\(\int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{5/2}} \, dx\) [198]

Optimal result

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

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 10.10 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.53 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{5/2}} \, dx=\frac {2 (a+b x)^{7/2} \left (7 B d-7 A e-\frac {(7 b B d-4 A b e-3 a B e) \left (\frac {b (d+e x)}{b d-a e}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {7}{2},\frac {9}{2},\frac {e (a+b x)}{-b d+a e}\right )}{b}\right )}{21 e (-b d+a e) (d+e x)^{3/2}} \] Input:

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

Output:

(2*(a + b*x)^(7/2)*(7*B*d - 7*A*e - ((7*b*B*d - 4*A*b*e - 3*a*B*e)*((b*(d 
+ e*x))/(b*d - a*e))^(3/2)*Hypergeometric2F1[3/2, 7/2, 9/2, (e*(a + b*x))/ 
(-(b*d) + a*e)])/b))/(21*e*(-(b*d) + a*e)*(d + e*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 221, 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.

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

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 57

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Output:

1/24*(b*x+a)^(1/2)*(-16*A*a^2*e^3*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-210* 
B*b^2*d^3*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+316*B*a*b*d*e^2*x*((e*x+d)*( 
b*x+a))^(1/2)*(b*e)^(1/2)+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))*b^3*d^2*e^2*x^2+24*A*b^2*e^3*x^2*((e*x+d 
)*(b*x+a))^(1/2)*(b*e)^(1/2)+210*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*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^2*d^3*e-48*B* 
a^2*e^3*x*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-32*B*a^2*d*e^2*((e*x+d)*(b*x 
+a))^(1/2)*(b*e)^(1/2)-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^2*d*e^3*x^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^2*d^2*e^2*x+54* 
B*a*b*e^3*x^2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-42*B*b^2*d*e^2*x^2*((e*x 
+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-112*A*a*b*e^3*x*((e*x+d)*(b*x+a))^(1/2)*(b* 
e)^(1/2)-280*B*b^2*d^2*e*x*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+160*A*b^2*d 
*e^2*x*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-80*A*a*b*d*e^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)/(b*e)^(1/2))*a*b^2*d*e^3*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^2*b*d*e^3*x+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^3 
*d^4+230*B*a*b*d^2*e*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+60*A*ln(1/2*(2...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 415 vs. \(2 (174) = 348\).

Time = 1.31 (sec) , antiderivative size = 855, normalized size of antiderivative = 4.03 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/48*(15*(7*B*b^2*d^4 - 2*(5*B*a*b + 2*A*b^2)*d^3*e + (3*B*a^2 + 4*A*a*b) 
*d^2*e^2 + (7*B*b^2*d^2*e^2 - 2*(5*B*a*b + 2*A*b^2)*d*e^3 + (3*B*a^2 + 4*A 
*a*b)*e^4)*x^2 + 2*(7*B*b^2*d^3*e - 2*(5*B*a*b + 2*A*b^2)*d^2*e^2 + (3*B*a 
^2 + 4*A*a*b)*d*e^3)*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^2*x + b*d*e + a*e^2)*sqrt(b*x + a)*sqrt(e*x + d)*sqrt 
(b/e) + 8*(b^2*d*e + a*b*e^2)*x) + 4*(6*B*b^2*e^3*x^3 - 105*B*b^2*d^3 - 8* 
A*a^2*e^3 + 5*(23*B*a*b + 12*A*b^2)*d^2*e - 8*(2*B*a^2 + 5*A*a*b)*d*e^2 - 
3*(7*B*b^2*d*e^2 - (9*B*a*b + 4*A*b^2)*e^3)*x^2 - 2*(70*B*b^2*d^2*e - (79* 
B*a*b + 40*A*b^2)*d*e^2 + 4*(3*B*a^2 + 7*A*a*b)*e^3)*x)*sqrt(b*x + a)*sqrt 
(e*x + d))/(e^6*x^2 + 2*d*e^5*x + d^2*e^4), -1/24*(15*(7*B*b^2*d^4 - 2*(5* 
B*a*b + 2*A*b^2)*d^3*e + (3*B*a^2 + 4*A*a*b)*d^2*e^2 + (7*B*b^2*d^2*e^2 - 
2*(5*B*a*b + 2*A*b^2)*d*e^3 + (3*B*a^2 + 4*A*a*b)*e^4)*x^2 + 2*(7*B*b^2*d^ 
3*e - 2*(5*B*a*b + 2*A*b^2)*d^2*e^2 + (3*B*a^2 + 4*A*a*b)*d*e^3)*x)*sqrt(- 
b/e)*arctan(1/2*(2*b*e*x + b*d + a*e)*sqrt(b*x + a)*sqrt(e*x + d)*sqrt(-b/ 
e)/(b^2*e*x^2 + a*b*d + (b^2*d + a*b*e)*x)) - 2*(6*B*b^2*e^3*x^3 - 105*B*b 
^2*d^3 - 8*A*a^2*e^3 + 5*(23*B*a*b + 12*A*b^2)*d^2*e - 8*(2*B*a^2 + 5*A*a* 
b)*d*e^2 - 3*(7*B*b^2*d*e^2 - (9*B*a*b + 4*A*b^2)*e^3)*x^2 - 2*(70*B*b^2*d 
^2*e - (79*B*a*b + 40*A*b^2)*d*e^2 + 4*(3*B*a^2 + 7*A*a*b)*e^3)*x)*sqrt(b* 
x + a)*sqrt(e*x + d))/(e^6*x^2 + 2*d*e^5*x + d^2*e^4)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(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(e*(a*e-b*d)>0)', see `assume?` f 
or more de
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 564 vs. \(2 (174) = 348\).

