*b^4*c^3*d*e^5 - b^5*c^2*e^6)*x^5 + (2*b^4*c^3*d^2*e^4 + 3*b^5 
*c^2*d*e^5 - 2*b^6*c*e^6)*x^4 + (4*b^5*c^2*d^2*e^4 - b^7*e^6)*x^3 + (2*b^6 
*c*d^2*e^4 - b^7*d*e^5)*x^2)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e)* 
x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)) - 2*(2*b^3*c^4*d^7 
 - 8*b^4*c^3*d^6*e + 12*b^5*c^2*d^5*e^2 - 8*b^6*c*d^4*e^3 + 2*b^7*d^3*e^4 
- (32*c^7*d^6*e - 96*b*c^6*d^5*e^2 + 76*b^2*c^5*d^4*e^3 + 8*b^3*c^4*d^3*e^ 
4 - 35*b^4*c^3*d^2*e^5 + 15*b^5*c^2*d*e^6)*x^4 - 2*(16*c^7*d^7 - 24*b*c^6* 
d^6*e - 34*b^2*c^5*d^5*e^2 + 61*b^3*c^4*d^4*e^3 - 4*b^4*c^3*d^3*e^4 - 30*b 
^5*c^2*d^2*e^5 + 15*b^6*c*d*e^6)*x^3 - 3*(16*b*c^6*d^7 - 44*b^2*c^5*d^6*e 
+ 26*b^3*c^4*d^5*e^2 + 16*b^4*c^3*d^4*e^3 - 14*b^5*c^2*d^3*e^4 - 5*b^6*c*d 
^2*e^5 + 5*b^7*d*e^6)*x^2 - 2*(6*b^2*c^5*d^7 - 19*b^3*c^4*d^6*e + 16*b^4*c 
^3*d^5*e^2 + 6*b^5*c^2*d^4*e^3 - 14*b^6*c*d^3*e^4 + 5*b^7*d^2*e^5)*x)*sqrt 
(c*x^2 + b*x))/((b^4*c^6*d^8*e - 4*b^5*c^5*d^7*e^2 + 6*b^6*c^4*d^6*e^3 - 4 
*b^7*c^3*d^5*e^4 + b^8*c^2*d^4*e^5)*x^5 + (b^4*c^6*d^9 - 2*b^5*c^5*d^8*e - 
 2*b^6*c^4*d^7*e^2 + 8*b^7*c^3*d^6*e^3 - 7*b^8*c^2*d^5*e^4 + 2*b^9*c*d^4*e 
^5)*x^4 + (2*b^5*c^5*d^9 - 7*b^6*c^4*d^8*e + 8*b^7*c^3*d^7*e^2 - 2*b^8*c^2 
*d^6*e^3 - 2*b^9*c*d^5*e^4 + b^10*d^4*e^5)*x^3 + (b^6*c^4*d^9 - 4*b^7*c^3* 
d^8*e + 6*b^8*c^2*d^7*e^2 - 4*b^9*c*d^6*e^3 + b^10*d^5*e^4)*x^2), -1/3*(15 
*((2*b^4*c^3*d*e^5 - b^5*c^2*e^6)*x^5 + (2*b^4*c^3*d^2*e^4 + 3*b^5*c^2*d*e 
^5 - 2*b^6*c*e^6)*x^4 + (4*b^5*c^2*d^2*e^4 - b^7*e^6)*x^3 + (2*b^6*c*d^...
 

Sympy [F]

\[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (d + e x\right )^{2}}\, dx \] Input:

integrate(1/(e*x+d)**2/(c*x**2+b*x)**(5/2),x)
 

Output:

Integral(1/((x*(b + c*x))**(5/2)*(d + e*x)**2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/(e*x+d)^2/(c*x^2+b*x)^(5/2),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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1457 vs. \(2 (322) = 644\).

