\(\int \frac {(A+B x) (d+e x)^{5/2}}{(b x+c x^2)^3} \, dx\) [88]

Optimal result

Integrand size = 26, antiderivative size = 373 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (b x+c x^2\right )^3} \, dx=-\frac {(c d-b e) \left (6 b B c d-12 A c^2 d-2 b^2 B e+9 A b c e\right ) \sqrt {d+e x}}{4 b^3 c (b+c x)^2}+\frac {\left (24 A c^3 d^2+b^3 B e^2-12 b c^2 d (B d+2 A e)+b^2 c e (7 B d+3 A e)\right ) \sqrt {d+e x}}{4 b^4 c (b+c x)}-\frac {(4 b B d-8 A c d+5 A b e) (d+e x)^{3/2}}{4 b^2 x (b+c x)^2}-\frac {A (d+e x)^{5/2}}{2 b x^2 (b+c x)^2}-\frac {\sqrt {d} \left (48 A c^2 d^2+5 b^2 e (4 B d+3 A e)-12 b c d (2 B d+5 A e)\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5}+\frac {\sqrt {c d-b e} \left (48 A c^3 d^2+b^3 B e^2-12 b c^2 d (2 B d+3 A e)+b^2 c e (8 B d+3 A e)\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 c^{3/2}} \] Output:

-1/4*(-b*e+c*d)*(9*A*b*c*e-12*A*c^2*d-2*B*b^2*e+6*B*b*c*d)*(e*x+d)^(1/2)/b 
^3/c/(c*x+b)^2+1/4*(24*A*c^3*d^2+b^3*B*e^2-12*b*c^2*d*(2*A*e+B*d)+b^2*c*e* 
(3*A*e+7*B*d))*(e*x+d)^(1/2)/b^4/c/(c*x+b)-1/4*(5*A*b*e-8*A*c*d+4*B*b*d)*( 
e*x+d)^(3/2)/b^2/x/(c*x+b)^2-1/2*A*(e*x+d)^(5/2)/b/x^2/(c*x+b)^2-1/4*d^(1/ 
2)*(48*A*c^2*d^2+5*b^2*e*(3*A*e+4*B*d)-12*b*c*d*(5*A*e+2*B*d))*arctanh((e* 
x+d)^(1/2)/d^(1/2))/b^5+1/4*(-b*e+c*d)^(1/2)*(48*A*c^3*d^2+b^3*B*e^2-12*b* 
c^2*d*(3*A*e+2*B*d)+b^2*c*e*(3*A*e+8*B*d))*arctanh(c^(1/2)*(e*x+d)^(1/2)/( 
-b*e+c*d)^(1/2))/b^5/c^(3/2)
 

Mathematica [A] (verified)

Time = 2.86 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.90 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (b x+c x^2\right )^3} \, dx=\frac {-\frac {b \sqrt {d+e x} \left (A c \left (-24 c^3 d^2 x^3+12 b c^2 d x^2 (-3 d+2 e x)+b^3 \left (2 d^2+9 d e x-5 e^2 x^2\right )+b^2 c x \left (-8 d^2+37 d e x-3 e^2 x^2\right )\right )+b B x \left (b^3 e^2 x+12 c^3 d^2 x^2+b c^2 d x (18 d-7 e x)+b^2 c \left (4 d^2-11 d e x-e^2 x^2\right )\right )\right )}{c x^2 (b+c x)^2}+\frac {\sqrt {-c d+b e} \left (48 A c^3 d^2+b^3 B e^2-12 b c^2 d (2 B d+3 A e)+b^2 c e (8 B d+3 A e)\right ) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{c^{3/2}}+\sqrt {d} \left (-48 A c^2 d^2-5 b^2 e (4 B d+3 A e)+12 b c d (2 B d+5 A e)\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5} \] Input:

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

Output:

