\(\int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{5/2}} \, dx\) [223]

Optimal result

Integrand size = 24, antiderivative size = 212 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{5/2}} \, dx=\frac {5 e (3 b B d+4 A b e-7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 b^4}-\frac {2 (3 b B d+5 A b e-8 a B e) (d+e x)^{3/2}}{3 b^3 \sqrt {a+b x}}+\frac {B e \sqrt {a+b x} (d+e x)^{3/2}}{2 b^3}-\frac {2 (A b-a B) (d+e x)^{5/2}}{3 b^2 (a+b x)^{3/2}}+\frac {5 \sqrt {e} (b d-a e) (3 b B d+4 A b e-7 a B e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{4 b^{9/2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.33 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.11 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{5/2}} \, dx=\frac {\sqrt {d+e x} \left (B \left (-105 a^3 e^2+5 a^2 b e (23 d-28 e x)+a b^2 \left (-16 d^2+158 d e x-21 e^2 x^2\right )+3 b^3 x \left (-8 d^2+9 d e x+2 e^2 x^2\right )\right )+4 A b \left (15 a^2 e^2-10 a b e (d-2 e x)+b^2 \left (-2 d^2-14 d e x+3 e^2 x^2\right )\right )\right )}{12 b^4 (a+b x)^{3/2}}+\frac {5 \sqrt {e} (-b d+a e) (3 b B d+4 A b e-7 a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \left (\sqrt {a-\frac {b d}{e}}-\sqrt {a+b x}\right )}\right )}{2 b^{9/2}} \] Input:

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

Output:

(Sqrt[d + e*x]*(B*(-105*a^3*e^2 + 5*a^2*b*e*(23*d - 28*e*x) + a*b^2*(-16*d 
^2 + 158*d*e*x - 21*e^2*x^2) + 3*b^3*x*(-8*d^2 + 9*d*e*x + 2*e^2*x^2)) + 4 
*A*b*(15*a^2*e^2 - 10*a*b*e*(d - 2*e*x) + b^2*(-2*d^2 - 14*d*e*x + 3*e^2*x 
^2))))/(12*b^4*(a + b*x)^(3/2)) + (5*Sqrt[e]*(-(b*d) + a*e)*(3*b*B*d + 4*A 
*b*e - 7*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/(Sqrt[e]*(Sqrt[a - (b*d)/e 
] - Sqrt[a + b*x]))])/(2*b^(9/2))
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {87, 57, 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.

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

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 57

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(-2*(A*b - a*B)*(d + e*x)^(7/2))/(3*b*(b*d - a*e)*(a + b*x)^(3/2)) + ((3*b 
*B*d + 4*A*b*e - 7*a*B*e)*((-2*(d + e*x)^(5/2))/(b*Sqrt[a + b*x]) + (5*e*( 
(Sqrt[a + b*x]*(d + e*x)^(3/2))/(2*b) + (3*(b*d - a*e)*((Sqrt[a + b*x]*Sqr 
t[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))/(3*b*(b*d - a*e))
 

Defintions of rubi rules used

rule 57

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & 
& GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m 
+ n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c 
, d, m, n, x]
 

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. \(1249\) vs. \(2(174)=348\).

Time = 0.27 (sec) , antiderivative size = 1250, normalized size of antiderivative = 5.90

method	result	size
default	\(\text {Expression too large to display}\)	\(1250\)

Input:

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

Output:

-1/24*(e*x+d)^(1/2)*(60*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*e^3*x^2-60*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*e^2*x^2-105*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^2 
*b^2*e^3*x^2-45*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^2*e*x^2+42*B*a*b^2*e^2*x^2*((e*x+d)*(b*x+a))^(1/ 
2)*(b*e)^(1/2)-54*B*b^3*d*e*x^2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-160*A* 
a*b^2*e^2*x*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+112*A*b^3*d*e*x*((e*x+d)*( 
b*x+a))^(1/2)*(b*e)^(1/2)+280*B*a^2*b*e^2*x*((e*x+d)*(b*x+a))^(1/2)*(b*e)^ 
(1/2)+60*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^3-316*B*a*b^2*d*e*x*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/ 
2)+80*A*a*b^2*d*e*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-230*B*a^2*b*d*e*((e* 
x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-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*b^3*d*e^2*x+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^2*b^2*d*e^ 
2*x-90*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+150*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*e^2*x^2-120*A*a^2*b*e^2*((e*x+d)* 
(b*x+a))^(1/2)*(b*e)^(1/2)+32*B*a*b^2*d^2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1 
/2)+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...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 431 vs. \(2 (174) = 348\).

