\(\int \frac {A+B x}{x^{5/2} (a+b x)^2} \, dx\) [360]

Optimal result

Integrand size = 18, antiderivative size = 107 \[ \int \frac {A+B x}{x^{5/2} (a+b x)^2} \, dx=-\frac {5 A b-3 a B}{3 a^2 b x^{3/2}}+\frac {5 A b-3 a B}{a^3 \sqrt {x}}+\frac {A b-a B}{a b x^{3/2} (a+b x)}+\frac {\sqrt {b} (5 A b-3 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {79, 53, 65, 211} \[ \int \frac {A+B x}{x^{5/2} (a+b x)^2} \, dx=\frac {\sqrt {b} (5 A b-3 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}}+\frac {5 A b-3 a B}{a^3 \sqrt {x}}-\frac {5 A b-3 a B}{3 a^2 b x^{3/2}}+\frac {A b-a B}{a b x^{3/2} (a+b x)} \]

[In]

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

[Out]

-1/3*(5*A*b - 3*a*B)/(a^2*b*x^(3/2)) + (5*A*b - 3*a*B)/(a^3*Sqrt[x]) + (A*b - a*B)/(a*b*x^(3/2)*(a + b*x)) + (
Sqrt[b]*(5*A*b - 3*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^(7/2)

Rule 53

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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}{a b x^{3/2} (a+b x)}-\frac {\left (-\frac {5 A b}{2}+\frac {3 a B}{2}\right ) \int \frac {1}{x^{5/2} (a+b x)} \, dx}{a b} \\ & = -\frac {5 A b-3 a B}{3 a^2 b x^{3/2}}+\frac {A b-a B}{a b x^{3/2} (a+b x)}-\frac {(5 A b-3 a B) \int \frac {1}{x^{3/2} (a+b x)} \, dx}{2 a^2} \\ & = -\frac {5 A b-3 a B}{3 a^2 b x^{3/2}}+\frac {5 A b-3 a B}{a^3 \sqrt {x}}+\frac {A b-a B}{a b x^{3/2} (a+b x)}+\frac {(b (5 A b-3 a B)) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{2 a^3} \\ & = -\frac {5 A b-3 a B}{3 a^2 b x^{3/2}}+\frac {5 A b-3 a B}{a^3 \sqrt {x}}+\frac {A b-a B}{a b x^{3/2} (a+b x)}+\frac {(b (5 A b-3 a B)) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{a^3} \\ & = -\frac {5 A b-3 a B}{3 a^2 b x^{3/2}}+\frac {5 A b-3 a B}{a^3 \sqrt {x}}+\frac {A b-a B}{a b x^{3/2} (a+b x)}+\frac {\sqrt {b} (5 A b-3 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.84 \[ \int \frac {A+B x}{x^{5/2} (a+b x)^2} \, dx=\frac {15 A b^2 x^2+a b x (10 A-9 B x)-2 a^2 (A+3 B x)}{3 a^3 x^{3/2} (a+b x)}+\frac {\sqrt {b} (5 A b-3 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}} \]

[In]

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

[Out]

(15*A*b^2*x^2 + a*b*x*(10*A - 9*B*x) - 2*a^2*(A + 3*B*x))/(3*a^3*x^(3/2)*(a + b*x)) + (Sqrt[b]*(5*A*b - 3*a*B)
*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^(7/2)

Maple [A] (verified)

Time = 1.08 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.72

method	result	size
risch	\(-\frac {2 \left (-6 A b x +3 B a x +A a \right )}{3 a^{3} x^{\frac {3}{2}}}+\frac {b \left (\frac {2 \left (\frac {A b}{2}-\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (5 A b -3 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )}{a^{3}}\)	\(77\)
derivativedivides	\(\frac {2 b \left (\frac {\left (\frac {A b}{2}-\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (5 A b -3 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{3}}-\frac {2 A}{3 a^{2} x^{\frac {3}{2}}}-\frac {2 \left (-2 A b +B a \right )}{a^{3} \sqrt {x}}\)	\(81\)
default	\(\frac {2 b \left (\frac {\left (\frac {A b}{2}-\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (5 A b -3 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{3}}-\frac {2 A}{3 a^{2} x^{\frac {3}{2}}}-\frac {2 \left (-2 A b +B a \right )}{a^{3} \sqrt {x}}\)	\(81\)

[In]

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

[Out]

