Optimal. Leaf size=198 \[ \frac {d (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{f (m+1)}+\frac {e (f x)^{m+3} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (m+3)}-\frac {b \sqrt {1-c^2 x^2} (f x)^{m+2} \left (c^2 d (m+3)^2+e (m+1) (m+2)\right ) \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{c f^2 (m+1) (m+2) (m+3)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b e \sqrt {c x-1} \sqrt {c x+1} (f x)^{m+2}}{c f^2 (m+3)^2} \]
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Rubi [A] time = 0.20, antiderivative size = 187, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5786, 460, 126, 365, 364} \[ \frac {d (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{f (m+1)}+\frac {e (f x)^{m+3} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (m+3)}-\frac {b \sqrt {1-c^2 x^2} (f x)^{m+2} \left (\frac {c^2 d}{m^2+3 m+2}+\frac {e}{(m+3)^2}\right ) \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{c f^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b e \sqrt {c x-1} \sqrt {c x+1} (f x)^{m+2}}{c f^2 (m+3)^2} \]
Antiderivative was successfully verified.
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Rule 126
Rule 364
Rule 365
Rule 460
Rule 5786
Rubi steps
\begin {align*} \int (f x)^m \left (d+e x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {d (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}+\frac {e (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}-\frac {(b c) \int \frac {(f x)^{1+m} \left (d (3+m)+e (1+m) x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{f \left (3+4 m+m^2\right )}\\ &=-\frac {b e (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x}}{c f^2 (3+m)^2}+\frac {d (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}+\frac {e (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}-\frac {\left (b \left (c^2 d (3+m)^2+e \left (2+3 m+m^2\right )\right )\right ) \int \frac {(f x)^{1+m}}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{c f (1+m) (3+m)^2}\\ &=-\frac {b e (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x}}{c f^2 (3+m)^2}+\frac {d (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}+\frac {e (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}-\frac {\left (b \left (c^2 d (3+m)^2+e \left (2+3 m+m^2\right )\right ) \sqrt {-1+c^2 x^2}\right ) \int \frac {(f x)^{1+m}}{\sqrt {-1+c^2 x^2}} \, dx}{c f (1+m) (3+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b e (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x}}{c f^2 (3+m)^2}+\frac {d (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}+\frac {e (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}-\frac {\left (b \left (c^2 d (3+m)^2+e \left (2+3 m+m^2\right )\right ) \sqrt {1-c^2 x^2}\right ) \int \frac {(f x)^{1+m}}{\sqrt {1-c^2 x^2}} \, dx}{c f (1+m) (3+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b e (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x}}{c f^2 (3+m)^2}+\frac {d (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}+\frac {e (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}-\frac {b \left (c^2 d (3+m)^2+e \left (2+3 m+m^2\right )\right ) (f x)^{2+m} \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{c f^2 (1+m) (2+m) (3+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A] time = 0.61, size = 186, normalized size = 0.94 \[ x (f x)^m \left (\frac {\frac {\left (d (m+3)+e (m+1) x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{m+1}-\frac {b c e x^3 \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {m+4}{2};\frac {m+6}{2};c^2 x^2\right )}{(m+4) \sqrt {c x-1} \sqrt {c x+1}}}{m+3}-\frac {b c d x \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{\left (m^2+3 m+2\right ) \sqrt {c x-1} \sqrt {c x+1}}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a e x^{2} + a d + {\left (b e x^{2} + b d\right )} \operatorname {arcosh}\left (c x\right )\right )} \left (f x\right )^{m}, x\right ) \]
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 \[ \int {\left (e x^{2} + d\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}\,{d x} \]
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 \[ \int \left (f x \right )^{m} \left (e \,x^{2}+d \right ) \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )\, dx \]
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 \[ \frac {a e f^{m} x^{3} x^{m}}{m + 3} + \frac {{\left (b e f^{m} {\left (m + 1\right )} x^{3} + b d f^{m} {\left (m + 3\right )} x\right )} x^{m} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{m^{2} + 4 \, m + 3} + \frac {\left (f x\right )^{m + 1} a d}{f {\left (m + 1\right )}} + \int \frac {{\left (b c e f^{m} {\left (m + 1\right )} x^{3} + b c d f^{m} {\left (m + 3\right )} x\right )} x^{m}}{{\left (m^{2} + 4 \, m + 3\right )} c^{3} x^{3} - {\left (m^{2} + 4 \, m + 3\right )} c x + {\left ({\left (m^{2} + 4 \, m + 3\right )} c^{2} x^{2} - m^{2} - 4 \, m - 3\right )} \sqrt {c x + 1} \sqrt {c x - 1}}\,{d x} - \int \frac {{\left (b c^{2} e f^{m} {\left (m + 1\right )} x^{4} + b c^{2} d f^{m} {\left (m + 3\right )} x^{2}\right )} x^{m}}{{\left (m^{2} + 4 \, m + 3\right )} c^{2} x^{2} - m^{2} - 4 \, m - 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (f\,x\right )}^m\,\left (e\,x^2+d\right ) \,d x \]
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 \[ \int \left (f x\right )^{m} \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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