Optimal. Leaf size=115 \[ -\frac {1}{7 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 x}{21 d^6 \sqrt {d^2-e^2 x^2}}+\frac {4 x}{21 d^4 \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {659, 192, 191} \begin {gather*} \frac {8 x}{21 d^6 \sqrt {d^2-e^2 x^2}}+\frac {4 x}{21 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 659
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx &=-\frac {1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {5 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{7 d}\\ &=-\frac {1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d^2 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}{7 d^2}\\ &=\frac {4 x}{21 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d^2 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}{21 d^4}\\ &=\frac {4 x}{21 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 x}{21 d^6 \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 93, normalized size = 0.81 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-6 d^5+9 d^4 e x+24 d^3 e^2 x^2+4 d^2 e^3 x^3-16 d e^4 x^4-8 e^5 x^5\right )}{21 d^6 e (d-e x)^2 (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.47, size = 93, normalized size = 0.81 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-6 d^5+9 d^4 e x+24 d^3 e^2 x^2+4 d^2 e^3 x^3-16 d e^4 x^4-8 e^5 x^5\right )}{21 d^6 e (d-e x)^2 (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 203, normalized size = 1.77 \begin {gather*} -\frac {6 \, e^{6} x^{6} + 12 \, d e^{5} x^{5} - 6 \, d^{2} e^{4} x^{4} - 24 \, d^{3} e^{3} x^{3} - 6 \, d^{4} e^{2} x^{2} + 12 \, d^{5} e x + 6 \, d^{6} + {\left (8 \, e^{5} x^{5} + 16 \, d e^{4} x^{4} - 4 \, d^{2} e^{3} x^{3} - 24 \, d^{3} e^{2} x^{2} - 9 \, d^{4} e x + 6 \, d^{5}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{21 \, {\left (d^{6} e^{7} x^{6} + 2 \, d^{7} e^{6} x^{5} - d^{8} e^{5} x^{4} - 4 \, d^{9} e^{4} x^{3} - d^{10} e^{3} x^{2} + 2 \, d^{11} e^{2} x + d^{12} e\right )}} \end {gather*}
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 \begin {gather*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 88, normalized size = 0.77 \begin {gather*} -\frac {\left (-e x +d \right ) \left (8 e^{5} x^{5}+16 e^{4} x^{4} d -4 e^{3} x^{3} d^{2}-24 e^{2} x^{2} d^{3}-9 x \,d^{4} e +6 d^{5}\right )}{21 \left (e x +d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d^{6} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 156, normalized size = 1.36 \begin {gather*} -\frac {1}{7 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d e^{3} x^{2} + 2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} e\right )}} - \frac {1}{7 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} e\right )}} + \frac {4 \, x}{21 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4}} + \frac {8 \, x}{21 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.64, size = 139, normalized size = 1.21 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {11\,x}{42\,d^4}-\frac {5}{28\,d^3\,e}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}-\frac {\sqrt {d^2-e^2\,x^2}}{28\,d^3\,e\,{\left (d+e\,x\right )}^4}-\frac {\sqrt {d^2-e^2\,x^2}}{14\,d^4\,e\,{\left (d+e\,x\right )}^3}+\frac {8\,x\,\sqrt {d^2-e^2\,x^2}}{21\,d^6\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}} \left (d + e x\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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