Optimal. Leaf size=372 \[ \frac {d^2 (f x)^{m+1} \left (a+b \text {sech}^{-1}(c x)\right )}{f (m+1)}+\frac {2 d e (f x)^{m+3} \left (a+b \text {sech}^{-1}(c x)\right )}{f^3 (m+3)}+\frac {e^2 (f x)^{m+5} \left (a+b \text {sech}^{-1}(c x)\right )}{f^5 (m+5)}-\frac {b e^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} (f x)^{m+3}}{c^2 f^3 (m+4) (m+5)}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} (f x)^{m+1} \left (c^4 d^2 (m+2) (m+3) (m+4) (m+5)+e (m+1)^2 \left (2 c^2 d \left (m^2+9 m+20\right )+e (m+3)^2\right )\right ) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};c^2 x^2\right )}{c^4 f (m+1)^2 (m+2) (m+3) (m+4) (m+5)}-\frac {b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} (f x)^{m+1} \left (2 c^2 d \left (m^2+9 m+20\right )+e (m+3)^2\right )}{c^4 f (m+2) (m+3) (m+4) (m+5)} \]
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Rubi [A] time = 0.43, antiderivative size = 352, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {270, 6301, 12, 1267, 459, 364} \[ \frac {d^2 (f x)^{m+1} \left (a+b \text {sech}^{-1}(c x)\right )}{f (m+1)}+\frac {2 d e (f x)^{m+3} \left (a+b \text {sech}^{-1}(c x)\right )}{f^3 (m+3)}+\frac {e^2 (f x)^{m+5} \left (a+b \text {sech}^{-1}(c x)\right )}{f^5 (m+5)}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} (f x)^{m+1} \left (\frac {e \left (2 c^2 d \left (m^2+9 m+20\right )+e (m+3)^2\right )}{c^4 (m+2) (m+3) (m+4) (m+5)}+\frac {d^2}{(m+1)^2}\right ) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};c^2 x^2\right )}{f}-\frac {b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} (f x)^{m+1} \left (2 c^2 d \left (m^2+9 m+20\right )+e (m+3)^2\right )}{c^4 f (m+2) (m+3) (m+4) (m+5)}-\frac {b e^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} (f x)^{m+3}}{c^2 f^3 (m+4) (m+5)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 270
Rule 364
Rule 459
Rule 1267
Rule 6301
Rubi steps
\begin {align*} \int (f x)^m \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx &=\frac {d^2 (f x)^{1+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f (1+m)}+\frac {2 d e (f x)^{3+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^5 (5+m)}+\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {(f x)^m \left (d^2 \left (15+8 m+m^2\right )+2 d e \left (5+6 m+m^2\right ) x^2+e^2 \left (3+4 m+m^2\right ) x^4\right )}{(1+m) (3+m) (5+m) \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {d^2 (f x)^{1+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f (1+m)}+\frac {2 d e (f x)^{3+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^5 (5+m)}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {(f x)^m \left (d^2 \left (15+8 m+m^2\right )+2 d e \left (5+6 m+m^2\right ) x^2+e^2 \left (3+4 m+m^2\right ) x^4\right )}{\sqrt {1-c^2 x^2}} \, dx}{15+23 m+9 m^2+m^3}\\ &=-\frac {b e^2 (f x)^{3+m} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{c^2 f^3 (4+m) (5+m)}+\frac {d^2 (f x)^{1+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f (1+m)}+\frac {2 d e (f x)^{3+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^5 (5+m)}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {(f x)^m \left (-c^2 d^2 (3+m) (4+m) (5+m)-e (1+m) \left (e (3+m)^2+2 c^2 d \left (20+9 m+m^2\right )\right ) x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{c^2 (4+m) \left (15+23 m+9 m^2+m^3\right )}\\ &=-\frac {b e \left (e (3+m)^2+2 c^2 d \left (20+9 m+m^2\right )\right ) (f x)^{1+m} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{c^4 f (2+m) (4+m) \left (15+8 m+m^2\right )}-\frac {b e^2 (f x)^{3+m} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{c^2 f^3 (4+m) (5+m)}+\frac {d^2 (f x)^{1+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f (1+m)}+\frac {2 d e (f x)^{3+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^5 (5+m)}+-\frac {\left (b \left (-c^4 d^2 (2+m) (3+m) (4+m) (5+m)-e (1+m)^2 \left (e (3+m)^2+2 c^2 d \left (20+9 m+m^2\right )\right )\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {(f x)^m}{\sqrt {1-c^2 x^2}} \, dx}{c^4 (2+m) (4+m) \left (15+23 m+9 m^2+m^3\right )}\\ &=-\frac {b e \left (e (3+m)^2+2 c^2 d \left (20+9 m+m^2\right )\right ) (f x)^{1+m} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{c^4 f (2+m) (4+m) \left (15+8 m+m^2\right )}-\frac {b e^2 (f x)^{3+m} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{c^2 f^3 (4+m) (5+m)}+\frac {d^2 (f x)^{1+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f (1+m)}+\frac {2 d e (f x)^{3+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^5 (5+m)}+\frac {b \left (c^4 d^2 (2+m) (3+m) (4+m) (5+m)+e (1+m)^2 \left (e (3+m)^2+2 c^2 d \left (20+9 m+m^2\right )\right )\right ) (f x)^{1+m} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};c^2 x^2\right )}{c^4 f (1+m) (2+m) (4+m) \left (15+23 m+9 m^2+m^3\right )}\\ \end {align*}
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Mathematica [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int (f x)^m \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a e^{2} x^{4} + 2 \, a d e x^{2} + a d^{2} + {\left (b e^{2} x^{4} + 2 \, b d e x^{2} + b d^{2}\right )} \operatorname {arsech}\left (c x\right )\right )} \left (f x\right )^{m}, x\right ) \]
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 \[ \int {\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} \left (f x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \left (f x \right )^{m} \left (e \,x^{2}+d \right )^{2} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )\, dx \]
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 \[ \frac {a e^{2} f^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, a d e f^{m} x^{3} x^{m}}{m + 3} + \frac {\left (f x\right )^{m + 1} a d^{2}}{f {\left (m + 1\right )}} + \frac {{\left ({\left (m^{2} + 4 \, m + 3\right )} b e^{2} f^{m} x^{5} x^{m} + 2 \, {\left (m^{2} + 6 \, m + 5\right )} b d e f^{m} x^{3} x^{m} + {\left (m^{2} + 8 \, m + 15\right )} b d^{2} f^{m} x x^{m}\right )} \log \left (\sqrt {c x + 1} \sqrt {-c x + 1} + 1\right ) - {\left ({\left (m^{2} + 4 \, m + 3\right )} b e^{2} f^{m} x^{5} x^{m} + 2 \, {\left (m^{2} + 6 \, m + 5\right )} b d e f^{m} x^{3} x^{m} + {\left (m^{2} + 8 \, m + 15\right )} b d^{2} f^{m} x x^{m}\right )} \log \relax (x)}{m^{3} + 9 \, m^{2} + 23 \, m + 15} - \int \frac {{\left (b c^{2} e^{2} f^{m} {\left (m + 5\right )} x^{2} \log \relax (c) - {\left (e^{2} f^{m} {\left (m + 5\right )} \log \relax (c) - e^{2} f^{m}\right )} b\right )} x^{4} x^{m}}{c^{2} {\left (m + 5\right )} x^{2} - m - 5}\,{d x} - \int \frac {2 \, {\left (b c^{2} d e f^{m} {\left (m + 3\right )} x^{2} \log \relax (c) - {\left (d e f^{m} {\left (m + 3\right )} \log \relax (c) - d e f^{m}\right )} b\right )} x^{2} x^{m}}{c^{2} {\left (m + 3\right )} x^{2} - m - 3}\,{d x} - \int \frac {{\left (b c^{2} d^{2} f^{m} {\left (m + 1\right )} x^{2} \log \relax (c) - {\left (d^{2} f^{m} {\left (m + 1\right )} \log \relax (c) - d^{2} f^{m}\right )} b\right )} x^{m}}{c^{2} {\left (m + 1\right )} x^{2} - m - 1}\,{d x} + \int \frac {{\left (m^{2} + 4 \, m + 3\right )} b c^{2} e^{2} f^{m} x^{6} x^{m} + 2 \, {\left (m^{2} + 6 \, m + 5\right )} b c^{2} d e f^{m} x^{4} x^{m} + {\left (m^{2} + 8 \, m + 15\right )} b c^{2} d^{2} f^{m} x^{2} x^{m}}{{\left (m^{3} + 9 \, m^{2} + 23 \, m + 15\right )} c^{2} x^{2} - m^{3} + {\left ({\left (m^{3} + 9 \, m^{2} + 23 \, m + 15\right )} c^{2} x^{2} - m^{3} - 9 \, m^{2} - 23 \, m - 15\right )} \sqrt {c x + 1} \sqrt {-c x + 1} - 9 \, m^{2} - 23 \, m - 15}\,{d x} \]
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 \[ \int {\left (f\,x\right )}^m\,{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
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 \[ \int \left (f x\right )^{m} \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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