\(\int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^{5/2}} \, dx\) [222]

Optimal result

Integrand size = 24, antiderivative size = 261 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^{5/2}} \, dx=\frac {35 e (b d-a e) (b B d+2 A b e-3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{8 b^5}+\frac {35 e (b B d+2 A b e-3 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^4}-\frac {2 (3 b B d+7 A b e-10 a B e) (d+e x)^{5/2}}{3 b^3 \sqrt {a+b x}}+\frac {B e \sqrt {a+b x} (d+e x)^{5/2}}{3 b^3}-\frac {2 (A b-a B) (d+e x)^{7/2}}{3 b^2 (a+b x)^{3/2}}+\frac {35 \sqrt {e} (b d-a e)^2 (b B d+2 A b e-3 a B e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 b^{11/2}} \] Output:

35/8*e*(-a*e+b*d)*(2*A*b*e-3*B*a*e+B*b*d)*(b*x+a)^(1/2)*(e*x+d)^(1/2)/b^5+ 
35/12*e*(2*A*b*e-3*B*a*e+B*b*d)*(b*x+a)^(1/2)*(e*x+d)^(3/2)/b^4-2/3*(7*A*b 
*e-10*B*a*e+3*B*b*d)*(e*x+d)^(5/2)/b^3/(b*x+a)^(1/2)+1/3*B*e*(b*x+a)^(1/2) 
*(e*x+d)^(5/2)/b^3-2/3*(A*b-B*a)*(e*x+d)^(7/2)/b^2/(b*x+a)^(3/2)+35/8*e^(1 
/2)*(-a*e+b*d)^2*(2*A*b*e-3*B*a*e+B*b*d)*arctanh(e^(1/2)*(b*x+a)^(1/2)/b^( 
1/2)/(e*x+d)^(1/2))/b^(11/2)
 

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.20 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^{5/2}} \, dx=\frac {\sqrt {d+e x} \left (-2 A b \left (105 a^3 e^3+35 a^2 b e^2 (-5 d+4 e x)+7 a b^2 e \left (8 d^2-34 d e x+3 e^2 x^2\right )+b^3 \left (8 d^3+80 d^2 e x-39 d e^2 x^2-6 e^3 x^3\right )\right )+B \left (315 a^4 e^3+210 a^3 b e^2 (-3 d+2 e x)+7 a^2 b^2 e \left (49 d^2-122 d e x+9 e^2 x^2\right )+b^4 x \left (-48 d^3+87 d^2 e x+38 d e^2 x^2+8 e^3 x^3\right )-2 a b^3 \left (16 d^3-239 d^2 e x+69 d e^2 x^2+9 e^3 x^3\right )\right )\right )}{24 b^5 (a+b x)^{3/2}}+\frac {35 \sqrt {e} (b d-a e)^2 (b B d+2 A b e-3 a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{8 b^{11/2}} \] Input:

Integrate[((A + B*x)*(d + e*x)^(7/2))/(a + b*x)^(5/2),x]
 

Output:

(Sqrt[d + e*x]*(-2*A*b*(105*a^3*e^3 + 35*a^2*b*e^2*(-5*d + 4*e*x) + 7*a*b^ 
2*e*(8*d^2 - 34*d*e*x + 3*e^2*x^2) + b^3*(8*d^3 + 80*d^2*e*x - 39*d*e^2*x^ 
2 - 6*e^3*x^3)) + B*(315*a^4*e^3 + 210*a^3*b*e^2*(-3*d + 2*e*x) + 7*a^2*b^ 
2*e*(49*d^2 - 122*d*e*x + 9*e^2*x^2) + b^4*x*(-48*d^3 + 87*d^2*e*x + 38*d* 
e^2*x^2 + 8*e^3*x^3) - 2*a*b^3*(16*d^3 - 239*d^2*e*x + 69*d*e^2*x^2 + 9*e^ 
3*x^3))))/(24*b^5*(a + b*x)^(3/2)) + (35*Sqrt[e]*(b*d - a*e)^2*(b*B*d + 2* 
A*b*e - 3*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a + b*x])]) 
/(8*b^(11/2))
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {87, 57, 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.

