\(\int \frac {(b x+c x^2)^{5/2}}{(d+e x)^5} \, dx\) [156]

Optimal result

Integrand size = 21, antiderivative size = 413 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^5} \, dx=\frac {5 \left (64 c^3 d^3-80 b c^2 d^2 e+16 b^2 c d e^2+b^3 e^3\right ) \sqrt {b x+c x^2}}{64 d^2 e^5 (c d-b e)}+\frac {5 \left (12 b c-\frac {16 c^2 d}{e}+\frac {b^2 e}{d}\right ) x^2 \sqrt {b x+c x^2}}{96 d e^2 (d+e x)^2}-\frac {5 (4 c d-b e) \left (24 c^2 d^2-22 b c d e-b^2 e^2\right ) x \sqrt {b x+c x^2}}{192 d^2 e^4 (c d-b e) (d+e x)}-\frac {5 (2 c d-b e) x \left (b x+c x^2\right )^{3/2}}{24 d e^2 (d+e x)^3}-\frac {\left (b x+c x^2\right )^{5/2}}{4 e (d+e x)^4}-\frac {5 c^{3/2} (2 c d-b e) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{e^6}+\frac {5 \left (128 c^4 d^4-256 b c^3 d^3 e+144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4\right ) \text {arctanh}\left (\frac {\sqrt {c d-b e} x}{\sqrt {d} \sqrt {b x+c x^2}}\right )}{64 d^{3/2} e^6 (c d-b e)^{3/2}} \] Output:

5/64*(b^3*e^3+16*b^2*c*d*e^2-80*b*c^2*d^2*e+64*c^3*d^3)*(c*x^2+b*x)^(1/2)/ 
d^2/e^5/(-b*e+c*d)+5/96*(12*b*c-16*c^2*d/e+b^2*e/d)*x^2*(c*x^2+b*x)^(1/2)/ 
d/e^2/(e*x+d)^2-5/192*(-b*e+4*c*d)*(-b^2*e^2-22*b*c*d*e+24*c^2*d^2)*x*(c*x 
^2+b*x)^(1/2)/d^2/e^4/(-b*e+c*d)/(e*x+d)-5/24*(-b*e+2*c*d)*x*(c*x^2+b*x)^( 
3/2)/d/e^2/(e*x+d)^3-1/4*(c*x^2+b*x)^(5/2)/e/(e*x+d)^4-5*c^(3/2)*(-b*e+2*c 
*d)*arctanh(c^(1/2)*x/(c*x^2+b*x)^(1/2))/e^6+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((-b*e+c*d)^(1/2) 
*x/d^(1/2)/(c*x^2+b*x)^(1/2))/d^(3/2)/e^6/(-b*e+c*d)^(3/2)
 

Mathematica [A] (verified)

Time = 12.13 (sec) , antiderivative size = 388, normalized size of antiderivative = 0.94 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^5} \, dx=\frac {\sqrt {x (b+c x)} \left (\frac {e \sqrt {x} \left (b^3 e^3 \left (15 d^3+55 d^2 e x+73 d e^2 x^2-15 e^3 x^3\right )+2 b^2 c d e^2 \left (120 d^3+435 d^2 e x+566 d e^2 x^2+323 e^3 x^3\right )+16 c^3 d^2 \left (60 d^4+210 d^3 e x+260 d^2 e^2 x^2+125 d e^3 x^3+12 e^4 x^4\right )-8 b c^2 d e \left (150 d^4+530 d^3 e x+665 d^2 e^2 x^2+327 d e^3 x^3+24 e^4 x^4\right )\right )}{d (c d-b e) (d+e x)^4}+\frac {960 c^{3/2} (-2 c d+b e) \text {arcsinh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {1+\frac {c x}{b}}}+\frac {15 \left (128 c^4 d^4-256 b c^3 d^3 e+144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4\right ) \text {arctanh}\left (\frac {\sqrt {c d-b e} \sqrt {x}}{\sqrt {d} \sqrt {b+c x}}\right )}{d^{3/2} (c d-b e)^{3/2} \sqrt {b+c x}}\right )}{192 e^6 \sqrt {x}} \] Input:

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

Output:

