\(\int \sqrt {a+b x} (A+B x) \sqrt {d+e x} \, dx\) [174]

Optimal result

Integrand size = 24, antiderivative size = 190 \[ \int \sqrt {a+b x} (A+B x) \sqrt {d+e x} \, dx=-\frac {(b d-a e) (b B d-2 A b e+a B e) \sqrt {a+b x} \sqrt {d+e x}}{8 b^2 e^2}-\frac {(b B d-2 A b e+a B e) (a+b x)^{3/2} \sqrt {d+e x}}{4 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{3/2}}{3 b e}+\frac {(b d-a e)^2 (b B d-2 A b e+a B e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 b^{5/2} e^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.84 \[ \int \sqrt {a+b x} (A+B x) \sqrt {d+e x} \, dx=\frac {\sqrt {a+b x} \sqrt {d+e x} \left (-3 a^2 B e^2+2 a b e (3 A e+B (d+e x))+b^2 \left (6 A e (d+2 e x)+B \left (-3 d^2+2 d e x+8 e^2 x^2\right )\right )\right )}{24 b^2 e^2}+\frac {(b d-a e)^2 (b B d-2 A b e+a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{8 b^{5/2} e^{5/2}} \] Input:

Integrate[Sqrt[a + b*x]*(A + B*x)*Sqrt[d + e*x],x]
 

Output:

(Sqrt[a + b*x]*Sqrt[d + e*x]*(-3*a^2*B*e^2 + 2*a*b*e*(3*A*e + B*(d + e*x)) 
 + b^2*(6*A*e*(d + 2*e*x) + B*(-3*d^2 + 2*d*e*x + 8*e^2*x^2))))/(24*b^2*e^ 
2) + ((b*d - a*e)^2*(b*B*d - 2*A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e* 
x])/(Sqrt[e]*Sqrt[a + b*x])])/(8*b^(5/2)*e^(5/2))
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.89, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {90, 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.

\(\displaystyle \int \sqrt {a+b x} (A+B x) \sqrt {d+e x} \, dx\)

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 221

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

Input:

Int[Sqrt[a + b*x]*(A + B*x)*Sqrt[d + e*x],x]
 

Output:

(B*(a + b*x)^(3/2)*(d + e*x)^(3/2))/(3*b*e) + ((2*A*b*e - B*(b*d + a*e))*( 
((a + b*x)^(3/2)*Sqrt[d + e*x])/(2*b) + ((b*d - a*e)*((Sqrt[a + b*x]*Sqrt[ 
d + e*x])/e - ((b*d - a*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d 
 + e*x])])/(Sqrt[b]*e^(3/2))))/(4*b)))/(2*b*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 90

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

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. \(635\) vs. \(2(158)=316\).

Time = 0.26 (sec) , antiderivative size = 636, normalized size of antiderivative = 3.35

method	result	size
default	\(-\frac {\sqrt {b x +a}\, \sqrt {e x +d}\, \left (-16 B \,b^{2} e^{2} x^{2} \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+6 A \ln \left (\frac {2 b e x +2 \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+a e +d b}{2 \sqrt {b e}}\right ) a^{2} b \,e^{3}-12 A \ln \left (\frac {2 b e x +2 \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+a e +d b}{2 \sqrt {b e}}\right ) a \,b^{2} d \,e^{2}+6 A \ln \left (\frac {2 b e x +2 \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+a e +d b}{2 \sqrt {b e}}\right ) b^{3} d^{2} e -24 A \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}\, b^{2} e^{2} x -3 B \ln \left (\frac {2 b e x +2 \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+a e +d b}{2 \sqrt {b e}}\right ) a^{3} e^{3}+3 B \ln \left (\frac {2 b e x +2 \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+a e +d b}{2 \sqrt {b e}}\right ) a^{2} b d \,e^{2}+3 B \ln \left (\frac {2 b e x +2 \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+a e +d b}{2 \sqrt {b e}}\right ) a \,b^{2} d^{2} e -3 B \ln \left (\frac {2 b e x +2 \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+a e +d b}{2 \sqrt {b e}}\right ) b^{3} d^{3}-4 B \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}\, a b \,e^{2} x -4 B \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}\, b^{2} d e x -12 A \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}\, a b \,e^{2}-12 A \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}\, b^{2} d e +6 B \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}\, a^{2} e^{2}-4 B \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}\, a b d e +6 B \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}\, b^{2} d^{2}\right )}{48 \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, b^{2} e^{2} \sqrt {b e}}\)	\(636\)