Time = 0.24 (sec) , antiderivative size = 564, normalized size of antiderivative = 2.66 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{5/2}} \, dx=\frac {{\left ({\left (3 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (B b^{5} d e^{6} {\left | b \right |} - B a b^{4} e^{7} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{4} d e^{7} - a b^{3} e^{8}} - \frac {7 \, B b^{6} d^{2} e^{5} {\left | b \right |} - 10 \, B a b^{5} d e^{6} {\left | b \right |} - 4 \, A b^{6} d e^{6} {\left | b \right |} + 3 \, B a^{2} b^{4} e^{7} {\left | b \right |} + 4 \, A a b^{5} e^{7} {\left | b \right |}}{b^{4} d e^{7} - a b^{3} e^{8}}\right )} - \frac {20 \, {\left (7 \, B b^{7} d^{3} e^{4} {\left | b \right |} - 17 \, B a b^{6} d^{2} e^{5} {\left | b \right |} - 4 \, A b^{7} d^{2} e^{5} {\left | b \right |} + 13 \, B a^{2} b^{5} d e^{6} {\left | b \right |} + 8 \, A a b^{6} d e^{6} {\left | b \right |} - 3 \, B a^{3} b^{4} e^{7} {\left | b \right |} - 4 \, A a^{2} b^{5} e^{7} {\left | b \right |}\right )}}{b^{4} d e^{7} - a b^{3} e^{8}}\right )} {\left (b x + a\right )} - \frac {15 \, {\left (7 \, B b^{8} d^{4} e^{3} {\left | b \right |} - 24 \, B a b^{7} d^{3} e^{4} {\left | b \right |} - 4 \, A b^{8} d^{3} e^{4} {\left | b \right |} + 30 \, B a^{2} b^{6} d^{2} e^{5} {\left | b \right |} + 12 \, A a b^{7} d^{2} e^{5} {\left | b \right |} - 16 \, B a^{3} b^{5} d e^{6} {\left | b \right |} - 12 \, A a^{2} b^{6} d e^{6} {\left | b \right |} + 3 \, B a^{4} b^{4} e^{7} {\left | b \right |} + 4 \, A a^{3} b^{5} e^{7} {\left | b \right |}\right )}}{b^{4} d e^{7} - a b^{3} e^{8}}\right )} \sqrt {b x + a}}{12 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {3}{2}}} - \frac {5 \, {\left (7 \, B b^{2} d^{2} {\left | b \right |} - 10 \, B a b d e {\left | b \right |} - 4 \, A b^{2} d e {\left | b \right |} + 3 \, 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 |}\right )}{4 \, \sqrt {b e} e^{4}} \] Input:

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

Output:

1/12*((3*(b*x + a)*(2*(B*b^5*d*e^6*abs(b) - B*a*b^4*e^7*abs(b))*(b*x + a)/ 
(b^4*d*e^7 - a*b^3*e^8) - (7*B*b^6*d^2*e^5*abs(b) - 10*B*a*b^5*d*e^6*abs(b 
) - 4*A*b^6*d*e^6*abs(b) + 3*B*a^2*b^4*e^7*abs(b) + 4*A*a*b^5*e^7*abs(b))/ 
(b^4*d*e^7 - a*b^3*e^8)) - 20*(7*B*b^7*d^3*e^4*abs(b) - 17*B*a*b^6*d^2*e^5 
*abs(b) - 4*A*b^7*d^2*e^5*abs(b) + 13*B*a^2*b^5*d*e^6*abs(b) + 8*A*a*b^6*d 
*e^6*abs(b) - 3*B*a^3*b^4*e^7*abs(b) - 4*A*a^2*b^5*e^7*abs(b))/(b^4*d*e^7 
- a*b^3*e^8))*(b*x + a) - 15*(7*B*b^8*d^4*e^3*abs(b) - 24*B*a*b^7*d^3*e^4* 
abs(b) - 4*A*b^8*d^3*e^4*abs(b) + 30*B*a^2*b^6*d^2*e^5*abs(b) + 12*A*a*b^7 
*d^2*e^5*abs(b) - 16*B*a^3*b^5*d*e^6*abs(b) - 12*A*a^2*b^6*d*e^6*abs(b) + 
3*B*a^4*b^4*e^7*abs(b) + 4*A*a^3*b^5*e^7*abs(b))/(b^4*d*e^7 - a*b^3*e^8))* 
sqrt(b*x + a)/(b^2*d + (b*x + a)*b*e - a*b*e)^(3/2) - 5/4*(7*B*b^2*d^2*abs 
(b) - 10*B*a*b*d*e*abs(b) - 4*A*b^2*d*e*abs(b) + 3*B*a^2*e^2*abs(b) + 4*A* 
a*b*e^2*abs(b))*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^4)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 817, normalized size of antiderivative = 3.85 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 64*sqrt(d + e*x)*sqrt(a + b*x)*a**3*e**4 - 448*sqrt(d + e*x)*sqrt(a + 
b*x)*a**2*b*d*e**3 - 640*sqrt(d + e*x)*sqrt(a + b*x)*a**2*b*e**4*x + 1400* 
sqrt(d + e*x)*sqrt(a + b*x)*a*b**2*d**2*e**2 + 1904*sqrt(d + e*x)*sqrt(a + 
 b*x)*a*b**2*d*e**3*x + 312*sqrt(d + e*x)*sqrt(a + b*x)*a*b**2*e**4*x**2 - 
 840*sqrt(d + e*x)*sqrt(a + b*x)*b**3*d**3*e - 1120*sqrt(d + e*x)*sqrt(a + 
 b*x)*b**3*d**2*e**2*x - 168*sqrt(d + e*x)*sqrt(a + b*x)*b**3*d*e**3*x**2 
+ 48*sqrt(d + e*x)*sqrt(a + b*x)*b**3*e**4*x**3 + 840*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* 
*2*e**2 + 1680*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**3*x + 840*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*e**4*x** 
2 - 1680*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**3*e - 3360*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**2*x - 1 
680*sqrt(e)*sqrt(b)*log((sqrt(e)*sqrt(a + b*x) + sqrt(b)*sqrt(d + e*x))/sq 
rt(a*e - b*d))*a*b**2*d*e**3*x**2 + 840*sqrt(e)*sqrt(b)*log((sqrt(e)*sqrt( 
a + b*x) + sqrt(b)*sqrt(d + e*x))/sqrt(a*e - b*d))*b**3*d**4 + 1680*sqrt(e 
)*sqrt(b)*log((sqrt(e)*sqrt(a + b*x) + sqrt(b)*sqrt(d + e*x))/sqrt(a*e - b 
*d))*b**3*d**3*e*x + 840*sqrt(e)*sqrt(b)*log((sqrt(e)*sqrt(a + b*x) + sqrt 
(b)*sqrt(d + e*x))/sqrt(a*e - b*d))*b**3*d**2*e**2*x**2 + 175*sqrt(e)*s...