\(\int \frac {(A+B x) (d+e x)^3}{(b x+c x^2)^{5/2}} \, dx\) [133]

Optimal result

Integrand size = 26, antiderivative size = 237 \[ \int \frac {(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 (b B-A c) (c d-b e)^3 x^2}{3 b^3 c^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 d^2 (7 A c d-3 b (B d+3 A e))}{3 b^3 \sqrt {b x+c x^2}}-\frac {2 A d^3}{3 b^2 x \sqrt {b x+c x^2}}+\frac {2 \left (16 A c^4 d^3-4 b^4 B e^3+6 b^2 c^2 d e (B d+A e)+b^3 c e^2 (3 B d+A e)-8 b c^3 d^2 (B d+3 A e)\right ) x}{3 b^4 c^2 \sqrt {b x+c x^2}}+\frac {2 B e^3 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{5/2}} \] Output:

-2/3*(-A*c+B*b)*(-b*e+c*d)^3*x^2/b^3/c^2/(c*x^2+b*x)^(3/2)+2/3*d^2*(7*A*c* 
d-3*b*(3*A*e+B*d))/b^3/(c*x^2+b*x)^(1/2)-2/3*A*d^3/b^2/x/(c*x^2+b*x)^(1/2) 
+2/3*(16*A*c^4*d^3-4*b^4*B*e^3+6*b^2*c^2*d*e*(A*e+B*d)+b^3*c*e^2*(A*e+3*B* 
d)-8*b*c^3*d^2*(3*A*e+B*d))*x/b^4/c^2/(c*x^2+b*x)^(1/2)+2*B*e^3*arctanh(c^ 
(1/2)*x/(c*x^2+b*x)^(1/2))/c^(5/2)
 

Mathematica [A] (verified)

Time = 0.97 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.06 \[ \int \frac {(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \left (\sqrt {c} \left (b B x \left (3 b^4 e^3 x+8 c^4 d^3 x^2+4 b^3 c e^3 x^2+6 b c^3 d^2 x (2 d-e x)+3 b^2 c^2 d \left (d^2-3 d e x-e^2 x^2\right )\right )-A c^2 \left (16 c^3 d^3 x^3+24 b c^2 d^2 x^2 (d-e x)+6 b^2 c d x \left (d^2-6 d e x+e^2 x^2\right )+b^3 \left (-d^3-9 d^2 e x+9 d e^2 x^2+e^3 x^3\right )\right )\right )+3 b^4 B e^3 x^{3/2} (b+c x)^{3/2} \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )\right )}{3 b^4 c^{5/2} (x (b+c x))^{3/2}} \] Input:

Integrate[((A + B*x)*(d + e*x)^3)/(b*x + c*x^2)^(5/2),x]
 

Output:

(-2*(Sqrt[c]*(b*B*x*(3*b^4*e^3*x + 8*c^4*d^3*x^2 + 4*b^3*c*e^3*x^2 + 6*b*c 
^3*d^2*x*(2*d - e*x) + 3*b^2*c^2*d*(d^2 - 3*d*e*x - e^2*x^2)) - A*c^2*(16* 
c^3*d^3*x^3 + 24*b*c^2*d^2*x^2*(d - e*x) + 6*b^2*c*d*x*(d^2 - 6*d*e*x + e^ 
2*x^2) + b^3*(-d^3 - 9*d^2*e*x + 9*d*e^2*x^2 + e^3*x^3))) + 3*b^4*B*e^3*x^ 
(3/2)*(b + c*x)^(3/2)*Log[-(Sqrt[c]*Sqrt[x]) + Sqrt[b + c*x]]))/(3*b^4*c^( 
5/2)*(x*(b + c*x))^(3/2))
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1233, 27, 1224, 1091, 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.