-2/3*(-6*A*b*x+3*B*a*x+A*a)/a^3/x^(3/2)+1/a^3*b*(2*(1/2*A*b-1/2*B*a)*x^(1/2)/(b*x+a)+(5*A*b-3*B*a)/(a*b)^(1/2)
*arctan(b*x^(1/2)/(a*b)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.45 \[ \int \frac {A+B x}{x^{5/2} (a+b x)^2} \, dx=\left [-\frac {3 \, {\left ({\left (3 \, B a b - 5 \, A b^{2}\right )} x^{3} + {\left (3 \, B a^{2} - 5 \, A a b\right )} x^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x + 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) + 2 \, {\left (2 \, A a^{2} + 3 \, {\left (3 \, B a b - 5 \, A b^{2}\right )} x^{2} + 2 \, {\left (3 \, B a^{2} - 5 \, A a b\right )} x\right )} \sqrt {x}}{6 \, {\left (a^{3} b x^{3} + a^{4} x^{2}\right )}}, \frac {3 \, {\left ({\left (3 \, B a b - 5 \, A b^{2}\right )} x^{3} + {\left (3 \, B a^{2} - 5 \, A a b\right )} x^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - {\left (2 \, A a^{2} + 3 \, {\left (3 \, B a b - 5 \, A b^{2}\right )} x^{2} + 2 \, {\left (3 \, B a^{2} - 5 \, A a b\right )} x\right )} \sqrt {x}}{3 \, {\left (a^{3} b x^{3} + a^{4} x^{2}\right )}}\right ] \]

[In]

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

[Out]

[-1/6*(3*((3*B*a*b - 5*A*b^2)*x^3 + (3*B*a^2 - 5*A*a*b)*x^2)*sqrt(-b/a)*log((b*x + 2*a*sqrt(x)*sqrt(-b/a) - a)
/(b*x + a)) + 2*(2*A*a^2 + 3*(3*B*a*b - 5*A*b^2)*x^2 + 2*(3*B*a^2 - 5*A*a*b)*x)*sqrt(x))/(a^3*b*x^3 + a^4*x^2)
, 1/3*(3*((3*B*a*b - 5*A*b^2)*x^3 + (3*B*a^2 - 5*A*a*b)*x^2)*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*sqrt(x))) - (2*A*
a^2 + 3*(3*B*a*b - 5*A*b^2)*x^2 + 2*(3*B*a^2 - 5*A*a*b)*x)*sqrt(x))/(a^3*b*x^3 + a^4*x^2)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 882 vs. \(2 (95) = 190\).

