Optimal. Leaf size=172 \[ \frac {16 x}{165 d^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{99 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{99 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {64 x}{495 d^7 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {128 x}{495 d^9 \sqrt {d^2-e^2 x^2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {673, 198, 197}
\begin {gather*} -\frac {8}{99 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {128 x}{495 d^9 \sqrt {d^2-e^2 x^2}}+\frac {64 x}{495 d^7 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 x}{165 d^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{99 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 198
Rule 673
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=-\frac {1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {8 \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{11 d}\\ &=-\frac {1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{99 d^2 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}{99 d^2}\\ &=-\frac {1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{99 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{99 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {16 \int \frac {1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{33 d^3}\\ &=\frac {16 x}{165 d^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{99 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{99 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {64 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{165 d^5}\\ &=\frac {16 x}{165 d^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{99 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{99 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {64 x}{495 d^7 \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}{495 d^7}\\ &=\frac {16 x}{165 d^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{99 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{99 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {64 x}{495 d^7 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {128 x}{495 d^9 \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.52, size = 126, normalized size = 0.73 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-125 d^8+120 d^7 e x+680 d^6 e^2 x^2+400 d^5 e^3 x^3-720 d^4 e^4 x^4-832 d^3 e^5 x^5+64 d^2 e^6 x^6+384 d e^7 x^7+128 e^8 x^8\right )}{495 d^9 e (d-e x)^3 (d+e x)^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(326\) vs.
\(2(148)=296\).
time = 0.48, size = 327, normalized size = 1.90
method | result | size |
gosper | \(-\frac {\left (-e x +d \right ) \left (-128 e^{8} x^{8}-384 e^{7} x^{7} d -64 d^{2} e^{6} x^{6}+832 e^{5} x^{5} d^{3}+720 e^{4} x^{4} d^{4}-400 e^{3} x^{3} d^{5}-680 d^{6} e^{2} x^{2}-120 d^{7} e x +125 d^{8}\right )}{495 \left (e x +d \right )^{2} d^{9} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\) | \(121\) |
trager | \(-\frac {\left (-128 e^{8} x^{8}-384 e^{7} x^{7} d -64 d^{2} e^{6} x^{6}+832 e^{5} x^{5} d^{3}+720 e^{4} x^{4} d^{4}-400 e^{3} x^{3} d^{5}-680 d^{6} e^{2} x^{2}-120 d^{7} e x +125 d^{8}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{495 d^{9} \left (e x +d \right )^{6} \left (-e x +d \right )^{3} e}\) | \(123\) |
default | \(\frac {-\frac {1}{11 d e \left (x +\frac {d}{e}\right )^{3} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {8 e \left (-\frac {1}{9 d e \left (x +\frac {d}{e}\right )^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {7 e \left (-\frac {1}{7 d e \left (x +\frac {d}{e}\right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {6 e \left (-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{10 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {-\frac {2 \left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right )}{15 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {4 \left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right )}{15 e^{2} d^{4} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}}{d^{2}}\right )}{7 d}\right )}{9 d}\right )}{11 d}}{e^{3}}\) | \(327\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 257, normalized size = 1.49 \begin {gather*} -\frac {1}{11 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d x^{3} e^{4} + 3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} x^{2} e^{3} + 3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x e^{2} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e\right )}} - \frac {8}{99 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} x^{2} e^{3} + 2 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x e^{2} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e\right )}} - \frac {8}{99 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x e^{2} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e\right )}} + \frac {16 \, x}{165 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5}} + \frac {64 \, x}{495 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{7}} + \frac {128 \, x}{495 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.27, size = 241, normalized size = 1.40 \begin {gather*} -\frac {125 \, x^{9} e^{9} + 375 \, d x^{8} e^{8} - 1000 \, d^{3} x^{6} e^{6} - 750 \, d^{4} x^{5} e^{5} + 750 \, d^{5} x^{4} e^{4} + 1000 \, d^{6} x^{3} e^{3} - 375 \, d^{8} x e - 125 \, d^{9} + {\left (128 \, x^{8} e^{8} + 384 \, d x^{7} e^{7} + 64 \, d^{2} x^{6} e^{6} - 832 \, d^{3} x^{5} e^{5} - 720 \, d^{4} x^{4} e^{4} + 400 \, d^{5} x^{3} e^{3} + 680 \, d^{6} x^{2} e^{2} + 120 \, d^{7} x e - 125 \, d^{8}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{495 \, {\left (d^{9} x^{9} e^{10} + 3 \, d^{10} x^{8} e^{9} - 8 \, d^{12} x^{6} e^{7} - 6 \, d^{13} x^{5} e^{6} + 6 \, d^{14} x^{4} e^{5} + 8 \, d^{15} x^{3} e^{4} - 3 \, d^{17} x e^{2} - d^{18} e\right )}} \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}} \left (d + e x\right )^{3}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.82, size = 213, normalized size = 1.24 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {64\,x}{495\,d^7}+\frac {67}{1584\,d^6\,e}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}+\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {631\,x}{2640\,d^5}-\frac {113}{528\,d^4\,e}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}-\frac {\sqrt {d^2-e^2\,x^2}}{88\,d^4\,e\,{\left (d+e\,x\right )}^6}-\frac {43\,\sqrt {d^2-e^2\,x^2}}{1584\,d^5\,e\,{\left (d+e\,x\right )}^5}-\frac {67\,\sqrt {d^2-e^2\,x^2}}{1584\,d^6\,e\,{\left (d+e\,x\right )}^4}+\frac {128\,x\,\sqrt {d^2-e^2\,x^2}}{495\,d^9\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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