Optimal. Leaf size=205 \[ -\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {128 x}{715 d^{10} \sqrt {d^2-e^2 x^2}}+\frac {64 x}{715 d^8 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {659, 192, 191} \[ \frac {128 x}{715 d^{10} \sqrt {d^2-e^2 x^2}}+\frac {64 x}{715 d^8 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 659
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {9 \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{13 d}\\ &=-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {72 \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 d^2}\\ &=-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {56 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 d^3}\\ &=-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {48 \int \frac {1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 d^4}\\ &=\frac {48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {192 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{715 d^6}\\ &=\frac {48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {64 x}{715 d^8 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {128 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{715 d^8}\\ &=\frac {48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {64 x}{715 d^8 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {128 x}{715 d^{10} \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 137, normalized size = 0.67 \[ \frac {\sqrt {d^2-e^2 x^2} \left (-180 d^9-5 d^8 e x+800 d^7 e^2 x^2+1080 d^6 e^3 x^3-320 d^5 e^4 x^4-1552 d^4 e^5 x^5-768 d^3 e^6 x^6+448 d^2 e^7 x^7+512 d e^8 x^8+128 e^9 x^9\right )}{715 d^{10} e (d-e x)^3 (d+e x)^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.33, size = 314, normalized size = 1.53 \[ -\frac {180 \, e^{10} x^{10} + 720 \, d e^{9} x^{9} + 540 \, d^{2} e^{8} x^{8} - 1440 \, d^{3} e^{7} x^{7} - 2520 \, d^{4} e^{6} x^{6} + 2520 \, d^{6} e^{4} x^{4} + 1440 \, d^{7} e^{3} x^{3} - 540 \, d^{8} e^{2} x^{2} - 720 \, d^{9} e x - 180 \, d^{10} + {\left (128 \, e^{9} x^{9} + 512 \, d e^{8} x^{8} + 448 \, d^{2} e^{7} x^{7} - 768 \, d^{3} e^{6} x^{6} - 1552 \, d^{4} e^{5} x^{5} - 320 \, d^{5} e^{4} x^{4} + 1080 \, d^{6} e^{3} x^{3} + 800 \, d^{7} e^{2} x^{2} - 5 \, d^{8} e x - 180 \, d^{9}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{715 \, {\left (d^{10} e^{11} x^{10} + 4 \, d^{11} e^{10} x^{9} + 3 \, d^{12} e^{9} x^{8} - 8 \, d^{13} e^{8} x^{7} - 14 \, d^{14} e^{7} x^{6} + 14 \, d^{16} e^{5} x^{4} + 8 \, d^{17} e^{4} x^{3} - 3 \, d^{18} e^{3} x^{2} - 4 \, d^{19} e^{2} x - d^{20} e\right )}} \]
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 = 132, normalized size = 0.64 \[ -\frac {\left (-e x +d \right ) \left (-128 e^{9} x^{9}-512 e^{8} x^{8} d -448 e^{7} x^{7} d^{2}+768 e^{6} x^{6} d^{3}+1552 e^{5} x^{5} d^{4}+320 x^{4} d^{5} e^{4}-1080 x^{3} d^{6} e^{3}-800 x^{2} d^{7} e^{2}+5 d^{8} x e +180 d^{9}\right )}{715 \left (e x +d \right )^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{10} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.50, size = 393, normalized size = 1.92 \[ -\frac {1}{13 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{5} x^{4} + 4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{4} x^{3} + 6 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3} x^{2} + 4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e\right )}} - \frac {9}{143 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{4} x^{3} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3} x^{2} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e\right )}} - \frac {8}{143 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3} x^{2} + 2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e\right )}} - \frac {8}{143 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e\right )}} + \frac {48 \, x}{715 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{6}} + \frac {64 \, x}{715 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{8}} + \frac {128 \, x}{715 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.12, size = 242, normalized size = 1.18 \[ \frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {64\,x}{715\,d^8}+\frac {189}{4576\,d^7\,e}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}+\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {1139\,x}{5720\,d^6}-\frac {427}{2288\,d^5\,e}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}-\frac {\sqrt {d^2-e^2\,x^2}}{104\,d^4\,e\,{\left (d+e\,x\right )}^7}-\frac {51\,\sqrt {d^2-e^2\,x^2}}{2288\,d^5\,e\,{\left (d+e\,x\right )}^6}-\frac {19\,\sqrt {d^2-e^2\,x^2}}{572\,d^6\,e\,{\left (d+e\,x\right )}^5}-\frac {189\,\sqrt {d^2-e^2\,x^2}}{4576\,d^7\,e\,{\left (d+e\,x\right )}^4}+\frac {128\,x\,\sqrt {d^2-e^2\,x^2}}{715\,d^{10}\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \]
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 \[ \int \frac {1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}} \left (d + e x\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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