(Sqrt[x*(b + c*x)]*((e*Sqrt[x]*(b^3*e^3*(15*d^3 + 55*d^2*e*x + 73*d*e^2*x^ 
2 - 15*e^3*x^3) + 2*b^2*c*d*e^2*(120*d^3 + 435*d^2*e*x + 566*d*e^2*x^2 + 3 
23*e^3*x^3) + 16*c^3*d^2*(60*d^4 + 210*d^3*e*x + 260*d^2*e^2*x^2 + 125*d*e 
^3*x^3 + 12*e^4*x^4) - 8*b*c^2*d*e*(150*d^4 + 530*d^3*e*x + 665*d^2*e^2*x^ 
2 + 327*d*e^3*x^3 + 24*e^4*x^4)))/(d*(c*d - b*e)*(d + e*x)^4) + (960*c^(3/ 
2)*(-2*c*d + b*e)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c 
*x)/b]) + (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)*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c 
*x])])/(d^(3/2)*(c*d - b*e)^(3/2)*Sqrt[b + c*x])))/(192*e^6*Sqrt[x])
 

Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.01, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1161, 1229, 27, 1230, 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.

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

\(\Big \downarrow \) 1161

\(\displaystyle \frac {5 \int \frac {(b+2 c x) \left (c x^2+b x\right )^{3/2}}{(d+e x)^4}dx}{8 e}-\frac {\left (b x+c x^2\right )^{5/2}}{4 e (d+e x)^4}\)

\(\Big \downarrow \) 1229

\(\displaystyle \frac {5 \left (-\frac {\int -\frac {\left (b \left (16 c^2 d^2-12 b c e d-b^2 e^2\right )+2 c \left (16 c^2 d^2-16 b c e d+b^2 e^2\right ) x\right ) \sqrt {c x^2+b x}}{2 (d+e x)^2}dx}{4 d e^2 (c d-b e)}-\frac {\left (b x+c x^2\right )^{3/2} \left (3 e x \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )+d \left (-b^2 e^2-12 b c d e+16 c^2 d^2\right )\right )}{12 d e^2 (d+e x)^3 (c d-b e)}\right )}{8 e}-\frac {\left (b x+c x^2\right )^{5/2}}{4 e (d+e x)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5 \left (\frac {\int \frac {\left (b \left (16 c^2 d^2-12 b c e d-b^2 e^2\right )+2 c \left (16 c^2 d^2-16 b c e d+b^2 e^2\right ) x\right ) \sqrt {c x^2+b x}}{(d+e x)^2}dx}{8 d e^2 (c d-b e)}-\frac {\left (b x+c x^2\right )^{3/2} \left (3 e x \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )+d \left (-b^2 e^2-12 b c d e+16 c^2 d^2\right )\right )}{12 d e^2 (d+e x)^3 (c d-b e)}\right )}{8 e}-\frac {\left (b x+c x^2\right )^{5/2}}{4 e (d+e x)^4}\)

\(\Big \downarrow \) 1230

\(\displaystyle \frac {5 \left (\frac {\frac {\sqrt {b x+c x^2} \left (b^3 e^3+2 c e x \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )+16 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )}{e^2 (d+e x)}-\frac {\int \frac {64 d (c d-b e) (2 c d-b e) x c^2+b \left (64 c^3 d^3-80 b c^2 e d^2+16 b^2 c e^2 d+b^3 e^3\right )}{(d+e x) \sqrt {c x^2+b x}}dx}{2 e^2}}{8 d e^2 (c d-b e)}-\frac {\left (b x+c x^2\right )^{3/2} \left (3 e x \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )+d \left (-b^2 e^2-12 b c d e+16 c^2 d^2\right )\right )}{12 d e^2 (d+e x)^3 (c d-b e)}\right )}{8 e}-\frac {\left (b x+c x^2\right )^{5/2}}{4 e (d+e x)^4}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {5 \left (\frac {\frac {\sqrt {b x+c x^2} \left (b^3 e^3+2 c e x \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )+16 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )}{e^2 (d+e x)}-\frac {\frac {64 c^2 d (c d-b e) (2 c d-b e) \int \frac {1}{\sqrt {c x^2+b x}}dx}{e}-\frac {\left (-b^4 e^4-16 b^3 c d e^3+144 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{2 e^2}}{8 d e^2 (c d-b e)}-\frac {\left (b x+c x^2\right )^{3/2} \left (3 e x \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )+d \left (-b^2 e^2-12 b c d e+16 c^2 d^2\right )\right )}{12 d e^2 (d+e x)^3 (c d-b e)}\right )}{8 e}-\frac {\left (b x+c x^2\right )^{5/2}}{4 e (d+e x)^4}\)

