\(\int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{3/2}} \, dx\) [214]

Optimal result

Integrand size = 24, antiderivative size = 215 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{3/2}} \, dx=\frac {5 (b d-a e) (b B d+6 A b e-7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{8 b^4}+\frac {5 (b B d+6 A b e-7 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^3}-\frac {2 (A b-a B) (d+e x)^{5/2}}{b^2 \sqrt {a+b x}}+\frac {B \sqrt {a+b x} (d+e x)^{5/2}}{3 b^2}+\frac {5 (b d-a e)^2 (b B d+6 A b e-7 a B e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 b^{9/2} \sqrt {e}} \] Output:

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

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.01 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{3/2}} \, dx=\frac {\sqrt {d+e x} \left (-6 A b \left (15 a^2 e^2+5 a b e (-5 d+e x)+b^2 \left (8 d^2-9 d e x-2 e^2 x^2\right )\right )+B \left (105 a^3 e^2+5 a^2 b e (-38 d+7 e x)+a b^2 \left (81 d^2-68 d e x-14 e^2 x^2\right )+b^3 x \left (33 d^2+26 d e x+8 e^2 x^2\right )\right )\right )}{24 b^4 \sqrt {a+b x}}+\frac {5 (b d-a e)^2 (b B d+6 A b e-7 a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{8 b^{9/2} \sqrt {e}} \] Input:

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

Output:

(Sqrt[d + e*x]*(-6*A*b*(15*a^2*e^2 + 5*a*b*e*(-5*d + e*x) + b^2*(8*d^2 - 9 
*d*e*x - 2*e^2*x^2)) + B*(105*a^3*e^2 + 5*a^2*b*e*(-38*d + 7*e*x) + a*b^2* 
(81*d^2 - 68*d*e*x - 14*e^2*x^2) + b^3*x*(33*d^2 + 26*d*e*x + 8*e^2*x^2))) 
)/(24*b^4*Sqrt[a + b*x]) + (5*(b*d - a*e)^2*(b*B*d + 6*A*b*e - 7*a*B*e)*Ar 
cTanh[(Sqrt[b]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a + b*x])])/(8*b^(9/2)*Sqrt[e] 
)
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {87, 60, 60, 60, 66, 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.

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

\(\Big \downarrow \) 87

\(\displaystyle \frac {(-7 a B e+6 A b e+b B d) \int \frac {(d+e x)^{5/2}}{\sqrt {a+b x}}dx}{b (b d-a e)}-\frac {2 (d+e x)^{7/2} (A b-a B)}{b \sqrt {a+b x} (b d-a e)}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(-7 a B e+6 A b e+b B d) \left (\frac {5 (b d-a e) \int \frac {(d+e x)^{3/2}}{\sqrt {a+b x}}dx}{6 b}+\frac {\sqrt {a+b x} (d+e x)^{5/2}}{3 b}\right )}{b (b d-a e)}-\frac {2 (d+e x)^{7/2} (A b-a B)}{b \sqrt {a+b x} (b d-a e)}\)

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

\(\displaystyle \frac {(-7 a B e+6 A b e+b B d) \left (\frac {5 (b d-a e) \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}}dx}{2 b}+\frac {\sqrt {a+b x} \sqrt {d+e x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (d+e x)^{3/2}}{2 b}\right )}{6 b}+\frac {\sqrt {a+b x} (d+e x)^{5/2}}{3 b}\right )}{b (b d-a e)}-\frac {2 (d+e x)^{7/2} (A b-a B)}{b \sqrt {a+b x} (b d-a e)}\)

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(-2*(A*b - a*B)*(d + e*x)^(7/2))/(b*(b*d - a*e)*Sqrt[a + b*x]) + ((b*B*d + 
 6*A*b*e - 7*a*B*e)*((Sqrt[a + b*x]*(d + e*x)^(5/2))/(3*b) + (5*(b*d - a*e 
)*((Sqrt[a + b*x]*(d + e*x)^(3/2))/(2*b) + (3*(b*d - a*e)*((Sqrt[a + b*x]* 
Sqrt[d + e*x])/b + ((b*d - a*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*S 
qrt[d + e*x])])/(b^(3/2)*Sqrt[e])))/(4*b)))/(6*b)))/(b*(b*d - a*e))
 

Defintions of rubi rules used

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1183\) vs. \(2(179)=358\).

