Optimal. Leaf size=503 \[ -\frac {3 b c \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+2} \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;c^2 x^2\right )}{f^2 (m+1) (m+2)^2 (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+1} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{f (m+4) \left (m^2+3 m+2\right ) \sqrt {1-c x} \sqrt {c x+1}}+\frac {3 \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{f \left (m^2+6 m+8\right )}+\frac {(c \text {d1} x+\text {d1})^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{f (m+4)}+\frac {b c^3 \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+4}}{f^4 (m+4)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+2}}{f^2 (m+2) (m+4) \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 b c \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+2}}{f^2 (m+2)^2 (m+4) \sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 0.99, antiderivative size = 513, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5745, 5743, 5763, 32, 14} \[ -\frac {3 b c \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+2} \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;c^2 x^2\right )}{f^2 (m+1) (m+2)^2 (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 \text {d1} \text {d2} \sqrt {1-c^2 x^2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+1} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{f (m+4) \left (m^2+3 m+2\right ) (1-c x) (c x+1)}+\frac {3 \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{f \left (m^2+6 m+8\right )}+\frac {(c \text {d1} x+\text {d1})^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{f (m+4)}+\frac {b c^3 \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+4}}{f^4 (m+4)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+2}}{f^2 (m+2) (m+4) \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 b c \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+2}}{f^2 (m+2)^2 (m+4) \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 32
Rule 5743
Rule 5745
Rule 5763
Rubi steps
\begin {align*} \int (f x)^m (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {(f x)^{1+m} (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{f (4+m)}+\frac {(3 \text {d1} \text {d2}) \int (f x)^m \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{4+m}+\frac {\left (b c \text {d1} \text {d2} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}\right ) \int (f x)^{1+m} \left (-1+c^2 x^2\right ) \, dx}{f (4+m) \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {3 \text {d1} \text {d2} (f x)^{1+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x} \left (a+b \cosh ^{-1}(c x)\right )}{f \left (8+6 m+m^2\right )}+\frac {(f x)^{1+m} (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{f (4+m)}+\frac {\left (b c \text {d1} \text {d2} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}\right ) \int \left (-(f x)^{1+m}+\frac {c^2 (f x)^{3+m}}{f^2}\right ) \, dx}{f (4+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 \text {d1} \text {d2} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}\right ) \int \frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{(2+m) (4+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 b c \text {d1} \text {d2} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}\right ) \int (f x)^{1+m} \, dx}{f (2+m) (4+m) \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {3 b c \text {d1} \text {d2} (f x)^{2+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}{f^2 (2+m)^2 (4+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c \text {d1} \text {d2} (f x)^{2+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}{f^2 (2+m) (4+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 \text {d1} \text {d2} (f x)^{4+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}{f^4 (4+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 \text {d1} \text {d2} (f x)^{1+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x} \left (a+b \cosh ^{-1}(c x)\right )}{f \left (8+6 m+m^2\right )}+\frac {(f x)^{1+m} (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{f (4+m)}+\frac {3 \text {d1} \text {d2} (f x)^{1+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x} \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right ) \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};c^2 x^2\right )}{f (1+m) (2+m) (4+m) (1-c x) (1+c x)}-\frac {3 b c \text {d1} \text {d2} (f x)^{2+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )}{f^2 (1+m) (2+m)^2 (4+m) \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A] time = 1.05, size = 288, normalized size = 0.57 \[ \frac {\text {d1} \text {d2} x \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^m \left (-\frac {3 b c x \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;c^2 x^2\right )}{(m+1) (m+2)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{(m+1) (m+2) (c x-1) (c x+1)}+\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{m+2}-(c x-1) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )+\frac {b c x \left (\frac {c^2 x^2}{m+4}-\frac {1}{m+2}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {3 b c x}{(m+2)^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{m+4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a c^{2} d_{1} d_{2} x^{2} - a d_{1} d_{2} + {\left (b c^{2} d_{1} d_{2} x^{2} - b d_{1} d_{2}\right )} \operatorname {arcosh}\left (c x\right )\right )} \sqrt {c d_{1} x + d_{1}} \sqrt {-c d_{2} x + d_{2}} \left (f x\right )^{m}, x\right ) \]
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 \[ \int {\left (c d_{1} x + d_{1}\right )}^{\frac {3}{2}} {\left (-c d_{2} x + d_{2}\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.60, size = 0, normalized size = 0.00 \[ \int \left (f x \right )^{m} \left (c \mathit {d1} x +\mathit {d1} \right )^{\frac {3}{2}} \left (-c \mathit {d2} x +\mathit {d2} \right )^{\frac {3}{2}} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )\, dx \]
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 \[ \int {\left (c d_{1} x + d_{1}\right )}^{\frac {3}{2}} {\left (-c d_{2} x + d_{2}\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (f\,x\right )}^m\,{\left (d_{1}+c\,d_{1}\,x\right )}^{3/2}\,{\left (d_{2}-c\,d_{2}\,x\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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