Input:

int((b*x+a)^(1/2)*(B*x+A)*(e*x+d)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/48*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(-16*B*b^2*e^2*x^2*((e*x+d)*(b*x+a))^(1/ 
2)*(b*e)^(1/2)+6*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^3-12*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^2+6*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^2*e-24*A*((e 
*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)*b^2*e^2*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))*a^3*e^3+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^2*b*d*e^2 
+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*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))*b^3*d^3-4*B*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)*a 
*b*e^2*x-4*B*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)*b^2*d*e*x-12*A*((e*x+d)*( 
b*x+a))^(1/2)*(b*e)^(1/2)*a*b*e^2-12*A*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2) 
*b^2*d*e+6*B*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)*a^2*e^2-4*B*((e*x+d)*(b*x 
+a))^(1/2)*(b*e)^(1/2)*a*b*d*e+6*B*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)*b^2 
*d^2)/((e*x+d)*(b*x+a))^(1/2)/b^2/e^2/(b*e)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 528, normalized size of antiderivative = 2.78 \[ \int \sqrt {a+b x} (A+B x) \sqrt {d+e x} \, dx=\left [-\frac {3 \, {\left (B b^{3} d^{3} - {\left (B a b^{2} + 2 \, A b^{3}\right )} d^{2} e - {\left (B a^{2} b - 4 \, A a b^{2}\right )} d e^{2} + {\left (B a^{3} - 2 \, A a^{2} b\right )} e^{3}\right )} \sqrt {b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} - 4 \, {\left (2 \, b e x + b d + a e\right )} \sqrt {b e} \sqrt {b x + a} \sqrt {e x + d} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) - 4 \, {\left (8 \, B b^{3} e^{3} x^{2} - 3 \, B b^{3} d^{2} e + 2 \, {\left (B a b^{2} + 3 \, A b^{3}\right )} d e^{2} - 3 \, {\left (B a^{2} b - 2 \, A a b^{2}\right )} e^{3} + 2 \, {\left (B b^{3} d e^{2} + {\left (B a b^{2} + 6 \, A b^{3}\right )} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{96 \, b^{3} e^{3}}, -\frac {3 \, {\left (B b^{3} d^{3} - {\left (B a b^{2} + 2 \, A b^{3}\right )} d^{2} e - {\left (B a^{2} b - 4 \, A a b^{2}\right )} d e^{2} + {\left (B a^{3} - 2 \, A a^{2} b\right )} e^{3}\right )} \sqrt {-b e} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {-b e} \sqrt {b x + a} \sqrt {e x + d}}{2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) - 2 \, {\left (8 \, B b^{3} e^{3} x^{2} - 3 \, B b^{3} d^{2} e + 2 \, {\left (B a b^{2} + 3 \, A b^{3}\right )} d e^{2} - 3 \, {\left (B a^{2} b - 2 \, A a b^{2}\right )} e^{3} + 2 \, {\left (B b^{3} d e^{2} + {\left (B a b^{2} + 6 \, A b^{3}\right )} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{48 \, b^{3} e^{3}}\right ] \] Input:

integrate((b*x+a)^(1/2)*(B*x+A)*(e*x+d)^(1/2),x, algorithm="fricas")
 

Output:

[-1/96*(3*(B*b^3*d^3 - (B*a*b^2 + 2*A*b^3)*d^2*e - (B*a^2*b - 4*A*a*b^2)*d 
*e^2 + (B*a^3 - 2*A*a^2*b)*e^3)*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^3*e^3*x^2 - 3*B*b^3*d^2*e + 
2*(B*a*b^2 + 3*A*b^3)*d*e^2 - 3*(B*a^2*b - 2*A*a*b^2)*e^3 + 2*(B*b^3*d*e^2 
 + (B*a*b^2 + 6*A*b^3)*e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^3*e^3), -1/ 
48*(3*(B*b^3*d^3 - (B*a*b^2 + 2*A*b^3)*d^2*e - (B*a^2*b - 4*A*a*b^2)*d*e^2 
 + (B*a^3 - 2*A*a^2*b)*e^3)*sqrt(-b*e)*arctan(1/2*(2*b*e*x + b*d + a*e)*sq 
rt(-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^3*e^3*x^2 - 3*B*b^3*d^2*e + 2*(B*a*b^2 + 3*A*b^3)*d 
*e^2 - 3*(B*a^2*b - 2*A*a*b^2)*e^3 + 2*(B*b^3*d*e^2 + (B*a*b^2 + 6*A*b^3)* 
e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^3*e^3)]
 

Sympy [F]

\[ \int \sqrt {a+b x} (A+B x) \sqrt {d+e x} \, dx=\int \left (A + B x\right ) \sqrt {a + b x} \sqrt {d + e x}\, dx \] Input:

integrate((b*x+a)**(1/2)*(B*x+A)*(e*x+d)**(1/2),x)
 

Output:

Integral((A + B*x)*sqrt(a + b*x)*sqrt(d + e*x), x)
 

Maxima [F(-2)]

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

integrate((b*x+a)^(1/2)*(B*x+A)*(e*x+d)^(1/2),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(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 557 vs. \(2 (158) = 316\).