(-((b*Sqrt[d + e*x]*(A*c*(-24*c^3*d^2*x^3 + 12*b*c^2*d*x^2*(-3*d + 2*e*x) 
+ b^3*(2*d^2 + 9*d*e*x - 5*e^2*x^2) + b^2*c*x*(-8*d^2 + 37*d*e*x - 3*e^2*x 
^2)) + b*B*x*(b^3*e^2*x + 12*c^3*d^2*x^2 + b*c^2*d*x*(18*d - 7*e*x) + b^2* 
c*(4*d^2 - 11*d*e*x - e^2*x^2))))/(c*x^2*(b + c*x)^2)) + (Sqrt[-(c*d) + b* 
e]*(48*A*c^3*d^2 + b^3*B*e^2 - 12*b*c^2*d*(2*B*d + 3*A*e) + b^2*c*e*(8*B*d 
 + 3*A*e))*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/c^(3/2) + S 
qrt[d]*(-48*A*c^2*d^2 - 5*b^2*e*(4*B*d + 3*A*e) + 12*b*c*d*(2*B*d + 5*A*e) 
)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5)
 

Rubi [A] (verified)

Time = 1.34 (sec) , antiderivative size = 358, normalized size of antiderivative = 0.96, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {1233, 27, 1234, 27, 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.

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

\(\Big \downarrow \) 1233

\(\displaystyle \frac {\int \frac {\sqrt {d+e x} \left (d \left (-2 B e b^2+6 B c d b+9 A c e b-12 A c^2 d\right )-e \left (-B e b^2-3 c (B d+A e) b+6 A c^2 d\right ) x\right )}{2 \left (c x^2+b x\right )^2}dx}{2 b^2 c}-\frac {(d+e x)^{3/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\sqrt {d+e x} \left (d \left (-2 B e b^2+6 B c d b+9 A c e b-12 A c^2 d\right )-e \left (-B e b^2-3 c (B d+A e) b+6 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^2}dx}{4 b^2 c}-\frac {(d+e x)^{3/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1234

\(\displaystyle \frac {-\frac {\int -\frac {c d \left (5 e (4 B d+3 A e) b^2-12 c d (2 B d+5 A e) b+48 A c^2 d^2\right )+e \left (B e^2 b^3+c e (7 B d+3 A e) b^2-12 c^2 d (B d+2 A e) b+24 A c^3 d^2\right ) x}{2 \sqrt {d+e x} \left (c x^2+b x\right )}dx}{b^2}-\frac {\sqrt {d+e x} \left (b d \left (9 A b c e-12 A c^2 d-2 b^2 B e+6 b B c d\right )-x \left (b^2 c e (3 A e+7 B d)-12 b c^2 d (2 A e+B d)+24 A c^3 d^2+b^3 B e^2\right )\right )}{b^2 \left (b x+c x^2\right )}}{4 b^2 c}-\frac {(d+e x)^{3/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {c d \left (5 e (4 B d+3 A e) b^2-12 c d (2 B d+5 A e) b+48 A c^2 d^2\right )+e \left (B e^2 b^3+c e (7 B d+3 A e) b^2-12 c^2 d (B d+2 A e) b+24 A c^3 d^2\right ) x}{\sqrt {d+e x} \left (c x^2+b x\right )}dx}{2 b^2}-\frac {\sqrt {d+e x} \left (b d \left (9 A b c e-12 A c^2 d-2 b^2 B e+6 b B c d\right )-x \left (b^2 c e (3 A e+7 B d)-12 b c^2 d (2 A e+B d)+24 A c^3 d^2+b^3 B e^2\right )\right )}{b^2 \left (b x+c x^2\right )}}{4 b^2 c}-\frac {(d+e x)^{3/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1197

