\(\int \frac {x^{5/2} (A+B x)}{(a+b x)^3} \, dx\) [364]

Optimal result

Integrand size = 18, antiderivative size = 147 \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^3} \, dx=\frac {5 (3 A b-7 a B) \sqrt {x}}{4 b^4}-\frac {5 (3 A b-7 a B) x^{3/2}}{12 a b^3}+\frac {(A b-a B) x^{7/2}}{2 a b (a+b x)^2}+\frac {(3 A b-7 a B) x^{5/2}}{4 a b^2 (a+b x)}-\frac {5 \sqrt {a} (3 A b-7 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{9/2}} \]

[Out]

-5/12*(3*A*b-7*B*a)*x^(3/2)/a/b^3+1/2*(A*b-B*a)*x^(7/2)/a/b/(b*x+a)^2+1/4*(3*A*b-7*B*a)*x^(5/2)/a/b^2/(b*x+a)-
5/4*(3*A*b-7*B*a)*arctan(b^(1/2)*x^(1/2)/a^(1/2))*a^(1/2)/b^(9/2)+5/4*(3*A*b-7*B*a)*x^(1/2)/b^4

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {79, 43, 52, 65, 211} \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^3} \, dx=-\frac {5 \sqrt {a} (3 A b-7 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{9/2}}+\frac {5 \sqrt {x} (3 A b-7 a B)}{4 b^4}-\frac {5 x^{3/2} (3 A b-7 a B)}{12 a b^3}+\frac {x^{5/2} (3 A b-7 a B)}{4 a b^2 (a+b x)}+\frac {x^{7/2} (A b-a B)}{2 a b (a+b x)^2} \]

[In]

Int[(x^(5/2)*(A + B*x))/(a + b*x)^3,x]

[Out]

(5*(3*A*b - 7*a*B)*Sqrt[x])/(4*b^4) - (5*(3*A*b - 7*a*B)*x^(3/2))/(12*a*b^3) + ((A*b - a*B)*x^(7/2))/(2*a*b*(a
+ b*x)^2) + ((3*A*b - 7*a*B)*x^(5/2))/(4*a*b^2*(a + b*x)) - (5*Sqrt[a]*(3*A*b - 7*a*B)*ArcTan[(Sqrt[b]*Sqrt[x
])/Sqrt[a]])/(4*b^(9/2))

Rule 43

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] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 52

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] + Dist[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] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(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] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps \begin{align*} \text {integral}& = \frac {(A b-a B) x^{7/2}}{2 a b (a+b x)^2}-\frac {\left (\frac {3 A b}{2}-\frac {7 a B}{2}\right ) \int \frac {x^{5/2}}{(a+b x)^2} \, dx}{2 a b} \\ & = \frac {(A b-a B) x^{7/2}}{2 a b (a+b x)^2}+\frac {(3 A b-7 a B) x^{5/2}}{4 a b^2 (a+b x)}-\frac {(5 (3 A b-7 a B)) \int \frac {x^{3/2}}{a+b x} \, dx}{8 a b^2} \\ & = -\frac {5 (3 A b-7 a B) x^{3/2}}{12 a b^3}+\frac {(A b-a B) x^{7/2}}{2 a b (a+b x)^2}+\frac {(3 A b-7 a B) x^{5/2}}{4 a b^2 (a+b x)}+\frac {(5 (3 A b-7 a B)) \int \frac {\sqrt {x}}{a+b x} \, dx}{8 b^3} \\ & = \frac {5 (3 A b-7 a B) \sqrt {x}}{4 b^4}-\frac {5 (3 A b-7 a B) x^{3/2}}{12 a b^3}+\frac {(A b-a B) x^{7/2}}{2 a b (a+b x)^2}+\frac {(3 A b-7 a B) x^{5/2}}{4 a b^2 (a+b x)}-\frac {(5 a (3 A b-7 a B)) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{8 b^4} \\ & = \frac {5 (3 A b-7 a B) \sqrt {x}}{4 b^4}-\frac {5 (3 A b-7 a B) x^{3/2}}{12 a b^3}+\frac {(A b-a B) x^{7/2}}{2 a b (a+b x)^2}+\frac {(3 A b-7 a B) x^{5/2}}{4 a b^2 (a+b x)}-\frac {(5 a (3 A b-7 a B)) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{4 b^4} \\ & = \frac {5 (3 A b-7 a B) \sqrt {x}}{4 b^4}-\frac {5 (3 A b-7 a B) x^{3/2}}{12 a b^3}+\frac {(A b-a B) x^{7/2}}{2 a b (a+b x)^2}+\frac {(3 A b-7 a B) x^{5/2}}{4 a b^2 (a+b x)}-\frac {5 \sqrt {a} (3 A b-7 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.75 \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^3} \, dx=\frac {\sqrt {x} \left (-105 a^3 B+a b^2 x (75 A-56 B x)+5 a^2 b (9 A-35 B x)+8 b^3 x^2 (3 A+B x)\right )}{12 b^4 (a+b x)^2}+\frac {5 \sqrt {a} (-3 A b+7 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{9/2}} \]