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

\(\Big \downarrow \) 1233

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1224

\(\displaystyle -\frac {-\frac {3 b^2 B e^3 \int \frac {1}{\sqrt {c x^2+b x}}dx}{c}-\frac {2 \left (b c d^2 \left (-4 b c (2 A e+B d)+8 A c^2 d+b^2 B e\right )+x \left (2 b^2 c^2 d e (4 A e+3 B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-3 b^4 B e^3+2 b^3 B c d e^2\right )\right )}{b^2 c \sqrt {b x+c x^2}}}{3 b^2 c}-\frac {2 (d+e x)^2 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1091

\(\displaystyle -\frac {-\frac {6 b^2 B e^3 \int \frac {1}{1-\frac {c x^2}{c x^2+b x}}d\frac {x}{\sqrt {c x^2+b x}}}{c}-\frac {2 \left (b c d^2 \left (-4 b c (2 A e+B d)+8 A c^2 d+b^2 B e\right )+x \left (2 b^2 c^2 d e (4 A e+3 B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-3 b^4 B e^3+2 b^3 B c d e^2\right )\right )}{b^2 c \sqrt {b x+c x^2}}}{3 b^2 c}-\frac {2 (d+e x)^2 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {-\frac {2 \left (b c d^2 \left (-4 b c (2 A e+B d)+8 A c^2 d+b^2 B e\right )+x \left (2 b^2 c^2 d e (4 A e+3 B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-3 b^4 B e^3+2 b^3 B c d e^2\right )\right )}{b^2 c \sqrt {b x+c x^2}}-\frac {6 b^2 B e^3 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{3/2}}}{3 b^2 c}-\frac {2 (d+e x)^2 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\)

Input:

Int[((A + B*x)*(d + e*x)^3)/(b*x + c*x^2)^(5/2),x]
 

Output:

(-2*(d + e*x)^2*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(3* 
b^2*c*(b*x + c*x^2)^(3/2)) - ((-2*(b*c*d^2*(8*A*c^2*d + b^2*B*e - 4*b*c*(B 
*d + 2*A*e)) + (16*A*c^4*d^3 + 2*b^3*B*c*d*e^2 - 3*b^4*B*e^3 - 8*b*c^3*d^2 
*(B*d + 3*A*e) + 2*b^2*c^2*d*e*(3*B*d + 4*A*e))*x))/(b^2*c*Sqrt[b*x + c*x^ 
2]) - (6*b^2*B*e^3*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/c^(3/2))/(3*b^2 
*c)
 

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 1224

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( 
x_)^2)^(p_), x_Symbol] :> Simp[(-(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - ( 
b^2*e*g - b*c*(e*f + d*g) + 2*c*(c*d*f - a*e*g))*x))*((a + b*x + c*x^2)^(p 
+ 1)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c 
*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(c*(p + 1)*(b^2 - 4*a*c))   Int[(a + 
b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, - 
1] &&  !(IntegerQ[p] && NeQ[a, 0] && NiceSqrtQ[b^2 - 4*a*c])
 

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])
 

Maple [A] (verified)

