Optimal. Leaf size=268 \[ \frac{b c m \sqrt{c^2 x^2+1} x^{m+2} \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{2}+1,\frac{m}{2}+1\right \},\left \{\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2\right \},-c^2 x^2\right )}{d \left (m^2+3 m+2\right ) \sqrt{c^2 d x^2+d}}-\frac{m \sqrt{c^2 x^2+1} x^{m+1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},-c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d (m+1) \sqrt{c^2 d x^2+d}}-\frac{b c \sqrt{c^2 x^2+1} x^{m+2} \text{Hypergeometric2F1}\left (1,\frac{m+2}{2},\frac{m+4}{2},-c^2 x^2\right )}{d (m+2) \sqrt{c^2 d x^2+d}}+\frac{x^{m+1} \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt{c^2 d x^2+d}} \]
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Rubi [A] time = 0.330701, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {5755, 5764, 5762, 364} \[ \frac{b c m \sqrt{c^2 x^2+1} x^{m+2} \, _3F_2\left (1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;-c^2 x^2\right )}{d \left (m^2+3 m+2\right ) \sqrt{c^2 d x^2+d}}-\frac{m \sqrt{c^2 x^2+1} x^{m+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d (m+1) \sqrt{c^2 d x^2+d}}+\frac{x^{m+1} \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt{c^2 d x^2+d}}-\frac{b c \sqrt{c^2 x^2+1} x^{m+2} \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};-c^2 x^2\right )}{d (m+2) \sqrt{c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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Rule 5755
Rule 5764
Rule 5762
Rule 364
Rubi steps
\begin{align*} \int \frac{x^m \left (a+b \sinh ^{-1}(c x)\right )}{\left (d+c^2 d x^2\right )^{3/2}} \, dx &=\frac{x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt{d+c^2 d x^2}}-\frac{m \int \frac{x^m \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{d+c^2 d x^2}} \, dx}{d}-\frac{\left (b c \sqrt{1+c^2 x^2}\right ) \int \frac{x^{1+m}}{1+c^2 x^2} \, dx}{d \sqrt{d+c^2 d x^2}}\\ &=\frac{x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt{d+c^2 d x^2}}-\frac{b c x^{2+m} \sqrt{1+c^2 x^2} \, _2F_1\left (1,\frac{2+m}{2};\frac{4+m}{2};-c^2 x^2\right )}{d (2+m) \sqrt{d+c^2 d x^2}}-\frac{\left (m \sqrt{1+c^2 x^2}\right ) \int \frac{x^m \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx}{d \sqrt{d+c^2 d x^2}}\\ &=\frac{x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt{d+c^2 d x^2}}-\frac{m x^{1+m} \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, _2F_1\left (\frac{1}{2},\frac{1+m}{2};\frac{3+m}{2};-c^2 x^2\right )}{d (1+m) \sqrt{d+c^2 d x^2}}-\frac{b c x^{2+m} \sqrt{1+c^2 x^2} \, _2F_1\left (1,\frac{2+m}{2};\frac{4+m}{2};-c^2 x^2\right )}{d (2+m) \sqrt{d+c^2 d x^2}}+\frac{b c m x^{2+m} \sqrt{1+c^2 x^2} \, _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 )}{d \left (2+3 m+m^2\right ) \sqrt{d+c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 0.224382, size = 206, normalized size = 0.77 \[ \frac{x^{m+1} \left (b c m x \sqrt{c^2 x^2+1} \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{2}+1,\frac{m}{2}+1\right \},\left \{\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2\right \},-c^2 x^2\right )-m (m+2) \sqrt{c^2 x^2+1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},-c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )+(m+1) \left ((m+2) \left (a+b \sinh ^{-1}(c x)\right )-b c x \sqrt{c^2 x^2+1} \text{Hypergeometric2F1}\left (1,\frac{m}{2}+1,\frac{m}{2}+2,-c^2 x^2\right )\right )\right )}{d (m+1) (m+2) \sqrt{c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.424, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m} \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) \left ({c}^{2}d{x}^{2}+d \right ) ^{-{\frac{3}{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{arsinh}\left (c x\right ) + a\right )} x^{m}}{{\left (c^{2} d x^{2} + d\right )}^{\frac{3}{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^{2} d x^{2} + d}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )} x^{m}}{c^{4} d^{2} x^{4} + 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m} \left (a + b \operatorname{asinh}{\left (c x \right )}\right )}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac{3}{2}}}\, dx \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{arsinh}\left (c x\right ) + a\right )} x^{m}}{{\left (c^{2} d x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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