[In]

Integrate[(x^(5/2)*(A + B*x))/(a + b*x)^3,x]

[Out]

(Sqrt[x]*(-105*a^3*B + a*b^2*x*(75*A - 56*B*x) + 5*a^2*b*(9*A - 35*B*x) + 8*b^3*x^2*(3*A + B*x)))/(12*b^4*(a +
b*x)^2) + (5*Sqrt[a]*(-3*A*b + 7*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*b^(9/2))

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.67

method	result	size
risch	\(\frac {2 \left (b B x +3 A b -9 B a \right ) \sqrt {x}}{3 b^{4}}-\frac {a \left (\frac {2 \left (-\frac {9}{8} b^{2} A +\frac {13}{8} a b B \right ) x^{\frac {3}{2}}-\frac {a \left (7 A b -11 B a \right ) \sqrt {x}}{4}}{\left (b x +a \right )^{2}}+\frac {5 \left (3 A b -7 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}}\right )}{b^{4}}\)	\(98\)
derivativedivides	\(\frac {\frac {2 b B \,x^{\frac {3}{2}}}{3}+2 A b \sqrt {x}-6 B a \sqrt {x}}{b^{4}}-\frac {2 a \left (\frac {\left (-\frac {9}{8} b^{2} A +\frac {13}{8} a b B \right ) x^{\frac {3}{2}}-\frac {a \left (7 A b -11 B a \right ) \sqrt {x}}{8}}{\left (b x +a \right )^{2}}+\frac {5 \left (3 A b -7 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{4}}\)	\(102\)
default	\(\frac {\frac {2 b B \,x^{\frac {3}{2}}}{3}+2 A b \sqrt {x}-6 B a \sqrt {x}}{b^{4}}-\frac {2 a \left (\frac {\left (-\frac {9}{8} b^{2} A +\frac {13}{8} a b B \right ) x^{\frac {3}{2}}-\frac {a \left (7 A b -11 B a \right ) \sqrt {x}}{8}}{\left (b x +a \right )^{2}}+\frac {5 \left (3 A b -7 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{4}}\)	\(102\)

