Optimal. Leaf size=123 \[ -\frac {x^2}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (d+2 e x)}{35 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 x}{105 d^5 e^2 \sqrt {d^2-e^2 x^2}}-\frac {4 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {855, 778, 192, 191} \[ -\frac {x^2}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 x}{105 d^5 e^2 \sqrt {d^2-e^2 x^2}}-\frac {4 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 (d+2 e x)}{35 d e^3 \left (d^2-e^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 778
Rule 855
Rubi steps
\begin {align*} \int \frac {x^2}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=-\frac {x^2}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {\int \frac {x (2 d+4 e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{7 d e}\\ &=-\frac {x^2}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (d+2 e x)}{35 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{35 d e^2}\\ &=-\frac {x^2}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (d+2 e x)}{35 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 x}{105 d^3 e^2 \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}{105 d^3 e^2}\\ &=-\frac {x^2}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (d+2 e x)}{35 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {8 x}{105 d^5 e^2 \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 104, normalized size = 0.85 \[ \frac {\sqrt {d^2-e^2 x^2} \left (6 d^6+6 d^5 e x-15 d^4 e^2 x^2+20 d^3 e^3 x^3+20 d^2 e^4 x^4-8 d e^5 x^5-8 e^6 x^6\right )}{105 d^5 e^3 (d-e x)^3 (d+e x)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.01, size = 238, normalized size = 1.93 \[ \frac {6 \, e^{7} x^{7} + 6 \, d e^{6} x^{6} - 18 \, d^{2} e^{5} x^{5} - 18 \, d^{3} e^{4} x^{4} + 18 \, d^{4} e^{3} x^{3} + 18 \, d^{5} e^{2} x^{2} - 6 \, d^{6} e x - 6 \, d^{7} + {\left (8 \, e^{6} x^{6} + 8 \, d e^{5} x^{5} - 20 \, d^{2} e^{4} x^{4} - 20 \, d^{3} e^{3} x^{3} + 15 \, d^{4} e^{2} x^{2} - 6 \, d^{5} e x - 6 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{105 \, {\left (d^{5} e^{10} x^{7} + d^{6} e^{9} x^{6} - 3 \, d^{7} e^{8} x^{5} - 3 \, d^{8} e^{7} x^{4} + 3 \, d^{9} e^{6} x^{3} + 3 \, d^{10} e^{5} x^{2} - d^{11} e^{4} x - d^{12} e^{3}\right )}} \]
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 = 92, normalized size = 0.75 \[ \frac {\left (-e x +d \right ) \left (-8 e^{6} x^{6}-8 e^{5} x^{5} d +20 e^{4} x^{4} d^{2}+20 x^{3} d^{3} e^{3}-15 x^{2} d^{4} e^{2}+6 d^{5} x e +6 d^{6}\right )}{105 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{5} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 133, normalized size = 1.08 \[ -\frac {d}{7 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{3}\right )}} - \frac {x}{35 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{2}} + \frac {1}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{3}} - \frac {4 \, x}{105 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} e^{2}} - \frac {8 \, x}{105 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{5} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.88, size = 161, normalized size = 1.31 \[ \frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {1}{56\,d^2\,e^3}-\frac {4\,x}{105\,d^3\,e^2}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}+\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {2}{35\,e^3}+\frac {3\,x}{70\,d\,e^2}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}-\frac {\sqrt {d^2-e^2\,x^2}}{56\,d^2\,e^3\,{\left (d+e\,x\right )}^4}-\frac {8\,x\,\sqrt {d^2-e^2\,x^2}}{105\,d^5\,e^2\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \]
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 \[ \int \frac {x^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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