\(\int \frac {(b x+c x^2)^{5/2}}{(d+e x)^2} \, dx\) [153]

Optimal result

Integrand size = 21, antiderivative size = 333 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^2} \, dx=-\frac {5 \left (64 c^3 d^3-112 b c^2 d^2 e+48 b^2 c d e^2-b^3 e^3\right ) \sqrt {b x+c x^2}}{64 c e^5}+\frac {5 \left (48 c^2 d^2-80 b c d e+31 b^2 e^2\right ) x \sqrt {b x+c x^2}}{96 e^4}-\frac {5 c (8 c d-7 b e) x^2 \sqrt {b x+c x^2}}{24 e^3}+\frac {5 c x \left (b x+c x^2\right )^{3/2}}{4 e^2}-\frac {\left (b x+c x^2\right )^{5/2}}{e (d+e x)}+\frac {5 \left (128 c^4 d^4-256 b c^3 d^3 e+144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{64 c^{3/2} e^6}-\frac {5 d^{3/2} (c d-b e)^{3/2} (2 c d-b e) \text {arctanh}\left (\frac {\sqrt {c d-b e} x}{\sqrt {d} \sqrt {b x+c x^2}}\right )}{e^6} \] Output:

-5/64*(-b^3*e^3+48*b^2*c*d*e^2-112*b*c^2*d^2*e+64*c^3*d^3)*(c*x^2+b*x)^(1/ 
2)/c/e^5+5/96*(31*b^2*e^2-80*b*c*d*e+48*c^2*d^2)*x*(c*x^2+b*x)^(1/2)/e^4-5 
/24*c*(-7*b*e+8*c*d)*x^2*(c*x^2+b*x)^(1/2)/e^3+5/4*c*x*(c*x^2+b*x)^(3/2)/e 
^2-(c*x^2+b*x)^(5/2)/e/(e*x+d)+5/64*(-b^4*e^4-16*b^3*c*d*e^3+144*b^2*c^2*d 
^2*e^2-256*b*c^3*d^3*e+128*c^4*d^4)*arctanh(c^(1/2)*x/(c*x^2+b*x)^(1/2))/c 
^(3/2)/e^6-5*d^(3/2)*(-b*e+c*d)^(3/2)*(-b*e+2*c*d)*arctanh((-b*e+c*d)^(1/2 
)*x/d^(1/2)/(c*x^2+b*x)^(1/2))/e^6
 

Mathematica [A] (verified)

Time = 10.96 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.04 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {\sqrt {x (b+c x)} \left (\frac {\sqrt {c} e \sqrt {x} \left (15 b^3 e^3 (d+e x)+2 b^2 c e^2 \left (-360 d^2-205 d e x+59 e^2 x^2\right )+8 b c^2 e \left (210 d^3+110 d^2 e x-35 d e^2 x^2+17 e^3 x^3\right )-16 c^3 \left (60 d^4+30 d^3 e x-10 d^2 e^2 x^2+5 d e^3 x^3-3 e^4 x^4\right )\right )}{d+e x}-\frac {15 \left (-128 c^4 d^4+256 b c^3 d^3 e-144 b^2 c^2 d^2 e^2+16 b^3 c d e^3+b^4 e^4\right ) \text {arcsinh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {1+\frac {c x}{b}}}-\frac {960 c^{3/2} d^{3/2} \sqrt {c d-b e} \left (2 c^2 d^2-3 b c d e+b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c d-b e} \sqrt {x}}{\sqrt {d} \sqrt {b+c x}}\right )}{\sqrt {b+c x}}\right )}{192 c^{3/2} e^6 \sqrt {x}} \] Input:

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

Output:

(Sqrt[x*(b + c*x)]*((Sqrt[c]*e*Sqrt[x]*(15*b^3*e^3*(d + e*x) + 2*b^2*c*e^2 
*(-360*d^2 - 205*d*e*x + 59*e^2*x^2) + 8*b*c^2*e*(210*d^3 + 110*d^2*e*x - 
35*d*e^2*x^2 + 17*e^3*x^3) - 16*c^3*(60*d^4 + 30*d^3*e*x - 10*d^2*e^2*x^2 
+ 5*d*e^3*x^3 - 3*e^4*x^4)))/(d + e*x) - (15*(-128*c^4*d^4 + 256*b*c^3*d^3 
*e - 144*b^2*c^2*d^2*e^2 + 16*b^3*c*d*e^3 + b^4*e^4)*ArcSinh[(Sqrt[c]*Sqrt 
[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c*x)/b]) - (960*c^(3/2)*d^(3/2)*Sqrt[c*d 
 - b*e]*(2*c^2*d^2 - 3*b*c*d*e + b^2*e^2)*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x] 
)/(Sqrt[d]*Sqrt[b + c*x])])/Sqrt[b + c*x]))/(192*c^(3/2)*e^6*Sqrt[x])
 

