\(\int \frac {(A+B x) (d+e x)^{3/2}}{(a+b x)^{3/2}} \, dx\) [215]

Optimal result

Integrand size = 24, antiderivative size = 165 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{(a+b x)^{3/2}} \, dx=\frac {3 (b B d+4 A b e-5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 b^3}-\frac {2 (A b-a B) (d+e x)^{3/2}}{b^2 \sqrt {a+b x}}+\frac {B \sqrt {a+b x} (d+e x)^{3/2}}{2 b^2}+\frac {3 (b d-a e) (b B d+4 A b e-5 a B e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{4 b^{7/2} \sqrt {e}} \] Output:

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

Mathematica [A] (verified)

Time = 0.98 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.99 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{(a+b x)^{3/2}} \, dx=\frac {\frac {\sqrt {b} \sqrt {d+e x} \left (4 A b (-2 b d+3 a e+b e x)+B \left (-15 a^2 e+a b (13 d-5 e x)+b^2 x (5 d+2 e x)\right )\right )}{\sqrt {a+b x}}+\frac {6 (-b d+a e) (b B d+4 A b e-5 a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \left (\sqrt {a-\frac {b d}{e}}-\sqrt {a+b x}\right )}\right )}{\sqrt {e}}}{4 b^{7/2}} \] Input:

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

Output:

((Sqrt[b]*Sqrt[d + e*x]*(4*A*b*(-2*b*d + 3*a*e + b*e*x) + B*(-15*a^2*e + a 
*b*(13*d - 5*e*x) + b^2*x*(5*d + 2*e*x))))/Sqrt[a + b*x] + (6*(-(b*d) + a* 
e)*(b*B*d + 4*A*b*e - 5*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/(Sqrt[e]*(S 
qrt[a - (b*d)/e] - Sqrt[a + b*x]))])/Sqrt[e])/(4*b^(7/2))
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.12, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {87, 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.

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

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(-2*(A*b - a*B)*(d + e*x)^(5/2))/(b*(b*d - a*e)*Sqrt[a + b*x]) + ((b*B*d + 
 4*A*b*e - 5*a*B*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]*Sqrt[d + e*x])])/(b^(3/2)*Sqrt[e])))/(4*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. \(739\) vs. \(2(135)=270\).

Time = 0.27 (sec) , antiderivative size = 740, normalized size of antiderivative = 4.48

method	result	size
default	\(-\frac {\sqrt {e x +d}\, \left (12 A \ln \left (\frac {2 b e x +2 \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+a e +d b}{2 \sqrt {b e}}\right ) a \,b^{2} e^{2} x -12 A \ln \left (\frac {2 b e x +2 \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+a e +d b}{2 \sqrt {b e}}\right ) b^{3} d e x -15 B \ln \left (\frac {2 b e x +2 \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+a e +d b}{2 \sqrt {b e}}\right ) a^{2} b \,e^{2} x +18 B \ln \left (\frac {2 b e x +2 \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+a e +d b}{2 \sqrt {b e}}\right ) a \,b^{2} d e x -3 B \ln \left (\frac {2 b e x +2 \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+a e +d b}{2 \sqrt {b e}}\right ) b^{3} d^{2} x -4 B \,b^{2} e \,x^{2} \sqrt {b e}\, \sqrt {\left (e x +d \right ) \left (b x +a \right )}+12 A \ln \left (\frac {2 b e x +2 \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+a e +d b}{2 \sqrt {b e}}\right ) a^{2} b \,e^{2}-12 A \ln \left (\frac {2 b e x +2 \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+a e +d b}{2 \sqrt {b e}}\right ) a \,b^{2} d e -8 A \,b^{2} e x \sqrt {b e}\, \sqrt {\left (e x +d \right ) \left (b x +a \right )}-15 B \ln \left (\frac {2 b e x +2 \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+a e +d b}{2 \sqrt {b e}}\right ) a^{3} e^{2}+18 B \ln \left (\frac {2 b e x +2 \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+a e +d b}{2 \sqrt {b e}}\right ) a^{2} b d e -3 B \ln \left (\frac {2 b e x +2 \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+a e +d b}{2 \sqrt {b e}}\right ) a \,b^{2} d^{2}+10 B a b e x \sqrt {b e}\, \sqrt {\left (e x +d \right ) \left (b x +a \right )}-10 B \,b^{2} d x \sqrt {b e}\, \sqrt {\left (e x +d \right ) \left (b x +a \right )}-24 A a b e \sqrt {b e}\, \sqrt {\left (e x +d \right ) \left (b x +a \right )}+16 A \,b^{2} d \sqrt {b e}\, \sqrt {\left (e x +d \right ) \left (b x +a \right )}+30 B \,a^{2} e \sqrt {b e}\, \sqrt {\left (e x +d \right ) \left (b x +a \right )}-26 B a b d \sqrt {b e}\, \sqrt {\left (e x +d \right ) \left (b x +a \right )}\right )}{8 \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}\, \sqrt {b x +a}\, b^{3}}\)	\(740\)

