\(\int \frac {(A+B x) (d+e x)^{5/2}}{a+b x} \, dx\) [149]

Optimal result

Integrand size = 22, antiderivative size = 164 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{a+b x} \, dx=\frac {2 (A b-a B) (b d-a e)^2 \sqrt {d+e x}}{b^4}+\frac {2 (A b-a B) (b d-a e) (d+e x)^{3/2}}{3 b^3}+\frac {2 (A b-a B) (d+e x)^{5/2}}{5 b^2}+\frac {2 B (d+e x)^{7/2}}{7 b e}-\frac {2 (A b-a B) (b d-a e)^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{9/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.13 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{a+b x} \, dx=\frac {2 \sqrt {d+e x} \left (-105 a^3 B e^3+35 a^2 b e^2 (7 B d+3 A e+B e x)-7 a b^2 e \left (5 A e (7 d+e x)+B \left (23 d^2+11 d e x+3 e^2 x^2\right )\right )+b^3 \left (15 B (d+e x)^3+7 A e \left (23 d^2+11 d e x+3 e^2 x^2\right )\right )\right )}{105 b^4 e}-\frac {2 (A b-a B) (-b d+a e)^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{b^{9/2}} \] Input:

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

Output:

(2*Sqrt[d + e*x]*(-105*a^3*B*e^3 + 35*a^2*b*e^2*(7*B*d + 3*A*e + B*e*x) - 
7*a*b^2*e*(5*A*e*(7*d + e*x) + B*(23*d^2 + 11*d*e*x + 3*e^2*x^2)) + b^3*(1 
5*B*(d + e*x)^3 + 7*A*e*(23*d^2 + 11*d*e*x + 3*e^2*x^2))))/(105*b^4*e) - ( 
2*(A*b - a*B)*(-(b*d) + a*e)^(5/2)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b 
*d) + a*e]])/b^(9/2)
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {90, 60, 60, 60, 73, 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.

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

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

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 [A] (verified)