Time = 1.06 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.99

method	result	size
pseudoelliptic	\(\frac {-\frac {2 \left (\left (-A \,e^{3}-3 B d \,e^{2}\right ) x^{3}+\left (-9 A d \,e^{2}-9 e B \,d^{2}\right ) x^{2}+\left (9 A \,d^{2} e +3 B \,d^{3}\right ) x +A \,d^{3}\right ) b^{3} c^{\frac {5}{2}}}{3}+2 \left (2 d \,b^{2} \left (\left (A \,e^{2}+B d e \right ) x^{2}+\left (-6 A d e -2 B \,d^{2}\right ) x +A \,d^{2}\right ) c^{\frac {7}{2}}+8 \left (\left (-A e -\frac {B d}{3}\right ) x +A d \right ) d^{2} x b \,c^{\frac {9}{2}}+\frac {16 A \,c^{\frac {11}{2}} d^{3} x^{2}}{3}+e^{3} B \left (-\frac {4 c^{\frac {3}{2}} x^{2}}{3}-b x \sqrt {c}+\sqrt {x \left (c x +b \right )}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right ) \left (c x +b \right )\right ) b^{4}\right ) x}{c^{\frac {5}{2}} x \left (c x +b \right ) \sqrt {x \left (c x +b \right )}\, b^{4}}\)	\(235\)
risch	\(-\frac {2 d^{2} \left (c x +b \right ) \left (9 A b e x -8 c x A d +3 B b d x +A b d \right )}{3 b^{4} x \sqrt {x \left (c x +b \right )}}+\frac {\frac {2 \left (A \,b^{3} c \,e^{3}-3 A b \,c^{3} d^{2} e +2 A \,c^{4} d^{3}-2 b^{4} B \,e^{3}+3 B \,b^{3} c d \,e^{2}-B b \,d^{3} c^{3}\right ) \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}}{c^{3} b \left (\frac {b}{c}+x \right )}+\frac {B \,e^{3} b^{3} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{c^{\frac {5}{2}}}-\frac {b \left (A \,b^{3} c \,e^{3}-3 A \,b^{2} c^{2} d \,e^{2}+3 A b \,c^{3} d^{2} e -A \,c^{4} d^{3}-b^{4} B \,e^{3}+3 B \,b^{3} c d \,e^{2}-3 B \,b^{2} d^{2} e \,c^{2}+B b \,d^{3} c^{3}\right ) \left (\frac {2 \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}}{3 b \left (\frac {b}{c}+x \right )^{2}}+\frac {4 c \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}}{3 b^{2} \left (\frac {b}{c}+x \right )}\right )}{c^{4}}}{b^{3}}\)	\(360\)
default	\(A \,d^{3} \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )+e^{2} \left (A e +3 B d \right ) \left (-\frac {x^{2}}{c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {b \left (-\frac {x}{2 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )}{4 c}\right )}{2 c}\right )+3 d e \left (A e +B d \right ) \left (-\frac {x}{2 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )}{4 c}\right )+d^{2} \left (3 A e +B d \right ) \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )+B \,e^{3} \left (-\frac {x^{3}}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {x^{2}}{c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {b \left (-\frac {x}{2 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )}{4 c}\right )}{2 c}\right )}{2 c}+\frac {-\frac {x}{c \sqrt {c \,x^{2}+b x}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x}}+\frac {2 c x +b}{b c \sqrt {c \,x^{2}+b x}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{c^{\frac {3}{2}}}}{c}\right )\)	\(616\)

Input:

int((B*x+A)*(e*x+d)^3/(c*x^2+b*x)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

