Optimal. Leaf size=198 \[ -\frac {b e (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x}}{c f^2 (3+m)^2}+\frac {d (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}+\frac {e (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}-\frac {b \left (e (1+m) (2+m)+c^2 d (3+m)^2\right ) (f x)^{2+m} \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{c f^2 (1+m) (2+m) (3+m)^2 \sqrt {-1+c x} \sqrt {1+c x}} \]
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Rubi [A]
time = 0.14, 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 = {5956, 471, 127,
372, 371} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 127
Rule 371
Rule 372
Rule 471
Rule 5956
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 [F]
time = 0.32, size = 0, normalized size = 0.00 \begin {gather*} \int (f x)^m \left (d+e x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 10.90, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 \left (f x\right )^{m} \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (f\,x\right )}^m\,\left (e\,x^2+d\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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