Optimal. Leaf size=92 \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{4 e^2 (a+b x) (d+e x)^4}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^2 (a+b x) (d+e x)^3} \]
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Rubi [A] time = 0.04, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {646, 43} \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{4 e^2 (a+b x) (d+e x)^4}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^2 (a+b x) (d+e x)^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rubi steps
\begin {align*} \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^5} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {a b+b^2 x}{(d+e x)^5} \, dx}{a b+b^2 x}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b (b d-a e)}{e (d+e x)^5}+\frac {b^2}{e (d+e x)^4}\right ) \, dx}{a b+b^2 x}\\ &=\frac {(b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^2 (a+b x) (d+e x)^4}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^2 (a+b x) (d+e x)^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 45, normalized size = 0.49 \[ -\frac {\sqrt {(a+b x)^2} (3 a e+b (d+4 e x))}{12 e^2 (a+b x) (d+e x)^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 61, normalized size = 0.66 \[ -\frac {4 \, b e x + b d + 3 \, a e}{12 \, {\left (e^{6} x^{4} + 4 \, d e^{5} x^{3} + 6 \, d^{2} e^{4} x^{2} + 4 \, d^{3} e^{3} x + d^{4} e^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 45, normalized size = 0.49 \[ -\frac {{\left (4 \, b x e \mathrm {sgn}\left (b x + a\right ) + b d \mathrm {sgn}\left (b x + a\right ) + 3 \, a e \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-2\right )}}{12 \, {\left (x e + d\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 42, normalized size = 0.46 \[ -\frac {\left (4 b e x +3 a e +b d \right ) \sqrt {\left (b x +a \right )^{2}}}{12 \left (e x +d \right )^{4} \left (b x +a \right ) e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.59, size = 41, normalized size = 0.45 \[ -\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (3\,a\,e+b\,d+4\,b\,e\,x\right )}{12\,e^2\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.46, size = 65, normalized size = 0.71 \[ \frac {- 3 a e - b d - 4 b e x}{12 d^{4} e^{2} + 48 d^{3} e^{3} x + 72 d^{2} e^{4} x^{2} + 48 d e^{5} x^{3} + 12 e^{6} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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