Optimal. Leaf size=53 \[ \frac{2 a^2 (a+b x)^{7/2}}{7 b^3}+\frac{2 (a+b x)^{11/2}}{11 b^3}-\frac{4 a (a+b x)^{9/2}}{9 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0124758, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ \frac{2 a^2 (a+b x)^{7/2}}{7 b^3}+\frac{2 (a+b x)^{11/2}}{11 b^3}-\frac{4 a (a+b x)^{9/2}}{9 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rubi steps
\begin{align*} \int x^2 (a+b x)^{5/2} \, dx &=\int \left (\frac{a^2 (a+b x)^{5/2}}{b^2}-\frac{2 a (a+b x)^{7/2}}{b^2}+\frac{(a+b x)^{9/2}}{b^2}\right ) \, dx\\ &=\frac{2 a^2 (a+b x)^{7/2}}{7 b^3}-\frac{4 a (a+b x)^{9/2}}{9 b^3}+\frac{2 (a+b x)^{11/2}}{11 b^3}\\ \end{align*}
Mathematica [A] time = 0.0264435, size = 35, normalized size = 0.66 \[ \frac{2 (a+b x)^{7/2} \left (8 a^2-28 a b x+63 b^2 x^2\right )}{693 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 32, normalized size = 0.6 \begin{align*}{\frac{126\,{b}^{2}{x}^{2}-56\,abx+16\,{a}^{2}}{693\,{b}^{3}} \left ( bx+a \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.02675, size = 55, normalized size = 1.04 \begin{align*} \frac{2 \,{\left (b x + a\right )}^{\frac{11}{2}}}{11 \, b^{3}} - \frac{4 \,{\left (b x + a\right )}^{\frac{9}{2}} a}{9 \, b^{3}} + \frac{2 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{2}}{7 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.50952, size = 146, normalized size = 2.75 \begin{align*} \frac{2 \,{\left (63 \, b^{5} x^{5} + 161 \, a b^{4} x^{4} + 113 \, a^{2} b^{3} x^{3} + 3 \, a^{3} b^{2} x^{2} - 4 \, a^{4} b x + 8 \, a^{5}\right )} \sqrt{b x + a}}{693 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 4.74117, size = 124, normalized size = 2.34 \begin{align*} \begin{cases} \frac{16 a^{5} \sqrt{a + b x}}{693 b^{3}} - \frac{8 a^{4} x \sqrt{a + b x}}{693 b^{2}} + \frac{2 a^{3} x^{2} \sqrt{a + b x}}{231 b} + \frac{226 a^{2} x^{3} \sqrt{a + b x}}{693} + \frac{46 a b x^{4} \sqrt{a + b x}}{99} + \frac{2 b^{2} x^{5} \sqrt{a + b x}}{11} & \text{for}\: b \neq 0 \\\frac{a^{\frac{5}{2}} x^{3}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.18155, size = 211, normalized size = 3.98 \begin{align*} \frac{2 \,{\left (\frac{33 \,{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2}\right )} a^{2}}{b^{2}} + \frac{22 \,{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3}\right )} a}{b^{2}} + \frac{315 \,{\left (b x + a\right )}^{\frac{11}{2}} - 1540 \,{\left (b x + a\right )}^{\frac{9}{2}} a + 2970 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{2} - 2772 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{3} + 1155 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{4}}{b^{2}}\right )}}{3465 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]