Optimal. Leaf size=447 \[ \frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^3}-\frac {8 b d^{7/2} \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{105 e^3}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e^2}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (29 c^2 d-25 e\right ) \left (d+e x^2\right )^{3/2}}{840 c^4 e^2}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) \sqrt {d+e x^2}}{1680 c^6 e^2}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (105 c^6 d^3-35 c^4 d^2 e+63 c^2 d e^2+75 e^3\right ) \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{1680 c^7 e^{5/2}} \]
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Rubi [A] time = 1.40, antiderivative size = 447, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {266, 43, 6301, 12, 1615, 154, 157, 63, 217, 203, 93, 207} \[ \frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^3}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) \sqrt {d+e x^2}}{1680 c^6 e^2}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-35 c^4 d^2 e+105 c^6 d^3+63 c^2 d e^2+75 e^3\right ) \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{1680 c^7 e^{5/2}}-\frac {8 b d^{7/2} \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{105 e^3}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e^2}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (29 c^2 d-25 e\right ) \left (d+e x^2\right )^{3/2}}{840 c^4 e^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 63
Rule 93
Rule 154
Rule 157
Rule 203
Rule 207
Rule 217
Rule 266
Rule 1615
Rule 6301
Rubi steps
\begin {align*} \int x^5 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx &=\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^3}+\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\left (d+e x^2\right )^{3/2} \left (8 d^2-12 d e x^2+15 e^2 x^4\right )}{105 e^3 x \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^3}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\left (d+e x^2\right )^{3/2} \left (8 d^2-12 d e x^2+15 e^2 x^4\right )}{x \sqrt {1-c^2 x^2}} \, dx}{105 e^3}\\ &=\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^3}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {(d+e x)^{3/2} \left (8 d^2-12 d e x+15 e^2 x^2\right )}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{210 e^3}\\ &=-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e^2}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^3}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {(d+e x)^{3/2} \left (-24 c^2 d^2 e+\frac {3}{2} \left (29 c^2 d-25 e\right ) e^2 x\right )}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{630 c^2 e^4}\\ &=\frac {b \left (29 c^2 d-25 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^4 e^2}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e^2}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^3}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {d+e x} \left (48 c^4 d^3 e-\frac {3}{4} e^2 \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) x\right )}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{1260 c^4 e^4}\\ &=\frac {b \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{1680 c^6 e^2}+\frac {b \left (29 c^2 d-25 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^4 e^2}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e^2}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^3}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {-48 c^6 d^4 e-\frac {3}{8} e^2 \left (105 c^6 d^3-35 c^4 d^2 e+63 c^2 d e^2+75 e^3\right ) x}{x \sqrt {1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{1260 c^6 e^4}\\ &=\frac {b \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{1680 c^6 e^2}+\frac {b \left (29 c^2 d-25 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^4 e^2}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e^2}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^3}+\frac {\left (4 b d^4 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{105 e^3}+\frac {\left (b \left (105 c^6 d^3-35 c^4 d^2 e+63 c^2 d e^2+75 e^3\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{3360 c^6 e^2}\\ &=\frac {b \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{1680 c^6 e^2}+\frac {b \left (29 c^2 d-25 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^4 e^2}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e^2}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^3}+\frac {\left (8 b d^4 