\(\int \frac {x^{7/2} (A+B x)}{(a+b x)^{5/2}} \, dx\) [335]

Optimal result

Integrand size = 20, antiderivative size = 185 \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^{5/2}} \, dx=-\frac {2 (A b-a B) x^{7/2}}{3 b^2 (a+b x)^{3/2}}-\frac {2 (7 A b-10 a B) x^{5/2}}{3 b^3 \sqrt {a+b x}}-\frac {35 a (2 A b-3 a B) \sqrt {x} \sqrt {a+b x}}{8 b^5}+\frac {35 (2 A b-3 a B) x^{3/2} \sqrt {a+b x}}{12 b^4}+\frac {B x^{5/2} \sqrt {a+b x}}{3 b^3}+\frac {35 a^2 (2 A b-3 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{11/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.77 \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {\sqrt {x} \left (315 a^4 B-210 a^3 b (A-2 B x)+4 b^4 x^3 (3 A+2 B x)-6 a b^3 x^2 (7 A+3 B x)+7 a^2 b^2 x (-40 A+9 B x)\right )}{24 b^5 (a+b x)^{3/2}}+\frac {35 a^2 (-2 A b+3 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}-\sqrt {a+b x}}\right )}{4 b^{11/2}} \] Input:

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

Output:

(Sqrt[x]*(315*a^4*B - 210*a^3*b*(A - 2*B*x) + 4*b^4*x^3*(3*A + 2*B*x) - 6* 
a*b^3*x^2*(7*A + 3*B*x) + 7*a^2*b^2*x*(-40*A + 9*B*x)))/(24*b^5*(a + b*x)^ 
(3/2)) + (35*a^2*(-2*A*b + 3*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/(Sqrt[a] - Sqr 
t[a + b*x])])/(4*b^(11/2))
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {87, 57, 60, 60, 60, 65, 219}

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.

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

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 57

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 65

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(2*(A*b - a*B)*x^(9/2))/(3*a*b*(a + b*x)^(3/2)) - ((2*A*b - 3*a*B)*((-2*x^ 
(7/2))/(b*Sqrt[a + b*x]) + (7*((x^(5/2)*Sqrt[a + b*x])/(3*b) - (5*a*((x^(3 
/2)*Sqrt[a + b*x])/(2*b) - (3*a*((Sqrt[x]*Sqrt[a + b*x])/b - (a*ArcTanh[(S 
qrt[b]*Sqrt[x])/Sqrt[a + b*x]])/b^(3/2)))/(4*b)))/(6*b)))/b))/(a*b)
 

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 65

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2   Sub 
st[Int[1/(b - d*x^2), x], x, Sqrt[b*x]/Sqrt[c + d*x]], x] /; FreeQ[{b, c, d 
}, x] &&  !GtQ[c, 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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.52

