Integrand size = 18, antiderivative size = 100 \[ \int \frac {A+B x}{\sqrt {x} (a+b x)^3} \, dx=\frac {(A b-a B) \sqrt {x}}{2 a b (a+b x)^2}+\frac {(3 A b+a B) \sqrt {x}}{4 a^2 b (a+b x)}+\frac {(3 A b+a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{5/2} b^{3/2}} \]
1/4*(3*A*b+B*a)*arctan(b^(1/2)*x^(1/2)/a^(1/2))/a^(5/2)/b^(3/2)+1/2*(A*b-B *a)*x^(1/2)/a/b/(b*x+a)^2+1/4*(3*A*b+B*a)*x^(1/2)/a^2/b/(b*x+a)
Time = 0.13 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.86 \[ \int \frac {A+B x}{\sqrt {x} (a+b x)^3} \, dx=-\frac {\sqrt {x} \left (-5 a A b+a^2 B-3 A b^2 x-a b B x\right )}{4 a^2 b (a+b x)^2}+\frac {(3 A b+a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{5/2} b^{3/2}} \]
-1/4*(Sqrt[x]*(-5*a*A*b + a^2*B - 3*A*b^2*x - a*b*B*x))/(a^2*b*(a + b*x)^2 ) + ((3*A*b + a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^(5/2)*b^(3/2))
Time = 0.19 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {87, 52, 73, 218}
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.
\(\displaystyle \int \frac {A+B x}{\sqrt {x} (a+b x)^3} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {(a B+3 A b) \int \frac {1}{\sqrt {x} (a+b x)^2}dx}{4 a b}+\frac {\sqrt {x} (A b-a B)}{2 a b (a+b x)^2}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {(a B+3 A b) \left (\frac {\int \frac {1}{\sqrt {x} (a+b x)}dx}{2 a}+\frac {\sqrt {x}}{a (a+b x)}\right )}{4 a b}+\frac {\sqrt {x} (A b-a B)}{2 a b (a+b x)^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {(a B+3 A b) \left (\frac {\int \frac {1}{a+b x}d\sqrt {x}}{a}+\frac {\sqrt {x}}{a (a+b x)}\right )}{4 a b}+\frac {\sqrt {x} (A b-a B)}{2 a b (a+b x)^2}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {(a B+3 A b) \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}}+\frac {\sqrt {x}}{a (a+b x)}\right )}{4 a b}+\frac {\sqrt {x} (A b-a B)}{2 a b (a+b x)^2}\) |
((A*b - a*B)*Sqrt[x])/(2*a*b*(a + b*x)^2) + ((3*A*b + a*B)*(Sqrt[x]/(a*(a + b*x)) + ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]]/(a^(3/2)*Sqrt[b])))/(4*a*b)
3.4.67.3.1 Defintions of rubi rules used
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] - Simp[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] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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]))))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Time = 1.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(\frac {\frac {\left (3 A b +B a \right ) x^{\frac {3}{2}}}{4 a^{2}}+\frac {\left (5 A b -B a \right ) \sqrt {x}}{4 a b}}{\left (b x +a \right )^{2}}+\frac {\left (3 A b +B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 a^{2} b \sqrt {a b}}\) | \(80\) |
