\(\int \frac {(a+b x)^{5/2} (A+B x)}{\sqrt {d+e x}} \, dx\) [196]

Optimal result

Integrand size = 24, antiderivative size = 246 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{\sqrt {d+e x}} \, dx=-\frac {5 (b d-a e)^2 (7 b B d-8 A b e+a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b e^4}+\frac {5 (b d-a e) (7 b B d-8 A b e+a B e) (a+b x)^{3/2} \sqrt {d+e x}}{96 b e^3}-\frac {(7 b B d-8 A b e+a B e) (a+b x)^{5/2} \sqrt {d+e x}}{24 b e^2}+\frac {B (a+b x)^{7/2} \sqrt {d+e x}}{4 b e}+\frac {5 (b d-a e)^3 (7 b B d-8 A b e+a B e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{64 b^{3/2} e^{9/2}} \] Output:

-5/64*(-a*e+b*d)^2*(-8*A*b*e+B*a*e+7*B*b*d)*(b*x+a)^(1/2)*(e*x+d)^(1/2)/b/ 
e^4+5/96*(-a*e+b*d)*(-8*A*b*e+B*a*e+7*B*b*d)*(b*x+a)^(3/2)*(e*x+d)^(1/2)/b 
/e^3-1/24*(-8*A*b*e+B*a*e+7*B*b*d)*(b*x+a)^(5/2)*(e*x+d)^(1/2)/b/e^2+1/4*B 
*(b*x+a)^(7/2)*(e*x+d)^(1/2)/b/e+5/64*(-a*e+b*d)^3*(-8*A*b*e+B*a*e+7*B*b*d 
)*arctanh(e^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(e*x+d)^(1/2))/b^(3/2)/e^(9/2)
 

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{\sqrt {d+e x}} \, dx=\frac {\sqrt {a+b x} \sqrt {d+e x} \left (15 a^3 B e^3+a^2 b e^2 (-191 B d+264 A e+118 B e x)+a b^2 e \left (16 A e (-20 d+13 e x)+B \left (265 d^2-172 d e x+136 e^2 x^2\right )\right )+b^3 \left (8 A e \left (15 d^2-10 d e x+8 e^2 x^2\right )+B \left (-105 d^3+70 d^2 e x-56 d e^2 x^2+48 e^3 x^3\right )\right )\right )}{192 b e^4}+\frac {5 (b d-a e)^3 (7 b B d-8 A b e+a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{64 b^{3/2} e^{9/2}} \] Input:

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

Output:

(Sqrt[a + b*x]*Sqrt[d + e*x]*(15*a^3*B*e^3 + a^2*b*e^2*(-191*B*d + 264*A*e 
 + 118*B*e*x) + a*b^2*e*(16*A*e*(-20*d + 13*e*x) + B*(265*d^2 - 172*d*e*x 
+ 136*e^2*x^2)) + b^3*(8*A*e*(15*d^2 - 10*d*e*x + 8*e^2*x^2) + B*(-105*d^3 
 + 70*d^2*e*x - 56*d*e^2*x^2 + 48*e^3*x^3))))/(192*b*e^4) + (5*(b*d - a*e) 
^3*(7*b*B*d - 8*A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/(Sqrt[e]*Sq 
rt[a + b*x])])/(64*b^(3/2)*e^(9/2))
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.85, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {90, 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.

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

\(\Big \downarrow \) 90

\(\displaystyle \frac {B (a+b x)^{7/2} \sqrt {d+e x}}{4 b e}-\frac {(a B e-8 A b e+7 b B d) \int \frac {(a+b x)^{5/2}}{\sqrt {d+e x}}dx}{8 b e}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {B (a+b x)^{7/2} \sqrt {d+e x}}{4 b e}-\frac {(a B e-8 A b e+7 b B d) \left (\frac {(a+b x)^{5/2} \sqrt {d+e x}}{3 e}-\frac {5 (b d-a e) \int \frac {(a+b x)^{3/2}}{\sqrt {d+e x}}dx}{6 e}\right )}{8 b e}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {B (a+b x)^{7/2} \sqrt {d+e x}}{4 b e}-\frac {(a B e-8 A b e+7 b B d) \left (\frac {(a+b x)^{5/2} \sqrt {d+e x}}{3 e}-\frac {5 (b d-a e) \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}}dx}{4 e}\right )}{6 e}\right )}{8 b e}\)

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(B*(a + b*x)^(7/2)*Sqrt[d + e*x])/(4*b*e) - ((7*b*B*d - 8*A*b*e + a*B*e)*( 
((a + b*x)^(5/2)*Sqrt[d + e*x])/(3*e) - (5*(b*d - a*e)*(((a + b*x)^(3/2)*S 
qrt[d + e*x])/(2*e) - (3*(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*e)))/(6*e)))/(8*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. \(967\) vs. \(2(208)=416\).

