Optimal. Leaf size=82 \[ -\frac {1}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 x}{15 d^5 \sqrt {d^2-e^2 x^2}}+\frac {4 x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {659, 192, 191} \[ \frac {8 x}{15 d^5 \sqrt {d^2-e^2 x^2}}+\frac {4 x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 659
Rubi steps
\begin {align*} \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx &=-\frac {1}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {4 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d}\\ &=\frac {4 x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^3}\\ &=\frac {4 x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 x}{15 d^5 \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 82, normalized size = 1.00 \[ -\frac {\sqrt {d^2-e^2 x^2} \left (3 d^4-12 d^3 e x-12 d^2 e^2 x^2+8 d e^3 x^3+8 e^4 x^4\right )}{15 d^5 e (d-e x)^2 (d+e x)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.92, size = 168, normalized size = 2.05 \[ -\frac {3 \, e^{5} x^{5} + 3 \, d e^{4} x^{4} - 6 \, d^{2} e^{3} x^{3} - 6 \, d^{3} e^{2} x^{2} + 3 \, d^{4} e x + 3 \, d^{5} + {\left (8 \, e^{4} x^{4} + 8 \, d e^{3} x^{3} - 12 \, d^{2} e^{2} x^{2} - 12 \, d^{3} e x + 3 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{5} e^{6} x^{5} + d^{6} e^{5} x^{4} - 2 \, d^{7} e^{4} x^{3} - 2 \, d^{8} e^{3} x^{2} + d^{9} e^{2} x + d^{10} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 70, normalized size = 0.85 \[ -\frac {\left (-e x +d \right ) \left (8 x^{4} e^{4}+8 x^{3} d \,e^{3}-12 d^{2} x^{2} e^{2}-12 d^{3} x e +3 d^{4}\right )}{15 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d^{5} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 85, normalized size = 1.04 \[ -\frac {1}{5 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} e\right )}} + \frac {4 \, x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3}} + \frac {8 \, x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.76, size = 78, normalized size = 0.95 \[ -\frac {\sqrt {d^2-e^2\,x^2}\,\left (3\,d^4-12\,d^3\,e\,x-12\,d^2\,e^2\,x^2+8\,d\,e^3\,x^3+8\,e^4\,x^4\right )}{15\,d^5\,e\,{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^2} \]
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 \[ \int \frac {1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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