Optimal. Leaf size=181 \[ -\frac {7}{99 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 x}{99 d^8 \sqrt {d^2-e^2 x^2}}+\frac {8 x}{99 d^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 181, 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 = {659, 192, 191} \begin {gather*} \frac {16 x}{99 d^8 \sqrt {d^2-e^2 x^2}}+\frac {8 x}{99 d^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{99 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}} \end {gather*}
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 )^{5/2}} \, dx &=-\frac {1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {7 \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{11 d}\\ &=-\frac {1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{99 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {14 \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{33 d^2}\\ &=-\frac {1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{99 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{33 d^3}\\ &=-\frac {1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{99 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{33 d^4}\\ &=\frac {8 x}{99 d^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{99 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^4 e (d+e x) \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}{99 d^6}\\ &=\frac {8 x}{99 d^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{99 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 x}{99 d^8 \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 115, normalized size = 0.64 \begin {gather*} -\frac {\sqrt {d^2-e^2 x^2} \left (28 d^7+13 d^6 e x-72 d^5 e^2 x^2-122 d^4 e^3 x^3-32 d^3 e^4 x^4+72 d^2 e^5 x^5+64 d e^6 x^6+16 e^7 x^7\right )}{99 d^8 e (d-e x)^2 (d+e x)^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.62, size = 115, normalized size = 0.64 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-28 d^7-13 d^6 e x+72 d^5 e^2 x^2+122 d^4 e^3 x^3+32 d^3 e^4 x^4-72 d^2 e^5 x^5-64 d e^6 x^6-16 e^7 x^7\right )}{99 d^8 e (d-e x)^2 (d+e x)^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 269, normalized size = 1.49 \begin {gather*} -\frac {28 \, e^{8} x^{8} + 112 \, d e^{7} x^{7} + 112 \, d^{2} e^{6} x^{6} - 112 \, d^{3} e^{5} x^{5} - 280 \, d^{4} e^{4} x^{4} - 112 \, d^{5} e^{3} x^{3} + 112 \, d^{6} e^{2} x^{2} + 112 \, d^{7} e x + 28 \, d^{8} + {\left (16 \, e^{7} x^{7} + 64 \, d e^{6} x^{6} + 72 \, d^{2} e^{5} x^{5} - 32 \, d^{3} e^{4} x^{4} - 122 \, d^{4} e^{3} x^{3} - 72 \, d^{5} e^{2} x^{2} + 13 \, d^{6} e x + 28 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{99 \, {\left (d^{8} e^{9} x^{8} + 4 \, d^{9} e^{8} x^{7} + 4 \, d^{10} e^{7} x^{6} - 4 \, d^{11} e^{6} x^{5} - 10 \, d^{12} e^{5} x^{4} - 4 \, d^{13} e^{4} x^{3} + 4 \, d^{14} e^{3} x^{2} + 4 \, d^{15} e^{2} x + d^{16} e\right )}} \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 110, normalized size = 0.61 \begin {gather*} -\frac {\left (-e x +d \right ) \left (16 e^{7} x^{7}+64 e^{6} x^{6} d +72 e^{5} x^{5} d^{2}-32 e^{4} x^{4} d^{3}-122 e^{3} x^{3} d^{4}-72 e^{2} x^{2} d^{5}+13 x \,d^{6} e +28 d^{7}\right )}{99 \left (e x +d \right )^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d^{8} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.46, size = 373, normalized size = 2.06 \begin {gather*} -\frac {1}{11 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d e^{5} x^{4} + 4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} e^{4} x^{3} + 6 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} e^{3} x^{2} + 4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} e\right )}} - \frac {7}{99 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} e^{4} x^{3} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} e^{3} x^{2} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} e\right )}} - \frac {2}{33 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} e^{3} x^{2} + 2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} e\right )}} - \frac {2}{33 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} e\right )}} + \frac {8 \, x}{99 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{6}} + \frac {16 \, x}{99 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.81, size = 197, normalized size = 1.09 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {215\,x}{1584\,d^6}-\frac {91}{792\,d^5\,e}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}-\frac {\sqrt {d^2-e^2\,x^2}}{44\,d^3\,e\,{\left (d+e\,x\right )}^6}-\frac {4\,\sqrt {d^2-e^2\,x^2}}{99\,d^4\,e\,{\left (d+e\,x\right )}^5}-\frac {79\,\sqrt {d^2-e^2\,x^2}}{1584\,d^5\,e\,{\left (d+e\,x\right )}^4}-\frac {29\,\sqrt {d^2-e^2\,x^2}}{528\,d^6\,e\,{\left (d+e\,x\right )}^3}+\frac {16\,x\,\sqrt {d^2-e^2\,x^2}}{99\,d^8\,\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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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 {5}{2}} \left (d + e x\right )^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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