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {1161, 1231, 25, 27, 1231, 27, 1269, 1091, 219, 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.

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

\(\Big \downarrow \) 1161

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

\(\Big \downarrow \) 1231

\(\displaystyle \frac {5 \left (-\frac {\int -\frac {c \left (b d (8 c d-7 b e)+\left (16 c^2 d^2-16 b c e d+b^2 e^2\right ) x\right ) \sqrt {c x^2+b x}}{d+e x}dx}{8 c e^2}-\frac {\left (b x+c x^2\right )^{3/2} (-7 b e+8 c d-6 c e x)}{12 e^2}\right )}{2 e}-\frac {\left (b x+c x^2\right )^{5/2}}{e (d+e x)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {5 \left (\frac {\int \frac {c \left (b d (8 c d-7 b e)+\left (16 c^2 d^2-16 b c e d+b^2 e^2\right ) x\right ) \sqrt {c x^2+b x}}{d+e x}dx}{8 c e^2}-\frac {\left (b x+c x^2\right )^{3/2} (-7 b e+8 c d-6 c e x)}{12 e^2}\right )}{2 e}-\frac {\left (b x+c x^2\right )^{5/2}}{e (d+e x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5 \left (\frac {\int \frac {\left (b d (8 c d-7 b e)+\left (16 c^2 d^2-16 b c e d+b^2 e^2\right ) x\right ) \sqrt {c x^2+b x}}{d+e x}dx}{8 e^2}-\frac {\left (b x+c x^2\right )^{3/2} (-7 b e+8 c d-6 c e x)}{12 e^2}\right )}{2 e}-\frac {\left (b x+c x^2\right )^{5/2}}{e (d+e x)}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {5 \left (\frac {-\frac {\int -\frac {b d \left (64 c^3 d^3-112 b c^2 e d^2+48 b^2 c e^2 d-b^3 e^3\right )+\left (128 c^4 d^4-256 b c^3 e d^3+144 b^2 c^2 e^2 d^2-16 b^3 c e^3 d-b^4 e^4\right ) x}{2 (d+e x) \sqrt {c x^2+b x}}dx}{4 c e^2}-\frac {\sqrt {b x+c x^2} \left (-b^3 e^3-2 c e x \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )+48 b^2 c d e^2-112 b c^2 d^2 e+64 c^3 d^3\right )}{4 c e^2}}{8 e^2}-\frac {\left (b x+c x^2\right )^{3/2} (-7 b e+8 c d-6 c e x)}{12 e^2}\right )}{2 e}-\frac {\left (b x+c x^2\right )^{5/2}}{e (d+e x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5 \left (\frac {\frac {\int \frac {b d \left (64 c^3 d^3-112 b c^2 e d^2+48 b^2 c e^2 d-b^3 e^3\right )+\left (128 c^4 d^4-256 b c^3 e d^3+144 b^2 c^2 e^2 d^2-16 b^3 c e^3 d-b^4 e^4\right ) x}{(d+e x) \sqrt {c x^2+b x}}dx}{8 c e^2}-\frac {\sqrt {b x+c x^2} \left (-b^3 e^3-2 c e x \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )+48 b^2 c d e^2-112 b c^2 d^2 e+64 c^3 d^3\right )}{4 c e^2}}{8 e^2}-\frac {\left (b x+c x^2\right )^{3/2} (-7 b e+8 c d-6 c e x)}{12 e^2}\right )}{2 e}-\frac {\left (b x+c x^2\right )^{5/2}}{e (d+e x)}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {5 \left (\frac {\frac {\frac {\left (-b^4 e^4-16 b^3 c d e^3+144 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) \int \frac {1}{\sqrt {c x^2+b x}}dx}{e}-\frac {64 c d^2 (c d-b e)^2 (2 c d-b e) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{8 c e^2}-\frac {\sqrt {b x+c x^2} \left (-b^3 e^3-2 c e x \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )+48 b^2 c d e^2-112 b c^2 d^2 e+64 c^3 d^3\right )}{4 c e^2}}{8 e^2}-\frac {\left (b x+c x^2\right )^{3/2} (-7 b e+8 c d-6 c e x)}{12 e^2}\right )}{2 e}-\frac {\left (b x+c x^2\right )^{5/2}}{e (d+e x)}\)

