Optimal. Leaf size=390 \[ -\frac{3 b c d x^{m+2} \sqrt{c^2 d x^2+d} \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{2}+1,\frac{m}{2}+1\right \},\left \{\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2\right \},-c^2 x^2\right )}{(m+1) (m+2)^2 (m+4) \sqrt{c^2 x^2+1}}+\frac{3 d x^{m+1} \sqrt{c^2 d x^2+d} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},-c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\left (m^3+7 m^2+14 m+8\right ) \sqrt{c^2 x^2+1}}+\frac{3 d x^{m+1} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{m^2+6 m+8}+\frac{x^{m+1} \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{m+4}-\frac{b c d x^{m+2} \sqrt{c^2 d x^2+d}}{\left (m^2+6 m+8\right ) \sqrt{c^2 x^2+1}}-\frac{3 b c d x^{m+2} \sqrt{c^2 d x^2+d}}{(m+2)^2 (m+4) \sqrt{c^2 x^2+1}}-\frac{b c^3 d x^{m+4} \sqrt{c^2 d x^2+d}}{(m+4)^2 \sqrt{c^2 x^2+1}} \]
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Rubi [A] time = 0.330163, antiderivative size = 390, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {5744, 5742, 5762, 30, 14} \[ -\frac{3 b c d x^{m+2} \sqrt{c^2 d x^2+d} \, _3F_2\left (1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;-c^2 x^2\right )}{(m+1) (m+2)^2 (m+4) \sqrt{c^2 x^2+1}}+\frac{3 d x^{m+1} \sqrt{c^2 d x^2+d} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\left (m^3+7 m^2+14 m+8\right ) \sqrt{c^2 x^2+1}}+\frac{3 d x^{m+1} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{m^2+6 m+8}+\frac{x^{m+1} \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{m+4}-\frac{b c d x^{m+2} \sqrt{c^2 d x^2+d}}{\left (m^2+6 m+8\right ) \sqrt{c^2 x^2+1}}-\frac{3 b c d x^{m+2} \sqrt{c^2 d x^2+d}}{(m+2)^2 (m+4) \sqrt{c^2 x^2+1}}-\frac{b c^3 d x^{m+4} \sqrt{c^2 d x^2+d}}{(m+4)^2 \sqrt{c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 5744
Rule 5742
Rule 5762
Rule 30
Rule 14
Rubi steps
\begin{align*} \int x^m \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac{x^{1+m} \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{4+m}+\frac{(3 d) \int x^m \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{4+m}-\frac{\left (b c d \sqrt{d+c^2 d x^2}\right ) \int x^{1+m} \left (1+c^2 x^2\right ) \, dx}{(4+m) \sqrt{1+c^2 x^2}}\\ &=\frac{3 d x^{1+m} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8+6 m+m^2}+\frac{x^{1+m} \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{4+m}-\frac{\left (b c d \sqrt{d+c^2 d x^2}\right ) \int \left (x^{1+m}+c^2 x^{3+m}\right ) \, dx}{(4+m) \sqrt{1+c^2 x^2}}+\frac{\left (3 d \sqrt{d+c^2 d x^2}\right ) \int \frac{x^m \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx}{(2+m) (4+m) \sqrt{1+c^2 x^2}}-\frac{\left (3 b c d \sqrt{d+c^2 d x^2}\right ) \int x^{1+m} \, dx}{(2+m) (4+m) \sqrt{1+c^2 x^2}}\\ &=-\frac{3 b c d x^{2+m} \sqrt{d+c^2 d x^2}}{(2+m)^2 (4+m) \sqrt{1+c^2 x^2}}-\frac{b c d x^{2+m} \sqrt{d+c^2 d x^2}}{\left (8+6 m+m^2\right ) \sqrt{1+c^2 x^2}}-\frac{b c^3 d x^{4+m} \sqrt{d+c^2 d x^2}}{(4+m)^2 \sqrt{1+c^2 x^2}}+\frac{3 d x^{1+m} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8+6 m+m^2}+\frac{x^{1+m} \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{4+m}+\frac{3 d x^{1+m} \sqrt{d+c^2 d 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 )}{\left (8+14 m+7 m^2+m^3\right ) \sqrt{1+c^2 x^2}}-\frac{3 b c d x^{2+m} \sqrt{d+c^2 d 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 )}{(1+m) (2+m)^2 (4+m) \sqrt{1+c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.528491, size = 233, normalized size = 0.6 \[ \frac{d x^{m+1} \sqrt{c^2 d x^2+d} \left (-\frac{3 \left (b c x \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{2}+1,\frac{m}{2}+1\right \},\left \{\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2\right \},-c^2 x^2\right )-(m+2) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},-c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )-(m+1) (m+2) \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )+b c (m+1) x\right )}{(m+1) (m+2)^2 \sqrt{c^2 x^2+1}}+\left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c x \left (c^2 (m+2) x^2+m+4\right )}{(m+2) (m+4) \sqrt{c^2 x^2+1}}\right )}{m+4} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.136, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ({c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}} \left ( a+b{\it Arcsinh} \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{\left (c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )} x^{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 ({\left (a c^{2} d x^{2} + a d +{\left (b c^{2} d x^{2} + b d\right )} \operatorname{arsinh}\left (c x\right )\right )} \sqrt{c^{2} d x^{2} + d} x^{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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