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {1}{-d+x^2} \, dx,x,\frac {\sqrt {d+e x^2}}{\sqrt {1-c^2 x^2}}\right )}{105 e^3}-\frac {\left (b \left (105 c^6 d^3-35 c^4 d^2 e+63 c^2 d e^2+75 e^3\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {d+\frac {e}{c^2}-\frac {e x^2}{c^2}}} \, dx,x,\sqrt {1-c^2 x^2}\right )}{1680 c^8 e^2}\\ &=\frac {b \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{1680 c^6 e^2}+\frac {b \left (29 c^2 d-25 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^4 e^2}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e^2}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^3}-\frac {8 b d^{7/2} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{105 e^3}-\frac {\left (b \left (105 c^6 d^3-35 c^4 d^2 e+63 c^2 d e^2+75 e^3\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\frac {e x^2}{c^2}} \, dx,x,\frac {\sqrt {1-c^2 x^2}}{\sqrt {d+e x^2}}\right )}{1680 c^8 e^2}\\ &=\frac {b \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{1680 c^6 e^2}+\frac {b \left (29 c^2 d-25 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^4 e^2}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e^2}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^3}-\frac {b \left (105 c^6 d^3-35 c^4 d^2 e+63 c^2 d e^2+75 e^3\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{1680 c^7 e^{5/2}}-\frac {8 b d^{7/2} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{105 e^3}\\ \end {align*}
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Mathematica [A] time = 3.10, size = 409, normalized size = 0.91 \[ \frac {\sqrt {d+e x^2} \left (16 a c^6 \left (8 d^3-4 d^2 e x^2+3 d e^2 x^4+15 e^3 x^6\right )+16 b c^6 \text {sech}^{-1}(c x) \left (8 d^3-4 d^2 e x^2+3 d e^2 x^4+15 e^3 x^6\right )-b e \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (c^4 \left (-41 d^2+22 d e x^2+40 e^2 x^4\right )+2 c^2 e \left (19 d+25 e x^2\right )+75 e^2\right )\right )}{1680 c^6 e^3}-\frac {b \sqrt {\frac {1-c x}{c x+1}} \sqrt {c^2 x^2-1} \left (128 c^9 d^{7/2} \sqrt {d+e x^2} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {c^2 x^2-1}}{\sqrt {d+e x^2}}\right )+\sqrt {c^2} \sqrt {e} \sqrt {c^2 d+e} \left (105 c^6 d^3-35 c^4 d^2 e+63 c^2 d e^2+75 e^3\right ) \sqrt {\frac {c^2 \left (d+e x^2\right )}{c^2 d+e}} \sinh ^{-1}\left (\frac {c \sqrt {e} \sqrt {c^2 x^2-1}}{\sqrt {c^2} \sqrt {c^2 d+e}}\right )\right )}{1680 c^9 e^3 (c x-1) \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 3.53, size = 1995, normalized size = 4.46 \[ \text {result too large to display} \]
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 \[ \int \sqrt {e x^{2} + d} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 3.72, size = 0, normalized size = 0.00 \[ \int x^{5} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right ) \sqrt {e \,x^{2}+d}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{105} \, {\left (\frac {15 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{4}}{e} - \frac {12 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d x^{2}}{e^{2}} + \frac {8 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{2}}{e^{3}}\right )} a + \frac {1}{105} \, b {\left (\frac {{\left (15 \, e^{3} x^{6} + 3 \, d e^{2} x^{4} - 4 \, d^{2} e x^{2} + 8 \, d^{3}\right )} \sqrt {e x^{2} + d} \log \left (\sqrt {c x + 1} \sqrt {-c x + 1} + 1\right )}{e^{3}} - 105 \, \int \frac {{\left (210 \, {\left (c^{2} e^{3} x^{6} - e^{3} x^{4}\right )} x^{5} \log \left (\sqrt {x}\right ) + 105 \, {\left (c^{2} e^{3} x^{6} \log \relax (c) - e^{3} x^{4} \log \relax (c)\right )} x^{5} + {\left (210 \, {\left (c^{2} e^{3} x^{6} - e^{3} x^{4}\right )} x^{5} \log \left (\sqrt {x}\right ) + {\left (15 \, {\left (7 \, e^{3} \log \relax (c) + e^{3}\right )} c^{2} x^{6} - 4 \, c^{2} d^{2} e x^{2} + 8 \, c^{2} d^{3} + 3 \, {\left (c^{2} d e^{2} - 35 \, e^{3} \log \relax (c)\right )} x^{4}\right )} x^{5}\right )} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (-c x + 1\right )\right )}\right )} \sqrt {e x^{2} + d}}{105 \, {\left (c^{2} e^{3} x^{6} - e^{3} x^{4} + {\left (c^{2} e^{3} x^{6} - e^{3} x^{4}\right )} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (-c x + 1\right )\right )}\right )}}\,{d x}\right )} \]
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 \[ \int x^5\,\sqrt {e\,x^2+d}\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{5} \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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