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

\(\Big \downarrow \) 87

\(\displaystyle \frac {(-3 a B e+2 A b e+b B d) \int \frac {(d+e x)^{7/2}}{(a+b x)^{3/2}}dx}{b (b d-a e)}-\frac {2 (d+e x)^{9/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}\)

\(\Big \downarrow \) 57

\(\displaystyle \frac {(-3 a B e+2 A b e+b B d) \left (\frac {7 e \int \frac {(d+e x)^{5/2}}{\sqrt {a+b x}}dx}{b}-\frac {2 (d+e x)^{7/2}}{b \sqrt {a+b x}}\right )}{b (b d-a e)}-\frac {2 (d+e x)^{9/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(-3 a B e+2 A b e+b B d) \left (\frac {7 e \left (\frac {5 (b d-a e) \int \frac {(d+e x)^{3/2}}{\sqrt {a+b x}}dx}{6 b}+\frac {\sqrt {a+b x} (d+e x)^{5/2}}{3 b}\right )}{b}-\frac {2 (d+e x)^{7/2}}{b \sqrt {a+b x}}\right )}{b (b d-a e)}-\frac {2 (d+e x)^{9/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(-3 a B e+2 A b e+b B d) \left (\frac {7 e \left (\frac {5 (b d-a e) \left (\frac {3 (b d-a e) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x}}dx}{4 b}+\frac {\sqrt {a+b x} (d+e x)^{3/2}}{2 b}\right )}{6 b}+\frac {\sqrt {a+b x} (d+e x)^{5/2}}{3 b}\right )}{b}-\frac {2 (d+e x)^{7/2}}{b \sqrt {a+b x}}\right )}{b (b d-a e)}-\frac {2 (d+e x)^{9/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(-3 a B e+2 A b e+b B d) \left (\frac {7 e \left (\frac {5 (b d-a e) \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}}dx}{2 b}+\frac {\sqrt {a+b x} \sqrt {d+e x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (d+e x)^{3/2}}{2 b}\right )}{6 b}+\frac {\sqrt {a+b x} (d+e x)^{5/2}}{3 b}\right )}{b}-\frac {2 (d+e x)^{7/2}}{b \sqrt {a+b x}}\right )}{b (b d-a e)}-\frac {2 (d+e x)^{9/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {(-3 a B e+2 A b e+b B d) \left (\frac {7 e \left (\frac {5 (b d-a e) \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \int \frac {1}{b-\frac {e (a+b x)}{d+e x}}d\frac {\sqrt {a+b x}}{\sqrt {d+e x}}}{b}+\frac {\sqrt {a+b x} \sqrt {d+e x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (d+e x)^{3/2}}{2 b}\right )}{6 b}+\frac {\sqrt {a+b x} (d+e x)^{5/2}}{3 b}\right )}{b}-\frac {2 (d+e x)^{7/2}}{b \sqrt {a+b x}}\right )}{b (b d-a e)}-\frac {2 (d+e x)^{9/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {(-3 a B e+2 A b e+b B d) \left (\frac {7 e \left (\frac {5 (b d-a e) \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{3/2} \sqrt {e}}+\frac {\sqrt {a+b x} \sqrt {d+e x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (d+e x)^{3/2}}{2 b}\right )}{6 b}+\frac {\sqrt {a+b x} (d+e x)^{5/2}}{3 b}\right )}{b}-\frac {2 (d+e x)^{7/2}}{b \sqrt {a+b x}}\right )}{b (b d-a e)}-\frac {2 (d+e x)^{9/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}\)

Input:

Int[((A + B*x)*(d + e*x)^(7/2))/(a + b*x)^(5/2),x]
 

Output:

(-2*(A*b - a*B)*(d + e*x)^(9/2))/(3*b*(b*d - a*e)*(a + b*x)^(3/2)) + ((b*B 
*d + 2*A*b*e - 3*a*B*e)*((-2*(d + e*x)^(7/2))/(b*Sqrt[a + b*x]) + (7*e*((S 
qrt[a + b*x]*(d + e*x)^(5/2))/(3*b) + (5*(b*d - a*e)*((Sqrt[a + b*x]*(d + 
e*x)^(3/2))/(2*b) + (3*(b*d - a*e)*((Sqrt[a + b*x]*Sqrt[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)))/(6*b)))/b))/(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. \(1881\) vs. \(2(217)=434\).

