Optimal. Leaf size=96 \[ \frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^2 (a+b x) (d+e x)^{5/2}}-\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^2 (a+b x) (d+e x)^{3/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {646, 43} \begin {gather*} \frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^2 (a+b x) (d+e x)^{5/2}}-\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^2 (a+b x) (d+e x)^{3/2}} \end {gather*}
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)^{7/2}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {a b+b^2 x}{(d+e x)^{7/2}} \, 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)^{7/2}}+\frac {b^2}{e (d+e x)^{5/2}}\right ) \, dx}{a b+b^2 x}\\ &=\frac {2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^2 (a+b x) (d+e x)^{5/2}}-\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^2 (a+b x) (d+e x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 48, normalized size = 0.50 \begin {gather*} -\frac {2 \sqrt {(a+b x)^2} (3 a e+2 b d+5 b e x)}{15 e^2 (a+b x) (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 25.48, size = 61, normalized size = 0.64 \begin {gather*} -\frac {2 \sqrt {\frac {(a e+b e x)^2}{e^2}} (3 a e+5 b (d+e x)-3 b d)}{15 e (d+e x)^{5/2} (a e+b e x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 58, normalized size = 0.60 \begin {gather*} -\frac {2 \, {\left (5 \, b e x + 2 \, b d + 3 \, a e\right )} \sqrt {e x + d}}{15 \, {\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 49, normalized size = 0.51 \begin {gather*} -\frac {2 \, {\left (5 \, {\left (x e + d\right )} b \mathrm {sgn}\left (b x + a\right ) - 3 \, b d \mathrm {sgn}\left (b x + a\right ) + 3 \, a e \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-2\right )}}{15 \, {\left (x e + d\right )}^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 43, normalized size = 0.45 \begin {gather*} -\frac {2 \left (5 b e x +3 a e +2 b d \right ) \sqrt {\left (b x +a \right )^{2}}}{15 \left (e x +d \right )^{\frac {5}{2}} \left (b x +a \right ) e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, size = 47, normalized size = 0.49 \begin {gather*} -\frac {2 \, {\left (5 \, b e x + 2 \, b d + 3 \, a e\right )}}{15 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )} \sqrt {e x + d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.94, size = 120, normalized size = 1.25 \begin {gather*} -\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (\frac {2\,x}{3\,e^3}+\frac {\frac {2\,a\,e}{5}+\frac {4\,b\,d}{15}}{b\,e^4}\right )}{x^3\,\sqrt {d+e\,x}+\frac {a\,d^2\,\sqrt {d+e\,x}}{b\,e^2}+\frac {x^2\,\left (a\,e^4+2\,b\,d\,e^3\right )\,\sqrt {d+e\,x}}{b\,e^4}+\frac {d\,x\,\left (2\,a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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