\(\displaystyle \frac {\frac {\int \frac {e \left (d (c d-b e) \left (B e b^2-12 c (B d+A e) b+24 A c^2 d\right )+\left (B e^2 b^3+c e (7 B d+3 A e) b^2-12 c^2 d (B d+2 A e) b+24 A c^3 d^2\right ) (d+e x)\right )}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{b^2}-\frac {\sqrt {d+e x} \left (b d \left (9 A b c e-12 A c^2 d-2 b^2 B e+6 b B c d\right )-x \left (b^2 c e (3 A e+7 B d)-12 b c^2 d (2 A e+B d)+24 A c^3 d^2+b^3 B e^2\right )\right )}{b^2 \left (b x+c x^2\right )}}{4 b^2 c}-\frac {(d+e x)^{3/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {e \int \frac {d (c d-b e) \left (B e b^2-12 c (B d+A e) b+24 A c^2 d\right )+\left (B e^2 b^3+c e (7 B d+3 A e) b^2-12 c^2 d (B d+2 A e) b+24 A c^3 d^2\right ) (d+e x)}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{b^2}-\frac {\sqrt {d+e x} \left (b d \left (9 A b c e-12 A c^2 d-2 b^2 B e+6 b B c d\right )-x \left (b^2 c e (3 A e+7 B d)-12 b c^2 d (2 A e+B d)+24 A c^3 d^2+b^3 B e^2\right )\right )}{b^2 \left (b x+c x^2\right )}}{4 b^2 c}-\frac {(d+e x)^{3/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {\frac {e \left (\frac {c^2 d \left (5 b^2 e (3 A e+4 B d)-12 b c d (5 A e+2 B d)+48 A c^2 d^2\right ) \int \frac {1}{c (d+e x)-c d}d\sqrt {d+e x}}{b e}-\frac {(c d-b e) \left (b^2 c e (3 A e+8 B d)-12 b c^2 d (3 A e+2 B d)+48 A c^3 d^2+b^3 B e^2\right ) \int \frac {1}{-c d+b e+c (d+e x)}d\sqrt {d+e x}}{b e}\right )}{b^2}-\frac {\sqrt {d+e x} \left (b d \left (9 A b c e-12 A c^2 d-2 b^2 B e+6 b B c d\right )-x \left (b^2 c e (3 A e+7 B d)-12 b c^2 d (2 A e+B d)+24 A c^3 d^2+b^3 B e^2\right )\right )}{b^2 \left (b x+c x^2\right )}}{4 b^2 c}-\frac {(d+e x)^{3/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {e \left (\frac {\sqrt {c d-b e} \left (b^2 c e (3 A e+8 B d)-12 b c^2 d (3 A e+2 B d)+48 A c^3 d^2+b^3 B e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b \sqrt {c} e}-\frac {c \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (5 b^2 e (3 A e+4 B d)-12 b c d (5 A e+2 B d)+48 A c^2 d^2\right )}{b e}\right )}{b^2}-\frac {\sqrt {d+e x} \left (b d \left (9 A b c e-12 A c^2 d-2 b^2 B e+6 b B c d\right )-x \left (b^2 c e (3 A e+7 B d)-12 b c^2 d (2 A e+B d)+24 A c^3 d^2+b^3 B e^2\right )\right )}{b^2 \left (b x+c x^2\right )}}{4 b^2 c}-\frac {(d+e x)^{3/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\)

Input:

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

Output:

-1/2*((d + e*x)^(3/2)*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x 
))/(b^2*c*(b*x + c*x^2)^2) + (-((Sqrt[d + e*x]*(b*d*(6*b*B*c*d - 12*A*c^2* 
d - 2*b^2*B*e + 9*A*b*c*e) - (24*A*c^3*d^2 + b^3*B*e^2 - 12*b*c^2*d*(B*d + 
 2*A*e) + b^2*c*e*(7*B*d + 3*A*e))*x))/(b^2*(b*x + c*x^2))) + (e*(-((c*Sqr 
t[d]*(48*A*c^2*d^2 + 5*b^2*e*(4*B*d + 3*A*e) - 12*b*c*d*(2*B*d + 5*A*e))*A 
rcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*e)) + (Sqrt[c*d - b*e]*(48*A*c^3*d^2 + b 
^3*B*e^2 - 12*b*c^2*d*(2*B*d + 3*A*e) + b^2*c*e*(8*B*d + 3*A*e))*ArcTanh[( 
Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*Sqrt[c]*e)))/b^2)/(4*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 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 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 1234