Time = 0.29 (sec) , antiderivative size = 1882, normalized size of antiderivative = 7.21

method	result	size
default	\(\text {Expression too large to display}\)	\(1882\)

Input:

int((B*x+A)*(e*x+d)^(7/2)/(b*x+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/48*(e*x+d)^(1/2)*(-560*A*a^2*b^2*e^3*x*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/ 
2)-320*A*b^4*d^2*e*x*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+840*B*a^3*b*e^3*x 
*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-315*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^5*e^4-276*B*a*b^3*d*e^2*x^2 
*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+952*A*a*b^3*d*e^2*x*((e*x+d)*(b*x+a)) 
^(1/2)*(b*e)^(1/2)-1708*B*a^2*b^2*d*e^2*x*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1 
/2)+956*B*a*b^3*d^2*e*x*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+420*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^ 
2*e^4*x-630*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*b*e^4*x-420*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^2*d*e^3+210*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^3*d^2*e^2+ 
735*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*b*d*e^3-525*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*b^2*d^2*e^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^3*d^3*e-32*A*b^4 
*d^3*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+630*B*a^4*e^3*((e*x+d)*(b*x+a))^( 
1/2)*(b*e)^(1/2)+210*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^4*b*e^4-420*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^4*d*e^3*x^2+735*B*ln(1/2...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 623 vs. \(2 (217) = 434\).

Time = 1.46 (sec) , antiderivative size = 1271, normalized size of antiderivative = 4.87 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(e*x+d)^(7/2)/(b*x+a)^(5/2),x, algorithm="fricas")
 

Output:

[1/96*(105*(B*a^2*b^3*d^3 - (5*B*a^3*b^2 - 2*A*a^2*b^3)*d^2*e + (7*B*a^4*b 
 - 4*A*a^3*b^2)*d*e^2 - (3*B*a^5 - 2*A*a^4*b)*e^3 + (B*b^5*d^3 - (5*B*a*b^ 
4 - 2*A*b^5)*d^2*e + (7*B*a^2*b^3 - 4*A*a*b^4)*d*e^2 - (3*B*a^3*b^2 - 2*A* 
a^2*b^3)*e^3)*x^2 + 2*(B*a*b^4*d^3 - (5*B*a^2*b^3 - 2*A*a*b^4)*d^2*e + (7* 
B*a^3*b^2 - 4*A*a^2*b^3)*d*e^2 - (3*B*a^4*b - 2*A*a^3*b^2)*e^3)*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)*sqrt(e*x + d)*sqrt(e/b) + 8*(b^2*d*e + a*b*e^2)*x 
) + 4*(8*B*b^4*e^3*x^4 - 16*(2*B*a*b^3 + A*b^4)*d^3 + 7*(49*B*a^2*b^2 - 16 
*A*a*b^3)*d^2*e - 70*(9*B*a^3*b - 5*A*a^2*b^2)*d*e^2 + 105*(3*B*a^4 - 2*A* 
a^3*b)*e^3 + 2*(19*B*b^4*d*e^2 - 3*(3*B*a*b^3 - 2*A*b^4)*e^3)*x^3 + 3*(29* 
B*b^4*d^2*e - 2*(23*B*a*b^3 - 13*A*b^4)*d*e^2 + 7*(3*B*a^2*b^2 - 2*A*a*b^3 
)*e^3)*x^2 - 2*(24*B*b^4*d^3 - (239*B*a*b^3 - 80*A*b^4)*d^2*e + 7*(61*B*a^ 
2*b^2 - 34*A*a*b^3)*d*e^2 - 70*(3*B*a^3*b - 2*A*a^2*b^2)*e^3)*x)*sqrt(b*x 
+ a)*sqrt(e*x + d))/(b^7*x^2 + 2*a*b^6*x + a^2*b^5), -1/48*(105*(B*a^2*b^3 
*d^3 - (5*B*a^3*b^2 - 2*A*a^2*b^3)*d^2*e + (7*B*a^4*b - 4*A*a^3*b^2)*d*e^2 
 - (3*B*a^5 - 2*A*a^4*b)*e^3 + (B*b^5*d^3 - (5*B*a*b^4 - 2*A*b^5)*d^2*e + 
(7*B*a^2*b^3 - 4*A*a*b^4)*d*e^2 - (3*B*a^3*b^2 - 2*A*a^2*b^3)*e^3)*x^2 + 2 
*(B*a*b^4*d^3 - (5*B*a^2*b^3 - 2*A*a*b^4)*d^2*e + (7*B*a^3*b^2 - 4*A*a^2*b 
^3)*d*e^2 - (3*B*a^4*b - 2*A*a^3*b^2)*e^3)*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...
 