Time = 0.26 (sec) , antiderivative size = 968, normalized size of antiderivative = 3.93

method	result	size
default	\(\frac {\sqrt {b x +a}\, \sqrt {e x +d}\, \left (96 B \,b^{3} e^{3} x^{3} \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+128 A \,b^{3} e^{3} x^{2} \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}-112 B \,b^{3} d \,e^{2} x^{2} \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}-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^{4} e^{4}+105 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^{4} d^{4}-344 B \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}\, a \,b^{2} d \,e^{2} x -360 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^{2} d \,e^{3}+360 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^{3} d^{2} e^{2}+416 A \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}\, a \,b^{2} e^{3} x -160 A \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}\, b^{3} d \,e^{2} x +236 B \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}\, a^{2} b \,e^{3} x +140 B \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}\, b^{3} d^{2} e x +30 B \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}\, a^{3} e^{3}-210 B \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}\, b^{3} d^{3}+120 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^{3} b \,e^{4}-120 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^{4} d^{3} e -640 A a \,b^{2} d \,e^{2} \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}+272 B a \,b^{2} e^{3} x^{2} \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}-300 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^{3} d^{3} e +528 A \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}\, a^{2} b \,e^{3}+240 A \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}\, b^{3} d^{2} e -60 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} b d \,e^{3}+270 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^{2} d^{2} e^{2}-382 B \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}\, a^{2} b d \,e^{2}+530 B \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}\, a \,b^{2} d^{2} e \right )}{384 b \,e^{4} \sqrt {\left (e x +d \right ) \left (b x +a \right )}\, \sqrt {b e}}\)	\(968\)

Input:

int((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/384*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(96*B*b^3*e^3*x^3*((e*x+d)*(b*x+a))^(1/2 
)*(b*e)^(1/2)+128*A*b^3*e^3*x^2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-112*B* 
b^3*d*e^2*x^2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(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^4*e^4+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))*b 
^4*d^4-344*B*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)*a*b^2*d*e^2*x-360*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^3+360*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^2*e^2+416*A*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2) 
*a*b^2*e^3*x-160*A*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)*b^3*d*e^2*x+236*B*( 
(e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)*a^2*b*e^3*x+140*B*((e*x+d)*(b*x+a))^(1/ 
2)*(b*e)^(1/2)*b^3*d^2*e*x+30*B*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)*a^3*e^ 
3-210*B*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)*b^3*d^3+120*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^4-120* 
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^3*e-640*A*a*b^2*d*e^2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+272*B* 
a*b^2*e^3*x^2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-300*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*e+528* 
A*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)*a^2*b*e^3+240*A*((e*x+d)*(b*x+a))^(1 
/2)*(b*e)^(1/2)*b^3*d^2*e-60*B*ln(1/2*(2*b*e*x+2*((e*x+d)*(b*x+a))^(1/2...
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 770, normalized size of antiderivative = 3.13 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{\sqrt {d+e x}} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(1/2),x, algorithm="fricas")
 

Output:

[1/768*(15*(7*B*b^4*d^4 - 4*(5*B*a*b^3 + 2*A*b^4)*d^3*e + 6*(3*B*a^2*b^2 + 
 4*A*a*b^3)*d^2*e^2 - 4*(B*a^3*b + 6*A*a^2*b^2)*d*e^3 - (B*a^4 - 8*A*a^3*b 
)*e^4)*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*(48*B*b^4*e^4*x^3 - 105*B*b^4*d^3*e + 5*(53*B*a*b^3 + 24*A*b 
^4)*d^2*e^2 - (191*B*a^2*b^2 + 320*A*a*b^3)*d*e^3 + 3*(5*B*a^3*b + 88*A*a^ 
2*b^2)*e^4 - 8*(7*B*b^4*d*e^3 - (17*B*a*b^3 + 8*A*b^4)*e^4)*x^2 + 2*(35*B* 
b^4*d^2*e^2 - 2*(43*B*a*b^3 + 20*A*b^4)*d*e^3 + (59*B*a^2*b^2 + 104*A*a*b^ 
3)*e^4)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^2*e^5), -1/384*(15*(7*B*b^4*d^4 
 - 4*(5*B*a*b^3 + 2*A*b^4)*d^3*e + 6*(3*B*a^2*b^2 + 4*A*a*b^3)*d^2*e^2 - 4 
*(B*a^3*b + 6*A*a^2*b^2)*d*e^3 - (B*a^4 - 8*A*a^3*b)*e^4)*sqrt(-b*e)*arcta 
n(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*(48*B*b^4*e^4*x^3 - 105*B*b^ 
4*d^3*e + 5*(53*B*a*b^3 + 24*A*b^4)*d^2*e^2 - (191*B*a^2*b^2 + 320*A*a*b^3 
)*d*e^3 + 3*(5*B*a^3*b + 88*A*a^2*b^2)*e^4 - 8*(7*B*b^4*d*e^3 - (17*B*a*b^ 
3 + 8*A*b^4)*e^4)*x^2 + 2*(35*B*b^4*d^2*e^2 - 2*(43*B*a*b^3 + 20*A*b^4)*d* 
e^3 + (59*B*a^2*b^2 + 104*A*a*b^3)*e^4)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b 
^2*e^5)]
 

Sympy [F]

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

integrate((b*x+a)**(5/2)*(B*x+A)/(e*x+d)**(1/2),x)
 

Output:

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

Maxima [F(-2)]

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

integrate((b*x+a)^(5/2)*(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(a*e-b*d>0)', see `assume?` for m 
ore detail
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.69 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{\sqrt {d+e x}} \, dx=\frac {{\left (\sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b x + a\right )} B}{b^{2} e} - \frac {7 \, B b^{3} d e^{5} + B a b^{2} e^{6} - 8 \, A b^{3} e^{6}}{b^{4} e^{7}}\right )} + \frac {5 \, {\left (7 \, B b^{4} d^{2} e^{4} - 6 \, B a b^{3} d e^{5} - 8 \, A b^{4} d e^{5} - B a^{2} b^{2} e^{6} + 8 \, A a b^{3} e^{6}\right )}}{b^{4} e^{7}}\right )} - \frac {15 \, {\left (7 \, B b^{5} d^{3} e^{3} - 13 \, B a b^{4} d^{2} e^{4} - 8 \, A b^{5} d^{2} e^{4} + 5 \, B a^{2} b^{3} d e^{5} + 16 \, A a b^{4} d e^{5} + B a^{3} b^{2} e^{6} - 8 \, A a^{2} b^{3} e^{6}\right )}}{b^{4} e^{7}}\right )} \sqrt {b x + a} - \frac {15 \, {\left (7 \, B b^{4} d^{4} - 20 \, B a b^{3} d^{3} e - 8 \, A b^{4} d^{3} e + 18 \, B a^{2} b^{2} d^{2} e^{2} + 24 \, A a b^{3} d^{2} e^{2} - 4 \, B a^{3} b d e^{3} - 24 \, A a^{2} b^{2} d e^{3} - B a^{4} e^{4} + 8 \, A a^{3} b e^{4}\right )} \log \left ({\left | -\sqrt {b e} \sqrt {b x + a} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b e} b e^{4}}\right )} b}{192 \, {\left | b \right |}} \] Input:

integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(1/2),x, algorithm="giac")
 

Output:

1/192*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(b*x + a)*(4*(b*x + a)*(6*(b 
*x + a)*B/(b^2*e) - (7*B*b^3*d*e^5 + B*a*b^2*e^6 - 8*A*b^3*e^6)/(b^4*e^7)) 
 + 5*(7*B*b^4*d^2*e^4 - 6*B*a*b^3*d*e^5 - 8*A*b^4*d*e^5 - B*a^2*b^2*e^6 + 
8*A*a*b^3*e^6)/(b^4*e^7)) - 15*(7*B*b^5*d^3*e^3 - 13*B*a*b^4*d^2*e^4 - 8*A 
*b^5*d^2*e^4 + 5*B*a^2*b^3*d*e^5 + 16*A*a*b^4*d*e^5 + B*a^3*b^2*e^6 - 8*A* 
a^2*b^3*e^6)/(b^4*e^7))*sqrt(b*x + a) - 15*(7*B*b^4*d^4 - 20*B*a*b^3*d^3*e 
 - 8*A*b^4*d^3*e + 18*B*a^2*b^2*d^2*e^2 + 24*A*a*b^3*d^2*e^2 - 4*B*a^3*b*d 
*e^3 - 24*A*a^2*b^2*d*e^3 - B*a^4*e^4 + 8*A*a^3*b*e^4)*log(abs(-sqrt(b*e)* 
sqrt(b*x + a) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/(sqrt(b*e)*b*e^4))*b 
/abs(b)
 