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

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 = 1.37 (sec) , antiderivative size = 360, normalized size of antiderivative = 0.97

method	result	size
pseudoelliptic	\(-\frac {-\frac {\left (3 A \,b^{2} e^{2} c -36 A b \,c^{2} d e +48 A \,c^{3} d^{2}+b^{3} B \,e^{2}+8 B \,b^{2} c d e -24 B b \,c^{2} d^{2}\right ) x^{2} \left (c x +b \right )^{2} \sqrt {d}\, \left (b e -c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{2}+\left (\frac {c d \left (15 A \,b^{2} e^{2}-60 A b c d e +48 A \,c^{2} d^{2}+20 B \,b^{2} d e -24 B b c \,d^{2}\right ) x^{2} \left (c x +b \right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2}+\sqrt {d}\, \left (-12 A \,c^{4} d^{2} x^{3}-18 \left (\left (-\frac {B x}{3}+A \right ) d -\frac {2 A e x}{3}\right ) d \,x^{2} b \,c^{3}-4 \left (\left (-\frac {9 B x}{4}+A \right ) d^{2}-\frac {37 \left (-\frac {7 B x}{37}+A \right ) e x d}{8}+\frac {3 A \,e^{2} x^{2}}{8}\right ) x \,b^{2} c^{2}+\left (\left (2 B x +A \right ) d^{2}+\frac {9 e \left (-\frac {11 B x}{9}+A \right ) x d}{2}-\frac {5 e^{2} x^{2} \left (\frac {B x}{5}+A \right )}{2}\right ) b^{3} c +\frac {B \,b^{4} e^{2} x^{2}}{2}\right ) b \sqrt {e x +d}\right ) \sqrt {c \left (b e -c d \right )}}{2 \sqrt {d}\, \sqrt {c \left (b e -c d \right )}\, b^{5} x^{2} \left (c x +b \right )^{2} c}\)	\(360\)
derivativedivides	\(2 e^{4} \left (\frac {\left (b e -c d \right ) \left (\frac {\left (\frac {3}{8} A \,b^{2} e^{2} c -\frac {3}{2} A b \,c^{2} d e +\frac {1}{8} b^{3} B \,e^{2}+B \,b^{2} c d e \right ) \left (e x +d \right )^{\frac {3}{2}}+\frac {b e \left (5 A \,b^{2} e^{2} c -17 A b \,c^{2} d e +12 A \,c^{3} d^{2}-b^{3} B \,e^{2}+9 B \,b^{2} c d e -8 B b \,c^{2} d^{2}\right ) \sqrt {e x +d}}{8 c}}{\left (\left (e x +d \right ) c +b e -c d \right )^{2}}+\frac {\left (3 A \,b^{2} e^{2} c -36 A b \,c^{2} d e +48 A \,c^{3} d^{2}+b^{3} B \,e^{2}+8 B \,b^{2} c d e -24 B b \,c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{8 c \sqrt {c \left (b e -c d \right )}}\right )}{e^{4} b^{5}}-\frac {d \left (\frac {\left (\frac {9}{8} A \,b^{2} e^{2}-\frac {3}{2} A b c d e +\frac {1}{2} B \,b^{2} d e \right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {7}{8} A \,b^{2} d \,e^{2}+\frac {3}{2} A b c \,d^{2} e -\frac {1}{2} B \,b^{2} d^{2} e \right ) \sqrt {e x +d}}{e^{2} x^{2}}+\frac {\left (15 A \,b^{2} e^{2}-60 A b c d e +48 A \,c^{2} d^{2}+20 B \,b^{2} d e -24 B b c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 \sqrt {d}}\right )}{b^{5} e^{4}}\right )\)	\(401\)