Time = 0.45 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.17

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {\left (a e -d b \right ) b}\, \left (\left (\frac {\left (e x +d \right )^{3} B}{7}+\frac {23 A \left (\frac {3}{23} e^{2} x^{2}+\frac {11}{23} d e x +d^{2}\right ) e}{15}\right ) b^{3}-\frac {7 a e \left (\frac {\left (3 e^{2} x^{2}+11 d e x +23 d^{2}\right ) B}{35}+A e \left (\frac {e x}{7}+d \right )\right ) b^{2}}{3}+a^{2} \left (\frac {\left (e x +7 d \right ) B}{3}+A e \right ) e^{2} b -B \,a^{3} e^{3}\right ) \sqrt {e x +d}-2 e \left (a e -d b \right )^{3} \left (A b -B a \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -d b \right ) b}}\right )}{e \,b^{4} \sqrt {\left (a e -d b \right ) b}}\)	\(192\)
risch	\(\frac {2 \left (15 b^{3} B \,x^{3} e^{3}+21 A \,b^{3} e^{3} x^{2}-21 B a \,b^{2} e^{3} x^{2}+45 B \,b^{3} d \,e^{2} x^{2}-35 A x a \,b^{2} e^{3}+77 A x \,b^{3} d \,e^{2}+35 B x \,a^{2} b \,e^{3}-77 B a \,b^{2} d \,e^{2} x +45 B x \,b^{3} d^{2} e +105 A \,a^{2} b \,e^{3}-245 A a \,b^{2} d \,e^{2}+161 A \,b^{3} d^{2} e -105 B \,a^{3} e^{3}+245 B \,a^{2} b d \,e^{2}-161 B a \,b^{2} d^{2} e +15 b^{3} B \,d^{3}\right ) \sqrt {e x +d}}{105 e \,b^{4}}-\frac {2 \left (A \,a^{3} b \,e^{3}-3 A \,a^{2} b^{2} d \,e^{2}+3 A a \,b^{3} d^{2} e -A \,b^{4} d^{3}-B \,a^{4} e^{3}+3 B \,a^{3} b d \,e^{2}-3 B \,a^{2} b^{2} d^{2} e +B a \,b^{3} d^{3}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -d b \right ) b}}\right )}{b^{4} \sqrt {\left (a e -d b \right ) b}}\)	\(319\)
derivativedivides	\(\frac {\frac {2 \left (\frac {b^{3} B \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {A \,b^{3} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {B a \,b^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {A a \,b^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {A \,b^{3} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {B \,a^{2} b \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {B a \,b^{2} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+A \,a^{2} b \,e^{3} \sqrt {e x +d}-2 A a \,b^{2} d \,e^{2} \sqrt {e x +d}+A \,b^{3} d^{2} e \sqrt {e x +d}-B \,a^{3} e^{3} \sqrt {e x +d}+2 B \,a^{2} b d \,e^{2} \sqrt {e x +d}-B a \,b^{2} d^{2} e \sqrt {e x +d}\right )}{b^{4}}-\frac {2 e \left (A \,a^{3} b \,e^{3}-3 A \,a^{2} b^{2} d \,e^{2}+3 A a \,b^{3} d^{2} e -A \,b^{4} d^{3}-B \,a^{4} e^{3}+3 B \,a^{3} b d \,e^{2}-3 B \,a^{2} b^{2} d^{2} e +B a \,b^{3} d^{3}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -d b \right ) b}}\right )}{b^{4} \sqrt {\left (a e -d b \right ) b}}}{e}\)	\(346\)
default	\(\frac {\frac {2 \left (\frac {b^{3} B \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {A \,b^{3} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {B a \,b^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {A a \,b^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {A \,b^{3} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {B \,a^{2} b \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {B a \,b^{2} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+A \,a^{2} b \,e^{3} \sqrt {e x +d}-2 A a \,b^{2} d \,e^{2} \sqrt {e x +d}+A \,b^{3} d^{2} e \sqrt {e x +d}-B \,a^{3} e^{3} \sqrt {e x +d}+2 B \,a^{2} b d \,e^{2} \sqrt {e x +d}-B a \,b^{2} d^{2} e \sqrt {e x +d}\right )}{b^{4}}-\frac {2 e \left (A \,a^{3} b \,e^{3}-3 A \,a^{2} b^{2} d \,e^{2}+3 A a \,b^{3} d^{2} e -A \,b^{4} d^{3}-B \,a^{4} e^{3}+3 B \,a^{3} b d \,e^{2}-3 B \,a^{2} b^{2} d^{2} e +B a \,b^{3} d^{3}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -d b \right ) b}}\right )}{b^{4} \sqrt {\left (a e -d b \right ) b}}}{e}\)	\(346\)

Input:

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

Output:

2*(((a*e-b*d)*b)^(1/2)*((1/7*(e*x+d)^3*B+23/15*A*(3/23*e^2*x^2+11/23*d*e*x 
+d^2)*e)*b^3-7/3*a*e*(1/35*(3*e^2*x^2+11*d*e*x+23*d^2)*B+A*e*(1/7*e*x+d))* 
b^2+a^2*(1/3*(e*x+7*d)*B+A*e)*e^2*b-B*a^3*e^3)*(e*x+d)^(1/2)-e*(a*e-b*d)^3 
*(A*b-B*a)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)))/((a*e-b*d)*b)^(1/2 
)/e/b^4
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 290 vs. \(2 (140) = 280\).