[In]

int(x^(5/2)*(B*x+A)/(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

2/3*(B*b*x+3*A*b-9*B*a)*x^(1/2)/b^4-a/b^4*(2*((-9/8*b^2*A+13/8*a*b*B)*x^(3/2)-1/8*a*(7*A*b-11*B*a)*x^(1/2))/(b
*x+a)^2+5/4*(3*A*b-7*B*a)/(a*b)^(1/2)*arctan(b*x^(1/2)/(a*b)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.37 \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^3} \, dx=\left [-\frac {15 \, {\left (7 \, B a^{3} - 3 \, A a^{2} b + {\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} x^{2} + 2 \, {\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} x\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x - 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) - 2 \, {\left (8 \, B b^{3} x^{3} - 105 \, B a^{3} + 45 \, A a^{2} b - 8 \, {\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} x^{2} - 25 \, {\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} x\right )} \sqrt {x}}{24 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, \frac {15 \, {\left (7 \, B a^{3} - 3 \, A a^{2} b + {\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} x^{2} + 2 \, {\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} x\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) + {\left (8 \, B b^{3} x^{3} - 105 \, B a^{3} + 45 \, A a^{2} b - 8 \, {\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} x^{2} - 25 \, {\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} x\right )} \sqrt {x}}{12 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \]

[In]

integrate(x^(5/2)*(B*x+A)/(b*x+a)^3,x, algorithm="fricas")

[Out]

[-1/24*(15*(7*B*a^3 - 3*A*a^2*b + (7*B*a*b^2 - 3*A*b^3)*x^2 + 2*(7*B*a^2*b - 3*A*a*b^2)*x)*sqrt(-a/b)*log((b*x
- 2*b*sqrt(x)*sqrt(-a/b) - a)/(b*x + a)) - 2*(8*B*b^3*x^3 - 105*B*a^3 + 45*A*a^2*b - 8*(7*B*a*b^2 - 3*A*b^3)*
x^2 - 25*(7*B*a^2*b - 3*A*a*b^2)*x)*sqrt(x))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4), 1/12*(15*(7*B*a^3 - 3*A*a^2*b +
(7*B*a*b^2 - 3*A*b^3)*x^2 + 2*(7*B*a^2*b - 3*A*a*b^2)*x)*sqrt(a/b)*arctan(b*sqrt(x)*sqrt(a/b)/a) + (8*B*b^3*x^
3 - 105*B*a^3 + 45*A*a^2*b - 8*(7*B*a*b^2 - 3*A*b^3)*x^2 - 25*(7*B*a^2*b - 3*A*a*b^2)*x)*sqrt(x))/(b^6*x^2 + 2
*a*b^5*x + a^2*b^4)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1496 vs. \(2 (139) = 278\).

Time = 21.31 (sec) , antiderivative size = 1496, normalized size of antiderivative = 10.18 \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^3} \, dx=\text {Too large to display} \]

[In]

integrate(x**(5/2)*(B*x+A)/(b*x+a)**3,x)

[Out]