Time = 0.28 (sec) , antiderivative size = 1184, normalized size of antiderivative = 5.51

method	result	size
default	\(\text {Expression too large to display}\)	\(1184\)

Input:

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

Output:

1/48*(e*x+d)^(1/2)*(-28*B*a*b^2*e^2*x^2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2 
)+52*B*b^3*d*e*x^2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-60*A*a*b^2*e^2*x*(( 
e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+108*A*b^3*d*e*x*((e*x+d)*(b*x+a))^(1/2)* 
(b*e)^(1/2)+70*B*a^2*b*e^2*x*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+90*A*ln(1 
/2*(2*b*e*x+2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+a*e+d*b)/(b*e)^(1/2))*a^ 
3*b*e^3+15*B*ln(1/2*(2*b*e*x+2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+a*e+d*b 
)/(b*e)^(1/2))*a*b^3*d^3-136*B*a*b^2*d*e*x*((e*x+d)*(b*x+a))^(1/2)*(b*e)^( 
1/2)+300*A*a*b^2*d*e*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-380*B*a^2*b*d*e*( 
(e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-180*A*ln(1/2*(2*b*e*x+2*((e*x+d)*(b*x+a 
))^(1/2)*(b*e)^(1/2)+a*e+d*b)/(b*e)^(1/2))*a*b^3*d*e^2*x+225*B*ln(1/2*(2*b 
*e*x+2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+a*e+d*b)/(b*e)^(1/2))*a^2*b^2*d 
*e^2*x-135*B*ln(1/2*(2*b*e*x+2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+a*e+d*b 
)/(b*e)^(1/2))*a*b^3*d^2*e*x-180*A*a^2*b*e^2*((e*x+d)*(b*x+a))^(1/2)*(b*e) 
^(1/2)+162*B*a*b^2*d^2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+90*A*ln(1/2*(2* 
b*e*x+2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+a*e+d*b)/(b*e)^(1/2))*a^2*b^2* 
e^3*x+90*A*ln(1/2*(2*b*e*x+2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+a*e+d*b)/ 
(b*e)^(1/2))*b^4*d^2*e*x-105*B*ln(1/2*(2*b*e*x+2*((e*x+d)*(b*x+a))^(1/2)*( 
b*e)^(1/2)+a*e+d*b)/(b*e)^(1/2))*a^3*b*e^3*x-180*A*ln(1/2*(2*b*e*x+2*((e*x 
+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+a*e+d*b)/(b*e)^(1/2))*a^2*b^2*d*e^2+90*A*ln 
(1/2*(2*b*e*x+2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+a*e+d*b)/(b*e)^(1/2...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 429 vs. \(2 (179) = 358\).

