Optimal. Leaf size=92 \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^2 (a+b x) (d+e x)^5}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^2 (a+b x) (d+e x)^4} \]
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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} \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^2 (a+b x) (d+e x)^5}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^2 (a+b x) (d+e x)^4} \]
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)^6} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {a b+b^2 x}{(d+e x)^6} \, 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)^6}+\frac {b^2}{e (d+e x)^5}\right ) \, dx}{a b+b^2 x}\\ &=\frac {(b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^2 (a+b x) (d+e x)^5}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^2 (a+b x) (d+e x)^4}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 45, normalized size = 0.49 \[ -\frac {\sqrt {(a+b x)^2} (4 a e+b (d+5 e x))}{20 e^2 (a+b x) (d+e x)^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 72, normalized size = 0.78 \[ -\frac {5 \, b e x + b d + 4 \, a e}{20 \, {\left (e^{7} x^{5} + 5 \, d e^{6} x^{4} + 10 \, d^{2} e^{5} x^{3} + 10 \, d^{3} e^{4} x^{2} + 5 \, d^{4} e^{3} x + d^{5} e^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 45, normalized size = 0.49 \[ -\frac {{\left (5 \, b x e \mathrm {sgn}\left (b x + a\right ) + b d \mathrm {sgn}\left (b x + a\right ) + 4 \, a e \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-2\right )}}{20 \, {\left (x e + d\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 42, normalized size = 0.46 \[ -\frac {\left (5 b e x +4 a e +b d \right ) \sqrt {\left (b x +a \right )^{2}}}{20 \left (e x +d \right )^{5} \left (b x +a \right ) e^{2}} \]
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 = 0.60, size = 41, normalized size = 0.45 \[ -\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (4\,a\,e+b\,d+5\,b\,e\,x\right )}{20\,e^2\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.56, size = 76, normalized size = 0.83 \[ \frac {- 4 a e - b d - 5 b e x}{20 d^{5} e^{2} + 100 d^{4} e^{3} x + 200 d^{3} e^{4} x^{2} + 200 d^{2} e^{5} x^{3} + 100 d e^{6} x^{4} + 20 e^{7} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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