Optimal. Leaf size=209 \[ -\frac {d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7}{1287 d e^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7}{1287 d^2 e^3 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {112 x}{6435 d^8 e^2 \sqrt {d^2-e^2 x^2}}+\frac {56 x}{6435 d^6 e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {14 x}{2145 d^4 e^2 \left (d^2-e^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.21, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1639, 793, 659, 192, 191} \[ -\frac {d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7}{1287 d e^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7}{1287 d^2 e^3 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {14 x}{2145 d^4 e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {56 x}{6435 d^6 e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {112 x}{6435 d^8 e^2 \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 659
Rule 793
Rule 1639
Rubi steps
\begin {align*} \int \frac {x^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {1}{8 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {\int \frac {3 d^2 e^2-5 d e^3 x}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{8 e^4}\\ &=-\frac {d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {1}{8 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(7 d) \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{104 e^2}\\ &=-\frac {d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {7 \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 e^2}\\ &=-\frac {d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7}{1287 d e^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {49 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{1287 d e^2}\\ &=-\frac {d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7}{1287 d e^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7}{1287 d^2 e^3 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {14 \int \frac {1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{429 d^2 e^2}\\ &=\frac {14 x}{2145 d^4 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7}{1287 d e^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7}{1287 d^2 e^3 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {56 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{2145 d^4 e^2}\\ &=\frac {14 x}{2145 d^4 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7}{1287 d e^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7}{1287 d^2 e^3 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {56 x}{6435 d^6 e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {112 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{6435 d^6 e^2}\\ &=\frac {14 x}{2145 d^4 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7}{1287 d e^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7}{1287 d^2 e^3 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {56 x}{6435 d^6 e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {112 x}{6435 d^8 e^2 \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 137, normalized size = 0.66 \[ \frac {\sqrt {d^2-e^2 x^2} \left (200 d^9+800 d^8 e x+700 d^7 e^2 x^2+945 d^6 e^3 x^3-280 d^5 e^4 x^4-1358 d^4 e^5 x^5-672 d^3 e^6 x^6+392 d^2 e^7 x^7+448 d e^8 x^8+112 e^9 x^9\right )}{6435 d^8 e^3 (d-e x)^3 (d+e x)^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.06, size = 317, normalized size = 1.52 \[ \frac {200 \, e^{10} x^{10} + 800 \, d e^{9} x^{9} + 600 \, d^{2} e^{8} x^{8} - 1600 \, d^{3} e^{7} x^{7} - 2800 \, d^{4} e^{6} x^{6} + 2800 \, d^{6} e^{4} x^{4} + 1600 \, d^{7} e^{3} x^{3} - 600 \, d^{8} e^{2} x^{2} - 800 \, d^{9} e x - 200 \, d^{10} - {\left (112 \, e^{9} x^{9} + 448 \, d e^{8} x^{8} + 392 \, d^{2} e^{7} x^{7} - 672 \, d^{3} e^{6} x^{6} - 1358 \, d^{4} e^{5} x^{5} - 280 \, d^{5} e^{4} x^{4} + 945 \, d^{6} e^{3} x^{3} + 700 \, d^{7} e^{2} x^{2} + 800 \, d^{8} e x + 200 \, d^{9}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{6435 \, {\left (d^{8} e^{13} x^{10} + 4 \, d^{9} e^{12} x^{9} + 3 \, d^{10} e^{11} x^{8} - 8 \, d^{11} e^{10} x^{7} - 14 \, d^{12} e^{9} x^{6} + 14 \, d^{14} e^{7} x^{4} + 8 \, d^{15} e^{6} x^{3} - 3 \, d^{16} e^{5} x^{2} - 4 \, d^{17} e^{4} x - d^{18} e^{3}\right )}} \]
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 = 132, normalized size = 0.63 \[ \frac {\left (-e x +d \right ) \left (112 e^{9} x^{9}+448 e^{8} x^{8} d +392 e^{7} x^{7} d^{2}-672 e^{6} x^{6} d^{3}-1358 e^{5} x^{5} d^{4}-280 x^{4} d^{5} e^{4}+945 x^{3} d^{6} e^{3}+700 x^{2} d^{7} e^{2}+800 d^{8} x e +200 d^{9}\right )}{6435 \left (e x +d \right )^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{8} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 401, normalized size = 1.92 \[ -\frac {d}{13 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{7} x^{4} + 4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{6} x^{3} + 6 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{5} x^{2} + 4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{4} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{3}\right )}} + \frac {17}{143 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6} x^{3} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{5} x^{2} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{4} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3}\right )}} - \frac {7}{1287 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{5} x^{2} + 2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{4} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3}\right )}} - \frac {7}{1287 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{4} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3}\right )}} + \frac {14 \, x}{2145 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2}} + \frac {56 \, x}{6435 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{6} e^{2}} + \frac {112 \, x}{6435 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{8} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.19, size = 252, normalized size = 1.21 \[ \frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {227}{6864\,d^3\,e^3}-\frac {353\,x}{17160\,d^4\,e^2}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}-\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {353}{41184\,d^5\,e^3}-\frac {56\,x}{6435\,d^6\,e^2}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}-\frac {\sqrt {d^2-e^2\,x^2}}{104\,d^2\,e^3\,{\left (d+e\,x\right )}^7}+\frac {\sqrt {d^2-e^2\,x^2}}{2288\,d^3\,e^3\,{\left (d+e\,x\right )}^6}+\frac {37\,\sqrt {d^2-e^2\,x^2}}{5148\,d^4\,e^3\,{\left (d+e\,x\right )}^5}+\frac {353\,\sqrt {d^2-e^2\,x^2}}{41184\,d^5\,e^3\,{\left (d+e\,x\right )}^4}+\frac {112\,x\,\sqrt {d^2-e^2\,x^2}}{6435\,d^8\,e^2\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \]
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 \[ \int \frac {x^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}} \left (d + e x\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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