2/c^(5/2)/(x*(c*x+b))^(1/2)*(-1/3*((-A*e^3-3*B*d*e^2)*x^3+(-9*A*d*e^2-9*B* 
d^2*e)*x^2+(9*A*d^2*e+3*B*d^3)*x+A*d^3)*b^3*c^(5/2)+(2*d*b^2*((A*e^2+B*d*e 
)*x^2+(-6*A*d*e-2*B*d^2)*x+A*d^2)*c^(7/2)+8*((-A*e-1/3*B*d)*x+A*d)*d^2*x*b 
*c^(9/2)+16/3*A*c^(11/2)*d^3*x^2+e^3*B*(-4/3*c^(3/2)*x^2-b*x*c^(1/2)+(x*(c 
*x+b))^(1/2)*arctanh((x*(c*x+b))^(1/2)/x/c^(1/2))*(c*x+b))*b^4)*x)/x/(c*x+ 
b)/b^4
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 672, normalized size of antiderivative = 2.84 \[ \int \frac {(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (B b^{4} c^{2} e^{3} x^{4} + 2 \, B b^{5} c e^{3} x^{3} + B b^{6} e^{3} x^{2}\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (A b^{3} c^{3} d^{3} + {\left (8 \, {\left (B b c^{5} - 2 \, A c^{6}\right )} d^{3} - 6 \, {\left (B b^{2} c^{4} - 4 \, A b c^{5}\right )} d^{2} e - 3 \, {\left (B b^{3} c^{3} + 2 \, A b^{2} c^{4}\right )} d e^{2} + {\left (4 \, B b^{4} c^{2} - A b^{3} c^{3}\right )} e^{3}\right )} x^{3} - 3 \, {\left (3 \, A b^{3} c^{3} d e^{2} - B b^{5} c e^{3} - 4 \, {\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{3} + 3 \, {\left (B b^{3} c^{3} - 4 \, A b^{2} c^{4}\right )} d^{2} e\right )} x^{2} + 3 \, {\left (3 \, A b^{3} c^{3} d^{2} e + {\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{3 \, {\left (b^{4} c^{5} x^{4} + 2 \, b^{5} c^{4} x^{3} + b^{6} c^{3} x^{2}\right )}}, -\frac {2 \, {\left (3 \, {\left (B b^{4} c^{2} e^{3} x^{4} + 2 \, B b^{5} c e^{3} x^{3} + B b^{6} e^{3} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x + b}\right ) + {\left (A b^{3} c^{3} d^{3} + {\left (8 \, {\left (B b c^{5} - 2 \, A c^{6}\right )} d^{3} - 6 \, {\left (B b^{2} c^{4} - 4 \, A b c^{5}\right )} d^{2} e - 3 \, {\left (B b^{3} c^{3} + 2 \, A b^{2} c^{4}\right )} d e^{2} + {\left (4 \, B b^{4} c^{2} - A b^{3} c^{3}\right )} e^{3}\right )} x^{3} - 3 \, {\left (3 \, A b^{3} c^{3} d e^{2} - B b^{5} c e^{3} - 4 \, {\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{3} + 3 \, {\left (B b^{3} c^{3} - 4 \, A b^{2} c^{4}\right )} d^{2} e\right )} x^{2} + 3 \, {\left (3 \, A b^{3} c^{3} d^{2} e + {\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d^{3}\right )} x\right )} \sqrt {c x^{2} + b x}\right )}}{3 \, {\left (b^{4} c^{5} x^{4} + 2 \, b^{5} c^{4} x^{3} + b^{6} c^{3} x^{2}\right )}}\right ] \] Input:

integrate((B*x+A)*(e*x+d)^3/(c*x^2+b*x)^(5/2),x, algorithm="fricas")
 

Output:

[1/3*(3*(B*b^4*c^2*e^3*x^4 + 2*B*b^5*c*e^3*x^3 + B*b^6*e^3*x^2)*sqrt(c)*lo 
g(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 2*(A*b^3*c^3*d^3 + (8*(B*b*c^ 
5 - 2*A*c^6)*d^3 - 6*(B*b^2*c^4 - 4*A*b*c^5)*d^2*e - 3*(B*b^3*c^3 + 2*A*b^ 
2*c^4)*d*e^2 + (4*B*b^4*c^2 - A*b^3*c^3)*e^3)*x^3 - 3*(3*A*b^3*c^3*d*e^2 - 
 B*b^5*c*e^3 - 4*(B*b^2*c^4 - 2*A*b*c^5)*d^3 + 3*(B*b^3*c^3 - 4*A*b^2*c^4) 
*d^2*e)*x^2 + 3*(3*A*b^3*c^3*d^2*e + (B*b^3*c^3 - 2*A*b^2*c^4)*d^3)*x)*sqr 
t(c*x^2 + b*x))/(b^4*c^5*x^4 + 2*b^5*c^4*x^3 + b^6*c^3*x^2), -2/3*(3*(B*b^ 
4*c^2*e^3*x^4 + 2*B*b^5*c*e^3*x^3 + B*b^6*e^3*x^2)*sqrt(-c)*arctan(sqrt(c* 
x^2 + b*x)*sqrt(-c)/(c*x + b)) + (A*b^3*c^3*d^3 + (8*(B*b*c^5 - 2*A*c^6)*d 
^3 - 6*(B*b^2*c^4 - 4*A*b*c^5)*d^2*e - 3*(B*b^3*c^3 + 2*A*b^2*c^4)*d*e^2 + 
 (4*B*b^4*c^2 - A*b^3*c^3)*e^3)*x^3 - 3*(3*A*b^3*c^3*d*e^2 - B*b^5*c*e^3 - 
 4*(B*b^2*c^4 - 2*A*b*c^5)*d^3 + 3*(B*b^3*c^3 - 4*A*b^2*c^4)*d^2*e)*x^2 + 
3*(3*A*b^3*c^3*d^2*e + (B*b^3*c^3 - 2*A*b^2*c^4)*d^3)*x)*sqrt(c*x^2 + b*x) 
)/(b^4*c^5*x^4 + 2*b^5*c^4*x^3 + b^6*c^3*x^2)]
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {\left (A + B x\right ) \left (d + e x\right )^{3}}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((B*x+A)*(e*x+d)**3/(c*x**2+b*x)**(5/2),x)
 