Mupad [F(-1)]

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

int(((A + B*x)*(a + b*x)^(5/2))/(d + e*x)^(1/2),x)
 

Output:

int(((A + B*x)*(a + b*x)^(5/2))/(d + e*x)^(1/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 471, normalized size of antiderivative = 1.91 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{\sqrt {d+e x}} \, dx=\frac {279 \sqrt {e x +d}\, \sqrt {b x +a}\, a^{3} b \,e^{4}-511 \sqrt {e x +d}\, \sqrt {b x +a}\, a^{2} b^{2} d \,e^{3}+326 \sqrt {e x +d}\, \sqrt {b x +a}\, a^{2} b^{2} e^{4} x +385 \sqrt {e x +d}\, \sqrt {b x +a}\, a \,b^{3} d^{2} e^{2}-252 \sqrt {e x +d}\, \sqrt {b x +a}\, a \,b^{3} d \,e^{3} x +200 \sqrt {e x +d}\, \sqrt {b x +a}\, a \,b^{3} e^{4} x^{2}-105 \sqrt {e x +d}\, \sqrt {b x +a}\, b^{4} d^{3} e +70 \sqrt {e x +d}\, \sqrt {b x +a}\, b^{4} d^{2} e^{2} x -56 \sqrt {e x +d}\, \sqrt {b x +a}\, b^{4} d \,e^{3} x^{2}+48 \sqrt {e x +d}\, \sqrt {b x +a}\, b^{4} e^{4} x^{3}+105 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) a^{4} e^{4}-420 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) a^{3} b d \,e^{3}+630 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) a^{2} b^{2} d^{2} e^{2}-420 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) a \,b^{3} d^{3} e +105 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) b^{4} d^{4}}{192 b \,e^{5}} \] Input:

int((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(1/2),x)
 

Output:

(279*sqrt(d + e*x)*sqrt(a + b*x)*a**3*b*e**4 - 511*sqrt(d + e*x)*sqrt(a + 
b*x)*a**2*b**2*d*e**3 + 326*sqrt(d + e*x)*sqrt(a + b*x)*a**2*b**2*e**4*x + 
 385*sqrt(d + e*x)*sqrt(a + b*x)*a*b**3*d**2*e**2 - 252*sqrt(d + e*x)*sqrt 
(a + b*x)*a*b**3*d*e**3*x + 200*sqrt(d + e*x)*sqrt(a + b*x)*a*b**3*e**4*x* 
*2 - 105*sqrt(d + e*x)*sqrt(a + b*x)*b**4*d**3*e + 70*sqrt(d + e*x)*sqrt(a 
 + b*x)*b**4*d**2*e**2*x - 56*sqrt(d + e*x)*sqrt(a + b*x)*b**4*d*e**3*x**2 
 + 48*sqrt(d + e*x)*sqrt(a + b*x)*b**4*e**4*x**3 + 105*sqrt(e)*sqrt(b)*log 
((sqrt(e)*sqrt(a + b*x) + sqrt(b)*sqrt(d + e*x))/sqrt(a*e - b*d))*a**4*e** 
4 - 420*sqrt(e)*sqrt(b)*log((sqrt(e)*sqrt(a + b*x) + sqrt(b)*sqrt(d + e*x) 
)/sqrt(a*e - b*d))*a**3*b*d*e**3 + 630*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**2*d**2*e**2 - 42 
0*sqrt(e)*sqrt(b)*log((sqrt(e)*sqrt(a + b*x) + sqrt(b)*sqrt(d + e*x))/sqrt 
(a*e - b*d))*a*b**3*d**3*e + 105*sqrt(e)*sqrt(b)*log((sqrt(e)*sqrt(a + b*x 
) + sqrt(b)*sqrt(d + e*x))/sqrt(a*e - b*d))*b**4*d**4)/(192*b*e**5)