Input:

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

Output:

-1/8*(e*x+d)^(1/2)*(12*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^2*e^2*x-12*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^3*d*e*x-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^2*b*e^2*x 
+18*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^2*d*e*x-3*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))*b^3*d^2*x-4*B*b^2*e*x^2*(b*e)^(1/2)*((e*x+d)*(b* 
x+a))^(1/2)+12*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*e^2-12*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^2*d*e-8*A*b^2*e*x*(b*e)^(1/2)*((e* 
x+d)*(b*x+a))^(1/2)-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^3*e^2+18*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*d*e-3*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^2*d^2+10*B*a 
*b*e*x*(b*e)^(1/2)*((e*x+d)*(b*x+a))^(1/2)-10*B*b^2*d*x*(b*e)^(1/2)*((e*x+ 
d)*(b*x+a))^(1/2)-24*A*a*b*e*(b*e)^(1/2)*((e*x+d)*(b*x+a))^(1/2)+16*A*b^2* 
d*(b*e)^(1/2)*((e*x+d)*(b*x+a))^(1/2)+30*B*a^2*e*(b*e)^(1/2)*((e*x+d)*(b*x 
+a))^(1/2)-26*B*a*b*d*(b*e)^(1/2)*((e*x+d)*(b*x+a))^(1/2))/((e*x+d)*(b*x+a 
))^(1/2)/(b*e)^(1/2)/(b*x+a)^(1/2)/b^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (135) = 270\).

Time = 0.53 (sec) , antiderivative size = 584, normalized size of antiderivative = 3.54 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{(a+b x)^{3/2}} \, dx=\left [-\frac {3 \, {\left (B a b^{2} d^{2} - 2 \, {\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} d e + {\left (5 \, B a^{3} - 4 \, A a^{2} b\right )} e^{2} + {\left (B b^{3} d^{2} - 2 \, {\left (3 \, B a b^{2} - 2 \, A b^{3}\right )} d e + {\left (5 \, B a^{2} b - 4 \, A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt {b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} - 4 \, {\left (2 \, b e x + b d + a e\right )} \sqrt {b e} \sqrt {b x + a} \sqrt {e x + d} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) - 4 \, {\left (2 \, B b^{3} e^{2} x^{2} + {\left (13 \, B a b^{2} - 8 \, A b^{3}\right )} d e - 3 \, {\left (5 \, B a^{2} b - 4 \, A a b^{2}\right )} e^{2} + {\left (5 \, B b^{3} d e - {\left (5 \, B a b^{2} - 4 \, A b^{3}\right )} e^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{16 \, {\left (b^{5} e x + a b^{4} e\right )}}, -\frac {3 \, {\left (B a b^{2} d^{2} - 2 \, {\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} d e + {\left (5 \, B a^{3} - 4 \, A a^{2} b\right )} e^{2} + {\left (B b^{3} d^{2} - 2 \, {\left (3 \, B a b^{2} - 2 \, A b^{3}\right )} d e + {\left (5 \, B a^{2} b - 4 \, A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt {-b e} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {-b e} \sqrt {b x + a} \sqrt {e x + d}}{2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) - 2 \, {\left (2 \, B b^{3} e^{2} x^{2} + {\left (13 \, B a b^{2} - 8 \, A b^{3}\right )} d e - 3 \, {\left (5 \, B a^{2} b - 4 \, A a b^{2}\right )} e^{2} + {\left (5 \, B b^{3} d e - {\left (5 \, B a b^{2} - 4 \, A b^{3}\right )} e^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{8 \, {\left (b^{5} e x + a b^{4} e\right )}}\right ] \] Input:

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

Output:

[-1/16*(3*(B*a*b^2*d^2 - 2*(3*B*a^2*b - 2*A*a*b^2)*d*e + (5*B*a^3 - 4*A*a^ 
2*b)*e^2 + (B*b^3*d^2 - 2*(3*B*a*b^2 - 2*A*b^3)*d*e + (5*B*a^2*b - 4*A*a*b 
^2)*e^2)*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*(2*B*b^3*e^2*x^2 + (13*B*a*b^2 - 8*A*b^3)*d*e - 3*(5*B* 
a^2*b - 4*A*a*b^2)*e^2 + (5*B*b^3*d*e - (5*B*a*b^2 - 4*A*b^3)*e^2)*x)*sqrt 
(b*x + a)*sqrt(e*x + d))/(b^5*e*x + a*b^4*e), -1/8*(3*(B*a*b^2*d^2 - 2*(3* 
B*a^2*b - 2*A*a*b^2)*d*e + (5*B*a^3 - 4*A*a^2*b)*e^2 + (B*b^3*d^2 - 2*(3*B 
*a*b^2 - 2*A*b^3)*d*e + (5*B*a^2*b - 4*A*a*b^2)*e^2)*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*(2*B*b^3*e^2*x^2 + (13*B*a*b^2 
 - 8*A*b^3)*d*e - 3*(5*B*a^2*b - 4*A*a*b^2)*e^2 + (5*B*b^3*d*e - (5*B*a*b^ 
2 - 4*A*b^3)*e^2)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^5*e*x + a*b^4*e)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((B*x+A)*(e*x+d)^(3/2)/(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. 327 vs. \(2 (135) = 270\).

