Optimal. Leaf size=49 \[ -\frac {a^2}{2 x \sqrt {c x^2}}-\frac {2 a b}{\sqrt {c x^2}}+\frac {b^2 x \log (x)}{\sqrt {c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 43} \[ -\frac {a^2}{2 x \sqrt {c x^2}}-\frac {2 a b}{\sqrt {c x^2}}+\frac {b^2 x \log (x)}{\sqrt {c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {(a+b x)^2}{x^2 \sqrt {c x^2}} \, dx &=\frac {x \int \frac {(a+b x)^2}{x^3} \, dx}{\sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {a^2}{x^3}+\frac {2 a b}{x^2}+\frac {b^2}{x}\right ) \, dx}{\sqrt {c x^2}}\\ &=-\frac {2 a b}{\sqrt {c x^2}}-\frac {a^2}{2 x \sqrt {c x^2}}+\frac {b^2 x \log (x)}{\sqrt {c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 35, normalized size = 0.71 \[ \frac {c x \left (2 b^2 x^2 \log (x)-a (a+4 b x)\right )}{2 \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 36, normalized size = 0.73 \[ \frac {{\left (2 \, b^{2} x^{2} \log \relax (x) - 4 \, a b x - a^{2}\right )} \sqrt {c x^{2}}}{2 \, c x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 34, normalized size = 0.69 \[ \frac {2 b^{2} x^{2} \ln \relax (x )-4 a b x -a^{2}}{2 \sqrt {c \,x^{2}}\, x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.29, size = 31, normalized size = 0.63 \[ \frac {b^{2} \log \relax (x)}{\sqrt {c}} - \frac {2 \, a b}{\sqrt {c} x} - \frac {a^{2}}{2 \, \sqrt {c} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a+b\,x\right )}^2}{x^2\,\sqrt {c\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b x\right )^{2}}{x^{2} \sqrt {c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________