default	\(2 e^{4} \left (\frac {\left (b e -c d \right ) \left (\frac {\left (\frac {3}{8} A \,b^{2} e^{2} c -\frac {3}{2} A b \,c^{2} d e +\frac {1}{8} b^{3} B \,e^{2}+B \,b^{2} c d e \right ) \left (e x +d \right )^{\frac {3}{2}}+\frac {b e \left (5 A \,b^{2} e^{2} c -17 A b \,c^{2} d e +12 A \,c^{3} d^{2}-b^{3} B \,e^{2}+9 B \,b^{2} c d e -8 B b \,c^{2} d^{2}\right ) \sqrt {e x +d}}{8 c}}{\left (\left (e x +d \right ) c +b e -c d \right )^{2}}+\frac {\left (3 A \,b^{2} e^{2} c -36 A b \,c^{2} d e +48 A \,c^{3} d^{2}+b^{3} B \,e^{2}+8 B \,b^{2} c d e -24 B b \,c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{8 c \sqrt {c \left (b e -c d \right )}}\right )}{e^{4} b^{5}}-\frac {d \left (\frac {\left (\frac {9}{8} A \,b^{2} e^{2}-\frac {3}{2} A b c d e +\frac {1}{2} B \,b^{2} d e \right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {7}{8} A \,b^{2} d \,e^{2}+\frac {3}{2} A b c \,d^{2} e -\frac {1}{2} B \,b^{2} d^{2} e \right ) \sqrt {e x +d}}{e^{2} x^{2}}+\frac {\left (15 A \,b^{2} e^{2}-60 A b c d e +48 A \,c^{2} d^{2}+20 B \,b^{2} d e -24 B b c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 \sqrt {d}}\right )}{b^{5} e^{4}}\right )\)	\(401\)
risch	\(-\frac {d \sqrt {e x +d}\, \left (9 A b e x -12 c x A d +4 B b d x +2 A b d \right )}{4 b^{4} x^{2}}-\frac {e \left (\frac {\sqrt {d}\, \left (15 A \,b^{2} e^{2}-60 A b c d e +48 A \,c^{2} d^{2}+20 B \,b^{2} d e -24 B b c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{e b}+\frac {\frac {8 \left (\left (-\frac {3}{8} A \,b^{3} c \,e^{3}+\frac {15}{8} A \,b^{2} c^{2} d \,e^{2}-\frac {3}{2} A b \,c^{3} d^{2} e -\frac {1}{8} b^{4} B \,e^{3}-\frac {7}{8} B \,b^{3} c d \,e^{2}+B \,b^{2} d^{2} e \,c^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}-\frac {b e \left (5 A \,b^{3} c \,e^{3}-22 A \,b^{2} c^{2} d \,e^{2}+29 A b \,c^{3} d^{2} e -12 A \,c^{4} d^{3}-b^{4} B \,e^{3}+10 B \,b^{3} c d \,e^{2}-17 B \,b^{2} d^{2} e \,c^{2}+8 B b \,d^{3} c^{3}\right ) \sqrt {e x +d}}{8 c}\right )}{\left (\left (e x +d \right ) c +b e -c d \right )^{2}}-\frac {\left (3 A \,b^{3} c \,e^{3}-39 A \,b^{2} c^{2} d \,e^{2}+84 A b \,c^{3} d^{2} e -48 A \,c^{4} d^{3}+b^{4} B \,e^{3}+7 B \,b^{3} c d \,e^{2}-32 B \,b^{2} d^{2} e \,c^{2}+24 B b \,d^{3} c^{3}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{c \sqrt {c \left (b e -c d \right )}}}{b e}\right )}{4 b^{4}}\)	\(442\)