Sympy [F]

\[ \int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^{5/2}} \, dx=\int \frac {\left (A + B x\right ) \left (d + e x\right )^{\frac {7}{2}}}{\left (a + b x\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((B*x+A)*(e*x+d)**(7/2)/(b*x+a)**(5/2),x)
 

Output:

Integral((A + B*x)*(d + e*x)**(7/2)/(a + b*x)**(5/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)*(e*x+d)^(7/2)/(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. 1814 vs. \(2 (217) = 434\).

Time = 0.65 (sec) , antiderivative size = 1814, normalized size of antiderivative = 6.95 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(e*x+d)^(7/2)/(b*x+a)^(5/2),x, algorithm="giac")
 

Output:

1/24*sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b* 
x + a)*B*e^3*abs(b)/b^7 + (19*B*b^21*d*e^6*abs(b) - 25*B*a*b^20*e^7*abs(b) 
 + 6*A*b^21*e^7*abs(b))/(b^27*e^4)) + 3*(29*B*b^22*d^2*e^5*abs(b) - 84*B*a 
*b^21*d*e^6*abs(b) + 26*A*b^22*d*e^6*abs(b) + 55*B*a^2*b^20*e^7*abs(b) - 2 
6*A*a*b^21*e^7*abs(b))/(b^27*e^4)) - 35/16*(sqrt(b*e)*B*b^3*d^3*abs(b) - 5 
*sqrt(b*e)*B*a*b^2*d^2*e*abs(b) + 2*sqrt(b*e)*A*b^3*d^2*e*abs(b) + 7*sqrt( 
b*e)*B*a^2*b*d*e^2*abs(b) - 4*sqrt(b*e)*A*a*b^2*d*e^2*abs(b) - 3*sqrt(b*e) 
*B*a^3*e^3*abs(b) + 2*sqrt(b*e)*A*a^2*b*e^3*abs(b))*log((sqrt(b*e)*sqrt(b* 
x + a) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2)/b^7 - 4/3*(3*sqrt(b*e)*B* 
b^8*d^6*abs(b) - 28*sqrt(b*e)*B*a*b^7*d^5*e*abs(b) + 10*sqrt(b*e)*A*b^8*d^ 
5*e*abs(b) + 95*sqrt(b*e)*B*a^2*b^6*d^4*e^2*abs(b) - 50*sqrt(b*e)*A*a*b^7* 
d^4*e^2*abs(b) - 160*sqrt(b*e)*B*a^3*b^5*d^3*e^3*abs(b) + 100*sqrt(b*e)*A* 
a^2*b^6*d^3*e^3*abs(b) + 145*sqrt(b*e)*B*a^4*b^4*d^2*e^4*abs(b) - 100*sqrt 
(b*e)*A*a^3*b^5*d^2*e^4*abs(b) - 68*sqrt(b*e)*B*a^5*b^3*d*e^5*abs(b) + 50* 
sqrt(b*e)*A*a^4*b^4*d*e^5*abs(b) + 13*sqrt(b*e)*B*a^6*b^2*e^6*abs(b) - 10* 
sqrt(b*e)*A*a^5*b^3*e^6*abs(b) - 6*sqrt(b*e)*(sqrt(b*e)*sqrt(b*x + a) - sq 
rt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*b^6*d^5*abs(b) + 48*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^5*d^4*e 
*abs(b) - 18*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^6*d^4*e*abs(b) - 132*sqrt(b*e)*(sqrt(b*e)*sqrt(b*x +...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^{5/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{7/2}}{{\left (a+b\,x\right )}^{5/2}} \,d x \] Input:

