Optimal. Leaf size=107 \[ -\frac {a^2}{4 b^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 a}{3 b^3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{2 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.05, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {646, 43} \begin {gather*} -\frac {a^2}{4 b^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 a}{3 b^3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{2 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {x^2}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \left (\frac {a^2}{b^7 (a+b x)^5}-\frac {2 a}{b^7 (a+b x)^4}+\frac {1}{b^7 (a+b x)^3}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {a^2}{4 b^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 a}{3 b^3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{2 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 44, normalized size = 0.41 \begin {gather*} \frac {-a^2-4 a b x-6 b^2 x^2}{12 b^3 (a+b x)^3 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.66, size = 248, normalized size = 2.32 \begin {gather*} \frac {2 \sqrt {b^2} \left (3 a^6+a^2 b^4 x^4+4 a b^5 x^5+6 b^6 x^6\right )+2 \sqrt {a^2+2 a b x+b^2 x^2} \left (3 a^5 b-3 a^4 b^2 x+3 a^3 b^3 x^2-3 a^2 b^4 x^3+2 a b^5 x^4-6 b^6 x^5\right )}{3 b^4 \sqrt {b^2} x^4 \sqrt {a^2+2 a b x+b^2 x^2} \left (-8 a^3 b^3-24 a^2 b^4 x-24 a b^5 x^2-8 b^6 x^3\right )+3 b^4 x^4 \left (8 a^4 b^4+32 a^3 b^5 x+48 a^2 b^6 x^2+32 a b^7 x^3+8 b^8 x^4\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 65, normalized size = 0.61 \begin {gather*} -\frac {6 \, b^{2} x^{2} + 4 \, a b x + a^{2}}{12 \, {\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 37, normalized size = 0.35 \begin {gather*} -\frac {\left (b x +a \right ) \left (6 b^{2} x^{2}+4 a b x +a^{2}\right )}{12 \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 47, normalized size = 0.44 \begin {gather*} -\frac {1}{2 \, b^{5} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {2 \, a}{3 \, b^{6} {\left (x + \frac {a}{b}\right )}^{3}} - \frac {a^{2}}{4 \, b^{7} {\left (x + \frac {a}{b}\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.24, size = 47, normalized size = 0.44 \begin {gather*} -\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (a^2+4\,a\,b\,x+6\,b^2\,x^2\right )}{12\,b^3\,{\left (a+b\,x\right )}^5} \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 (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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