Optimal. Leaf size=71 \[ -\frac {a \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {646, 43} \[ -\frac {a \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 646
Rubi steps
\begin {align*} \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{x^5} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {a b+b^2 x}{x^5} \, dx}{a b+b^2 x}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a b}{x^5}+\frac {b^2}{x^4}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {a \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 33, normalized size = 0.46 \[ -\frac {\sqrt {(a+b x)^2} (3 a+4 b x)}{12 x^4 (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.58, size = 13, normalized size = 0.18 \[ -\frac {4 \, b x + 3 \, a}{12 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.19, size = 40, normalized size = 0.56 \[ -\frac {b^{4} \mathrm {sgn}\left (b x + a\right )}{12 \, a^{3}} - \frac {4 \, b x \mathrm {sgn}\left (b x + a\right ) + 3 \, a \mathrm {sgn}\left (b x + a\right )}{12 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 30, normalized size = 0.42 \[ -\frac {\left (4 b x +3 a \right ) \sqrt {\left (b x +a \right )^{2}}}{12 \left (b x +a \right ) x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.38, size = 138, normalized size = 1.94 \[ \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{4}}{2 \, a^{4}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{3}}{2 \, a^{3} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2}}{2 \, a^{4} x^{2}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b}{12 \, a^{3} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}}}{4 \, a^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.16, size = 29, normalized size = 0.41 \[ -\frac {\left (3\,a+4\,b\,x\right )\,\sqrt {{\left (a+b\,x\right )}^2}}{12\,x^4\,\left (a+b\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.15, size = 14, normalized size = 0.20 \[ \frac {- 3 a - 4 b x}{12 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________