default | \(\frac {\frac {\left (3 A b +B a \right ) x^{\frac {3}{2}}}{4 a^{2}}+\frac {\left (5 A b -B a \right ) \sqrt {x}}{4 a b}}{\left (b x +a \right )^{2}}+\frac {\left (3 A b +B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 a^{2} b \sqrt {a b}}\) | \(80\) |
2*(1/8*(3*A*b+B*a)/a^2*x^(3/2)+1/8*(5*A*b-B*a)/a/b*x^(1/2))/(b*x+a)^2+1/4* (3*A*b+B*a)/a^2/b/(a*b)^(1/2)*arctan(b*x^(1/2)/(a*b)^(1/2))
Time = 0.24 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.91 \[ \int \frac {A+B x}{\sqrt {x} (a+b x)^3} \, dx=\left [-\frac {{\left (B a^{3} + 3 \, A a^{2} b + {\left (B a b^{2} + 3 \, A b^{3}\right )} x^{2} + 2 \, {\left (B a^{2} b + 3 \, A a b^{2}\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a - 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) + 2 \, {\left (B a^{3} b - 5 \, A a^{2} b^{2} - {\left (B a^{2} b^{2} + 3 \, A a b^{3}\right )} x\right )} \sqrt {x}}{8 \, {\left (a^{3} b^{4} x^{2} + 2 \, a^{4} b^{3} x + a^{5} b^{2}\right )}}, -\frac {{\left (B a^{3} + 3 \, A a^{2} b + {\left (B a b^{2} + 3 \, A b^{3}\right )} x^{2} + 2 \, {\left (B a^{2} b + 3 \, A a b^{2}\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (B a^{3} b - 5 \, A a^{2} b^{2} - {\left (B a^{2} b^{2} + 3 \, A a b^{3}\right )} x\right )} \sqrt {x}}{4 \, {\left (a^{3} b^{4} x^{2} + 2 \, a^{4} b^{3} x + a^{5} b^{2}\right )}}\right ] \]
[-1/8*((B*a^3 + 3*A*a^2*b + (B*a*b^2 + 3*A*b^3)*x^2 + 2*(B*a^2*b + 3*A*a*b ^2)*x)*sqrt(-a*b)*log((b*x - a - 2*sqrt(-a*b)*sqrt(x))/(b*x + a)) + 2*(B*a ^3*b - 5*A*a^2*b^2 - (B*a^2*b^2 + 3*A*a*b^3)*x)*sqrt(x))/(a^3*b^4*x^2 + 2* a^4*b^3*x + a^5*b^2), -1/4*((B*a^3 + 3*A*a^2*b + (B*a*b^2 + 3*A*b^3)*x^2 + 2*(B*a^2*b + 3*A*a*b^2)*x)*sqrt(a*b)*arctan(sqrt(a*b)/(b*sqrt(x))) + (B*a ^3*b - 5*A*a^2*b^2 - (B*a^2*b^2 + 3*A*a*b^3)*x)*sqrt(x))/(a^3*b^4*x^2 + 2* a^4*b^3*x + a^5*b^2)]
Leaf count of result is larger than twice the leaf count of optimal. 1345 vs. \(2 (90) = 180\).
Time = 7.73 (sec) , antiderivative size = 1345, normalized size of antiderivative = 13.45 \[ \int \frac {A+B x}{\sqrt {x} (a+b x)^3} \, dx=\text {Too large to display} \]