Output:

Integral((A + B*x)*(d + e*x)**3/(x*(b + c*x))**(5/2), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 550 vs. \(2 (215) = 430\).

Time = 0.05 (sec) , antiderivative size = 550, normalized size of antiderivative = 2.32 \[ \int \frac {(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {1}{3} \, B e^{3} x {\left (\frac {3 \, x^{2}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {b x}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c^{2}} - \frac {2 \, x}{\sqrt {c x^{2} + b x} b c} - \frac {1}{\sqrt {c x^{2} + b x} c^{2}}\right )} - \frac {4 \, A c d^{3} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}} + \frac {32 \, A c^{2} d^{3} x}{3 \, \sqrt {c x^{2} + b x} b^{4}} - \frac {4 \, B e^{3} x}{3 \, \sqrt {c x^{2} + b x} c^{2}} + \frac {B e^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{c^{\frac {5}{2}}} - \frac {2 \, A d^{3}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} + \frac {16 \, A c d^{3}}{3 \, \sqrt {c x^{2} + b x} b^{3}} - \frac {2 \, \sqrt {c x^{2} + b x} B e^{3}}{3 \, b c^{2}} - \frac {{\left (3 \, B d e^{2} + A e^{3}\right )} x^{2}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {4 \, {\left (B d^{2} e + A d e^{2}\right )} x}{\sqrt {c x^{2} + b x} b^{2}} + \frac {2 \, {\left (B d^{3} + 3 \, A d^{2} e\right )} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} - \frac {{\left (3 \, B d e^{2} + A e^{3}\right )} b x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} c^{2}} - \frac {2 \, {\left (B d^{2} e + A d e^{2}\right )} x}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {2 \, {\left (3 \, B d e^{2} + A e^{3}\right )} x}{3 \, \sqrt {c x^{2} + b x} b c} - \frac {16 \, {\left (B d^{3} + 3 \, A d^{2} e\right )} c x}{3 \, \sqrt {c x^{2} + b x} b^{3}} - \frac {8 \, {\left (B d^{3} + 3 \, A d^{2} e\right )}}{3 \, \sqrt {c x^{2} + b x} b^{2}} + \frac {3 \, B d e^{2} + A e^{3}}{3 \, \sqrt {c x^{2} + b x} c^{2}} + \frac {2 \, {\left (B d^{2} e + A d e^{2}\right )}}{\sqrt {c x^{2} + b x} b c} \] Input:

integrate((B*x+A)*(e*x+d)^3/(c*x^2+b*x)^(5/2),x, algorithm="maxima")
 

Output:

-1/3*B*e^3*x*(3*x^2/((c*x^2 + b*x)^(3/2)*c) + b*x/((c*x^2 + b*x)^(3/2)*c^2 
) - 2*x/(sqrt(c*x^2 + b*x)*b*c) - 1/(sqrt(c*x^2 + b*x)*c^2)) - 4/3*A*c*d^3 
*x/((c*x^2 + b*x)^(3/2)*b^2) + 32/3*A*c^2*d^3*x/(sqrt(c*x^2 + b*x)*b^4) - 
4/3*B*e^3*x/(sqrt(c*x^2 + b*x)*c^2) + B*e^3*log(2*c*x + b + 2*sqrt(c*x^2 + 
 b*x)*sqrt(c))/c^(5/2) - 2/3*A*d^3/((c*x^2 + b*x)^(3/2)*b) + 16/3*A*c*d^3/ 
(sqrt(c*x^2 + b*x)*b^3) - 2/3*sqrt(c*x^2 + b*x)*B*e^3/(b*c^2) - (3*B*d*e^2 
 + A*e^3)*x^2/((c*x^2 + b*x)^(3/2)*c) + 4*(B*d^2*e + A*d*e^2)*x/(sqrt(c*x^ 
2 + b*x)*b^2) + 2/3*(B*d^3 + 3*A*d^2*e)*x/((c*x^2 + b*x)^(3/2)*b) - 1/3*(3 
*B*d*e^2 + A*e^3)*b*x/((c*x^2 + b*x)^(3/2)*c^2) - 2*(B*d^2*e + A*d*e^2)*x/ 
((c*x^2 + b*x)^(3/2)*c) + 2/3*(3*B*d*e^2 + A*e^3)*x/(sqrt(c*x^2 + b*x)*b*c 
) - 16/3*(B*d^3 + 3*A*d^2*e)*c*x/(sqrt(c*x^2 + b*x)*b^3) - 8/3*(B*d^3 + 3* 
A*d^2*e)/(sqrt(c*x^2 + b*x)*b^2) + 1/3*(3*B*d*e^2 + A*e^3)/(sqrt(c*x^2 + b 
*x)*c^2) + 2*(B*d^2*e + A*d*e^2)/(sqrt(c*x^2 + b*x)*b*c)
 

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.22 \[ \int \frac {(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {B e^{3} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{c^{\frac {5}{2}}} - \frac {2 \, {\left (\frac {A d^{3}}{b} + {\left (x {\left (\frac {{\left (8 \, B b c^{4} d^{3} - 16 \, A c^{5} d^{3} - 6 \, B b^{2} c^{3} d^{2} e + 24 \, A b c^{4} d^{2} e - 3 \, B b^{3} c^{2} d e^{2} - 6 \, A b^{2} c^{3} d e^{2} + 4 \, B b^{4} c e^{3} - A b^{3} c^{2} e^{3}\right )} x}{b^{4} c^{2}} + \frac {3 \, {\left (4 \, B b^{2} c^{3} d^{3} - 8 \, A b c^{4} d^{3} - 3 \, B b^{3} c^{2} d^{2} e + 12 \, A b^{2} c^{3} d^{2} e - 3 \, A b^{3} c^{2} d e^{2} + B b^{5} e^{3}\right )}}{b^{4} c^{2}}\right )} + \frac {3 \, {\left (B b^{3} c^{2} d^{3} - 2 \, A b^{2} c^{3} d^{3} + 3 \, A b^{3} c^{2} d^{2} e\right )}}{b^{4} c^{2}}\right )} x\right )}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \] Input:

integrate((B*x+A)*(e*x+d)^3/(c*x^2+b*x)^(5/2),x, algorithm="giac")
 

Output:

-B*e^3*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) + b))/c^(5/2) - 2 
/3*(A*d^3/b + (x*((8*B*b*c^4*d^3 - 16*A*c^5*d^3 - 6*B*b^2*c^3*d^2*e + 24*A 
*b*c^4*d^2*e - 3*B*b^3*c^2*d*e^2 - 6*A*b^2*c^3*d*e^2 + 4*B*b^4*c*e^3 - A*b 
^3*c^2*e^3)*x/(b^4*c^2) + 3*(4*B*b^2*c^3*d^3 - 8*A*b*c^4*d^3 - 3*B*b^3*c^2 
*d^2*e + 12*A*b^2*c^3*d^2*e - 3*A*b^3*c^2*d*e^2 + B*b^5*e^3)/(b^4*c^2)) + 
3*(B*b^3*c^2*d^3 - 2*A*b^2*c^3*d^3 + 3*A*b^3*c^2*d^2*e)/(b^4*c^2))*x)/(c*x 
^2 + b*x)^(3/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^3}{{\left (c\,x^2+b\,x\right )}^{5/2}} \,d x \] Input:

int(((A + B*x)*(d + e*x)^3)/(b*x + c*x^2)^(5/2),x)
 