Time = 0.09 (sec) , antiderivative size = 591, normalized size of antiderivative = 3.60 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{a+b x} \, dx=\left [-\frac {105 \, {\left ({\left (B a b^{2} - A b^{3}\right )} d^{2} e - 2 \, {\left (B a^{2} b - A a b^{2}\right )} d e^{2} + {\left (B a^{3} - A a^{2} b\right )} e^{3}\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) - 2 \, {\left (15 \, B b^{3} e^{3} x^{3} + 15 \, B b^{3} d^{3} - 161 \, {\left (B a b^{2} - A b^{3}\right )} d^{2} e + 245 \, {\left (B a^{2} b - A a b^{2}\right )} d e^{2} - 105 \, {\left (B a^{3} - A a^{2} b\right )} e^{3} + 3 \, {\left (15 \, B b^{3} d e^{2} - 7 \, {\left (B a b^{2} - A b^{3}\right )} e^{3}\right )} x^{2} + {\left (45 \, B b^{3} d^{2} e - 77 \, {\left (B a b^{2} - A b^{3}\right )} d e^{2} + 35 \, {\left (B a^{2} b - A a b^{2}\right )} e^{3}\right )} x\right )} \sqrt {e x + d}}{105 \, b^{4} e}, \frac {2 \, {\left (105 \, {\left ({\left (B a b^{2} - A b^{3}\right )} d^{2} e - 2 \, {\left (B a^{2} b - A a b^{2}\right )} d e^{2} + {\left (B a^{3} - A a^{2} b\right )} e^{3}\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) + {\left (15 \, B b^{3} e^{3} x^{3} + 15 \, B b^{3} d^{3} - 161 \, {\left (B a b^{2} - A b^{3}\right )} d^{2} e + 245 \, {\left (B a^{2} b - A a b^{2}\right )} d e^{2} - 105 \, {\left (B a^{3} - A a^{2} b\right )} e^{3} + 3 \, {\left (15 \, B b^{3} d e^{2} - 7 \, {\left (B a b^{2} - A b^{3}\right )} e^{3}\right )} x^{2} + {\left (45 \, B b^{3} d^{2} e - 77 \, {\left (B a b^{2} - A b^{3}\right )} d e^{2} + 35 \, {\left (B a^{2} b - A a b^{2}\right )} e^{3}\right )} x\right )} \sqrt {e x + d}\right )}}{105 \, b^{4} e}\right ] \] Input:

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

Output:

[-1/105*(105*((B*a*b^2 - A*b^3)*d^2*e - 2*(B*a^2*b - A*a*b^2)*d*e^2 + (B*a 
^3 - A*a^2*b)*e^3)*sqrt((b*d - a*e)/b)*log((b*e*x + 2*b*d - a*e - 2*sqrt(e 
*x + d)*b*sqrt((b*d - a*e)/b))/(b*x + a)) - 2*(15*B*b^3*e^3*x^3 + 15*B*b^3 
*d^3 - 161*(B*a*b^2 - A*b^3)*d^2*e + 245*(B*a^2*b - A*a*b^2)*d*e^2 - 105*( 
B*a^3 - A*a^2*b)*e^3 + 3*(15*B*b^3*d*e^2 - 7*(B*a*b^2 - A*b^3)*e^3)*x^2 + 
(45*B*b^3*d^2*e - 77*(B*a*b^2 - A*b^3)*d*e^2 + 35*(B*a^2*b - A*a*b^2)*e^3) 
*x)*sqrt(e*x + d))/(b^4*e), 2/105*(105*((B*a*b^2 - A*b^3)*d^2*e - 2*(B*a^2 
*b - A*a*b^2)*d*e^2 + (B*a^3 - A*a^2*b)*e^3)*sqrt(-(b*d - a*e)/b)*arctan(- 
sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) + (15*B*b^3*e^3*x^3 + 15 
*B*b^3*d^3 - 161*(B*a*b^2 - A*b^3)*d^2*e + 245*(B*a^2*b - A*a*b^2)*d*e^2 - 
 105*(B*a^3 - A*a^2*b)*e^3 + 3*(15*B*b^3*d*e^2 - 7*(B*a*b^2 - A*b^3)*e^3)* 
x^2 + (45*B*b^3*d^2*e - 77*(B*a*b^2 - A*b^3)*d*e^2 + 35*(B*a^2*b - A*a*b^2 
)*e^3)*x)*sqrt(e*x + d))/(b^4*e)]
 

