Optimal. Leaf size=96 \[ \frac {a^2 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 b^3}-\frac {2 a \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b^3}+\frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{6 b^3} \]
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Rubi [A]
time = 0.02, antiderivative size = 107, normalized size of antiderivative = 1.11, number of steps
used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {659}
\begin {gather*} \frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^3}-\frac {2 a \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^4}{5 b^3}+\frac {a^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^3}{4 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 659
Rubi steps
\begin {align*} \int x^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a^2 \left (a b+b^2 x\right )^3}{b^2}-\frac {2 a \left (a b+b^2 x\right )^4}{b^3}+\frac {\left (a b+b^2 x\right )^5}{b^4}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {a^2 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 b^3}-\frac {2 a (a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{5 b^3}+\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 55, normalized size = 0.57 \begin {gather*} \frac {x^3 \sqrt {(a+b x)^2} \left (20 a^3+45 a^2 b x+36 a b^2 x^2+10 b^3 x^3\right )}{60 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.49, size = 52, normalized size = 0.54
method | result | size |
gosper | \(\frac {x^{3} \left (10 b^{3} x^{3}+36 a \,b^{2} x^{2}+45 a^{2} b x +20 a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{60 \left (b x +a \right )^{3}}\) | \(52\) |
default | \(\frac {x^{3} \left (10 b^{3} x^{3}+36 a \,b^{2} x^{2}+45 a^{2} b x +20 a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{60 \left (b x +a \right )^{3}}\) | \(52\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{3} x^{6}}{6 b x +6 a}+\frac {3 \sqrt {\left (b x +a \right )^{2}}\, a \,b^{2} x^{5}}{5 \left (b x +a \right )}+\frac {3 \sqrt {\left (b x +a \right )^{2}}\, a^{2} b \,x^{4}}{4 \left (b x +a \right )}+\frac {\sqrt {\left (b x +a \right )^{2}}\, a^{3} x^{3}}{3 b x +3 a}\) | \(100\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 102, normalized size = 1.06 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} x}{4 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3}}{4 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} x}{6 \, b^{2}} - \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a}{30 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.44, size = 35, normalized size = 0.36 \begin {gather*} \frac {1}{6} \, b^{3} x^{6} + \frac {3}{5} \, a b^{2} x^{5} + \frac {3}{4} \, a^{2} b x^{4} + \frac {1}{3} \, a^{3} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.70, size = 73, normalized size = 0.76 \begin {gather*} \frac {1}{6} \, b^{3} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{5} \, a b^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{4} \, a^{2} b x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, a^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {a^{6} \mathrm {sgn}\left (b x + a\right )}{60 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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