\(\Big \downarrow \) 1091

\(\displaystyle \frac {5 \left (\frac {\frac {\sqrt {b x+c x^2} \left (b^3 e^3+2 c e x \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )+16 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )}{e^2 (d+e x)}-\frac {\frac {128 c^2 d (c d-b e) (2 c d-b e) \int \frac {1}{1-\frac {c x^2}{c x^2+b x}}d\frac {x}{\sqrt {c x^2+b x}}}{e}-\frac {\left (-b^4 e^4-16 b^3 c d e^3+144 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{2 e^2}}{8 d e^2 (c d-b e)}-\frac {\left (b x+c x^2\right )^{3/2} \left (3 e x \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )+d \left (-b^2 e^2-12 b c d e+16 c^2 d^2\right )\right )}{12 d e^2 (d+e x)^3 (c d-b e)}\right )}{8 e}-\frac {\left (b x+c x^2\right )^{5/2}}{4 e (d+e x)^4}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {5 \left (\frac {\frac {\sqrt {b x+c x^2} \left (b^3 e^3+2 c e x \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )+16 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )}{e^2 (d+e x)}-\frac {\frac {128 c^{3/2} d \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (c d-b e) (2 c d-b e)}{e}-\frac {\left (-b^4 e^4-16 b^3 c d e^3+144 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{2 e^2}}{8 d e^2 (c d-b e)}-\frac {\left (b x+c x^2\right )^{3/2} \left (3 e x \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )+d \left (-b^2 e^2-12 b c d e+16 c^2 d^2\right )\right )}{12 d e^2 (d+e x)^3 (c d-b e)}\right )}{8 e}-\frac {\left (b x+c x^2\right )^{5/2}}{4 e (d+e x)^4}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {5 \left (\frac {\frac {\sqrt {b x+c x^2} \left (b^3 e^3+2 c e x \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )+16 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )}{e^2 (d+e x)}-\frac {\frac {2 \left (-b^4 e^4-16 b^3 c d e^3+144 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) \int \frac {1}{4 d (c d-b e)-\frac {(b d+(2 c d-b e) x)^2}{c x^2+b x}}d\left (-\frac {b d+(2 c d-b e) x}{\sqrt {c x^2+b x}}\right )}{e}+\frac {128 c^{3/2} d \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (c d-b e) (2 c d-b e)}{e}}{2 e^2}}{8 d e^2 (c d-b e)}-\frac {\left (b x+c x^2\right )^{3/2} \left (3 e x \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )+d \left (-b^2 e^2-12 b c d e+16 c^2 d^2\right )\right )}{12 d e^2 (d+e x)^3 (c d-b e)}\right )}{8 e}-\frac {\left (b x+c x^2\right )^{5/2}}{4 e (d+e x)^4}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {5 \left (\frac {\frac {\sqrt {b x+c x^2} \left (b^3 e^3+2 c e x \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )+16 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )}{e^2 (d+e x)}-\frac {\frac {128 c^{3/2} d \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (c d-b e) (2 c d-b e)}{e}-\frac {\left (-b^4 e^4-16 b^3 c d e^3+144 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{\sqrt {d} e \sqrt {c d-b e}}}{2 e^2}}{8 d e^2 (c d-b e)}-\frac {\left (b x+c x^2\right )^{3/2} \left (3 e x \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )+d \left (-b^2 e^2-12 b c d e+16 c^2 d^2\right )\right )}{12 d e^2 (d+e x)^3 (c d-b e)}\right )}{8 e}-\frac {\left (b x+c x^2\right )^{5/2}}{4 e (d+e x)^4}\)

Input:

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

Output:

-1/4*(b*x + c*x^2)^(5/2)/(e*(d + e*x)^4) + (5*(-1/12*((d*(16*c^2*d^2 - 12* 
b*c*d*e - b^2*e^2) + 3*e*(8*c^2*d^2 - 8*b*c*d*e + b^2*e^2)*x)*(b*x + c*x^2 
)^(3/2))/(d*e^2*(c*d - b*e)*(d + e*x)^3) + (((64*c^3*d^3 - 80*b*c^2*d^2*e 
+ 16*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])/(e^2*(d + e*x)) - ((128*c^(3/2)*d*(c*d - b*e)*(2*c*d - 
b*e)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/e - ((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[(b*d + (2 
*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(Sqrt[d]*e* 
Sqrt[c*d - b*e]))/(2*e^2))/(8*d*e^2*(c*d - b*e))))/(8*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 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 1229