Piecewise((zoo*(2*A*sqrt(x) + 2*B*x**(3/2)/3), Eq(a, 0) & Eq(b, 0)), ((2*A*x**(7/2)/7 + 2*B*x**(9/2)/9)/a**3,
Eq(b, 0)), ((2*A*sqrt(x) + 2*B*x**(3/2)/3)/b**3, Eq(a, 0)), (-45*A*a**3*b*log(sqrt(x) - sqrt(-a/b))/(24*a**2*b
**5*sqrt(-a/b) + 48*a*b**6*x*sqrt(-a/b) + 24*b**7*x**2*sqrt(-a/b)) + 45*A*a**3*b*log(sqrt(x) + sqrt(-a/b))/(24
*a**2*b**5*sqrt(-a/b) + 48*a*b**6*x*sqrt(-a/b) + 24*b**7*x**2*sqrt(-a/b)) + 90*A*a**2*b**2*sqrt(x)*sqrt(-a/b)/
(24*a**2*b**5*sqrt(-a/b) + 48*a*b**6*x*sqrt(-a/b) + 24*b**7*x**2*sqrt(-a/b)) - 90*A*a**2*b**2*x*log(sqrt(x) -
sqrt(-a/b))/(24*a**2*b**5*sqrt(-a/b) + 48*a*b**6*x*sqrt(-a/b) + 24*b**7*x**2*sqrt(-a/b)) + 90*A*a**2*b**2*x*lo
g(sqrt(x) + sqrt(-a/b))/(24*a**2*b**5*sqrt(-a/b) + 48*a*b**6*x*sqrt(-a/b) + 24*b**7*x**2*sqrt(-a/b)) + 150*A*a
*b**3*x**(3/2)*sqrt(-a/b)/(24*a**2*b**5*sqrt(-a/b) + 48*a*b**6*x*sqrt(-a/b) + 24*b**7*x**2*sqrt(-a/b)) - 45*A*
a*b**3*x**2*log(sqrt(x) - sqrt(-a/b))/(24*a**2*b**5*sqrt(-a/b) + 48*a*b**6*x*sqrt(-a/b) + 24*b**7*x**2*sqrt(-a
/b)) + 45*A*a*b**3*x**2*log(sqrt(x) + sqrt(-a/b))/(24*a**2*b**5*sqrt(-a/b) + 48*a*b**6*x*sqrt(-a/b) + 24*b**7*
x**2*sqrt(-a/b)) + 48*A*b**4*x**(5/2)*sqrt(-a/b)/(24*a**2*b**5*sqrt(-a/b) + 48*a*b**6*x*sqrt(-a/b) + 24*b**7*x
**2*sqrt(-a/b)) + 105*B*a**4*log(sqrt(x) - sqrt(-a/b))/(24*a**2*b**5*sqrt(-a/b) + 48*a*b**6*x*sqrt(-a/b) + 24*
b**7*x**2*sqrt(-a/b)) - 105*B*a**4*log(sqrt(x) + sqrt(-a/b))/(24*a**2*b**5*sqrt(-a/b) + 48*a*b**6*x*sqrt(-a/b)
+ 24*b**7*x**2*sqrt(-a/b)) - 210*B*a**3*b*sqrt(x)*sqrt(-a/b)/(24*a**2*b**5*sqrt(-a/b) + 48*a*b**6*x*sqrt(-a/b
) + 24*b**7*x**2*sqrt(-a/b)) + 210*B*a**3*b*x*log(sqrt(x) - sqrt(-a/b))/(24*a**2*b**5*sqrt(-a/b) + 48*a*b**6*x
*sqrt(-a/b) + 24*b**7*x**2*sqrt(-a/b)) - 210*B*a**3*b*x*log(sqrt(x) + sqrt(-a/b))/(24*a**2*b**5*sqrt(-a/b) + 4
8*a*b**6*x*sqrt(-a/b) + 24*b**7*x**2*sqrt(-a/b)) - 350*B*a**2*b**2*x**(3/2)*sqrt(-a/b)/(24*a**2*b**5*sqrt(-a/b
) + 48*a*b**6*x*sqrt(-a/b) + 24*b**7*x**2*sqrt(-a/b)) + 105*B*a**2*b**2*x**2*log(sqrt(x) - sqrt(-a/b))/(24*a**
2*b**5*sqrt(-a/b) + 48*a*b**6*x*sqrt(-a/b) + 24*b**7*x**2*sqrt(-a/b)) - 105*B*a**2*b**2*x**2*log(sqrt(x) + sqr
t(-a/b))/(24*a**2*b**5*sqrt(-a/b) + 48*a*b**6*x*sqrt(-a/b) + 24*b**7*x**2*sqrt(-a/b)) - 112*B*a*b**3*x**(5/2)*
sqrt(-a/b)/(24*a**2*b**5*sqrt(-a/b) + 48*a*b**6*x*sqrt(-a/b) + 24*b**7*x**2*sqrt(-a/b)) + 16*B*b**4*x**(7/2)*s
qrt(-a/b)/(24*a**2*b**5*sqrt(-a/b) + 48*a*b**6*x*sqrt(-a/b) + 24*b**7*x**2*sqrt(-a/b)), True))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.84 \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^3} \, dx=-\frac {{\left (13 \, B a^{2} b - 9 \, A a b^{2}\right )} x^{\frac {3}{2}} + {\left (11 \, B a^{3} - 7 \, A a^{2} b\right )} \sqrt {x}}{4 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} + \frac {5 \, {\left (7 \, B a^{2} - 3 \, A a b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} b^{4}} + \frac {2 \, {\left (B b x^{\frac {3}{2}} - 3 \, {\left (3 \, B a - A b\right )} \sqrt {x}\right )}}{3 \, b^{4}} \]