int(((A + B*x)*(d + e*x)^(7/2))/(a + b*x)^(5/2),x)
 

Output:

int(((A + B*x)*(d + e*x)^(7/2))/(a + b*x)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.77 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^{5/2}} \, dx=\frac {-840 \sqrt {e}\, \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) a^{3} e^{3}+2520 \sqrt {e}\, \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) a^{2} b d \,e^{2}-2520 \sqrt {e}\, \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) a \,b^{2} d^{2} e +840 \sqrt {e}\, \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) b^{3} d^{3}+525 \sqrt {e}\, \sqrt {b}\, \sqrt {b x +a}\, a^{3} e^{3}-1575 \sqrt {e}\, \sqrt {b}\, \sqrt {b x +a}\, a^{2} b d \,e^{2}+1575 \sqrt {e}\, \sqrt {b}\, \sqrt {b x +a}\, a \,b^{2} d^{2} e -525 \sqrt {e}\, \sqrt {b}\, \sqrt {b x +a}\, b^{3} d^{3}+840 \sqrt {e x +d}\, a^{3} b \,e^{3}-2240 \sqrt {e x +d}\, a^{2} b^{2} d \,e^{2}+280 \sqrt {e x +d}\, a^{2} b^{2} e^{3} x +1848 \sqrt {e x +d}\, a \,b^{3} d^{2} e -784 \sqrt {e x +d}\, a \,b^{3} d \,e^{2} x -112 \sqrt {e x +d}\, a \,b^{3} e^{3} x^{2}-384 \sqrt {e x +d}\, b^{4} d^{3}+696 \sqrt {e x +d}\, b^{4} d^{2} e x +304 \sqrt {e x +d}\, b^{4} d \,e^{2} x^{2}+64 \sqrt {e x +d}\, b^{4} e^{3} x^{3}}{192 \sqrt {b x +a}\, b^{5}} \] Input:

int((B*x+A)*(e*x+d)^(7/2)/(b*x+a)^(5/2),x)
 

Output:

( - 840*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**3*e**3 + 2520*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**2*b*d*e**2 - 2520*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**2*d**2*e + 840*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**3*d**3 + 525*sqrt(e)*sqrt(b)*sqrt(a + b*x)*a**3*e**3 
- 1575*sqrt(e)*sqrt(b)*sqrt(a + b*x)*a**2*b*d*e**2 + 1575*sqrt(e)*sqrt(b)* 
sqrt(a + b*x)*a*b**2*d**2*e - 525*sqrt(e)*sqrt(b)*sqrt(a + b*x)*b**3*d**3 
+ 840*sqrt(d + e*x)*a**3*b*e**3 - 2240*sqrt(d + e*x)*a**2*b**2*d*e**2 + 28 
0*sqrt(d + e*x)*a**2*b**2*e**3*x + 1848*sqrt(d + e*x)*a*b**3*d**2*e - 784* 
sqrt(d + e*x)*a*b**3*d*e**2*x - 112*sqrt(d + e*x)*a*b**3*e**3*x**2 - 384*s 
qrt(d + e*x)*b**4*d**3 + 696*sqrt(d + e*x)*b**4*d**2*e*x + 304*sqrt(d + e* 
x)*b**4*d*e**2*x**2 + 64*sqrt(d + e*x)*b**4*e**3*x**3)/(192*sqrt(a + b*x)* 
b**5)