Optimal. Leaf size=51 \[ \frac {2 a^2 \sqrt {a+b x}}{b^3}+\frac {2 (a+b x)^{5/2}}{5 b^3}-\frac {4 a (a+b x)^{3/2}}{3 b^3} \]
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Rubi [A] time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.00, 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} \[ \frac {2 a^2 \sqrt {a+b x}}{b^3}+\frac {2 (a+b x)^{5/2}}{5 b^3}-\frac {4 a (a+b x)^{3/2}}{3 b^3} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {a+b x}} \, dx &=\int \left (\frac {a^2}{b^2 \sqrt {a+b x}}-\frac {2 a \sqrt {a+b x}}{b^2}+\frac {(a+b x)^{3/2}}{b^2}\right ) \, dx\\ &=\frac {2 a^2 \sqrt {a+b x}}{b^3}-\frac {4 a (a+b x)^{3/2}}{3 b^3}+\frac {2 (a+b x)^{5/2}}{5 b^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 35, normalized size = 0.69 \[ \frac {2 \sqrt {a+b x} \left (8 a^2-4 a b x+3 b^2 x^2\right )}{15 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 31, normalized size = 0.61 \[ \frac {2 \, {\left (3 \, b^{2} x^{2} - 4 \, a b x + 8 \, a^{2}\right )} \sqrt {b x + a}}{15 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.34, size = 37, normalized size = 0.73 \[ \frac {2 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )}}{15 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 32, normalized size = 0.63 \[ \frac {2 \sqrt {b x +a}\, \left (3 b^{2} x^{2}-4 a b x +8 a^{2}\right )}{15 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 41, normalized size = 0.80 \[ \frac {2 \, {\left (b x + a\right )}^{\frac {5}{2}}}{5 \, b^{3}} - \frac {4 \, {\left (b x + a\right )}^{\frac {3}{2}} a}{3 \, b^{3}} + \frac {2 \, \sqrt {b x + a} a^{2}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 37, normalized size = 0.73 \[ \frac {6\,{\left (a+b\,x\right )}^{5/2}-20\,a\,{\left (a+b\,x\right )}^{3/2}+30\,a^2\,\sqrt {a+b\,x}}{15\,b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.77, size = 600, normalized size = 11.76 \[ \frac {16 a^{\frac {21}{2}} \sqrt {1 + \frac {b x}{a}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} - \frac {16 a^{\frac {21}{2}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} + \frac {40 a^{\frac {19}{2}} b x \sqrt {1 + \frac {b x}{a}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} - \frac {48 a^{\frac {19}{2}} b x}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} + \frac {30 a^{\frac {17}{2}} b^{2} x^{2} \sqrt {1 + \frac {b x}{a}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} - \frac {48 a^{\frac {17}{2}} b^{2} x^{2}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} + \frac {10 a^{\frac {15}{2}} b^{3} x^{3} \sqrt {1 + \frac {b x}{a}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} - \frac {16 a^{\frac {15}{2}} b^{3} x^{3}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} + \frac {10 a^{\frac {13}{2}} b^{4} x^{4} \sqrt {1 + \frac {b x}{a}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} + \frac {6 a^{\frac {11}{2}} b^{5} x^{5} \sqrt {1 + \frac {b x}{a}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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