[In]

integrate(x^(5/2)*(B*x+A)/(b*x+a)^3,x, algorithm="maxima")

[Out]

-1/4*((13*B*a^2*b - 9*A*a*b^2)*x^(3/2) + (11*B*a^3 - 7*A*a^2*b)*sqrt(x))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4) + 5/4
*(7*B*a^2 - 3*A*a*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^4) + 2/3*(B*b*x^(3/2) - 3*(3*B*a - A*b)*sqrt(x))
/b^4

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.81 \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^3} \, dx=\frac {5 \, {\left (7 \, B a^{2} - 3 \, A a b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} b^{4}} - \frac {13 \, B a^{2} b x^{\frac {3}{2}} - 9 \, A a b^{2} x^{\frac {3}{2}} + 11 \, B a^{3} \sqrt {x} - 7 \, A a^{2} b \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} b^{4}} + \frac {2 \, {\left (B b^{6} x^{\frac {3}{2}} - 9 \, B a b^{5} \sqrt {x} + 3 \, A b^{6} \sqrt {x}\right )}}{3 \, b^{9}} \]

[In]

integrate(x^(5/2)*(B*x+A)/(b*x+a)^3,x, algorithm="giac")

[Out]

5/4*(7*B*a^2 - 3*A*a*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^4) - 1/4*(13*B*a^2*b*x^(3/2) - 9*A*a*b^2*x^(3
/2) + 11*B*a^3*sqrt(x) - 7*A*a^2*b*sqrt(x))/((b*x + a)^2*b^4) + 2/3*(B*b^6*x^(3/2) - 9*B*a*b^5*sqrt(x) + 3*A*b
^6*sqrt(x))/b^9

Mupad [B] (verification not implemented)

Time = 0.58 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.97 \[ \int \frac {x^{5/2} (A+B x)}{(a+b x)^3} \, dx=\frac {x^{3/2}\,\left (\frac {9\,A\,a\,b^2}{4}-\frac {13\,B\,a^2\,b}{4}\right )-\sqrt {x}\,\left (\frac {11\,B\,a^3}{4}-\frac {7\,A\,a^2\,b}{4}\right )}{a^2\,b^4+2\,a\,b^5\,x+b^6\,x^2}+\sqrt {x}\,\left (\frac {2\,A}{b^3}-\frac {6\,B\,a}{b^4}\right )+\frac {2\,B\,x^{3/2}}{3\,b^3}+\frac {5\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}\,\sqrt {x}\,\left (3\,A\,b-7\,B\,a\right )}{7\,B\,a^2-3\,A\,a\,b}\right )\,\left (3\,A\,b-7\,B\,a\right )}{4\,b^{9/2}} \]

[In]

int((x^(5/2)*(A + B*x))/(a + b*x)^3,x)

[Out]

(x^(3/2)*((9*A*a*b^2)/4 - (13*B*a^2*b)/4) - x^(1/2)*((11*B*a^3)/4 - (7*A*a^2*b)/4))/(a^2*b^4 + b^6*x^2 + 2*a*b
^5*x) + x^(1/2)*((2*A)/b^3 - (6*B*a)/b^4) + (2*B*x^(3/2))/(3*b^3) + (5*a^(1/2)*atan((a^(1/2)*b^(1/2)*x^(1/2)*(
3*A*b - 7*B*a))/(7*B*a^2 - 3*A*a*b))*(3*A*b - 7*B*a))/(4*b^(9/2))