\(\Big \downarrow \) 1091

\(\displaystyle \frac {5 \left (\frac {\frac {\frac {2 \left (-b^4 e^4-16 b^3 c d e^3+144 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) \int \frac {1}{1-\frac {c x^2}{c x^2+b x}}d\frac {x}{\sqrt {c x^2+b x}}}{e}-\frac {64 c d^2 (c d-b e)^2 (2 c d-b e) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{8 c e^2}-\frac {\sqrt {b x+c x^2} \left (-b^3 e^3-2 c e x \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )+48 b^2 c d e^2-112 b c^2 d^2 e+64 c^3 d^3\right )}{4 c e^2}}{8 e^2}-\frac {\left (b x+c x^2\right )^{3/2} (-7 b e+8 c d-6 c e x)}{12 e^2}\right )}{2 e}-\frac {\left (b x+c x^2\right )^{5/2}}{e (d+e x)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {5 \left (\frac {\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (-b^4 e^4-16 b^3 c d e^3+144 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right )}{\sqrt {c} e}-\frac {64 c d^2 (c d-b e)^2 (2 c d-b e) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{8 c e^2}-\frac {\sqrt {b x+c x^2} \left (-b^3 e^3-2 c e x \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )+48 b^2 c d e^2-112 b c^2 d^2 e+64 c^3 d^3\right )}{4 c e^2}}{8 e^2}-\frac {\left (b x+c x^2\right )^{3/2} (-7 b e+8 c d-6 c e x)}{12 e^2}\right )}{2 e}-\frac {\left (b x+c x^2\right )^{5/2}}{e (d+e x)}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {5 \left (\frac {\frac {\frac {128 c d^2 (2 c d-b e) (c d-b e)^2 \int \frac {1}{4 d (c d-b e)-\frac {(b d+(2 c d-b e) x)^2}{c x^2+b x}}d\left (-\frac {b d+(2 c d-b e) x}{\sqrt {c x^2+b x}}\right )}{e}+\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (-b^4 e^4-16 b^3 c d e^3+144 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right )}{\sqrt {c} e}}{8 c e^2}-\frac {\sqrt {b x+c x^2} \left (-b^3 e^3-2 c e x \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )+48 b^2 c d e^2-112 b c^2 d^2 e+64 c^3 d^3\right )}{4 c e^2}}{8 e^2}-\frac {\left (b x+c x^2\right )^{3/2} (-7 b e+8 c d-6 c e x)}{12 e^2}\right )}{2 e}-\frac {\left (b x+c x^2\right )^{5/2}}{e (d+e x)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {5 \left (\frac {\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (-b^4 e^4-16 b^3 c d e^3+144 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right )}{\sqrt {c} e}-\frac {64 c d^{3/2} (c d-b e)^{3/2} (2 c d-b e) \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{e}}{8 c e^2}-\frac {\sqrt {b x+c x^2} \left (-b^3 e^3-2 c e x \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )+48 b^2 c d e^2-112 b c^2 d^2 e+64 c^3 d^3\right )}{4 c e^2}}{8 e^2}-\frac {\left (b x+c x^2\right )^{3/2} (-7 b e+8 c d-6 c e x)}{12 e^2}\right )}{2 e}-\frac {\left (b x+c x^2\right )^{5/2}}{e (d+e x)}\)

Input:

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

Output:

-((b*x + c*x^2)^(5/2)/(e*(d + e*x))) + (5*(-1/12*((8*c*d - 7*b*e - 6*c*e*x 
)*(b*x + c*x^2)^(3/2))/e^2 + (-1/4*((64*c^3*d^3 - 112*b*c^2*d^2*e + 48*b^2 
*c*d*e^2 - b^3*e^3 - 2*c*e*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2)*x)*Sqrt[b*x 
 + c*x^2])/(c*e^2) + ((2*(128*c^4*d^4 - 256*b*c^3*d^3*e + 144*b^2*c^2*d^2* 
e^2 - 16*b^3*c*d*e^3 - b^4*e^4)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(S 
qrt[c]*e) - (64*c*d^(3/2)*(c*d - b*e)^(3/2)*(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])])/e)/(8*c*e^ 
2))/(8*e^2)))/(2*e)
 

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 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 1091

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[Int[1/(1 
 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{b, c}, x]
 

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 1161

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 1))), x] - Si 
mp[p/(e*(m + 1))   Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^(p - 
 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && GtQ[p, 0] && (IntegerQ[p] || 
 LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, 
 c, d, e, m, p, x]
 

rule 1231

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) 
 - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ 
(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 
 2*p + 2))   Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* 
a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* 
c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c 
^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x 
] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  !R 
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Integer 
Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1269

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

Maple [A] (verified)

Time = 0.89 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.98

method	result	size
pseudoelliptic	\(-\frac {5 \left (64 d^{2} \left (e x +d \right ) \left (b^{3} e^{3} c^{\frac {3}{2}}-4 b^{2} c^{\frac {5}{2}} d \,e^{2}+5 b \,d^{2} e \,c^{\frac {7}{2}}-2 c^{\frac {9}{2}} d^{3}\right ) \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )+\left (\left (b^{4} e^{4}+16 d \,e^{3} b^{3} c -144 d^{2} e^{2} b^{2} c^{2}+256 d^{3} e b \,c^{3}-128 d^{4} c^{4}\right ) \left (e x +d \right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )-e \left (\left (\frac {16}{5} e^{4} x^{4}-\frac {16}{3} d \,e^{3} x^{3}+\frac {32}{3} d^{2} e^{2} x^{2}-32 d^{3} e x -64 d^{4}\right ) c^{\frac {7}{2}}+e \left (\left (\frac {136}{15} e^{3} x^{3}-\frac {56}{3} d \,e^{2} x^{2}+\frac {176}{3} d^{2} e x +112 d^{3}\right ) c^{\frac {5}{2}}+e \left (\left (\frac {118}{15} e^{2} x^{2}-\frac {82}{3} d e x -48 d^{2}\right ) c^{\frac {3}{2}}+b e \sqrt {c}\, \left (e x +d \right )\right ) b \right ) b \right ) \sqrt {x \left (c x +b \right )}\right ) \sqrt {d \left (b e -c d \right )}\right )}{64 c^{\frac {3}{2}} \sqrt {d \left (b e -c d \right )}\, e^{6} \left (e x +d \right )}\)	\(328\)
risch	\(\frac {\left (48 c^{3} e^{3} x^{3}+136 e^{3} x^{2} b \,c^{2}-128 c^{3} d \,e^{2} x^{2}+118 x \,b^{2} c \,e^{3}-416 b \,c^{2} d \,e^{2} x +288 c^{3} d^{2} e x +15 b^{3} e^{3}-528 d \,e^{2} b^{2} c +1296 d^{2} e b \,c^{2}-768 d^{3} c^{3}\right ) x \left (c x +b \right )}{192 c \,e^{5} \sqrt {x \left (c x +b \right )}}-\frac {\frac {5 \left (b^{4} e^{4}+16 d \,e^{3} b^{3} c -144 d^{2} e^{2} b^{2} c^{2}+256 d^{3} e b \,c^{3}-128 d^{4} c^{4}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{e \sqrt {c}}+\frac {384 c \,d^{2} \left (b^{3} e^{3}-4 d \,e^{2} b^{2} c +5 d^{2} e b \,c^{2}-2 d^{3} c^{3}\right ) \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {128 c \,d^{3} \left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right ) \left (\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )}-\frac {\left (b e -2 c d \right ) e \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{e^{3}}}{128 e^{5} c}\)	\(659\)
default	\(\text {Expression too large to display}\)	\(1455\)

Input:

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

Output:

-5/64/c^(3/2)/(d*(b*e-c*d))^(1/2)*(64*d^2*(e*x+d)*(b^3*e^3*c^(3/2)-4*b^2*c 
^(5/2)*d*e^2+5*b*d^2*e*c^(7/2)-2*c^(9/2)*d^3)*arctan((x*(c*x+b))^(1/2)/x*d 
/(d*(b*e-c*d))^(1/2))+((b^4*e^4+16*b^3*c*d*e^3-144*b^2*c^2*d^2*e^2+256*b*c 
^3*d^3*e-128*c^4*d^4)*(e*x+d)*arctanh((x*(c*x+b))^(1/2)/x/c^(1/2))-e*((16/ 
5*e^4*x^4-16/3*d*e^3*x^3+32/3*d^2*e^2*x^2-32*d^3*e*x-64*d^4)*c^(7/2)+e*((1 
36/15*e^3*x^3-56/3*d*e^2*x^2+176/3*d^2*e*x+112*d^3)*c^(5/2)+e*((118/15*e^2 
*x^2-82/3*d*e*x-48*d^2)*c^(3/2)+b*e*c^(1/2)*(e*x+d))*b)*b)*(x*(c*x+b))^(1/ 
2))*(d*(b*e-c*d))^(1/2))/e^6/(e*x+d)
 

Fricas [A] (verification not implemented)

Time = 0.78 (sec) , antiderivative size = 1868, normalized size of antiderivative = 5.61 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/384*(15*(128*c^4*d^5 - 256*b*c^3*d^4*e + 144*b^2*c^2*d^3*e^2 - 16*b^3* 
c*d^2*e^3 - b^4*d*e^4 + (128*c^4*d^4*e - 256*b*c^3*d^3*e^2 + 144*b^2*c^2*d 
^2*e^3 - 16*b^3*c*d*e^4 - b^4*e^5)*x)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 
 + b*x)*sqrt(c)) - 960*(2*c^4*d^4 - 3*b*c^3*d^3*e + b^2*c^2*d^2*e^2 + (2*c 
^4*d^3*e - 3*b*c^3*d^2*e^2 + b^2*c^2*d*e^3)*x)*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*(48*c^4*e^5*x^4 - 960*c^4*d^4*e + 1680*b*c^3*d^3*e^2 - 720*b^2*c^2*d^2 
*e^3 + 15*b^3*c*d*e^4 - 8*(10*c^4*d*e^4 - 17*b*c^3*e^5)*x^3 + 2*(80*c^4*d^ 
2*e^3 - 140*b*c^3*d*e^4 + 59*b^2*c^2*e^5)*x^2 - 5*(96*c^4*d^3*e^2 - 176*b* 
c^3*d^2*e^3 + 82*b^2*c^2*d*e^4 - 3*b^3*c*e^5)*x)*sqrt(c*x^2 + b*x))/(c^2*e 
^7*x + c^2*d*e^6), 1/384*(1920*(2*c^4*d^4 - 3*b*c^3*d^3*e + b^2*c^2*d^2*e^ 
2 + (2*c^4*d^3*e - 3*b*c^3*d^2*e^2 + b^2*c^2*d*e^3)*x)*sqrt(-c*d^2 + b*d*e 
)*arctan(sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/(c*d*x + b*d)) - 15*(128*c 
^4*d^5 - 256*b*c^3*d^4*e + 144*b^2*c^2*d^3*e^2 - 16*b^3*c*d^2*e^3 - b^4*d* 
e^4 + (128*c^4*d^4*e - 256*b*c^3*d^3*e^2 + 144*b^2*c^2*d^2*e^3 - 16*b^3*c* 
d*e^4 - b^4*e^5)*x)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sqrt(c)) + 
 2*(48*c^4*e^5*x^4 - 960*c^4*d^4*e + 1680*b*c^3*d^3*e^2 - 720*b^2*c^2*d^2* 
e^3 + 15*b^3*c*d*e^4 - 8*(10*c^4*d*e^4 - 17*b*c^3*e^5)*x^3 + 2*(80*c^4*d^2 
*e^3 - 140*b*c^3*d*e^4 + 59*b^2*c^2*e^5)*x^2 - 5*(96*c^4*d^3*e^2 - 176*b*c 
^3*d^2*e^3 + 82*b^2*c^2*d*e^4 - 3*b^3*c*e^5)*x)*sqrt(c*x^2 + b*x))/(c^2...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((c*x^2+b*x)^(5/2)/(e*x+d)^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 [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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