Optimal. Leaf size=209 \[ \frac {d^2}{13 e^4 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {30 d}{143 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {21}{143 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4}{1001 d e^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {64 x}{5005 d^7 e^3 \sqrt {d^2-e^2 x^2}}-\frac {32 x}{5005 d^5 e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {24 x}{5005 d^3 e^3 \left (d^2-e^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.31, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ \frac {d^2}{13 e^4 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {30 d}{143 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {21}{143 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4}{1001 d e^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {24 x}{5005 d^3 e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32 x}{5005 d^5 e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {64 x}{5005 d^7 e^3 \sqrt {d^2-e^2 x^2}} \]
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^3}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {1}{7 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {\int \frac {2 d^3 e^2-3 d^2 e^3 x-12 d e^4 x^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{7 e^5}\\ &=-\frac {3 d}{14 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {1}{7 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {\int \frac {-20 d^3 e^6+36 d^2 e^7 x}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{56 e^9}\\ &=\frac {d^2}{13 e^4 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {3 d}{14 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {1}{7 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\left (9 d^2\right ) \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{182 e^3}\\ &=\frac {d^2}{13 e^4 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {30 d}{143 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {1}{7 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {(36 d) \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{1001 e^3}\\ &=\frac {d^2}{13 e^4 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {30 d}{143 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {21}{143 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 e^3}\\ &=\frac {d^2}{13 e^4 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {30 d}{143 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {21}{143 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4}{1001 d e^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {24 \int \frac {1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{1001 d e^3}\\ &=-\frac {24 x}{5005 d^3 e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {d^2}{13 e^4 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {30 d}{143 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {21}{143 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4}{1001 d e^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {96 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5005 d^3 e^3}\\ &=-\frac {24 x}{5005 d^3 e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {d^2}{13 e^4 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {30 d}{143 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {21}{143 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4}{1001 d e^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32 x}{5005 d^5 e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {64 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{5005 d^5 e^3}\\ &=-\frac {24 x}{5005 d^3 e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {d^2}{13 e^4 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {30 d}{143 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {21}{143 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4}{1001 d e^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32 x}{5005 d^5 e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {64 x}{5005 d^7 e^3 \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 137, normalized size = 0.66 \[ \frac {\sqrt {d^2-e^2 x^2} \left (90 d^9+360 d^8 e x+315 d^7 e^2 x^2-540 d^6 e^3 x^3+160 d^5 e^4 x^4+776 d^4 e^5 x^5+384 d^3 e^6 x^6-224 d^2 e^7 x^7-256 d e^8 x^8-64 e^9 x^9\right )}{5005 d^7 e^4 (d-e x)^3 (d+e x)^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.92, size = 316, normalized size = 1.51 \[ \frac {90 \, e^{10} x^{10} + 360 \, d e^{9} x^{9} + 270 \, d^{2} e^{8} x^{8} - 720 \, d^{3} e^{7} x^{7} - 1260 \, d^{4} e^{6} x^{6} + 1260 \, d^{6} e^{4} x^{4} + 720 \, d^{7} e^{3} x^{3} - 270 \, d^{8} e^{2} x^{2} - 360 \, d^{9} e x - 90 \, d^{10} + {\left (64 \, e^{9} x^{9} + 256 \, d e^{8} x^{8} + 224 \, d^{2} e^{7} x^{7} - 384 \, d^{3} e^{6} x^{6} - 776 \, d^{4} e^{5} x^{5} - 160 \, d^{5} e^{4} x^{4} + 540 \, d^{6} e^{3} x^{3} - 315 \, d^{7} e^{2} x^{2} - 360 \, d^{8} e x - 90 \, d^{9}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{5005 \, {\left (d^{7} e^{14} x^{10} + 4 \, d^{8} e^{13} x^{9} + 3 \, d^{9} e^{12} x^{8} - 8 \, d^{10} e^{11} x^{7} - 14 \, d^{11} e^{10} x^{6} + 14 \, d^{13} e^{8} x^{4} + 8 \, d^{14} e^{7} x^{3} - 3 \, d^{15} e^{6} x^{2} - 4 \, d^{16} e^{5} x - d^{17} e^{4}\right )}} \]
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 \[ \frac {\left (-e x +d \right ) \left (-64 e^{9} x^{9}-256 e^{8} x^{8} d -224 e^{7} x^{7} d^{2}+384 e^{6} x^{6} d^{3}+776 e^{5} x^{5} d^{4}+160 x^{4} d^{5} e^{4}-540 x^{3} d^{6} e^{3}+315 x^{2} d^{7} e^{2}+360 d^{8} x e +90 d^{9}\right )}{5005 \left (e x +d \right )^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{7} e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.50, size = 399, normalized size = 1.91 \[ \frac {d^{2}}{13 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{8} x^{4} + 4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{7} x^{3} + 6 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{6} x^{2} + 4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{5} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{4}\right )}} - \frac {30 \, d}{143 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{7} x^{3} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{6} x^{2} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{5} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{4}\right )}} + \frac {21}{143 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6} x^{2} + 2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{5} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{4}\right )}} + \frac {4}{1001 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{5} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{4}\right )}} - \frac {24 \, x}{5005 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3}} - \frac {32 \, x}{5005 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} e^{3}} - \frac {64 \, x}{5005 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{7} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.22, size = 252, normalized size = 1.21 \[ \frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {107}{4004\,d^2\,e^4}-\frac {1139\,x}{80080\,d^3\,e^3}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}-\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {23}{32032\,d^4\,e^4}+\frac {32\,x}{5005\,d^5\,e^3}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}+\frac {\sqrt {d^2-e^2\,x^2}}{104\,d\,e^4\,{\left (d+e\,x\right )}^7}-\frac {27\,\sqrt {d^2-e^2\,x^2}}{2288\,d^2\,e^4\,{\left (d+e\,x\right )}^6}-\frac {15\,\sqrt {d^2-e^2\,x^2}}{2288\,d^3\,e^4\,{\left (d+e\,x\right )}^5}+\frac {23\,\sqrt {d^2-e^2\,x^2}}{32032\,d^4\,e^4\,{\left (d+e\,x\right )}^4}-\frac {64\,x\,\sqrt {d^2-e^2\,x^2}}{5005\,d^7\,e^3\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \]
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 \[ \int \frac {x^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}} \left (d + e x\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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