Piecewise((zoo*(-2*A/(5*x**(5/2)) - 2*B/(3*x**(3/2))), Eq(a, 0) & Eq(b, 0) ), ((2*A*sqrt(x) + 2*B*x**(3/2)/3)/a**3, Eq(b, 0)), ((-2*A/(5*x**(5/2)) - 2*B/(3*x**(3/2)))/b**3, Eq(a, 0)), (3*A*a**2*b*log(sqrt(x) - sqrt(-a/b))/( 8*a**4*b**2*sqrt(-a/b) + 16*a**3*b**3*x*sqrt(-a/b) + 8*a**2*b**4*x**2*sqrt (-a/b)) - 3*A*a**2*b*log(sqrt(x) + sqrt(-a/b))/(8*a**4*b**2*sqrt(-a/b) + 1 6*a**3*b**3*x*sqrt(-a/b) + 8*a**2*b**4*x**2*sqrt(-a/b)) + 10*A*a*b**2*sqrt (x)*sqrt(-a/b)/(8*a**4*b**2*sqrt(-a/b) + 16*a**3*b**3*x*sqrt(-a/b) + 8*a** 2*b**4*x**2*sqrt(-a/b)) + 6*A*a*b**2*x*log(sqrt(x) - sqrt(-a/b))/(8*a**4*b **2*sqrt(-a/b) + 16*a**3*b**3*x*sqrt(-a/b) + 8*a**2*b**4*x**2*sqrt(-a/b)) - 6*A*a*b**2*x*log(sqrt(x) + sqrt(-a/b))/(8*a**4*b**2*sqrt(-a/b) + 16*a**3 *b**3*x*sqrt(-a/b) + 8*a**2*b**4*x**2*sqrt(-a/b)) + 6*A*b**3*x**(3/2)*sqrt (-a/b)/(8*a**4*b**2*sqrt(-a/b) + 16*a**3*b**3*x*sqrt(-a/b) + 8*a**2*b**4*x **2*sqrt(-a/b)) + 3*A*b**3*x**2*log(sqrt(x) - sqrt(-a/b))/(8*a**4*b**2*sqr t(-a/b) + 16*a**3*b**3*x*sqrt(-a/b) + 8*a**2*b**4*x**2*sqrt(-a/b)) - 3*A*b **3*x**2*log(sqrt(x) + sqrt(-a/b))/(8*a**4*b**2*sqrt(-a/b) + 16*a**3*b**3* x*sqrt(-a/b) + 8*a**2*b**4*x**2*sqrt(-a/b)) + B*a**3*log(sqrt(x) - sqrt(-a /b))/(8*a**4*b**2*sqrt(-a/b) + 16*a**3*b**3*x*sqrt(-a/b) + 8*a**2*b**4*x** 2*sqrt(-a/b)) - B*a**3*log(sqrt(x) + sqrt(-a/b))/(8*a**4*b**2*sqrt(-a/b) + 16*a**3*b**3*x*sqrt(-a/b) + 8*a**2*b**4*x**2*sqrt(-a/b)) - 2*B*a**2*b*sqr t(x)*sqrt(-a/b)/(8*a**4*b**2*sqrt(-a/b) + 16*a**3*b**3*x*sqrt(-a/b) + 8...
Time = 0.29 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x}{\sqrt {x} (a+b x)^3} \, dx=\frac {{\left (B a b + 3 \, A b^{2}\right )} x^{\frac {3}{2}} - {\left (B a^{2} - 5 \, A a b\right )} \sqrt {x}}{4 \, {\left (a^{2} b^{3} x^{2} + 2 \, a^{3} b^{2} x + a^{4} b\right )}} + \frac {{\left (B a + 3 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{2} b} \]
1/4*((B*a*b + 3*A*b^2)*x^(3/2) - (B*a^2 - 5*A*a*b)*sqrt(x))/(a^2*b^3*x^2 + 2*a^3*b^2*x + a^4*b) + 1/4*(B*a + 3*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqr t(a*b)*a^2*b)
Time = 0.31 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.82 \[ \int \frac {A+B x}{\sqrt {x} (a+b x)^3} \, dx=\frac {{\left (B a + 3 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{2} b} + \frac {B a b x^{\frac {3}{2}} + 3 \, A b^{2} x^{\frac {3}{2}} - B a^{2} \sqrt {x} + 5 \, A a b \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} a^{2} b} \]
1/4*(B*a + 3*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^2*b) + 1/4*(B*a *b*x^(3/2) + 3*A*b^2*x^(3/2) - B*a^2*sqrt(x) + 5*A*a*b*sqrt(x))/((b*x + a) ^2*a^2*b)
Time = 0.13 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.84 \[ \int \frac {A+B x}{\sqrt {x} (a+b x)^3} \, dx=\frac {\frac {x^{3/2}\,\left (3\,A\,b+B\,a\right )}{4\,a^2}+\frac {\sqrt {x}\,\left (5\,A\,b-B\,a\right )}{4\,a\,b}}{a^2+2\,a\,b\,x+b^2\,x^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (3\,A\,b+B\,a\right )}{4\,a^{5/2}\,b^{3/2}} \]