Output:

int(((A + B*x)*(d + e*x)^3)/(b*x + c*x^2)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 2.18 (sec) , antiderivative size = 741, normalized size of antiderivative = 3.13 \[ \int \frac {(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

int((B*x+A)*(e*x+d)^3/(c*x^2+b*x)^(5/2),x)
 

Output:

(2*(3*sqrt(c)*sqrt(b + c*x)*log((sqrt(b + c*x) + sqrt(x)*sqrt(c))/sqrt(b)) 
*b**6*e**3*x**2 + 3*sqrt(c)*sqrt(b + c*x)*log((sqrt(b + c*x) + sqrt(x)*sqr 
t(c))/sqrt(b))*b**5*c*e**3*x**3 + 5*sqrt(c)*sqrt(b + c*x)*a*b**4*c*e**3*x* 
*2 - 18*sqrt(c)*sqrt(b + c*x)*a*b**3*c**2*d*e**2*x**2 + 5*sqrt(c)*sqrt(b + 
 c*x)*a*b**3*c**2*e**3*x**3 + 24*sqrt(c)*sqrt(b + c*x)*a*b**2*c**3*d**2*e* 
x**2 - 18*sqrt(c)*sqrt(b + c*x)*a*b**2*c**3*d*e**2*x**3 - 16*sqrt(c)*sqrt( 
b + c*x)*a*b*c**4*d**3*x**2 + 24*sqrt(c)*sqrt(b + c*x)*a*b*c**4*d**2*e*x** 
3 - 16*sqrt(c)*sqrt(b + c*x)*a*c**5*d**3*x**3 - 4*sqrt(c)*sqrt(b + c*x)*b* 
*6*e**3*x**2 + 15*sqrt(c)*sqrt(b + c*x)*b**5*c*d*e**2*x**2 - 4*sqrt(c)*sqr 
t(b + c*x)*b**5*c*e**3*x**3 - 18*sqrt(c)*sqrt(b + c*x)*b**4*c**2*d**2*e*x* 
*2 + 15*sqrt(c)*sqrt(b + c*x)*b**4*c**2*d*e**2*x**3 + 8*sqrt(c)*sqrt(b + c 
*x)*b**3*c**3*d**3*x**2 - 18*sqrt(c)*sqrt(b + c*x)*b**3*c**3*d**2*e*x**3 + 
 8*sqrt(c)*sqrt(b + c*x)*b**2*c**4*d**3*x**3 - sqrt(x)*a*b**3*c**3*d**3 - 
9*sqrt(x)*a*b**3*c**3*d**2*e*x + 9*sqrt(x)*a*b**3*c**3*d*e**2*x**2 + sqrt( 
x)*a*b**3*c**3*e**3*x**3 + 6*sqrt(x)*a*b**2*c**4*d**3*x - 36*sqrt(x)*a*b** 
2*c**4*d**2*e*x**2 + 6*sqrt(x)*a*b**2*c**4*d*e**2*x**3 + 24*sqrt(x)*a*b*c* 
*5*d**3*x**2 - 24*sqrt(x)*a*b*c**5*d**2*e*x**3 + 16*sqrt(x)*a*c**6*d**3*x* 
*3 - 3*sqrt(x)*b**6*c*e**3*x**2 - 4*sqrt(x)*b**5*c**2*e**3*x**3 - 3*sqrt(x 
)*b**4*c**3*d**3*x + 9*sqrt(x)*b**4*c**3*d**2*e*x**2 + 3*sqrt(x)*b**4*c**3 
*d*e**2*x**3 - 12*sqrt(x)*b**3*c**4*d**3*x**2 + 6*sqrt(x)*b**3*c**4*d**...