Optimal. Leaf size=71 \[ -\frac {b d-a e}{4 b^2 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}-\frac {e}{3 b^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {640, 607} \[ -\frac {b d-a e}{4 b^2 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}-\frac {e}{3 b^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 607
Rule 640
Rubi steps
\begin {align*} \int \frac {d+e x}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=-\frac {e}{3 b^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}+\frac {\left (2 b^2 d-2 a b e\right ) \int \frac {1}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx}{2 b^2}\\ &=-\frac {e}{3 b^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}-\frac {b d-a e}{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 = 0.02, size = 39, normalized size = 0.55 \[ \frac {-a e-3 b d-4 b e x}{12 b^2 (a+b x)^3 \sqrt {(a+b x)^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 61, normalized size = 0.86 \[ -\frac {4 \, b e x + 3 \, b d + a e}{12 \, {\left (b^{6} x^{4} + 4 \, a b^{5} x^{3} + 6 \, a^{2} b^{4} x^{2} + 4 \, a^{3} b^{3} x + a^{4} b^{2}\right )}} \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 33, normalized size = 0.46 \[ -\frac {\left (b x +a \right ) \left (4 b e x +a e +3 b d \right )}{12 \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 56, normalized size = 0.79 \[ -\frac {e}{3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2}} - \frac {d}{4 \, b^{5} {\left (x + \frac {a}{b}\right )}^{4}} + \frac {a e}{4 \, b^{6} {\left (x + \frac {a}{b}\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.65, size = 43, normalized size = 0.61 \[ -\frac {\left (a\,e+3\,b\,d+4\,b\,e\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{12\,b^2\,{\left (a+b\,x\right )}^5} \]
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 \[ \int \frac {d + e x}{\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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