Optimal. Leaf size=817 \[ \frac {(c x \text {d1}+\text {d1})^{5/2} (\text {d2}-c \text {d2} x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) (f x)^{m+1}}{f (m+6)}+\frac {5 \text {d1} \text {d2} (c x \text {d1}+\text {d1})^{3/2} (\text {d2}-c \text {d2} x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) (f x)^{m+1}}{f (m+4) (m+6)}+\frac {15 \text {d1}^2 \text {d2}^2 \sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \left (a+b \cosh ^{-1}(c x)\right ) (f x)^{m+1}}{f (m+6) \left (m^2+6 m+8\right )}+\frac {15 \text {d1}^2 \text {d2}^2 \sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \left (a+b \cosh ^{-1}(c x)\right ) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};c^2 x^2\right ) (f x)^{m+1}}{f (m+4) (m+6) \left (m^2+3 m+2\right ) \sqrt {1-c x} \sqrt {c x+1}}-\frac {15 b c \text {d1}^2 \text {d2}^2 \sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} 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 ) (f x)^{m+2}}{f^2 (m+1) (m+2)^2 (m+4) (m+6) \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \text {d1}^2 \text {d2}^2 \sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+2}}{f^2 (m+2) (m+6) \sqrt {c x-1} \sqrt {c x+1}}-\frac {5 b c \text {d1}^2 \text {d2}^2 \sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+2}}{f^2 (m+2) (m+4) (m+6) \sqrt {c x-1} \sqrt {c x+1}}-\frac {15 b c \text {d1}^2 \text {d2}^2 \sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+2}}{f^2 (m+2)^2 (m+4) (m+6) \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b c^3 \text {d1}^2 \text {d2}^2 \sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+4}}{f^4 (m+4) (m+6) \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b c^3 \text {d1}^2 \text {d2}^2 \sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+4}}{f^4 (m+4)^2 (m+6) \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 \text {d1}^2 \text {d2}^2 \sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+6}}{f^6 (m+6)^2 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 1.58, antiderivative size = 827, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {5745, 5743, 5763, 32, 14, 270} \[ \frac {(c x \text {d1}+\text {d1})^{5/2} (\text {d2}-c \text {d2} x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) (f x)^{m+1}}{f (m+6)}+\frac {5 \text {d1} \text {d2} (c x \text {d1}+\text {d1})^{3/2} (\text {d2}-c \text {d2} x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) (f x)^{m+1}}{f (m+4) (m+6)}+\frac {15 \text {d1}^2 \text {d2}^2 \sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \left (a+b \cosh ^{-1}(c x)\right ) (f x)^{m+1}}{f (m+6) \left (m^2+6 m+8\right )}+\frac {15 \text {d1}^2 \text {d2}^2 \sqrt {c x \text {d1}+\text {d1}} \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 {m+1}{2};\frac {m+3}{2};c^2 x^2\right ) (f x)^{m+1}}{f (m+4) (m+6) \left (m^2+3 m+2\right ) (1-c x) (c x+1)}-\frac {15 b c \text {d1}^2 \text {d2}^2 \sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} 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 ) (f x)^{m+2}}{f^2 (m+1) (m+2)^2 (m+4) (m+6) \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \text {d1}^2 \text {d2}^2 \sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+2}}{f^2 (m+2) (m+6) \sqrt {c x-1} \sqrt {c x+1}}-\frac {5 b c \text {d1}^2 \text {d2}^2 \sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+2}}{f^2 (m+2) (m+4) (m+6) \sqrt {c x-1} \sqrt {c x+1}}-\frac {15 b c \text {d1}^2 \text {d2}^2 \sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+2}}{f^2 (m+2)^2 (m+4) (m+6) \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b c^3 \text {d1}^2 \text {d2}^2 \sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+4}}{f^4 (m+4) (m+6) \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b c^3 \text {d1}^2 \text {d2}^2 \sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+4}}{f^4 (m+4)^2 (m+6) \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 \text {d1}^2 \text {d2}^2 \sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+6}}{f^6 (m+6)^2 \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 32
Rule 270
Rule 5743
Rule 5745
Rule 5763
Rubi steps
\begin {align*} \int (f x)^m (\text {d1}+c \text {d1} x)^{5/2} (\text {d2}-c \text {d2} x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {(f x)^{1+m} (\text {d1}+c \text {d1} x)^{5/2} (\text {d2}-c \text {d2} x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{f (6+m)}+\frac {(5 \text {d1} \text {d2}) \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}{6+m}-\frac {\left (b c \text {d1}^2 \text {d2}^2 \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 )^2 \, dx}{f (6+m) \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {5 \text {d1} \text {d2} (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) (6+m)}+\frac {(f x)^{1+m} (\text {d1}+c \text {d1} x)^{5/2} (\text {d2}-c \text {d2} x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{f (6+m)}+\frac {\left (15 \text {d1}^2 \text {d2}^2\right ) \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) (6+m)}-\frac {\left (b c \text {d1}^2 \text {d2}^2 \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}\right ) \int \left ((f x)^{1+m}-\frac {2 c^2 (f x)^{3+m}}{f^2}+\frac {c^4 (f x)^{5+m}}{f^4}\right ) \, dx}{f (6+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b