Input:

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

Output:

-1/2/d^(1/2)*(-1/2*(3*A*b^2*c*e^2-36*A*b*c^2*d*e+48*A*c^3*d^2+B*b^3*e^2+8* 
B*b^2*c*d*e-24*B*b*c^2*d^2)*x^2*(c*x+b)^2*d^(1/2)*(b*e-c*d)*arctan(c*(e*x+ 
d)^(1/2)/(c*(b*e-c*d))^(1/2))+(1/2*c*d*(15*A*b^2*e^2-60*A*b*c*d*e+48*A*c^2 
*d^2+20*B*b^2*d*e-24*B*b*c*d^2)*x^2*(c*x+b)^2*arctanh((e*x+d)^(1/2)/d^(1/2 
))+d^(1/2)*(-12*A*c^4*d^2*x^3-18*((-1/3*B*x+A)*d-2/3*A*e*x)*d*x^2*b*c^3-4* 
((-9/4*B*x+A)*d^2-37/8*(-7/37*B*x+A)*e*x*d+3/8*A*e^2*x^2)*x*b^2*c^2+((2*B* 
x+A)*d^2+9/2*e*(-11/9*B*x+A)*x*d-5/2*e^2*x^2*(1/5*B*x+A))*b^3*c+1/2*B*b^4* 
e^2*x^2)*b*(e*x+d)^(1/2))*(c*(b*e-c*d))^(1/2))/(c*(b*e-c*d))^(1/2)/b^5/x^2 
/(c*x+b)^2/c
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 677 vs. \(2 (337) = 674\).

Time = 1.88 (sec) , antiderivative size = 2748, normalized size of antiderivative = 7.37 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/8*(((24*(B*b*c^4 - 2*A*c^5)*d^2 - 4*(2*B*b^2*c^3 - 9*A*b*c^4)*d*e - (B 
*b^3*c^2 + 3*A*b^2*c^3)*e^2)*x^4 + 2*(24*(B*b^2*c^3 - 2*A*b*c^4)*d^2 - 4*( 
2*B*b^3*c^2 - 9*A*b^2*c^3)*d*e - (B*b^4*c + 3*A*b^3*c^2)*e^2)*x^3 + (24*(B 
*b^3*c^2 - 2*A*b^2*c^3)*d^2 - 4*(2*B*b^4*c - 9*A*b^3*c^2)*d*e - (B*b^5 + 3 
*A*b^4*c)*e^2)*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)) - ((15*A*b^2*c^3*e^2 - 24*(B*b* 
c^4 - 2*A*c^5)*d^2 + 20*(B*b^2*c^3 - 3*A*b*c^4)*d*e)*x^4 + 2*(15*A*b^3*c^2 
*e^2 - 24*(B*b^2*c^3 - 2*A*b*c^4)*d^2 + 20*(B*b^3*c^2 - 3*A*b^2*c^3)*d*e)* 
x^3 + (15*A*b^4*c*e^2 - 24*(B*b^3*c^2 - 2*A*b^2*c^3)*d^2 + 20*(B*b^4*c - 3 
*A*b^3*c^2)*d*e)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) 
 + 2*(2*A*b^4*c*d^2 + (12*(B*b^2*c^3 - 2*A*b*c^4)*d^2 - (7*B*b^3*c^2 - 24* 
A*b^2*c^3)*d*e - (B*b^4*c + 3*A*b^3*c^2)*e^2)*x^3 + (18*(B*b^3*c^2 - 2*A*b 
^2*c^3)*d^2 - (11*B*b^4*c - 37*A*b^3*c^2)*d*e + (B*b^5 - 5*A*b^4*c)*e^2)*x 
^2 + (9*A*b^4*c*d*e + 4*(B*b^4*c - 2*A*b^3*c^2)*d^2)*x)*sqrt(e*x + d))/(b^ 
5*c^3*x^4 + 2*b^6*c^2*x^3 + b^7*c*x^2), -1/8*(2*((24*(B*b*c^4 - 2*A*c^5)*d 
^2 - 4*(2*B*b^2*c^3 - 9*A*b*c^4)*d*e - (B*b^3*c^2 + 3*A*b^2*c^3)*e^2)*x^4 
+ 2*(24*(B*b^2*c^3 - 2*A*b*c^4)*d^2 - 4*(2*B*b^3*c^2 - 9*A*b^2*c^3)*d*e - 
(B*b^4*c + 3*A*b^3*c^2)*e^2)*x^3 + (24*(B*b^3*c^2 - 2*A*b^2*c^3)*d^2 - 4*( 
2*B*b^4*c - 9*A*b^3*c^2)*d*e - (B*b^5 + 3*A*b^4*c)*e^2)*x^2)*sqrt(-(c*d - 
b*e)/c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) - ((1...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((B*x+A)*(e*x+d)^(5/2)/(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. 817 vs. \(2 (337) = 674\).