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/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* 
d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 
- b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 
)*(m + 2)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 
)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + 
p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c 
*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( 
m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g 
}, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 
0]
 

rule 1230

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - 
 d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p 
+ 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a 
+ b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m 
+ 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, 
 x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 
1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ 
[m + 2*p + 1, 0] && (IntegerQ[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 = 1.13 (sec) , antiderivative size = 377, normalized size of antiderivative = 0.91

method	result	size
pseudoelliptic	\(-\frac {5 \left (\left (e x +d \right )^{4} \left (b^{4} e^{4} \sqrt {c}+16 b^{3} d \,e^{3} c^{\frac {3}{2}}-144 b^{2} c^{\frac {5}{2}} d^{2} e^{2}+256 b \,c^{\frac {7}{2}} d^{3} e -128 c^{\frac {9}{2}} d^{4}\right ) \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )+\sqrt {d \left (b e -c d \right )}\, \left (-64 c^{2} d \left (e x +d \right )^{4} \left (b e -c d \right ) \left (b e -2 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )+e \sqrt {x \left (c x +b \right )}\, \left (-80 e \left (\frac {4}{25} e^{4} x^{4}+\frac {109}{50} d \,e^{3} x^{3}+\frac {133}{30} d^{2} e^{2} x^{2}+\frac {53}{15} d^{3} e x +d^{4}\right ) b d \,c^{\frac {5}{2}}+64 \left (\frac {1}{5} e^{4} x^{4}+\frac {25}{12} d \,e^{3} x^{3}+\frac {13}{3} d^{2} e^{2} x^{2}+\frac {7}{2} d^{3} e x +d^{4}\right ) d^{2} c^{\frac {7}{2}}+e^{2} \left (\left (\frac {646}{15} d \,e^{3} x^{3}+\frac {1132}{15} d^{2} e^{2} x^{2}+58 d^{3} e x +16 d^{4}\right ) c^{\frac {3}{2}}+b e \sqrt {c}\, \left (\frac {11}{3} d^{2} e x +\frac {73}{15} d \,e^{2} x^{2}-e^{3} x^{3}+d^{3}\right )\right ) b^{2}\right )\right )\right )}{64 \sqrt {c}\, \sqrt {d \left (b e -c d \right )}\, \left (e x +d \right )^{4} e^{6} \left (b e -c d \right ) d}\)	\(377\)
risch	\(\text {Expression too large to display}\)	\(3293\)
default	\(\text {Expression too large to display}\)	\(6669\)

Input:

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

Output:

-5/64/c^(1/2)*((e*x+d)^4*(b^4*e^4*c^(1/2)+16*b^3*d*e^3*c^(3/2)-144*b^2*c^( 
5/2)*d^2*e^2+256*b*c^(7/2)*d^3*e-128*c^(9/2)*d^4)*arctan((x*(c*x+b))^(1/2) 
/x*d/(d*(b*e-c*d))^(1/2))+(d*(b*e-c*d))^(1/2)*(-64*c^2*d*(e*x+d)^4*(b*e-c* 
d)*(b*e-2*c*d)*arctanh((x*(c*x+b))^(1/2)/x/c^(1/2))+e*(x*(c*x+b))^(1/2)*(- 
80*e*(4/25*e^4*x^4+109/50*d*e^3*x^3+133/30*d^2*e^2*x^2+53/15*d^3*e*x+d^4)* 
b*d*c^(5/2)+64*(1/5*e^4*x^4+25/12*d*e^3*x^3+13/3*d^2*e^2*x^2+7/2*d^3*e*x+d 
^4)*d^2*c^(7/2)+e^2*((646/15*d*e^3*x^3+1132/15*d^2*e^2*x^2+58*d^3*e*x+16*d 
^4)*c^(3/2)+b*e*c^(1/2)*(11/3*d^2*e*x+73/15*d*e^2*x^2-e^3*x^3+d^3))*b^2))) 
/(d*(b*e-c*d))^(1/2)/(e*x+d)^4/e^6/(b*e-c*d)/d
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1133 vs. \(2 (375) = 750\).

Time = 0.77 (sec) , antiderivative size = 4553, normalized size of antiderivative = 11.02 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^5} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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