Sympy [A] (verification not implemented)

Time = 3.96 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.57 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{a+b x} \, dx=\begin {cases} \frac {2 \left (\frac {B \left (d + e x\right )^{\frac {7}{2}}}{7 b} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (A b e - B a e\right )}{5 b^{2}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (- A a b e^{2} + A b^{2} d e + B a^{2} e^{2} - B a b d e\right )}{3 b^{3}} + \frac {\sqrt {d + e x} \left (A a^{2} b e^{3} - 2 A a b^{2} d e^{2} + A b^{3} d^{2} e - B a^{3} e^{3} + 2 B a^{2} b d e^{2} - B a b^{2} d^{2} e\right )}{b^{4}} + \frac {e \left (- A b + B a\right ) \left (a e - b d\right )^{3} \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e - b d}{b}}} \right )}}{b^{5} \sqrt {\frac {a e - b d}{b}}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (\frac {B x}{b} - \frac {\left (- A b + B a\right ) \left (\begin {cases} \frac {x}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b x \right )}}{b} & \text {otherwise} \end {cases}\right )}{b}\right ) & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((2*(B*(d + e*x)**(7/2)/(7*b) + (d + e*x)**(5/2)*(A*b*e - B*a*e)/ 
(5*b**2) + (d + e*x)**(3/2)*(-A*a*b*e**2 + A*b**2*d*e + B*a**2*e**2 - B*a* 
b*d*e)/(3*b**3) + sqrt(d + e*x)*(A*a**2*b*e**3 - 2*A*a*b**2*d*e**2 + A*b** 
3*d**2*e - B*a**3*e**3 + 2*B*a**2*b*d*e**2 - B*a*b**2*d**2*e)/b**4 + e*(-A 
*b + B*a)*(a*e - b*d)**3*atan(sqrt(d + e*x)/sqrt((a*e - b*d)/b))/(b**5*sqr 
t((a*e - b*d)/b)))/e, Ne(e, 0)), (d**(5/2)*(B*x/b - (-A*b + B*a)*Piecewise 
((x/a, Eq(b, 0)), (log(a + b*x)/b, True))/b), True))
 

Maxima [F(-2)]

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

integrate((B*x+A)*(e*x+d)^(5/2)/(b*x+a),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. 371 vs. \(2 (140) = 280\).

Time = 0.13 (sec) , antiderivative size = 371, normalized size of antiderivative = 2.26 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{a+b x} \, dx=-\frac {2 \, {\left (B a b^{3} d^{3} - A b^{4} d^{3} - 3 \, B a^{2} b^{2} d^{2} e + 3 \, A a b^{3} d^{2} e + 3 \, B a^{3} b d e^{2} - 3 \, A a^{2} b^{2} d e^{2} - B a^{4} e^{3} + A a^{3} b e^{3}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{4}} + \frac {2 \, {\left (15 \, {\left (e x + d\right )}^{\frac {7}{2}} B b^{6} e^{6} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} B a b^{5} e^{7} + 21 \, {\left (e x + d\right )}^{\frac {5}{2}} A b^{6} e^{7} - 35 \, {\left (e x + d\right )}^{\frac {3}{2}} B a b^{5} d e^{7} + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} A b^{6} d e^{7} - 105 \, \sqrt {e x + d} B a b^{5} d^{2} e^{7} + 105 \, \sqrt {e x + d} A b^{6} d^{2} e^{7} + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{2} b^{4} e^{8} - 35 \, {\left (e x + d\right )}^{\frac {3}{2}} A a b^{5} e^{8} + 210 \, \sqrt {e x + d} B a^{2} b^{4} d e^{8} - 210 \, \sqrt {e x + d} A a b^{5} d e^{8} - 105 \, \sqrt {e x + d} B a^{3} b^{3} e^{9} + 105 \, \sqrt {e x + d} A a^{2} b^{4} e^{9}\right )}}{105 \, b^{7} e^{7}} \] Input:

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

Output:

-2*(B*a*b^3*d^3 - A*b^4*d^3 - 3*B*a^2*b^2*d^2*e + 3*A*a*b^3*d^2*e + 3*B*a^ 
3*b*d*e^2 - 3*A*a^2*b^2*d*e^2 - B*a^4*e^3 + A*a^3*b*e^3)*arctan(sqrt(e*x + 
 d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^4) + 2/105*(15*(e*x + 
d)^(7/2)*B*b^6*e^6 - 21*(e*x + d)^(5/2)*B*a*b^5*e^7 + 21*(e*x + d)^(5/2)*A 
*b^6*e^7 - 35*(e*x + d)^(3/2)*B*a*b^5*d*e^7 + 35*(e*x + d)^(3/2)*A*b^6*d*e 
^7 - 105*sqrt(e*x + d)*B*a*b^5*d^2*e^7 + 105*sqrt(e*x + d)*A*b^6*d^2*e^7 + 
 35*(e*x + d)^(3/2)*B*a^2*b^4*e^8 - 35*(e*x + d)^(3/2)*A*a*b^5*e^8 + 210*s 
qrt(e*x + d)*B*a^2*b^4*d*e^8 - 210*sqrt(e*x + d)*A*a*b^5*d*e^8 - 105*sqrt( 
e*x + d)*B*a^3*b^3*e^9 + 105*sqrt(e*x + d)*A*a^2*b^4*e^9)/(b^7*e^7)
 

Mupad [B] (verification not implemented)

Time = 1.00 (sec) , antiderivative size = 330, normalized size of antiderivative = 2.01 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{a+b x} \, dx=\left (\frac {2\,A\,e-2\,B\,d}{5\,b\,e}-\frac {2\,B\,\left (a\,e^2-b\,d\,e\right )}{5\,b^2\,e^2}\right )\,{\left (d+e\,x\right )}^{5/2}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\left (A\,b-B\,a\right )\,{\left (a\,e-b\,d\right )}^{5/2}\,\sqrt {d+e\,x}}{B\,a^4\,e^3-3\,B\,a^3\,b\,d\,e^2-A\,a^3\,b\,e^3+3\,B\,a^2\,b^2\,d^2\,e+3\,A\,a^2\,b^2\,d\,e^2-B\,a\,b^3\,d^3-3\,A\,a\,b^3\,d^2\,e+A\,b^4\,d^3}\right )\,\left (A\,b-B\,a\right )\,{\left (a\,e-b\,d\right )}^{5/2}}{b^{9/2}}+\frac {2\,B\,{\left (d+e\,x\right )}^{7/2}}{7\,b\,e}+\frac {\left (\frac {2\,A\,e-2\,B\,d}{b\,e}-\frac {2\,B\,\left (a\,e^2-b\,d\,e\right )}{b^2\,e^2}\right )\,{\left (a\,e^2-b\,d\,e\right )}^2\,\sqrt {d+e\,x}}{b^2\,e^2}-\frac {\left (\frac {2\,A\,e-2\,B\,d}{b\,e}-\frac {2\,B\,\left (a\,e^2-b\,d\,e\right )}{b^2\,e^2}\right )\,\left (a\,e^2-b\,d\,e\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,b\,e} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.23 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{a+b x} \, dx=\frac {2 \sqrt {e x +d}\, \left (e^{3} x^{3}+3 d \,e^{2} x^{2}+3 d^{2} e x +d^{3}\right )}{7 e} \] Input:

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

Output:

(2*sqrt(d + e*x)*(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e**3*x**3))/(7*e)