Time = 0.20 (sec) , antiderivative size = 557, normalized size of antiderivative = 2.93 \[ \int \sqrt {a+b x} (A+B x) \sqrt {d+e x} \, dx=-\frac {\frac {24 \, {\left (\frac {{\left (b^{2} d - a b e\right )} \log \left ({\left | -\sqrt {b e} \sqrt {b x + a} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b e}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \sqrt {b x + a}\right )} A a {\left | b \right |}}{b^{2}} - \frac {6 \, {\left (\sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} {\left (2 \, b x + 2 \, a + \frac {b d e - 5 \, a e^{2}}{e^{2}}\right )} \sqrt {b x + a} + \frac {{\left (b^{3} d^{2} + 2 \, a b^{2} d e - 3 \, a^{2} b e^{2}\right )} \log \left ({\left | -\sqrt {b e} \sqrt {b x + a} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b e} e}\right )} B a {\left | b \right |}}{b^{3}} - \frac {6 \, {\left (\sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} {\left (2 \, b x + 2 \, a + \frac {b d e - 5 \, a e^{2}}{e^{2}}\right )} \sqrt {b x + a} + \frac {{\left (b^{3} d^{2} + 2 \, a b^{2} d e - 3 \, a^{2} b e^{2}\right )} \log \left ({\left | -\sqrt {b e} \sqrt {b x + a} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b e} e}\right )} A {\left | b \right |}}{b^{2}} - \frac {{\left (\sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} {\left (2 \, {\left (4 \, b x + 4 \, a + \frac {b d e^{3} - 13 \, a e^{4}}{e^{4}}\right )} {\left (b x + a\right )} - \frac {3 \, {\left (b^{2} d^{2} e^{2} + 2 \, a b d e^{3} - 11 \, a^{2} e^{4}\right )}}{e^{4}}\right )} \sqrt {b x + a} - \frac {3 \, {\left (b^{4} d^{3} + a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - 5 \, a^{3} b e^{3}\right )} \log \left ({\left | -\sqrt {b e} \sqrt {b x + a} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b e} e^{2}}\right )} B {\left | b \right |}}{b^{3}}}{24 \, b} \] Input:

integrate((b*x+a)^(1/2)*(B*x+A)*(e*x+d)^(1/2),x, algorithm="giac")
 

Output:

-1/24*(24*((b^2*d - a*b*e)*log(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d + 
 (b*x + a)*b*e - a*b*e)))/sqrt(b*e) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e)* 
sqrt(b*x + a))*A*a*abs(b)/b^2 - 6*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2* 
b*x + 2*a + (b*d*e - 5*a*e^2)/e^2)*sqrt(b*x + a) + (b^3*d^2 + 2*a*b^2*d*e 
- 3*a^2*b*e^2)*log(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d + (b*x + a)*b 
*e - a*b*e)))/(sqrt(b*e)*e))*B*a*abs(b)/b^3 - 6*(sqrt(b^2*d + (b*x + a)*b* 
e - a*b*e)*(2*b*x + 2*a + (b*d*e - 5*a*e^2)/e^2)*sqrt(b*x + a) + (b^3*d^2 
+ 2*a*b^2*d*e - 3*a^2*b*e^2)*log(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d 
 + (b*x + a)*b*e - a*b*e)))/(sqrt(b*e)*e))*A*abs(b)/b^2 - (sqrt(b^2*d + (b 
*x + a)*b*e - a*b*e)*(2*(4*b*x + 4*a + (b*d*e^3 - 13*a*e^4)/e^4)*(b*x + a) 
 - 3*(b^2*d^2*e^2 + 2*a*b*d*e^3 - 11*a^2*e^4)/e^4)*sqrt(b*x + a) - 3*(b^4* 
d^3 + a*b^3*d^2*e + 3*a^2*b^2*d*e^2 - 5*a^3*b*e^3)*log(abs(-sqrt(b*e)*sqrt 
(b*x + a) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/(sqrt(b*e)*e^2))*B*abs(b 
)/b^3)/b
 

Mupad [B] (verification not implemented)

Time = 44.92 (sec) , antiderivative size = 1207, normalized size of antiderivative = 6.35 \[ \int \sqrt {a+b x} (A+B x) \sqrt {d+e x} \, dx=\text {Too large to display} \] Input:

int((A + B*x)*(a + b*x)^(1/2)*(d + e*x)^(1/2),x)
 

Output:

A*(x/2 + (a*e + b*d)/(4*b*e))*(a + b*x)^(1/2)*(d + e*x)^(1/2) - ((((a + b* 
x)^(1/2) - a^(1/2))*((B*b^6*d^3)/4 + (B*a^3*b^3*e^3)/4 - (B*a^2*b^4*d*e^2) 
/4 - (B*a*b^5*d^2*e)/4))/(e^8*((d + e*x)^(1/2) - d^(1/2))) - (((a + b*x)^( 
1/2) - a^(1/2))^3*((17*B*b^5*d^3)/12 + (17*B*a^3*b^2*e^3)/12 + (101*B*a^2* 
b^3*d*e^2)/4 + (101*B*a*b^4*d^2*e)/4))/(e^7*((d + e*x)^(1/2) - d^(1/2))^3) 
 - (((a + b*x)^(1/2) - a^(1/2))^7*((19*B*a^3*e^3)/2 + (19*B*b^3*d^3)/2 + ( 
269*B*a*b^2*d^2*e)/2 + (269*B*a^2*b*d*e^2)/2))/(e^5*((d + e*x)^(1/2) - d^( 
1/2))^7) - (((a + b*x)^(1/2) - a^(1/2))^5*((19*B*b^4*d^3)/2 + (19*B*a^3*b* 
e^3)/2 + (269*B*a^2*b^2*d*e^2)/2 + (269*B*a*b^3*d^2*e)/2))/(e^6*((d + e*x) 
^(1/2) - d^(1/2))^5) + (((a + b*x)^(1/2) - a^(1/2))^11*((B*a^3*e^3)/4 + (B 
*b^3*d^3)/4 - (B*a*b^2*d^2*e)/4 - (B*a^2*b*d*e^2)/4))/(b^2*e^3*((d + e*x)^ 
(1/2) - d^(1/2))^11) - (((a + b*x)^(1/2) - a^(1/2))^9*((17*B*a^3*e^3)/12 + 
 (17*B*b^3*d^3)/12 + (101*B*a*b^2*d^2*e)/4 + (101*B*a^2*b*d*e^2)/4))/(b*e^ 
4*((d + e*x)^(1/2) - d^(1/2))^9) + (a^(1/2)*d^(1/2)*((a + b*x)^(1/2) - a^( 
1/2))^4*(32*B*b^4*d^2 + 32*B*a^2*b^2*e^2 + 96*B*a*b^3*d*e))/(e^6*((d + e*x 
)^(1/2) - d^(1/2))^4) + (8*B*a^(3/2)*d^(3/2)*((a + b*x)^(1/2) - a^(1/2))^1 
0)/(e^2*((d + e*x)^(1/2) - d^(1/2))^10) + (a^(1/2)*d^(1/2)*((a + b*x)^(1/2 
) - a^(1/2))^6*(64*B*b^3*d^2 + 64*B*a^2*b*e^2 + (656*B*a*b^2*d*e)/3))/(e^5 
*((d + e*x)^(1/2) - d^(1/2))^6) + (a^(1/2)*d^(1/2)*((a + b*x)^(1/2) - a^(1 
/2))^8*(32*B*a^2*e^2 + 32*B*b^2*d^2 + 96*B*a*b*d*e))/(e^4*((d + e*x)^(1...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.68 \[ \int \sqrt {a+b x} (A+B x) \sqrt {d+e x} \, dx=\frac {3 \sqrt {e x +d}\, \sqrt {b x +a}\, a^{2} b \,e^{3}+8 \sqrt {e x +d}\, \sqrt {b x +a}\, a \,b^{2} d \,e^{2}+14 \sqrt {e x +d}\, \sqrt {b x +a}\, a \,b^{2} e^{3} x -3 \sqrt {e x +d}\, \sqrt {b x +a}\, b^{3} d^{2} e +2 \sqrt {e x +d}\, \sqrt {b x +a}\, b^{3} d \,e^{2} x +8 \sqrt {e x +d}\, \sqrt {b x +a}\, b^{3} e^{3} x^{2}-3 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) a^{3} e^{3}+9 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) a^{2} b d \,e^{2}-9 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) a \,b^{2} d^{2} e +3 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) b^{3} d^{3}}{24 b^{2} e^{3}} \] Input:

int((b*x+a)^(1/2)*(B*x+A)*(e*x+d)^(1/2),x)
 

Output:

(3*sqrt(d + e*x)*sqrt(a + b*x)*a**2*b*e**3 + 8*sqrt(d + e*x)*sqrt(a + b*x) 
*a*b**2*d*e**2 + 14*sqrt(d + e*x)*sqrt(a + b*x)*a*b**2*e**3*x - 3*sqrt(d + 
 e*x)*sqrt(a + b*x)*b**3*d**2*e + 2*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 - 3*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 + 9*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 - 9*sqrt(e)*sqrt(b)*log((sqrt(e)*sqrt(a + b 
*x) + sqrt(b)*sqrt(d + e*x))/sqrt(a*e - b*d))*a*b**2*d**2*e + 3*sqrt(e)*sq 
rt(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**2*e**3)