Time = 14.76 (sec) , antiderivative size = 882, normalized size of antiderivative = 8.24 \[ \int \frac {A+B x}{x^{5/2} (a+b x)^2} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{5 x^{\frac {5}{2}}}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 A}{3 x^{\frac {3}{2}}} - \frac {2 B}{\sqrt {x}}}{a^{2}} & \text {for}\: b = 0 \\\frac {- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{5 x^{\frac {5}{2}}}}{b^{2}} & \text {for}\: a = 0 \\- \frac {4 A a^{2} \sqrt {- \frac {a}{b}}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} + \frac {15 A a b x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} - \frac {15 A a b x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} + \frac {20 A a b x \sqrt {- \frac {a}{b}}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} + \frac {15 A b^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} - \frac {15 A b^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} + \frac {30 A b^{2} x^{2} \sqrt {- \frac {a}{b}}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} - \frac {9 B a^{2} x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} + \frac {9 B a^{2} x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} - \frac {12 B a^{2} x \sqrt {- \frac {a}{b}}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} - \frac {9 B a b x^{\frac {5}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} + \frac {9 B a b x^{\frac {5}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} - \frac {18 B a b x^{2} \sqrt {- \frac {a}{b}}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((zoo*(-2*A/(7*x**(7/2)) - 2*B/(5*x**(5/2))), Eq(a, 0) & Eq(b, 0)), ((-2*A/(3*x**(3/2)) - 2*B/sqrt(x)
)/a**2, Eq(b, 0)), ((-2*A/(7*x**(7/2)) - 2*B/(5*x**(5/2)))/b**2, Eq(a, 0)), (-4*A*a**2*sqrt(-a/b)/(6*a**4*x**(
3/2)*sqrt(-a/b) + 6*a**3*b*x**(5/2)*sqrt(-a/b)) + 15*A*a*b*x**(3/2)*log(sqrt(x) - sqrt(-a/b))/(6*a**4*x**(3/2)
*sqrt(-a/b) + 6*a**3*b*x**(5/2)*sqrt(-a/b)) - 15*A*a*b*x**(3/2)*log(sqrt(x) + sqrt(-a/b))/(6*a**4*x**(3/2)*sqr
t(-a/b) + 6*a**3*b*x**(5/2)*sqrt(-a/b)) + 20*A*a*b*x*sqrt(-a/b)/(6*a**4*x**(3/2)*sqrt(-a/b) + 6*a**3*b*x**(5/2
)*sqrt(-a/b)) + 15*A*b**2*x**(5/2)*log(sqrt(x) - sqrt(-a/b))/(6*a**4*x**(3/2)*sqrt(-a/b) + 6*a**3*b*x**(5/2)*s
qrt(-a/b)) - 15*A*b**2*x**(5/2)*log(sqrt(x) + sqrt(-a/b))/(6*a**4*x**(3/2)*sqrt(-a/b) + 6*a**3*b*x**(5/2)*sqrt
(-a/b)) + 30*A*b**2*x**2*sqrt(-a/b)/(6*a**4*x**(3/2)*sqrt(-a/b) + 6*a**3*b*x**(5/2)*sqrt(-a/b)) - 9*B*a**2*x**
(3/2)*log(sqrt(x) - sqrt(-a/b))/(6*a**4*x**(3/2)*sqrt(-a/b) + 6*a**3*b*x**(5/2)*sqrt(-a/b)) + 9*B*a**2*x**(3/2
)*log(sqrt(x) + sqrt(-a/b))/(6*a**4*x**(3/2)*sqrt(-a/b) + 6*a**3*b*x**(5/2)*sqrt(-a/b)) - 12*B*a**2*x*sqrt(-a/
b)/(6*a**4*x**(3/2)*sqrt(-a/b) + 6*a**3*b*x**(5/2)*sqrt(-a/b)) - 9*B*a*b*x**(5/2)*log(sqrt(x) - sqrt(-a/b))/(6
*a**4*x**(3/2)*sqrt(-a/b) + 6*a**3*b*x**(5/2)*sqrt(-a/b)) + 9*B*a*b*x**(5/2)*log(sqrt(x) + sqrt(-a/b))/(6*a**4
*x**(3/2)*sqrt(-a/b) + 6*a**3*b*x**(5/2)*sqrt(-a/b)) - 18*B*a*b*x**2*sqrt(-a/b)/(6*a**4*x**(3/2)*sqrt(-a/b) +
6*a**3*b*x**(5/2)*sqrt(-a/b)), True))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.87 \[ \int \frac {A+B x}{x^{5/2} (a+b x)^2} \, dx=-\frac {2 \, A a^{2} + 3 \, {\left (3 \, B a b - 5 \, A b^{2}\right )} x^{2} + 2 \, {\left (3 \, B a^{2} - 5 \, A a b\right )} x}{3 \, {\left (a^{3} b x^{\frac {5}{2}} + a^{4} x^{\frac {3}{2}}\right )}} - \frac {{\left (3 \, B a b - 5 \, A b^{2}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{3}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.79 \[ \int \frac {A+B x}{x^{5/2} (a+b x)^2} \, dx=-\frac {{\left (3 \, B a b - 5 \, A b^{2}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{3}} - \frac {B a b \sqrt {x} - A b^{2} \sqrt {x}}{{\left (b x + a\right )} a^{3}} - \frac {2 \, {\left (3 \, B a x - 6 \, A b x + A a\right )}}{3 \, a^{3} x^{\frac {3}{2}}} \]

[In]

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

[Out]

-(3*B*a*b - 5*A*b^2)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^3) - (B*a*b*sqrt(x) - A*b^2*sqrt(x))/((b*x + a)*
a^3) - 2/3*(3*B*a*x - 6*A*b*x + A*a)/(a^3*x^(3/2))

Mupad [B] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.76 \[ \int \frac {A+B x}{x^{5/2} (a+b x)^2} \, dx=\frac {\frac {2\,x\,\left (5\,A\,b-3\,B\,a\right )}{3\,a^2}-\frac {2\,A}{3\,a}+\frac {b\,x^2\,\left (5\,A\,b-3\,B\,a\right )}{a^3}}{a\,x^{3/2}+b\,x^{5/2}}+\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (5\,A\,b-3\,B\,a\right )}{a^{7/2}} \]

[In]

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

[Out]

((2*x*(5*A*b - 3*B*a))/(3*a^2) - (2*A)/(3*a) + (b*x^2*(5*A*b - 3*B*a))/a^3)/(a*x^(3/2) + b*x^(5/2)) + (b^(1/2)
*atan((b^(1/2)*x^(1/2))/a^(1/2))*(5*A*b - 3*B*a))/a^(7/2)