Optimal. Leaf size=128 \[ \frac {2 (f x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 f}-\frac {8 b c (f x)^{5/2} \sqrt {1-c x} \left (a+b \cosh ^{-1}(c x)\right ) \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {9}{4};c^2 x^2\right )}{15 f^2 \sqrt {-1+c x}}-\frac {16 b^2 c^2 (f x)^{7/2} \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};c^2 x^2\right )}{105 f^3} \]
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Rubi [A]
time = 0.20, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5883, 5949}
\begin {gather*} -\frac {16 b^2 c^2 (f x)^{7/2} \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};c^2 x^2\right )}{105 f^3}-\frac {8 b c \sqrt {1-c x} (f x)^{5/2} \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {9}{4};c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{15 f^2 \sqrt {c x-1}}+\frac {2 (f x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 5883
Rule 5949
Rubi steps
\begin {align*} \int \sqrt {f x} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx &=\frac {2 (f x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 f}-\frac {(4 b c) \int \frac {(f x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 f}\\ &=\frac {2 (f x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 f}-\frac {8 b c (f x)^{5/2} \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right ) \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {9}{4};c^2 x^2\right )}{15 f^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b^2 c^2 (f x)^{7/2} \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};c^2 x^2\right )}{105 f^3}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 118, normalized size = 0.92 \begin {gather*} \frac {2}{105} x \sqrt {f x} \left (35 \left (a+b \cosh ^{-1}(c x)\right )^2-4 b c x \left (\frac {7 \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right ) \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {9}{4};c^2 x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+2 b c x \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};c^2 x^2\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2} \sqrt {f x}\, dx\]
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 \sqrt {f x} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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 )}^2\,\sqrt {f\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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