Optimal. Leaf size=504 \[ -\frac{b c (2-m) m \sqrt{c x-1} \sqrt{c x+1} (f x)^{m+2} \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{2}+1,\frac{m}{2}+1\right \},\left \{\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2\right \},c^2 x^2\right )}{3 \text{d1}^2 \text{d2}^2 f^2 (m+1) (m+2) \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x}}-\frac{(2-m) m \sqrt{1-c^2 x^2} (f x)^{m+1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 \text{d1}^2 \text{d2}^2 f (m+1) \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x}}+\frac{b c (2-m) \sqrt{c x-1} \sqrt{c x+1} (f x)^{m+2} \text{Hypergeometric2F1}\left (1,\frac{m+2}{2},\frac{m+4}{2},c^2 x^2\right )}{3 \text{d1}^2 \text{d2}^2 f^2 (m+2) \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x}}+\frac{b c \sqrt{c x-1} \sqrt{c x+1} (f x)^{m+2} \text{Hypergeometric2F1}\left (2,\frac{m+2}{2},\frac{m+4}{2},c^2 x^2\right )}{3 \text{d1}^2 \text{d2}^2 f^2 (m+2) \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x}}+\frac{(2-m) (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 \text{d1}^2 \text{d2}^2 f \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x}}+\frac{(f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 \text{d1} \text{d2} f (c \text{d1} x+\text{d1})^{3/2} (\text{d2}-c \text{d2} x)^{3/2}} \]
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Rubi [A] time = 1.45193, antiderivative size = 504, normalized size of antiderivative = 1., 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 5756
Rule 5765
Rule 5763
Rule 364
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*}
Mathematica [F] time = 2.48476, size = 0, normalized size = 0. \[ \int \frac{(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{(\text{d1}+c \text{d1} x)^{5/2} (\text{d2}-c \text{d2} x)^{5/2}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.677, size = 0, normalized size = 0. \begin{align*} \int{ \left ( fx \right ) ^{m} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) \left ( c{\it d1}\,x+{\it d1} \right ) ^{-{\frac{5}{2}}} \left ( -c{\it d2}\,x+{\it d2} \right ) ^{-{\frac{5}{2}}}}\, 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 \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} \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 (-\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 ) \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] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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