Time = 1.26 (sec) , antiderivative size = 887, normalized size of antiderivative = 4.18 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/48*(15*(3*B*a^2*b^2*d^2 - 2*(5*B*a^3*b - 2*A*a^2*b^2)*d*e + (7*B*a^4 - 
 4*A*a^3*b)*e^2 + (3*B*b^4*d^2 - 2*(5*B*a*b^3 - 2*A*b^4)*d*e + (7*B*a^2*b^ 
2 - 4*A*a*b^3)*e^2)*x^2 + 2*(3*B*a*b^3*d^2 - 2*(5*B*a^2*b^2 - 2*A*a*b^3)*d 
*e + (7*B*a^3*b - 4*A*a^2*b^2)*e^2)*x)*sqrt(e/b)*log(8*b^2*e^2*x^2 + b^2*d 
^2 + 6*a*b*d*e + a^2*e^2 - 4*(2*b^2*e*x + b^2*d + a*b*e)*sqrt(b*x + a)*sqr 
t(e*x + d)*sqrt(e/b) + 8*(b^2*d*e + a*b*e^2)*x) - 4*(6*B*b^3*e^2*x^3 - 8*( 
2*B*a*b^2 + A*b^3)*d^2 + 5*(23*B*a^2*b - 8*A*a*b^2)*d*e - 15*(7*B*a^3 - 4* 
A*a^2*b)*e^2 + 3*(9*B*b^3*d*e - (7*B*a*b^2 - 4*A*b^3)*e^2)*x^2 - 2*(12*B*b 
^3*d^2 - (79*B*a*b^2 - 28*A*b^3)*d*e + 10*(7*B*a^2*b - 4*A*a*b^2)*e^2)*x)* 
sqrt(b*x + a)*sqrt(e*x + d))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4), -1/24*(15*(3 
*B*a^2*b^2*d^2 - 2*(5*B*a^3*b - 2*A*a^2*b^2)*d*e + (7*B*a^4 - 4*A*a^3*b)*e 
^2 + (3*B*b^4*d^2 - 2*(5*B*a*b^3 - 2*A*b^4)*d*e + (7*B*a^2*b^2 - 4*A*a*b^3 
)*e^2)*x^2 + 2*(3*B*a*b^3*d^2 - 2*(5*B*a^2*b^2 - 2*A*a*b^3)*d*e + (7*B*a^3 
*b - 4*A*a^2*b^2)*e^2)*x)*sqrt(-e/b)*arctan(1/2*(2*b*e*x + b*d + a*e)*sqrt 
(b*x + a)*sqrt(e*x + d)*sqrt(-e/b)/(b*e^2*x^2 + a*d*e + (b*d*e + a*e^2)*x) 
) - 2*(6*B*b^3*e^2*x^3 - 8*(2*B*a*b^2 + A*b^3)*d^2 + 5*(23*B*a^2*b - 8*A*a 
*b^2)*d*e - 15*(7*B*a^3 - 4*A*a^2*b)*e^2 + 3*(9*B*b^3*d*e - (7*B*a*b^2 - 4 
*A*b^3)*e^2)*x^2 - 2*(12*B*b^3*d^2 - (79*B*a*b^2 - 28*A*b^3)*d*e + 10*(7*B 
*a^2*b - 4*A*a*b^2)*e^2)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^6*x^2 + 2*a*b^ 
5*x + a^2*b^4)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((B*x+A)*(e*x+d)^(5/2)/(b*x+a)^(5/2),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. 1401 vs. \(2 (174) = 348\).