Time = 0.39 (sec) , antiderivative size = 1457, normalized size of antiderivative = 4.19 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x+d)^2/(c*x^2+b*x)^(5/2),x, algorithm="giac")
 

Output:

1/6*((30*b^4*c^(3/2)*d*e^7*log(abs(2*c*d*e - b*e^2 - 2*sqrt(c*d^2 - b*d*e) 
*sqrt(c)*abs(e))) - 15*b^5*sqrt(c)*e^8*log(abs(2*c*d*e - b*e^2 - 2*sqrt(c* 
d^2 - b*d*e)*sqrt(c)*abs(e))) - 64*sqrt(c*d^2 - b*d*e)*c^5*d^4*e^2*abs(e) 
+ 128*sqrt(c*d^2 - b*d*e)*b*c^4*d^3*e^3*abs(e) - 24*sqrt(c*d^2 - b*d*e)*b^ 
2*c^3*d^2*e^4*abs(e) - 40*sqrt(c*d^2 - b*d*e)*b^3*c^2*d*e^5*abs(e) + 30*sq 
rt(c*d^2 - b*d*e)*b^4*c*e^6*abs(e))*sgn(1/(e*x + d))*sgn(e)/(sqrt(c*d^2 - 
b*d*e)*b^4*c^(7/2)*d^6*abs(e) - 3*sqrt(c*d^2 - b*d*e)*b^5*c^(5/2)*d^5*e*ab 
s(e) + 3*sqrt(c*d^2 - b*d*e)*b^6*c^(3/2)*d^4*e^2*abs(e) - sqrt(c*d^2 - b*d 
*e)*b^7*sqrt(c)*d^3*e^3*abs(e)) + 2*((32*c^6*d^4*e^13 - 64*b*c^5*d^3*e^14 
+ 12*b^2*c^4*d^2*e^15 + 20*b^3*c^3*d*e^16 - 15*b^4*c^2*e^17)/(b^4*c^3*d^6* 
e^11*sgn(1/(e*x + d))*sgn(e) - 3*b^5*c^2*d^5*e^12*sgn(1/(e*x + d))*sgn(e) 
+ 3*b^6*c*d^4*e^13*sgn(1/(e*x + d))*sgn(e) - b^7*d^3*e^14*sgn(1/(e*x + d)) 
*sgn(e)) - (6*(16*c^6*d^5*e^14 - 40*b*c^5*d^4*e^15 + 22*b^2*c^4*d^3*e^16 + 
 7*b^3*c^3*d^2*e^17 - 15*b^4*c^2*d*e^18 + 5*b^5*c*e^19)/(b^4*c^3*d^6*e^11* 
sgn(1/(e*x + d))*sgn(e) - 3*b^5*c^2*d^5*e^12*sgn(1/(e*x + d))*sgn(e) + 3*b 
^6*c*d^4*e^13*sgn(1/(e*x + d))*sgn(e) - b^7*d^3*e^14*sgn(1/(e*x + d))*sgn( 
e)) - (3*(32*c^6*d^6*e^15 - 96*b*c^5*d^5*e^16 + 80*b^2*c^4*d^4*e^17 - 46*b 
^4*c^2*d^2*e^19 + 30*b^5*c*d*e^20 - 5*b^6*e^21)/(b^4*c^3*d^6*e^11*sgn(1/(e 
*x + d))*sgn(e) - 3*b^5*c^2*d^5*e^12*sgn(1/(e*x + d))*sgn(e) + 3*b^6*c*d^4 
*e^13*sgn(1/(e*x + d))*sgn(e) - b^7*d^3*e^14*sgn(1/(e*x + d))*sgn(e)) -...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (c\,x^2+b\,x\right )}^{5/2}\,{\left (d+e\,x\right )}^2} \,d x \] Input:

int(1/((b*x + c*x^2)^(5/2)*(d + e*x)^2),x)
 

Output:

int(1/((b*x + c*x^2)^(5/2)*(d + e*x)^2), x)
 

Reduce [F]

\[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (e x +d \right )^{2} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}d x \] Input:

int(1/(e*x+d)^2/(c*x^2+b*x)^(5/2),x)
 

Output:

int(1/(e*x+d)^2/(c*x^2+b*x)^(5/2),x)