Optimal. Leaf size=232 \[ \frac {1}{6} d x^6 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2} \left (4 c^2 d+9 e\right )}{120 c^8}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2} \left (8 c^2 d+9 e\right )}{72 c^8}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (4 c^2 d+3 e\right )}{24 c^8}+\frac {b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (1-c^2 x^2\right )^{7/2}}{56 c^8} \]
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Rubi [A] time = 0.16, antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {14, 6301, 12, 446, 77} \[ \frac {1}{6} d x^6 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2} \left (4 c^2 d+9 e\right )}{120 c^8}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2} \left (8 c^2 d+9 e\right )}{72 c^8}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (4 c^2 d+3 e\right )}{24 c^8}+\frac {b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (1-c^2 x^2\right )^{7/2}}{56 c^8} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 77
Rule 446
Rule 6301
Rubi steps
\begin {align*} \int x^5 \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx &=\frac {1}{6} d x^6 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \text {sech}^{-1}(c x)\right )+\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {x^5 \left (4 d+3 e x^2\right )}{24 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {1}{6} d x^6 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{24} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {x^5 \left (4 d+3 e x^2\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {1}{6} d x^6 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{48} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {x^2 (4 d+3 e x)}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=\frac {1}{6} d x^6 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{48} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \left (\frac {4 c^2 d+3 e}{c^6 \sqrt {1-c^2 x}}+\frac {\left (-8 c^2 d-9 e\right ) \sqrt {1-c^2 x}}{c^6}+\frac {\left (4 c^2 d+9 e\right ) \left (1-c^2 x\right )^{3/2}}{c^6}-\frac {3 e \left (1-c^2 x\right )^{5/2}}{c^6}\right ) \, dx,x,x^2\right )\\ &=-\frac {b \left (4 c^2 d+3 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{24 c^8}+\frac {b \left (8 c^2 d+9 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{72 c^8}-\frac {b \left (4 c^2 d+9 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \left (1-c^2 x^2\right )^{5/2}}{120 c^8}+\frac {b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \left (1-c^2 x^2\right )^{7/2}}{56 c^8}+\frac {1}{6} d x^6 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \text {sech}^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.22, size = 126, normalized size = 0.54 \[ \frac {1}{24} a x^6 \left (4 d+3 e x^2\right )-\frac {b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (c^6 \left (84 d x^4+45 e x^6\right )+2 c^4 \left (56 d x^2+27 e x^4\right )+8 c^2 \left (28 d+9 e x^2\right )+144 e\right )}{2520 c^8}+\frac {1}{24} b x^6 \text {sech}^{-1}(c x) \left (4 d+3 e x^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 168, normalized size = 0.72 \[ \frac {315 \, a c^{7} e x^{8} + 420 \, a c^{7} d x^{6} + 105 \, {\left (3 \, b c^{7} e x^{8} + 4 \, b c^{7} d x^{6}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (45 \, b c^{6} e x^{7} + 6 \, {\left (14 \, b c^{6} d + 9 \, b c^{4} e\right )} x^{5} + 8 \, {\left (14 \, b c^{4} d + 9 \, b c^{2} e\right )} x^{3} + 16 \, {\left (14 \, b c^{2} d + 9 \, b e\right )} x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{2520 \, c^{7}} \]
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 \[ \int {\left (e x^{2} + d\right )} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 150, normalized size = 0.65 \[ \frac {\frac {a \left (\frac {1}{8} e \,c^{8} x^{8}+\frac {1}{6} c^{8} d \,x^{6}\right )}{c^{2}}+\frac {b \left (\frac {\mathrm {arcsech}\left (c x \right ) e \,c^{8} x^{8}}{8}+\frac {\mathrm {arcsech}\left (c x \right ) c^{8} x^{6} d}{6}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (45 c^{6} e \,x^{6}+84 c^{6} d \,x^{4}+54 c^{4} e \,x^{4}+112 c^{4} d \,x^{2}+72 c^{2} x^{2} e +224 c^{2} d +144 e \right )}{2520}\right )}{c^{2}}}{c^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 177, normalized size = 0.76 \[ \frac {1}{8} \, a e x^{8} + \frac {1}{6} \, a d x^{6} + \frac {1}{90} \, {\left (15 \, x^{6} \operatorname {arsech}\left (c x\right ) - \frac {3 \, c^{4} x^{5} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} - 10 \, c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{5}}\right )} b d + \frac {1}{280} \, {\left (35 \, x^{8} \operatorname {arsech}\left (c x\right ) + \frac {5 \, c^{6} x^{7} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {7}{2}} - 21 \, c^{4} x^{5} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} + 35 \, c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} - 35 \, x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{7}}\right )} b e \]
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 \[ \int x^5\,\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 15.66, size = 228, normalized size = 0.98 \[ \begin {cases} \frac {a d x^{6}}{6} + \frac {a e x^{8}}{8} + \frac {b d x^{6} \operatorname {asech}{\left (c x \right )}}{6} + \frac {b e x^{8} \operatorname {asech}{\left (c x \right )}}{8} - \frac {b d x^{4} \sqrt {- c^{2} x^{2} + 1}}{30 c^{2}} - \frac {b e x^{6} \sqrt {- c^{2} x^{2} + 1}}{56 c^{2}} - \frac {2 b d x^{2} \sqrt {- c^{2} x^{2} + 1}}{45 c^{4}} - \frac {3 b e x^{4} \sqrt {- c^{2} x^{2} + 1}}{140 c^{4}} - \frac {4 b d \sqrt {- c^{2} x^{2} + 1}}{45 c^{6}} - \frac {b e x^{2} \sqrt {- c^{2} x^{2} + 1}}{35 c^{6}} - \frac {2 b e \sqrt {- c^{2} x^{2} + 1}}{35 c^{8}} & \text {for}\: c \neq 0 \\\left (a + \infty b\right ) \left (\frac {d x^{6}}{6} + \frac {e x^{8}}{8}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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