Optimal. Leaf size=63 \[ \frac {a}{4 b^2 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}-\frac {1}{3 b^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {640, 607} \begin {gather*} \frac {a}{4 b^2 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}-\frac {1}{3 b^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 607
Rule 640
Rubi steps
\begin {align*} \int \frac {x}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=-\frac {1}{3 b^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}-\frac {a \int \frac {1}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx}{b}\\ &=-\frac {1}{3 b^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}+\frac {a}{4 b^2 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 0.52 \begin {gather*} \frac {-a-4 b x}{12 b^2 (a+b x)^3 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.68, size = 219, normalized size = 3.48 \begin {gather*} \frac {-2 \left (3 a^5 b-a b^5 x^4-4 b^6 x^5\right )-2 \sqrt {b^2} \sqrt {a^2+2 a b x+b^2 x^2} \left (3 a^4-3 a^3 b x+3 a^2 b^2 x^2-3 a b^3 x^3+4 b^4 x^4\right )}{3 x^4 \sqrt {a^2+2 a b x+b^2 x^2} \left (-8 a^3 b^7-24 a^2 b^8 x-24 a b^9 x^2-8 b^{10} x^3\right )+3 \sqrt {b^2} x^4 \left (8 a^4 b^6+32 a^3 b^7 x+48 a^2 b^8 x^2+32 a b^9 x^3+8 b^{10} x^4\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 54, normalized size = 0.86 \begin {gather*} -\frac {4 \, b x + a}{12 \, {\left (b^{6} x^{4} + 4 \, a b^{5} x^{3} + 6 \, a^{2} b^{4} x^{2} + 4 \, a^{3} b^{3} x + a^{4} b^{2}\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 = 26, normalized size = 0.41 \begin {gather*} -\frac {\left (b x +a \right ) \left (4 b x +a \right )}{12 \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.24, size = 39, normalized size = 0.62 \begin {gather*} -\frac {1}{3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2}} + \frac {a}{4 \, b^{6} {\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 = 0.22, size = 36, normalized size = 0.57 \begin {gather*} -\frac {\left (a+4\,b\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{12\,b^2\,{\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}{\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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