Optimal. Leaf size=66 \[ -\frac{a^2}{8 c^2 x^7 \sqrt{c x^2}}-\frac{2 a b}{7 c^2 x^6 \sqrt{c x^2}}-\frac{b^2}{6 c^2 x^5 \sqrt{c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0134873, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 43} \[ -\frac{a^2}{8 c^2 x^7 \sqrt{c x^2}}-\frac{2 a b}{7 c^2 x^6 \sqrt{c x^2}}-\frac{b^2}{6 c^2 x^5 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^2}{x^4 \left (c x^2\right )^{5/2}} \, dx &=\frac{x \int \frac{(a+b x)^2}{x^9} \, dx}{c^2 \sqrt{c x^2}}\\ &=\frac{x \int \left (\frac{a^2}{x^9}+\frac{2 a b}{x^8}+\frac{b^2}{x^7}\right ) \, dx}{c^2 \sqrt{c x^2}}\\ &=-\frac{a^2}{8 c^2 x^7 \sqrt{c x^2}}-\frac{2 a b}{7 c^2 x^6 \sqrt{c x^2}}-\frac{b^2}{6 c^2 x^5 \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0093735, size = 35, normalized size = 0.53 \[ \frac{-21 a^2-48 a b x-28 b^2 x^2}{168 x^3 \left (c x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 32, normalized size = 0.5 \begin{align*} -{\frac{28\,{b}^{2}{x}^{2}+48\,abx+21\,{a}^{2}}{168\,{x}^{3}} \left ( c{x}^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.01, size = 45, normalized size = 0.68 \begin{align*} -\frac{b^{2}}{6 \, c^{\frac{5}{2}} x^{6}} - \frac{2 \, a b}{7 \, c^{\frac{5}{2}} x^{7}} - \frac{a^{2}}{8 \, c^{\frac{5}{2}} x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.5617, size = 85, normalized size = 1.29 \begin{align*} -\frac{{\left (28 \, b^{2} x^{2} + 48 \, a b x + 21 \, a^{2}\right )} \sqrt{c x^{2}}}{168 \, c^{3} x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.83628, size = 61, normalized size = 0.92 \begin{align*} - \frac{a^{2}}{8 c^{\frac{5}{2}} x^{3} \left (x^{2}\right )^{\frac{5}{2}}} - \frac{2 a b}{7 c^{\frac{5}{2}} x^{2} \left (x^{2}\right )^{\frac{5}{2}}} - \frac{b^{2}}{6 c^{\frac{5}{2}} x \left (x^{2}\right )^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]