Optimal. Leaf size=51 \[ -\frac {2 x}{3 b \left (a x+b x^2\right )^{3/2}}+\frac {2 (a+2 b x)}{3 a^2 b \sqrt {a x+b x^2}} \]
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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 = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {666, 627}
\begin {gather*} \frac {2 (a+2 b x)}{3 a^2 b \sqrt {a x+b x^2}}-\frac {2 x}{3 b \left (a x+b x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 627
Rule 666
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a x+b x^2\right )^{5/2}} \, dx &=-\frac {2 x}{3 b \left (a x+b x^2\right )^{3/2}}-\frac {\int \frac {1}{\left (a x+b x^2\right )^{3/2}} \, dx}{3 b}\\ &=-\frac {2 x}{3 b \left (a x+b x^2\right )^{3/2}}+\frac {2 (a+2 b x)}{3 a^2 b \sqrt {a x+b x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 29, normalized size = 0.57 \begin {gather*} \frac {2 x^2 (3 a+2 b x)}{3 a^2 (x (a+b x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(93\) vs.
\(2(43)=86\).
time = 0.52, size = 94, normalized size = 1.84
method | result | size |
trager | \(\frac {2 \left (2 b x +3 a \right ) \sqrt {b \,x^{2}+a x}}{3 a^{2} \left (b x +a \right )^{2}}\) | \(32\) |
gosper | \(\frac {2 x^{3} \left (b x +a \right ) \left (2 b x +3 a \right )}{3 a^{2} \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}\) | \(33\) |
default | \(-\frac {x}{2 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}-\frac {a \left (-\frac {1}{3 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}-\frac {a \left (-\frac {2 \left (2 b x +a \right )}{3 a^{2} \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}+\frac {16 b \left (2 b x +a \right )}{3 a^{4} \sqrt {b \,x^{2}+a x}}\right )}{2 b}\right )}{4 b}\) | \(94\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 54, normalized size = 1.06 \begin {gather*} \frac {4 \, x}{3 \, \sqrt {b x^{2} + a x} a^{2}} - \frac {2 \, x}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} b} + \frac {2}{3 \, \sqrt {b x^{2} + a x} a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.84, size = 44, normalized size = 0.86 \begin {gather*} \frac {2 \, \sqrt {b x^{2} + a x} {\left (2 \, b x + 3 \, a\right )}}{3 \, {\left (a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}\right )}} \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 \frac {x^{2}}{\left (x \left (a + b x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.31, size = 61, normalized size = 1.20 \begin {gather*} \frac {2 \, {\left (3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} b + 2 \, a \sqrt {b}\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} + a\right )}^{3} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.21, size = 31, normalized size = 0.61 \begin {gather*} \frac {2\,\sqrt {b\,x^2+a\,x}\,\left (3\,a+2\,b\,x\right )}{3\,a^2\,{\left (a+b\,x\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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