Optimal. Leaf size=32 \[ -\frac{a^2}{3 x^3}-\frac{4 a b}{5 x^{5/2}}-\frac{b^2}{2 x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0136014, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac{a^2}{3 x^3}-\frac{4 a b}{5 x^{5/2}}-\frac{b^2}{2 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b \sqrt{x}\right )^2}{x^4} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^2}{x^7} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{a^2}{x^7}+\frac{2 a b}{x^6}+\frac{b^2}{x^5}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a^2}{3 x^3}-\frac{4 a b}{5 x^{5/2}}-\frac{b^2}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0108061, size = 28, normalized size = 0.88 \[ -\frac{10 a^2+24 a b \sqrt{x}+15 b^2 x}{30 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.001, size = 25, normalized size = 0.8 \begin{align*} -{\frac{{a}^{2}}{3\,{x}^{3}}}-{\frac{4\,ab}{5}{x}^{-{\frac{5}{2}}}}-{\frac{{b}^{2}}{2\,{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.981268, size = 32, normalized size = 1. \begin{align*} -\frac{15 \, b^{2} x + 24 \, a b \sqrt{x} + 10 \, a^{2}}{30 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.40874, size = 65, normalized size = 2.03 \begin{align*} -\frac{15 \, b^{2} x + 24 \, a b \sqrt{x} + 10 \, a^{2}}{30 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.922679, size = 29, normalized size = 0.91 \begin{align*} - \frac{a^{2}}{3 x^{3}} - \frac{4 a b}{5 x^{\frac{5}{2}}} - \frac{b^{2}}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.13645, size = 32, normalized size = 1. \begin{align*} -\frac{15 \, b^{2} x + 24 \, a b \sqrt{x} + 10 \, a^{2}}{30 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]