Optimal. Leaf size=209 \[ -\frac {14 e^2 \sqrt {d^2-e^2 x^2}}{3 d^8 x}+\frac {2 e^3 (45 d+53 e x)}{15 d^8 \sqrt {d^2-e^2 x^2}}-\frac {7 e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^8}-\frac {e \sqrt {d^2-e^2 x^2}}{d^7 x^2}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^6 x^3}+\frac {e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.47, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1805, 1807, 807, 266, 63, 208} \[ \frac {2 e^3 (45 d+53 e x)}{15 d^8 \sqrt {d^2-e^2 x^2}}+\frac {e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {14 e^2 \sqrt {d^2-e^2 x^2}}{3 d^8 x}-\frac {e \sqrt {d^2-e^2 x^2}}{d^7 x^2}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^6 x^3}-\frac {7 e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^8} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 1805
Rule 1807
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{x^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {-5 d^2-10 d e x-10 e^2 x^2-\frac {10 e^3 x^3}{d}-\frac {8 e^4 x^4}{d^2}}{x^4 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2}\\ &=\frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {15 d^2+30 d e x+45 e^2 x^2+\frac {60 e^3 x^3}{d}+\frac {46 e^4 x^4}{d^2}}{x^4 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4}\\ &=\frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^3 (45 d+53 e x)}{15 d^8 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-15 d^2-30 d e x-60 e^2 x^2-\frac {90 e^3 x^3}{d}}{x^4 \sqrt {d^2-e^2 x^2}} \, dx}{15 d^6}\\ &=\frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^3 (45 d+53 e x)}{15 d^8 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^6 x^3}+\frac {\int \frac {90 d^3 e+210 d^2 e^2 x+270 d e^3 x^2}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{45 d^8}\\ &=\frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^3 (45 d+53 e x)}{15 d^8 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^6 x^3}-\frac {e \sqrt {d^2-e^2 x^2}}{d^7 x^2}-\frac {\int \frac {-420 d^4 e^2-630 d^3 e^3 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{90 d^{10}}\\ &=\frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^3 (45 d+53 e x)}{15 d^8 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^6 x^3}-\frac {e \sqrt {d^2-e^2 x^2}}{d^7 x^2}-\frac {14 e^2 \sqrt {d^2-e^2 x^2}}{3 d^8 x}+\frac {\left (7 e^3\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^7}\\ &=\frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^3 (45 d+53 e x)}{15 d^8 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^6 x^3}-\frac {e \sqrt {d^2-e^2 x^2}}{d^7 x^2}-\frac {14 e^2 \sqrt {d^2-e^2 x^2}}{3 d^8 x}+\frac {\left (7 e^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{2 d^7}\\ &=\frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^3 (45 d+53 e x)}{15 d^8 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^6 x^3}-\frac {e \sqrt {d^2-e^2 x^2}}{d^7 x^2}-\frac {14 e^2 \sqrt {d^2-e^2 x^2}}{3 d^8 x}-\frac {(7 e) \operatorname {Subst}\left (\int \frac {1}{\frac {d^2}{e^2}-\frac {x^2}{e^2}} \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{d^7}\\ &=\frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^3 (45 d+53 e x)}{15 d^8 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^6 x^3}-\frac {e \sqrt {d^2-e^2 x^2}}{d^7 x^2}-\frac {14 e^2 \sqrt {d^2-e^2 x^2}}{3 d^8 x}-\frac {7 e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^8}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 105, normalized size = 0.50 \[ \frac {-5 d^8-55 d^6 e^2 x^2+330 d^4 e^4 x^4-440 d^2 e^6 x^6+6 d^5 e^3 x^3 \, _2F_1\left (-\frac {5}{2},2;-\frac {3}{2};1-\frac {e^2 x^2}{d^2}\right )+176 e^8 x^8}{15 d^8 x^3 \left (d^2-e^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.04, size = 227, normalized size = 1.09 \[ \frac {116 \, e^{7} x^{7} - 232 \, d e^{6} x^{6} + 232 \, d^{3} e^{4} x^{4} - 116 \, d^{4} e^{3} x^{3} + 105 \, {\left (e^{7} x^{7} - 2 \, d e^{6} x^{6} + 2 \, d^{3} e^{4} x^{4} - d^{4} e^{3} x^{3}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (176 \, e^{6} x^{6} - 247 \, d e^{5} x^{5} - 122 \, d^{2} e^{4} x^{4} + 246 \, d^{3} e^{3} x^{3} - 40 \, d^{4} e^{2} x^{2} - 5 \, d^{5} e x - 5 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{8} e^{4} x^{7} - 2 \, d^{9} e^{3} x^{6} + 2 \, d^{11} e x^{4} - d^{12} x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 325, normalized size = 1.56 \[ -\frac {\sqrt {-x^{2} e^{2} + d^{2}} {\left ({\left ({\left ({\left (2 \, x {\left (\frac {53 \, x e^{8}}{d^{8}} + \frac {45 \, e^{7}}{d^{7}}\right )} - \frac {235 \, e^{6}}{d^{6}}\right )} x - \frac {200 \, e^{5}}{d^{5}}\right )} x + \frac {135 \, e^{4}}{d^{4}}\right )} x + \frac {116 \, e^{3}}{d^{3}}\right )}}{15 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} + \frac {x^{3} {\left (\frac {6 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{6}}{x} + \frac {57 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{4}}{x^{2}} + e^{8}\right )} e}{24 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{8}} - \frac {7 \, e^{3} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right )}{d^{8}} - \frac {{\left (\frac {57 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{16} e^{16}}{x} + \frac {6 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{16} e^{14}}{x^{2}} + \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{16} e^{12}}{x^{3}}\right )} e^{\left (-15\right )}}{24 \, d^{24}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 249, normalized size = 1.19 \[ \frac {22 e^{4} x}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d^{4}}+\frac {7 e^{3}}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d^{3}}-\frac {11 e^{2}}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d^{2} x}+\frac {88 e^{4} x}{15 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{6}}-\frac {e}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d \,x^{2}}+\frac {7 e^{3}}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{5}}-\frac {7 e^{3} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}\, d^{7}}+\frac {176 e^{4} x}{15 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{8}}-\frac {1}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} x^{3}}+\frac {7 e^{3}}{\sqrt {-e^{2} x^{2}+d^{2}}\, d^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 243, normalized size = 1.16 \[ \frac {22 \, e^{4} x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4}} + \frac {7 \, e^{3}}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3}} + \frac {88 \, e^{4} x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{6}} + \frac {7 \, e^{3}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5}} - \frac {11 \, e^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} x} + \frac {176 \, e^{4} x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{8}} - \frac {7 \, e^{3} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{d^{8}} + \frac {7 \, e^{3}}{\sqrt {-e^{2} x^{2} + d^{2}} d^{7}} - \frac {e}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d x^{2}} - \frac {1}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (d+e\,x\right )}^2}{x^4\,{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]
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 \[ \int \frac {\left (d + e x\right )^{2}}{x^{4} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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