Time = 0.29 (sec) , antiderivative size = 817, normalized size of antiderivative = 2.19 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/4*(24*B*b*c*d^3 - 48*A*c^2*d^3 - 20*B*b^2*d^2*e + 60*A*b*c*d^2*e - 15*A 
*b^2*d*e^2)*arctan(sqrt(e*x + d)/sqrt(-d))/(b^5*sqrt(-d)) + 1/4*(24*B*b*c^ 
3*d^3 - 48*A*c^4*d^3 - 32*B*b^2*c^2*d^2*e + 84*A*b*c^3*d^2*e + 7*B*b^3*c*d 
*e^2 - 39*A*b^2*c^2*d*e^2 + B*b^4*e^3 + 3*A*b^3*c*e^3)*arctan(sqrt(e*x + d 
)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c^2*d + b*c*e)*b^5*c) - 1/4*(12*(e*x + d) 
^(7/2)*B*b*c^3*d^2*e - 24*(e*x + d)^(7/2)*A*c^4*d^2*e - 36*(e*x + d)^(5/2) 
*B*b*c^3*d^3*e + 72*(e*x + d)^(5/2)*A*c^4*d^3*e + 36*(e*x + d)^(3/2)*B*b*c 
^3*d^4*e - 72*(e*x + d)^(3/2)*A*c^4*d^4*e - 12*sqrt(e*x + d)*B*b*c^3*d^5*e 
 + 24*sqrt(e*x + d)*A*c^4*d^5*e - 7*(e*x + d)^(7/2)*B*b^2*c^2*d*e^2 + 24*( 
e*x + d)^(7/2)*A*b*c^3*d*e^2 + 39*(e*x + d)^(5/2)*B*b^2*c^2*d^2*e^2 - 108* 
(e*x + d)^(5/2)*A*b*c^3*d^2*e^2 - 57*(e*x + d)^(3/2)*B*b^2*c^2*d^3*e^2 + 1 
44*(e*x + d)^(3/2)*A*b*c^3*d^3*e^2 + 25*sqrt(e*x + d)*B*b^2*c^2*d^4*e^2 - 
60*sqrt(e*x + d)*A*b*c^3*d^4*e^2 - (e*x + d)^(7/2)*B*b^3*c*e^3 - 3*(e*x + 
d)^(7/2)*A*b^2*c^2*e^3 - 8*(e*x + d)^(5/2)*B*b^3*c*d*e^3 + 46*(e*x + d)^(5 
/2)*A*b^2*c^2*d*e^3 + 23*(e*x + d)^(3/2)*B*b^3*c*d^2*e^3 - 91*(e*x + d)^(3 
/2)*A*b^2*c^2*d^2*e^3 - 14*sqrt(e*x + d)*B*b^3*c*d^3*e^3 + 48*sqrt(e*x + d 
)*A*b^2*c^2*d^3*e^3 + (e*x + d)^(5/2)*B*b^4*e^4 - 5*(e*x + d)^(5/2)*A*b^3* 
c*e^4 - 2*(e*x + d)^(3/2)*B*b^4*d*e^4 + 19*(e*x + d)^(3/2)*A*b^3*c*d*e^4 + 
 sqrt(e*x + d)*B*b^4*d^2*e^4 - 12*sqrt(e*x + d)*A*b^3*c*d^2*e^4)/(((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)
 

Mupad [B] (verification not implemented)

Time = 13.49 (sec) , antiderivative size = 7001, normalized size of antiderivative = 18.77 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