Time = 0.27 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.98 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{(a+b x)^{3/2}} \, dx=\frac {1}{4} \, \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \sqrt {b x + a} {\left (\frac {2 \, {\left (b x + a\right )} B e {\left | b \right |}}{b^{5}} + \frac {5 \, B b^{10} d e^{2} {\left | b \right |} - 9 \, B a b^{9} e^{3} {\left | b \right |} + 4 \, A b^{10} e^{3} {\left | b \right |}}{b^{14} e^{2}}\right )} - \frac {3 \, {\left (B b^{2} d^{2} {\left | b \right |} - 6 \, B a b d e {\left | b \right |} + 4 \, A b^{2} d e {\left | b \right |} + 5 \, B a^{2} e^{2} {\left | b \right |} - 4 \, A a b e^{2} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2}\right )}{8 \, \sqrt {b e} b^{4}} + \frac {4 \, {\left (B a b^{2} d^{2} e {\left | b \right |} - A b^{3} d^{2} e {\left | b \right |} - 2 \, B a^{2} b d e^{2} {\left | b \right |} + 2 \, A a b^{2} d e^{2} {\left | b \right |} + B a^{3} e^{3} {\left | b \right |} - A a^{2} b e^{3} {\left | b \right |}\right )}}{{\left (b^{2} d - a b e - {\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2}\right )} \sqrt {b e} b^{3}} \] Input:

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

Output:

1/4*sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*B*e*abs 
(b)/b^5 + (5*B*b^10*d*e^2*abs(b) - 9*B*a*b^9*e^3*abs(b) + 4*A*b^10*e^3*abs 
(b))/(b^14*e^2)) - 3/8*(B*b^2*d^2*abs(b) - 6*B*a*b*d*e*abs(b) + 4*A*b^2*d* 
e*abs(b) + 5*B*a^2*e^2*abs(b) - 4*A*a*b*e^2*abs(b))*log((sqrt(b*e)*sqrt(b* 
x + a) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2)/(sqrt(b*e)*b^4) + 4*(B*a* 
b^2*d^2*e*abs(b) - A*b^3*d^2*e*abs(b) - 2*B*a^2*b*d*e^2*abs(b) + 2*A*a*b^2 
*d*e^2*abs(b) + B*a^3*e^3*abs(b) - A*a^2*b*e^3*abs(b))/((b^2*d - a*b*e - ( 
sqrt(b*e)*sqrt(b*x + a) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2)*sqrt(b*e 
)*b^3)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.20 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{(a+b x)^{3/2}} \, dx=\frac {-3 \sqrt {e x +d}\, \sqrt {b x +a}\, a b \,e^{2}+5 \sqrt {e x +d}\, \sqrt {b x +a}\, b^{2} d e +2 \sqrt {e x +d}\, \sqrt {b x +a}\, b^{2} e^{2} x +3 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) a^{2} e^{2}-6 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) a b d e +3 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) b^{2} d^{2}}{4 b^{3} e} \] Input:

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

Output:

( - 3*sqrt(d + e*x)*sqrt(a + b*x)*a*b*e**2 + 5*sqrt(d + e*x)*sqrt(a + b*x) 
*b**2*d*e + 2*sqrt(d + e*x)*sqrt(a + b*x)*b**2*e**2*x + 3*sqrt(e)*sqrt(b)* 
log((sqrt(e)*sqrt(a + b*x) + sqrt(b)*sqrt(d + e*x))/sqrt(a*e - b*d))*a**2* 
e**2 - 6*sqrt(e)*sqrt(b)*log((sqrt(e)*sqrt(a + b*x) + sqrt(b)*sqrt(d + e*x 
))/sqrt(a*e - b*d))*a*b*d*e + 3*sqrt(e)*sqrt(b)*log((sqrt(e)*sqrt(a + b*x) 
 + sqrt(b)*sqrt(d + e*x))/sqrt(a*e - b*d))*b**2*d**2)/(4*b**3*e)