Time = 0.53 (sec) , antiderivative size = 872, normalized size of antiderivative = 4.06 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/96*(15*(B*a*b^3*d^3 - 3*(3*B*a^2*b^2 - 2*A*a*b^3)*d^2*e + 3*(5*B*a^3*b 
- 4*A*a^2*b^2)*d*e^2 - (7*B*a^4 - 6*A*a^3*b)*e^3 + (B*b^4*d^3 - 3*(3*B*a*b 
^3 - 2*A*b^4)*d^2*e + 3*(5*B*a^2*b^2 - 4*A*a*b^3)*d*e^2 - (7*B*a^3*b - 6*A 
*a^2*b^2)*e^3)*x)*sqrt(b*e)*log(8*b^2*e^2*x^2 + b^2*d^2 + 6*a*b*d*e + a^2* 
e^2 + 4*(2*b*e*x + b*d + a*e)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 8*(b 
^2*d*e + a*b*e^2)*x) + 4*(8*B*b^4*e^3*x^3 + 3*(27*B*a*b^3 - 16*A*b^4)*d^2* 
e - 10*(19*B*a^2*b^2 - 15*A*a*b^3)*d*e^2 + 15*(7*B*a^3*b - 6*A*a^2*b^2)*e^ 
3 + 2*(13*B*b^4*d*e^2 - (7*B*a*b^3 - 6*A*b^4)*e^3)*x^2 + (33*B*b^4*d^2*e - 
 2*(34*B*a*b^3 - 27*A*b^4)*d*e^2 + 5*(7*B*a^2*b^2 - 6*A*a*b^3)*e^3)*x)*sqr 
t(b*x + a)*sqrt(e*x + d))/(b^6*e*x + a*b^5*e), -1/48*(15*(B*a*b^3*d^3 - 3* 
(3*B*a^2*b^2 - 2*A*a*b^3)*d^2*e + 3*(5*B*a^3*b - 4*A*a^2*b^2)*d*e^2 - (7*B 
*a^4 - 6*A*a^3*b)*e^3 + (B*b^4*d^3 - 3*(3*B*a*b^3 - 2*A*b^4)*d^2*e + 3*(5* 
B*a^2*b^2 - 4*A*a*b^3)*d*e^2 - (7*B*a^3*b - 6*A*a^2*b^2)*e^3)*x)*sqrt(-b*e 
)*arctan(1/2*(2*b*e*x + b*d + a*e)*sqrt(-b*e)*sqrt(b*x + a)*sqrt(e*x + d)/ 
(b^2*e^2*x^2 + a*b*d*e + (b^2*d*e + a*b*e^2)*x)) - 2*(8*B*b^4*e^3*x^3 + 3* 
(27*B*a*b^3 - 16*A*b^4)*d^2*e - 10*(19*B*a^2*b^2 - 15*A*a*b^3)*d*e^2 + 15* 
(7*B*a^3*b - 6*A*a^2*b^2)*e^3 + 2*(13*B*b^4*d*e^2 - (7*B*a*b^3 - 6*A*b^4)* 
e^3)*x^2 + (33*B*b^4*d^2*e - 2*(34*B*a*b^3 - 27*A*b^4)*d*e^2 + 5*(7*B*a^2* 
b^2 - 6*A*a*b^3)*e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^6*e*x + a*b^5*e)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((B*x+A)*(e*x+d)^(5/2)/(b*x+a)^(3/2),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(a*e-b*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. 479 vs. \(2 (179) = 358\).

Time = 0.39 (sec) , antiderivative size = 479, normalized size of antiderivative = 2.23 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{3/2}} \, dx=\frac {1}{24} \, \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )} B e^{2} {\left | b \right |}}{b^{6}} + \frac {13 \, B b^{18} d e^{5} {\left | b \right |} - 19 \, B a b^{17} e^{6} {\left | b \right |} + 6 \, A b^{18} e^{6} {\left | b \right |}}{b^{23} e^{4}}\right )} + \frac {3 \, {\left (11 \, B b^{19} d^{2} e^{4} {\left | b \right |} - 40 \, B a b^{18} d e^{5} {\left | b \right |} + 18 \, A b^{19} d e^{5} {\left | b \right |} + 29 \, B a^{2} b^{17} e^{6} {\left | b \right |} - 18 \, A a b^{18} e^{6} {\left | b \right |}\right )}}{b^{23} e^{4}}\right )} - \frac {5 \, {\left (B b^{3} d^{3} {\left | b \right |} - 9 \, B a b^{2} d^{2} e {\left | b \right |} + 6 \, A b^{3} d^{2} e {\left | b \right |} + 15 \, B a^{2} b d e^{2} {\left | b \right |} - 12 \, A a b^{2} d e^{2} {\left | b \right |} - 7 \, B a^{3} e^{3} {\left | b \right |} + 6 \, A a^{2} b e^{3} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2}\right )}{16 \, \sqrt {b e} b^{5}} + \frac {4 \, {\left (B a b^{3} d^{3} e {\left | b \right |} - A b^{4} d^{3} e {\left | b \right |} - 3 \, B a^{2} b^{2} d^{2} e^{2} {\left | b \right |} + 3 \, A a b^{3} d^{2} e^{2} {\left | b \right |} + 3 \, B a^{3} b d e^{3} {\left | b \right |} - 3 \, A a^{2} b^{2} d e^{3} {\left | b \right |} - B a^{4} e^{4} {\left | b \right |} + A a^{3} b e^{4} {\left | b \right |}\right )}}{{\left (b^{2} d - a b e - {\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2}\right )} \sqrt {b e} b^{4}} \] Input:

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

Output:

1/24*sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b* 
x + a)*B*e^2*abs(b)/b^6 + (13*B*b^18*d*e^5*abs(b) - 19*B*a*b^17*e^6*abs(b) 
 + 6*A*b^18*e^6*abs(b))/(b^23*e^4)) + 3*(11*B*b^19*d^2*e^4*abs(b) - 40*B*a 
*b^18*d*e^5*abs(b) + 18*A*b^19*d*e^5*abs(b) + 29*B*a^2*b^17*e^6*abs(b) - 1 
8*A*a*b^18*e^6*abs(b))/(b^23*e^4)) - 5/16*(B*b^3*d^3*abs(b) - 9*B*a*b^2*d^ 
2*e*abs(b) + 6*A*b^3*d^2*e*abs(b) + 15*B*a^2*b*d*e^2*abs(b) - 12*A*a*b^2*d 
*e^2*abs(b) - 7*B*a^3*e^3*abs(b) + 6*A*a^2*b*e^3*abs(b))*log((sqrt(b*e)*sq 
rt(b*x + a) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2)/(sqrt(b*e)*b^5) + 4* 
(B*a*b^3*d^3*e*abs(b) - A*b^4*d^3*e*abs(b) - 3*B*a^2*b^2*d^2*e^2*abs(b) + 
3*A*a*b^3*d^2*e^2*abs(b) + 3*B*a^3*b*d*e^3*abs(b) - 3*A*a^2*b^2*d*e^3*abs( 
b) - B*a^4*e^4*abs(b) + A*a^3*b*e^4*abs(b))/((b^2*d - a*b*e - (sqrt(b*e)*s 
qrt(b*x + a) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2)*sqrt(b*e)*b^4)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.49 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{3/2}} \, dx=\frac {15 \sqrt {e x +d}\, \sqrt {b x +a}\, a^{2} b \,e^{3}-40 \sqrt {e x +d}\, \sqrt {b x +a}\, a \,b^{2} d \,e^{2}-10 \sqrt {e x +d}\, \sqrt {b x +a}\, a \,b^{2} e^{3} x +33 \sqrt {e x +d}\, \sqrt {b x +a}\, b^{3} d^{2} e +26 \sqrt {e x +d}\, \sqrt {b x +a}\, b^{3} d \,e^{2} x +8 \sqrt {e x +d}\, \sqrt {b x +a}\, b^{3} e^{3} x^{2}-15 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) a^{3} e^{3}+45 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) a^{2} b d \,e^{2}-45 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) a \,b^{2} d^{2} e +15 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) b^{3} d^{3}}{24 b^{4} e} \] Input:

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

Output:

(15*sqrt(d + e*x)*sqrt(a + b*x)*a**2*b*e**3 - 40*sqrt(d + e*x)*sqrt(a + b* 
x)*a*b**2*d*e**2 - 10*sqrt(d + e*x)*sqrt(a + b*x)*a*b**2*e**3*x + 33*sqrt( 
d + e*x)*sqrt(a + b*x)*b**3*d**2*e + 26*sqrt(d + e*x)*sqrt(a + b*x)*b**3*d 
*e**2*x + 8*sqrt(d + e*x)*sqrt(a + b*x)*b**3*e**3*x**2 - 15*sqrt(e)*sqrt(b 
)*log((sqrt(e)*sqrt(a + b*x) + sqrt(b)*sqrt(d + e*x))/sqrt(a*e - b*d))*a** 
3*e**3 + 45*sqrt(e)*sqrt(b)*log((sqrt(e)*sqrt(a + b*x) + sqrt(b)*sqrt(d + 
e*x))/sqrt(a*e - b*d))*a**2*b*d*e**2 - 45*sqrt(e)*sqrt(b)*log((sqrt(e)*sqr 
t(a + b*x) + sqrt(b)*sqrt(d + e*x))/sqrt(a*e - b*d))*a*b**2*d**2*e + 15*sq 
rt(e)*sqrt(b)*log((sqrt(e)*sqrt(a + b*x) + sqrt(b)*sqrt(d + e*x))/sqrt(a*e 
 - b*d))*b**3*d**3)/(24*b**4*e)