Time = 0.49 (sec) , antiderivative size = 1401, normalized size of antiderivative = 6.61 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(e*x+d)^(5/2)/(b*x+a)^(5/2),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^2*a 
bs(b)/b^6 + (9*B*b^12*d*e^3*abs(b) - 13*B*a*b^11*e^4*abs(b) + 4*A*b^12*e^4 
*abs(b))/(b^17*e^2)) - 5/8*(3*sqrt(b*e)*B*b^2*d^2*abs(b) - 10*sqrt(b*e)*B* 
a*b*d*e*abs(b) + 4*sqrt(b*e)*A*b^2*d*e*abs(b) + 7*sqrt(b*e)*B*a^2*e^2*abs( 
b) - 4*sqrt(b*e)*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)/b^6 - 4/3*(3*sqrt(b*e)*B*b^7*d^5*abs(b) - 
22*sqrt(b*e)*B*a*b^6*d^4*e*abs(b) + 7*sqrt(b*e)*A*b^7*d^4*e*abs(b) + 58*sq 
rt(b*e)*B*a^2*b^5*d^3*e^2*abs(b) - 28*sqrt(b*e)*A*a*b^6*d^3*e^2*abs(b) - 7 
2*sqrt(b*e)*B*a^3*b^4*d^2*e^3*abs(b) + 42*sqrt(b*e)*A*a^2*b^5*d^2*e^3*abs( 
b) + 43*sqrt(b*e)*B*a^4*b^3*d*e^4*abs(b) - 28*sqrt(b*e)*A*a^3*b^4*d*e^4*ab 
s(b) - 10*sqrt(b*e)*B*a^5*b^2*e^5*abs(b) + 7*sqrt(b*e)*A*a^4*b^3*e^5*abs(b 
) - 6*sqrt(b*e)*(sqrt(b*e)*sqrt(b*x + a) - sqrt(b^2*d + (b*x + a)*b*e - a* 
b*e))^2*B*b^5*d^4*abs(b) + 36*sqrt(b*e)*(sqrt(b*e)*sqrt(b*x + a) - sqrt(b^ 
2*d + (b*x + a)*b*e - a*b*e))^2*B*a*b^4*d^3*e*abs(b) - 12*sqrt(b*e)*(sqrt( 
b*e)*sqrt(b*x + a) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*A*b^5*d^3*e*ab 
s(b) - 72*sqrt(b*e)*(sqrt(b*e)*sqrt(b*x + a) - sqrt(b^2*d + (b*x + a)*b*e 
- a*b*e))^2*B*a^2*b^3*d^2*e^2*abs(b) + 36*sqrt(b*e)*(sqrt(b*e)*sqrt(b*x + 
a) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*A*a*b^4*d^2*e^2*abs(b) + 60*sq 
rt(b*e)*(sqrt(b*e)*sqrt(b*x + a) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2* 
B*a^3*b^2*d*e^3*abs(b) - 36*sqrt(b*e)*(sqrt(b*e)*sqrt(b*x + a) - sqrt(b...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.43 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^{5/2}} \, dx=\frac {15 \sqrt {e}\, \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) a^{2} e^{2}-30 \sqrt {e}\, \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) a b d e +15 \sqrt {e}\, \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) b^{2} d^{2}-10 \sqrt {e}\, \sqrt {b}\, \sqrt {b x +a}\, a^{2} e^{2}+20 \sqrt {e}\, \sqrt {b}\, \sqrt {b x +a}\, a b d e -10 \sqrt {e}\, \sqrt {b}\, \sqrt {b x +a}\, b^{2} d^{2}-15 \sqrt {e x +d}\, a^{2} b \,e^{2}+25 \sqrt {e x +d}\, a \,b^{2} d e -5 \sqrt {e x +d}\, a \,b^{2} e^{2} x -8 \sqrt {e x +d}\, b^{3} d^{2}+9 \sqrt {e x +d}\, b^{3} d e x +2 \sqrt {e x +d}\, b^{3} e^{2} x^{2}}{4 \sqrt {b x +a}\, b^{4}} \] Input:

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

Output:

(15*sqrt(e)*sqrt(b)*sqrt(a + b*x)*log((sqrt(e)*sqrt(a + b*x) + sqrt(b)*sqr 
t(d + e*x))/sqrt(a*e - b*d))*a**2*e**2 - 30*sqrt(e)*sqrt(b)*sqrt(a + b*x)* 
log((sqrt(e)*sqrt(a + b*x) + sqrt(b)*sqrt(d + e*x))/sqrt(a*e - b*d))*a*b*d 
*e + 15*sqrt(e)*sqrt(b)*sqrt(a + b*x)*log((sqrt(e)*sqrt(a + b*x) + sqrt(b) 
*sqrt(d + e*x))/sqrt(a*e - b*d))*b**2*d**2 - 10*sqrt(e)*sqrt(b)*sqrt(a + b 
*x)*a**2*e**2 + 20*sqrt(e)*sqrt(b)*sqrt(a + b*x)*a*b*d*e - 10*sqrt(e)*sqrt 
(b)*sqrt(a + b*x)*b**2*d**2 - 15*sqrt(d + e*x)*a**2*b*e**2 + 25*sqrt(d + e 
*x)*a*b**2*d*e - 5*sqrt(d + e*x)*a*b**2*e**2*x - 8*sqrt(d + e*x)*b**3*d**2 
 + 9*sqrt(d + e*x)*b**3*d*e*x + 2*sqrt(d + e*x)*b**3*e**2*x**2)/(4*sqrt(a 
+ b*x)*b**4)