Optimal. Leaf size=302 \[ -\frac{b c \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+2} \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{2}+1,\frac{m}{2}+1\right \},\left \{\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2\right \},c^2 x^2\right )}{f^2 (m+1) (m+2)^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{\sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{f \left (m^2+3 m+2\right ) \sqrt{1-c x} \sqrt{c x+1}}+\frac{\sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{f (m+2)}-\frac{b c \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+2}}{f^2 (m+2)^2 \sqrt{c x-1} \sqrt{c x+1}} \]
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Rubi [A] time = 0.557134, antiderivative size = 312, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086, Rules used = {5743, 5763, 32} \[ -\frac{b c \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+2} \, _3F_2\left (1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right )}{f^2 (m+1) (m+2)^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{\sqrt{1-c^2 x^2} \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{f \left (m^2+3 m+2\right ) (1-c x) (c x+1)}+\frac{\sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{f (m+2)}-\frac{b c \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+2}}{f^2 (m+2)^2 \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 5743
Rule 5763
Rule 32
Rubi steps
\begin{align*} \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 &=\frac{(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)}-\frac{\left (\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) \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b c \sqrt{\text{d1}+c \text{d1} x} \sqrt{\text{d2}-c \text{d2} x}\right ) \int (f x)^{1+m} \, dx}{f (2+m) \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c (f x)^{2+m} \sqrt{\text{d1}+c \text{d1} x} \sqrt{\text{d2}-c \text{d2} x}}{f^2 (2+m)^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{(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)}+\frac{(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 \left (2+3 m+m^2\right ) (1-c x) (1+c x)}-\frac{b c (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 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 0.21783, size = 229, normalized size = 0.76 \[ \frac{x \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^m \left (-b c x \sqrt{c x-1} \sqrt{c x+1} \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{2}+1,\frac{m}{2}+1\right \},\left \{\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2\right \},c^2 x^2\right )-(m+2) \sqrt{1-c^2 x^2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+(m+1) \left (a (m+2) \left (c^2 x^2-1\right )+b (m+2) \left (c^2 x^2-1\right ) \cosh ^{-1}(c x)-b c x \sqrt{c x-1} \sqrt{c x+1}\right )\right )}{(m+1) (m+2)^2 (c x-1) (c x+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.584, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{m}\sqrt{c{\it d1}\,x+{\it d1}}\sqrt{-c{\it d2}\,x+{\it d2}} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\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}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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