method	result	size
risch	\(-\frac {\left (-8 b^{2} B \,x^{2}-12 A \,b^{2} x +34 B a b x +66 a b A -123 a^{2} B \right ) \sqrt {x}\, \sqrt {b x +a}}{24 b^{5}}+\frac {a^{2} \left (70 A \sqrt {b}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )-\frac {105 B a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{\sqrt {b}}-\frac {32 \left (4 A b -5 B a \right ) \sqrt {b \left (x +\frac {a}{b}\right )^{2}-\left (x +\frac {a}{b}\right ) a}}{b \left (x +\frac {a}{b}\right )}+\frac {16 a^{2} \left (A b -B a \right ) \left (\frac {2 \sqrt {b \left (x +\frac {a}{b}\right )^{2}-\left (x +\frac {a}{b}\right ) a}}{3 a \left (x +\frac {a}{b}\right )^{2}}+\frac {4 b \sqrt {b \left (x +\frac {a}{b}\right )^{2}-\left (x +\frac {a}{b}\right ) a}}{3 a^{2} \left (x +\frac {a}{b}\right )}\right )}{b^{2}}\right ) \sqrt {x \left (b x +a \right )}}{16 b^{5} \sqrt {x}\, \sqrt {b x +a}}\)	\(281\)
default	\(\frac {\left (16 B \,b^{\frac {9}{2}} x^{4} \sqrt {x \left (b x +a \right )}+24 A \,b^{\frac {9}{2}} x^{3} \sqrt {x \left (b x +a \right )}-36 B a \,b^{\frac {7}{2}} x^{3} \sqrt {x \left (b x +a \right )}+210 A \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{2} b^{3} x^{2}-84 A a \,b^{\frac {7}{2}} x^{2} \sqrt {x \left (b x +a \right )}-315 B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{3} b^{2} x^{2}+126 B \,a^{2} b^{\frac {5}{2}} x^{2} \sqrt {x \left (b x +a \right )}+420 A \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{3} b^{2} x -560 A \sqrt {x \left (b x +a \right )}\, b^{\frac {5}{2}} a^{2} x -630 B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{4} b x +840 B \sqrt {x \left (b x +a \right )}\, b^{\frac {3}{2}} a^{3} x +210 A \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{4} b -420 A \sqrt {x \left (b x +a \right )}\, b^{\frac {3}{2}} a^{3}-315 B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{5}+630 B \sqrt {x \left (b x +a \right )}\, \sqrt {b}\, a^{4}\right ) \sqrt {x}}{48 b^{\frac {11}{2}} \sqrt {x \left (b x +a \right )}\, \left (b x +a \right )^{\frac {3}{2}}}\)	\(406\)

Input:

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

Output:

-1/24*(-8*B*b^2*x^2-12*A*b^2*x+34*B*a*b*x+66*A*a*b-123*B*a^2)*x^(1/2)*(b*x 
+a)^(1/2)/b^5+1/16*a^2/b^5*(70*A*b^(1/2)*ln((1/2*a+b*x)/b^(1/2)+(b*x^2+a*x 
)^(1/2))-105*B*a*ln((1/2*a+b*x)/b^(1/2)+(b*x^2+a*x)^(1/2))/b^(1/2)-32*(4*A 
*b-5*B*a)/b/(x+a/b)*(b*(x+a/b)^2-(x+a/b)*a)^(1/2)+16*a^2*(A*b-B*a)/b^2*(2/ 
3/a/(x+a/b)^2*(b*(x+a/b)^2-(x+a/b)*a)^(1/2)+4/3*b/a^2/(x+a/b)*(b*(x+a/b)^2 
-(x+a/b)*a)^(1/2)))*(x*(b*x+a))^(1/2)/x^(1/2)/(b*x+a)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 421, normalized size of antiderivative = 2.28 \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^{5/2}} \, dx=\left [-\frac {105 \, {\left (3 \, B a^{5} - 2 \, A a^{4} b + {\left (3 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} x^{2} + 2 \, {\left (3 \, B a^{4} b - 2 \, A a^{3} b^{2}\right )} x\right )} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (8 \, B b^{5} x^{4} + 315 \, B a^{4} b - 210 \, A a^{3} b^{2} - 6 \, {\left (3 \, B a b^{4} - 2 \, A b^{5}\right )} x^{3} + 21 \, {\left (3 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} x^{2} + 140 \, {\left (3 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{48 \, {\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}}, \frac {105 \, {\left (3 \, B a^{5} - 2 \, A a^{4} b + {\left (3 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} x^{2} + 2 \, {\left (3 \, B a^{4} b - 2 \, A a^{3} b^{2}\right )} x\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {b x + a}}\right ) + {\left (8 \, B b^{5} x^{4} + 315 \, B a^{4} b - 210 \, A a^{3} b^{2} - 6 \, {\left (3 \, B a b^{4} - 2 \, A b^{5}\right )} x^{3} + 21 \, {\left (3 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} x^{2} + 140 \, {\left (3 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{24 \, {\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}}\right ] \] Input:

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

Output:

[-1/48*(105*(3*B*a^5 - 2*A*a^4*b + (3*B*a^3*b^2 - 2*A*a^2*b^3)*x^2 + 2*(3* 
B*a^4*b - 2*A*a^3*b^2)*x)*sqrt(b)*log(2*b*x + 2*sqrt(b*x + a)*sqrt(b)*sqrt 
(x) + a) - 2*(8*B*b^5*x^4 + 315*B*a^4*b - 210*A*a^3*b^2 - 6*(3*B*a*b^4 - 2 
*A*b^5)*x^3 + 21*(3*B*a^2*b^3 - 2*A*a*b^4)*x^2 + 140*(3*B*a^3*b^2 - 2*A*a^ 
2*b^3)*x)*sqrt(b*x + a)*sqrt(x))/(b^8*x^2 + 2*a*b^7*x + a^2*b^6), 1/24*(10 
5*(3*B*a^5 - 2*A*a^4*b + (3*B*a^3*b^2 - 2*A*a^2*b^3)*x^2 + 2*(3*B*a^4*b - 
2*A*a^3*b^2)*x)*sqrt(-b)*arctan(sqrt(-b)*sqrt(x)/sqrt(b*x + a)) + (8*B*b^5 
*x^4 + 315*B*a^4*b - 210*A*a^3*b^2 - 6*(3*B*a*b^4 - 2*A*b^5)*x^3 + 21*(3*B 
*a^2*b^3 - 2*A*a*b^4)*x^2 + 140*(3*B*a^3*b^2 - 2*A*a^2*b^3)*x)*sqrt(b*x + 
a)*sqrt(x))/(b^8*x^2 + 2*a*b^7*x + a^2*b^6)]
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (145) = 290\).

Time = 0.04 (sec) , antiderivative size = 416, normalized size of antiderivative = 2.25 \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {B x^{6}}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} b} - \frac {3 \, B a x^{5}}{4 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} b^{2}} + \frac {A x^{5}}{2 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} b} + \frac {35 \, B a^{3} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b} + \frac {a x}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b^{2}} - \frac {2 \, x}{\sqrt {b x^{2} + a x} a b} - \frac {1}{\sqrt {b x^{2} + a x} b^{2}}\right )}}{16 \, b^{3}} - \frac {35 \, A a^{2} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b} + \frac {a x}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b^{2}} - \frac {2 \, x}{\sqrt {b x^{2} + a x} a b} - \frac {1}{\sqrt {b x^{2} + a x} b^{2}}\right )}}{24 \, b^{2}} + \frac {21 \, B a^{2} x^{4}}{8 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} b^{3}} - \frac {7 \, A a x^{4}}{4 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} b^{2}} + \frac {35 \, B a^{3} x}{4 \, \sqrt {b x^{2} + a x} b^{5}} - \frac {35 \, A a^{2} x}{6 \, \sqrt {b x^{2} + a x} b^{4}} - \frac {105 \, B a^{3} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{16 \, b^{\frac {11}{2}}} + \frac {35 \, A a^{2} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{8 \, b^{\frac {9}{2}}} + \frac {35 \, \sqrt {b x^{2} + a x} B a^{2}}{8 \, b^{5}} - \frac {35 \, \sqrt {b x^{2} + a x} A a}{12 \, b^{4}} \] Input:

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

Output:

1/3*B*x^6/((b*x^2 + a*x)^(3/2)*b) - 3/4*B*a*x^5/((b*x^2 + a*x)^(3/2)*b^2) 
+ 1/2*A*x^5/((b*x^2 + a*x)^(3/2)*b) + 35/16*B*a^3*x*(3*x^2/((b*x^2 + a*x)^ 
(3/2)*b) + a*x/((b*x^2 + a*x)^(3/2)*b^2) - 2*x/(sqrt(b*x^2 + a*x)*a*b) - 1 
/(sqrt(b*x^2 + a*x)*b^2))/b^3 - 35/24*A*a^2*x*(3*x^2/((b*x^2 + a*x)^(3/2)* 
b) + a*x/((b*x^2 + a*x)^(3/2)*b^2) - 2*x/(sqrt(b*x^2 + a*x)*a*b) - 1/(sqrt 
(b*x^2 + a*x)*b^2))/b^2 + 21/8*B*a^2*x^4/((b*x^2 + a*x)^(3/2)*b^3) - 7/4*A 
*a*x^4/((b*x^2 + a*x)^(3/2)*b^2) + 35/4*B*a^3*x/(sqrt(b*x^2 + a*x)*b^5) - 
35/6*A*a^2*x/(sqrt(b*x^2 + a*x)*b^4) - 105/16*B*a^3*log(2*b*x + a + 2*sqrt 
(b*x^2 + a*x)*sqrt(b))/b^(11/2) + 35/8*A*a^2*log(2*b*x + a + 2*sqrt(b*x^2 
+ a*x)*sqrt(b))/b^(9/2) + 35/8*sqrt(b*x^2 + a*x)*B*a^2/b^5 - 35/12*sqrt(b* 
x^2 + a*x)*A*a/b^4
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 370 vs. \(2 (145) = 290\).

Time = 15.57 (sec) , antiderivative size = 370, normalized size of antiderivative = 2.00 \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {1}{24} \, \sqrt {{\left (b x + a\right )} b - a b} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )} B {\left | b \right |}}{b^{7}} - \frac {25 \, B a b^{20} {\left | b \right |} - 6 \, A b^{21} {\left | b \right |}}{b^{27}}\right )} + \frac {3 \, {\left (55 \, B a^{2} b^{20} {\left | b \right |} - 26 \, A a b^{21} {\left | b \right |}\right )}}{b^{27}}\right )} + \frac {35 \, {\left (3 \, B a^{3} {\left | b \right |} - 2 \, A a^{2} b {\left | b \right |}\right )} \log \left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{16 \, b^{\frac {13}{2}}} + \frac {4 \, {\left (15 \, B a^{4} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} {\left | b \right |} + 24 \, B a^{5} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b {\left | b \right |} - 12 \, A a^{3} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} b {\left | b \right |} + 13 \, B a^{6} b^{2} {\left | b \right |} - 18 \, A a^{4} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{2} {\left | b \right |} - 10 \, A a^{5} b^{3} {\left | b \right |}\right )}}{3 \, {\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} b^{\frac {11}{2}}} \] Input:

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

Output:

1/24*sqrt((b*x + a)*b - a*b)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)*B*abs 
(b)/b^7 - (25*B*a*b^20*abs(b) - 6*A*b^21*abs(b))/b^27) + 3*(55*B*a^2*b^20* 
abs(b) - 26*A*a*b^21*abs(b))/b^27) + 35/16*(3*B*a^3*abs(b) - 2*A*a^2*b*abs 
(b))*log((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2)/b^(13/2) + 4 
/3*(15*B*a^4*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^4*abs(b) + 
24*B*a^5*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b*abs(b) - 12 
*A*a^3*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^4*b*abs(b) + 13*B 
*a^6*b^2*abs(b) - 18*A*a^4*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b 
))^2*b^2*abs(b) - 10*A*a^5*b^3*abs(b))/(((sqrt(b*x + a)*sqrt(b) - sqrt((b* 
x + a)*b - a*b))^2 + a*b)^3*b^(11/2))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.53 \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {-840 \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{3}+525 \sqrt {b}\, \sqrt {b x +a}\, a^{3}+840 \sqrt {x}\, a^{3} b +280 \sqrt {x}\, a^{2} b^{2} x -112 \sqrt {x}\, a \,b^{3} x^{2}+64 \sqrt {x}\, b^{4} x^{3}}{192 \sqrt {b x +a}\, b^{5}} \] Input:

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

Output:

( - 840*sqrt(b)*sqrt(a + b*x)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a 
))*a**3 + 525*sqrt(b)*sqrt(a + b*x)*a**3 + 840*sqrt(x)*a**3*b + 280*sqrt(x 
)*a**2*b**2*x - 112*sqrt(x)*a*b**3*x**2 + 64*sqrt(x)*b**4*x**3)/(192*sqrt( 
a + b*x)*b**5)