c \text {d1}^2 \text {d2}^2 \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) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c \text {d1}^2 \text {d2}^2 (f x)^{2+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}{f^2 (2+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 \text {d1}^2 \text {d2}^2 (f x)^{4+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}{f^4 (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 \text {d1}^2 \text {d2}^2 (f x)^{6+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}{f^6 (6+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {15 \text {d1}^2 \text {d2}^2 (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 (2+m) (4+m) (6+m)}+\frac {5 \text {d1} \text {d2} (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) (6+m)}+\frac {(f x)^{1+m} (\text {d1}+c \text {d1} x)^{5/2} (\text {d2}-c \text {d2} x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{f (6+m)}+\frac {\left (5 b c \text {d1}^2 \text {d2}^2 \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) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (15 \text {d1}^2 \text {d2}^2 \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) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (15 b c \text {d1}^2 \text {d2}^2 \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) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c \text {d1}^2 \text {d2}^2 (f x)^{2+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}{f^2 (2+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {15 b c \text {d1}^2 \text {d2}^2 (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) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 b c \text {d1}^2 \text {d2}^2 (f x)^{2+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}{f^2 (2+m) (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b c^3 \text {d1}^2 \text {d2}^2 (f x)^{4+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}{f^4 (4+m)^2 (6+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 \text {d1}^2 \text {d2}^2 (f x)^{4+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}{f^4 (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 \text {d1}^2 \text {d2}^2 (f x)^{6+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}{f^6 (6+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {15 \text {d1}^2 \text {d2}^2 (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 (2+m) (4+m) (6+m)}+\frac {5 \text {d1} \text {d2} (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) (6+m)}+\frac {(f x)^{1+m} (\text {d1}+c \text {d1} x)^{5/2} (\text {d2}-c \text {d2} x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{f (6+m)}+\frac {15 \text {d1}^2 \text {d2}^2 (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) (6+m) (1-c x) (1+c x)}-\frac {15 b c \text {d1}^2 \text {d2}^2 (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) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A] time = 2.35, size = 387, normalized size = 0.47 \[ \frac {\text {d1}^2 \text {d2}^2 x \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^m \left (\frac {5 \left (\frac {3 \left (-b c x \sqrt {c x-1} \sqrt {c x+1} \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;c^2 x^2\right )-(m+2) \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+(m+1) \left (a (m+2) \left (c^2 x^2-1\right )+b (m+2) \left (c^2 x^2-1\right ) \cosh ^{-1}(c x)-b c x \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{(m+1) (m+2)^2 (c x-1) (c x+1)}-(c x-1) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )+\frac {b c x \left (\frac {c^2 x^2}{m+4}-\frac {1}{m+2}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )}{m+4}+\left (c^2 x^2-1\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac {b c x \left (\frac {c^4 x^4}{m+6}-\frac {2 c^2 x^2}{m+4}+\frac {1}{m+2}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )}{m+6} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a c^{4} d_{1}^{2} d_{2}^{2} x^{4} - 2 \, a c^{2} d_{1}^{2} d_{2}^{2} x^{2} + a d_{1}^{2} d_{2}^{2} + {\left (b c^{4} d_{1}^{2} d_{2}^{2} x^{4} - 2 \, b c^{2} d_{1}^{2} d_{2}^{2} x^{2} + b d_{1}^{2} d_{2}^{2}\right )} \operatorname {arcosh}\left (c x\right )\right )} \sqrt {c d_{1} x + d_{1}} \sqrt {-c d_{2} x + d_{2}} \left (f x\right )^{m}, x\right ) \]
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 \[ \int {\left (c d_{1} x + d_{1}\right )}^{\frac {5}{2}} {\left (-c d_{2} x + d_{2}\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.97, size = 0, normalized size = 0.00 \[ \int \left (f x \right )^{m} \left (c \mathit {d1} x +\mathit {d1} \right )^{\frac {5}{2}} \left (-c \mathit {d2} x +\mathit {d2} \right )^{\frac {5}{2}} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )\, dx \]
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 \[ \int {\left (c d_{1} x + d_{1}\right )}^{\frac {5}{2}} {\left (-c d_{2} x + d_{2}\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}\,{d x} \]
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 \[ \int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (f\,x\right )}^m\,{\left (d_{1}+c\,d_{1}\,x\right )}^{5/2}\,{\left (d_{2}-c\,d_{2}\,x\right )}^{5/2} \,d x \]
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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