Optimal. Leaf size=504 \[ -\frac {b c (2-m) m \sqrt {c x-1} \sqrt {c x+1} (f x)^{m+2} \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;c^2 x^2\right )}{3 \text {d1}^2 \text {d2}^2 f^2 (m+1) (m+2) \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}-\frac {(2-m) m \sqrt {1-c^2 x^2} (f x)^{m+1} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 \text {d1}^2 \text {d2}^2 f (m+1) \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}+\frac {(2-m) (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 \text {d1}^2 \text {d2}^2 f \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}+\frac {(f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 \text {d1} \text {d2} f (c \text {d1} x+\text {d1})^{3/2} (\text {d2}-c \text {d2} x)^{3/2}}+\frac {b c (2-m) \sqrt {c x-1} \sqrt {c x+1} (f x)^{m+2} \, _2F_1\left (1,\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{3 \text {d1}^2 \text {d2}^2 f^2 (m+2) \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} (f x)^{m+2} \, _2F_1\left (2,\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{3 \text {d1}^2 \text {d2}^2 f^2 (m+2) \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}} \]
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Rubi [A] time = 1.45, antiderivative size = 504, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {5756, 5765, 5763, 364} \[ -\frac {b c (2-m) m \sqrt {c x-1} \sqrt {c x+1} (f x)^{m+2} \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;c^2 x^2\right )}{3 \text {d1}^2 \text {d2}^2 f^2 (m+1) (m+2) \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}-\frac {(2-m) m \sqrt {1-c^2 x^2} (f x)^{m+1} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 \text {d1}^2 \text {d2}^2 f (m+1) \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}+\frac {(2-m) (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 \text {d1}^2 \text {d2}^2 f \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}+\frac {(f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 \text {d1} \text {d2} f (c \text {d1} x+\text {d1})^{3/2} (\text {d2}-c \text {d2} x)^{3/2}}+\frac {b c (2-m) \sqrt {c x-1} \sqrt {c x+1} (f x)^{m+2} \, _2F_1\left (1,\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{3 \text {d1}^2 \text {d2}^2 f^2 (m+2) \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} (f x)^{m+2} \, _2F_1\left (2,\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{3 \text {d1}^2 \text {d2}^2 f^2 (m+2) \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 5756
Rule 5763
Rule 5765
Rubi steps
\begin {align*} \int \frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{(\text {d1}+c \text {d1} x)^{5/2} (\text {d2}-c \text {d2} x)^{5/2}} \, dx &=\frac {(f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{3 \text {d1} \text {d2} f (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2}}+\frac {(2-m) \int \frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{(\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2}} \, dx}{3 \text {d1} \text {d2}}+\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {(f x)^{1+m}}{\left (-1+c^2 x^2\right )^2} \, dx}{3 \text {d1}^2 \text {d2}^2 f \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}\\ &=\frac {(f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{3 \text {d1} \text {d2} f (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2}}+\frac {(2-m) (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{3 \text {d1}^2 \text {d2}^2 f \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}+\frac {b c (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x} \, _2F_1\left (2,\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{3 \text {d1}^2 \text {d2}^2 f^2 (2+m) \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}-\frac {((2-m) m) \int \frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}} \, dx}{3 \text {d1}^2 \text {d2}^2}-\frac {\left (b c (2-m) \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {(f x)^{1+m}}{-1+c^2 x^2} \, dx}{3 \text {d1}^2 \text {d2}^2 f \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}\\ &=\frac {(f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{3 \text {d1} \text {d2} f (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2}}+\frac {(2-m) (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{3 \text {d1}^2 \text {d2}^2 f \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}+\frac {b c (2-m) (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x} \, _2F_1\left (1,\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{3 \text {d1}^2 \text {d2}^2 f^2 (2+m) \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}+\frac {b c (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x} \, _2F_1\left (2,\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{3 \text {d1}^2 \text {d2}^2 f^2 (2+m) \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}-\frac {\left ((2-m) m \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 \text {d1}^2 \text {d2}^2 \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}\\ &=\frac {(f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{3 \text {d1} \text {d2} f (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2}}+\frac {(2-m) (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{3 \text {d1}^2 \text {d2}^2 f \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}-\frac {(2-m) m (f x)^{1+m} \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 )}{3 \text {d1}^2 \text {d2}^2 f (1+m) \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}+\frac {b c (2-m) (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x} \, _2F_1\left (1,\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{3 \text {d1}^2 \text {d2}^2 f^2 (2+m) \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}+\frac {b c (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x} \, _2F_1\left (2,\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{3 \text {d1}^2 \text {d2}^2 f^2 (2+m) \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}-\frac {b c (2-m) m (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c 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 )}{3 \text {d1}^2 \text {d2}^2 f^2 (1+m) (2+m) \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}\\ \end {align*}
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Mathematica [F] time = 2.70, size = 0, normalized size = 0.00 \[ \int \frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{(\text {d1}+c \text {d1} x)^{5/2} (\text {d2}-c \text {d2} x)^{5/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {c d_{1} x + d_{1}} \sqrt {-c d_{2} x + d_{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{c^{6} d_{1}^{3} d_{2}^{3} x^{6} - 3 \, c^{4} d_{1}^{3} d_{2}^{3} x^{4} + 3 \, c^{2} d_{1}^{3} d_{2}^{3} x^{2} - d_{1}^{3} d_{2}^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{{\left (c d_{1} x + d_{1}\right )}^{\frac {5}{2}} {\left (-c d_{2} x + d_{2}\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.97, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x \right )^{m} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}{\left (c \mathit {d1} x +\mathit {d1} \right )^{\frac {5}{2}} \left (-c \mathit {d2} x +\mathit {d2} \right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{{\left (c d_{1} x + d_{1}\right )}^{\frac {5}{2}} {\left (-c d_{2} x + d_{2}\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (f\,x\right )}^m}{{\left (d_{1}+c\,d_{1}\,x\right )}^{5/2}\,{\left (d_{2}-c\,d_{2}\,x\right )}^{5/2}} \,d x \]
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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