Optimal. Leaf size=139 \[ -\frac {1}{9 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {16 x}{45 d^8 \sqrt {d^2-e^2 x^2}}+\frac {8 x}{45 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 x}{15 d^4 \left (d^2-e^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {659, 192, 191} \[ \frac {16 x}{45 d^8 \sqrt {d^2-e^2 x^2}}+\frac {8 x}{45 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 x}{15 d^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{9 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 659
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=-\frac {1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {7 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{9 d}\\ &=-\frac {1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{9 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 \int \frac {1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{3 d^2}\\ &=\frac {2 x}{15 d^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{9 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {8 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{15 d^4}\\ &=\frac {2 x}{15 d^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{9 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {8 x}{45 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{45 d^6}\\ &=\frac {2 x}{15 d^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{9 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {8 x}{45 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 x}{45 d^8 \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 115, normalized size = 0.83 \[ \frac {\sqrt {d^2-e^2 x^2} \left (-10 d^7+25 d^6 e x+60 d^5 e^2 x^2-10 d^4 e^3 x^3-80 d^3 e^4 x^4-24 d^2 e^5 x^5+32 d e^6 x^6+16 e^7 x^7\right )}{45 d^8 e (d-e x)^3 (d+e x)^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.33, size = 248, normalized size = 1.78 \[ -\frac {10 \, e^{8} x^{8} + 20 \, d e^{7} x^{7} - 20 \, d^{2} e^{6} x^{6} - 60 \, d^{3} e^{5} x^{5} + 60 \, d^{5} e^{3} x^{3} + 20 \, d^{6} e^{2} x^{2} - 20 \, d^{7} e x - 10 \, d^{8} + {\left (16 \, e^{7} x^{7} + 32 \, d e^{6} x^{6} - 24 \, d^{2} e^{5} x^{5} - 80 \, d^{3} e^{4} x^{4} - 10 \, d^{4} e^{3} x^{3} + 60 \, d^{5} e^{2} x^{2} + 25 \, d^{6} e x - 10 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{45 \, {\left (d^{8} e^{9} x^{8} + 2 \, d^{9} e^{8} x^{7} - 2 \, d^{10} e^{7} x^{6} - 6 \, d^{11} e^{6} x^{5} + 6 \, d^{13} e^{4} x^{3} + 2 \, d^{14} e^{3} x^{2} - 2 \, d^{15} e^{2} x - d^{16} e\right )}} \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 110, normalized size = 0.79 \[ -\frac {\left (-e x +d \right ) \left (-16 e^{7} x^{7}-32 e^{6} x^{6} d +24 e^{5} x^{5} d^{2}+80 e^{4} x^{4} d^{3}+10 e^{3} x^{3} d^{4}-60 e^{2} x^{2} d^{5}-25 x \,d^{6} e +10 d^{7}\right )}{45 \left (e x +d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{8} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.45, size = 176, normalized size = 1.27 \[ -\frac {1}{9 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{3} x^{2} + 2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e\right )}} - \frac {1}{9 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e\right )}} + \frac {2 \, x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4}} + \frac {8 \, x}{45 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{6}} + \frac {16 \, x}{45 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.69, size = 184, normalized size = 1.32 \[ \frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {31\,x}{120\,d^4}-\frac {5}{24\,d^3\,e}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}+\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {8\,x}{45\,d^6}+\frac {5}{144\,d^5\,e}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}-\frac {\sqrt {d^2-e^2\,x^2}}{72\,d^4\,e\,{\left (d+e\,x\right )}^5}-\frac {5\,\sqrt {d^2-e^2\,x^2}}{144\,d^5\,e\,{\left (d+e\,x\right )}^4}+\frac {16\,x\,\sqrt {d^2-e^2\,x^2}}{45\,d^8\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \]
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 \[ \int \frac {1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}} \left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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