(((d + e*x)^(7/2)*(B*b^3*e^3 + 3*A*b^2*c*e^3 + 24*A*c^3*d^2*e - 24*A*b*c^2 
*d*e^2 - 12*B*b*c^2*d^2*e + 7*B*b^2*c*d*e^2))/(4*b^4) - ((d + e*x)^(5/2)*( 
B*b^4*e^4 - 5*A*b^3*c*e^4 + 72*A*c^4*d^3*e - 108*A*b*c^3*d^2*e^2 + 46*A*b^ 
2*c^2*d*e^3 + 39*B*b^2*c^2*d^2*e^2 - 36*B*b*c^3*d^3*e - 8*B*b^3*c*d*e^3))/ 
(4*b^4*c) - ((d + e*x)^(1/2)*(24*A*c^4*d^5*e + B*b^4*d^2*e^4 - 60*A*b*c^3* 
d^4*e^2 - 12*A*b^3*c*d^2*e^4 - 14*B*b^3*c*d^3*e^3 + 48*A*b^2*c^2*d^3*e^3 + 
 25*B*b^2*c^2*d^4*e^2 - 12*B*b*c^3*d^5*e))/(4*b^4*c) + ((d + e*x)^(3/2)*(7 
2*A*c^4*d^4*e + 2*B*b^4*d*e^4 - 144*A*b*c^3*d^3*e^2 - 23*B*b^3*c*d^2*e^3 + 
 91*A*b^2*c^2*d^2*e^3 + 57*B*b^2*c^2*d^3*e^2 - 19*A*b^3*c*d*e^4 - 36*B*b*c 
^3*d^4*e))/(4*b^4*c))/(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) - (d^(1/2) 
*atan(((d^(1/2)*((d^(1/2)*((12*A*b^12*c^3*d*e^5 - B*b^13*c^2*d*e^5 + 24*A* 
b^10*c^5*d^3*e^3 - 36*A*b^11*c^4*d^2*e^4 - 12*B*b^11*c^4*d^3*e^3 + 13*B*b^ 
12*c^3*d^2*e^4)/(b^12*c) - (d^(1/2)*(64*b^11*c^3*e^3 - 128*b^10*c^4*d*e^2) 
*(d + e*x)^(1/2)*(15*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 + 20*B*b^2*d* 
e - 60*A*b*c*d*e))/(64*b^13*c))*(15*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^ 
2 + 20*B*b^2*d*e - 60*A*b*c*d*e))/(8*b^5) - ((d + e*x)^(1/2)*(B^2*b^8*e^8 
+ 9*A^2*b^6*c^2*e^8 + 4608*A^2*c^8*d^6*e^2 + 15840*A^2*b^2*c^6*d^4*e^4 - 8 
640*A^2*b^3*c^5*d^3*e^5 + 2250*A^2*b^4*c^4*d^2*e^6 + 1152*B^2*b^2*c^6*d...
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 2012, normalized size of antiderivative = 5.39 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int((B*x+A)*(e*x+d)^(5/2)/(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) 
))*a*b**4*c*e**2*x**2 - 72*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/ 
(sqrt(c)*sqrt(b*e - c*d)))*a*b**3*c**2*d*e*x**2 + 12*sqrt(c)*sqrt(b*e - c* 
d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*a*b**3*c**2*e**2*x**3 
 + 96*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c 
*d)))*a*b**2*c**3*d**2*x**2 - 144*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e 
*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*a*b**2*c**3*d*e*x**3 + 6*sqrt(c)*sqrt(b* 
e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*a*b**2*c**3*e** 
2*x**4 + 192*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt( 
b*e - c*d)))*a*b*c**4*d**2*x**3 - 72*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d 
+ e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*a*b*c**4*d*e*x**4 + 96*sqrt(c)*sqrt(b 
*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*a*c**5*d**2*x* 
*4 + 2*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - 
c*d)))*b**6*e**2*x**2 + 16*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/ 
(sqrt(c)*sqrt(b*e - c*d)))*b**5*c*d*e*x**2 + 4*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**2*x**3 - 48*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*x**2 + 32*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*e*x**3 + 2*sqrt(c)*sqrt(b*e - c*d)*atan((s 
qrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**4*c**2*e**2*x**4 - 96*sqr...