Optimal. Leaf size=313 \[ -\frac{3 b c d^3 \left (35 m^3+455 m^2+1813 m+2161\right ) x^{m+2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},-c^2 x^2\right )}{(m+1) (m+2) (m+3)^2 (m+5)^2 (m+7)^2}+\frac{3 c^2 d^3 x^{m+3} \left (a+b \sinh ^{-1}(c x)\right )}{m+3}+\frac{3 c^4 d^3 x^{m+5} \left (a+b \sinh ^{-1}(c x)\right )}{m+5}+\frac{c^6 d^3 x^{m+7} \left (a+b \sinh ^{-1}(c x)\right )}{m+7}+\frac{d^3 x^{m+1} \left (a+b \sinh ^{-1}(c x)\right )}{m+1}-\frac{b c d^3 \left (m^4+27 m^3+284 m^2+1329 m+2271\right ) \sqrt{c^2 x^2+1} x^{m+2}}{(m+3)^2 (m+5)^2 (m+7)^2}-\frac{b c^3 d^3 (m+9) (2 m+13) \sqrt{c^2 x^2+1} x^{m+4}}{(m+5)^2 (m+7)^2}-\frac{b c^5 d^3 \sqrt{c^2 x^2+1} x^{m+6}}{(m+7)^2} \]
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Rubi [A] time = 2.17226, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {270, 5730, 12, 1809, 1267, 459, 364} \[ \frac{3 c^2 d^3 x^{m+3} \left (a+b \sinh ^{-1}(c x)\right )}{m+3}+\frac{3 c^4 d^3 x^{m+5} \left (a+b \sinh ^{-1}(c x)\right )}{m+5}+\frac{c^6 d^3 x^{m+7} \left (a+b \sinh ^{-1}(c x)\right )}{m+7}+\frac{d^3 x^{m+1} \left (a+b \sinh ^{-1}(c x)\right )}{m+1}-\frac{3 b c d^3 \left (35 m^3+455 m^2+1813 m+2161\right ) x^{m+2} \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};-c^2 x^2\right )}{(m+1) (m+2) (m+3)^2 (m+5)^2 (m+7)^2}-\frac{b c d^3 \left (m^4+27 m^3+284 m^2+1329 m+2271\right ) \sqrt{c^2 x^2+1} x^{m+2}}{(m+3)^2 (m+5)^2 (m+7)^2}-\frac{b c^3 d^3 (m+9) (2 m+13) \sqrt{c^2 x^2+1} x^{m+4}}{(m+5)^2 (m+7)^2}-\frac{b c^5 d^3 \sqrt{c^2 x^2+1} x^{m+6}}{(m+7)^2} \]
Antiderivative was successfully verified.
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Rule 270
Rule 5730
Rule 12
Rule 1809
Rule 1267
Rule 459
Rule 364
Rubi steps
\begin{align*} \int x^m \left (d+c^2 d x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac{d^3 x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{1+m}+\frac{3 c^2 d^3 x^{3+m} \left (a+b \sinh ^{-1}(c x)\right )}{3+m}+\frac{3 c^4 d^3 x^{5+m} \left (a+b \sinh ^{-1}(c x)\right )}{5+m}+\frac{c^6 d^3 x^{7+m} \left (a+b \sinh ^{-1}(c x)\right )}{7+m}-(b c) \int \frac{d^3 x^{1+m} \left (\frac{1}{1+m}+\frac{3 c^2 x^2}{3+m}+\frac{3 c^4 x^4}{5+m}+\frac{c^6 x^6}{7+m}\right )}{\sqrt{1+c^2 x^2}} \, dx\\ &=\frac{d^3 x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{1+m}+\frac{3 c^2 d^3 x^{3+m} \left (a+b \sinh ^{-1}(c x)\right )}{3+m}+\frac{3 c^4 d^3 x^{5+m} \left (a+b \sinh ^{-1}(c x)\right )}{5+m}+\frac{c^6 d^3 x^{7+m} \left (a+b \sinh ^{-1}(c x)\right )}{7+m}-\left (b c d^3\right ) \int \frac{x^{1+m} \left (\frac{1}{1+m}+\frac{3 c^2 x^2}{3+m}+\frac{3 c^4 x^4}{5+m}+\frac{c^6 x^6}{7+m}\right )}{\sqrt{1+c^2 x^2}} \, dx\\ &=-\frac{b c^5 d^3 x^{6+m} \sqrt{1+c^2 x^2}}{(7+m)^2}+\frac{d^3 x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{1+m}+\frac{3 c^2 d^3 x^{3+m} \left (a+b \sinh ^{-1}(c x)\right )}{3+m}+\frac{3 c^4 d^3 x^{5+m} \left (a+b \sinh ^{-1}(c x)\right )}{5+m}+\frac{c^6 d^3 x^{7+m} \left (a+b \sinh ^{-1}(c x)\right )}{7+m}-\frac{\left (b d^3\right ) \int \frac{x^{1+m} \left (\frac{c^2 (7+m)}{1+m}+\frac{3 c^4 (7+m) x^2}{3+m}+\frac{c^6 (9+m) (13+2 m) x^4}{(5+m) (7+m)}\right )}{\sqrt{1+c^2 x^2}} \, dx}{c (7+m)}\\ &=-\frac{b c^3 d^3 (9+m) (13+2 m) x^{4+m} \sqrt{1+c^2 x^2}}{(5+m)^2 (7+m)^2}-\frac{b c^5 d^3 x^{6+m} \sqrt{1+c^2 x^2}}{(7+m)^2}+\frac{d^3 x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{1+m}+\frac{3 c^2 d^3 x^{3+m} \left (a+b \sinh ^{-1}(c x)\right )}{3+m}+\frac{3 c^4 d^3 x^{5+m} \left (a+b \sinh ^{-1}(c x)\right )}{5+m}+\frac{c^6 d^3 x^{7+m} \left (a+b \sinh ^{-1}(c x)\right )}{7+m}-\frac{\left (b d^3\right ) \int \frac{x^{1+m} \left (\frac{c^4 (5+m) (7+m)}{1+m}+\frac{c^6 \left (2271+1329 m+284 m^2+27 m^3+m^4\right ) x^2}{(3+m) (5+m) (7+m)}\right )}{\sqrt{1+c^2 x^2}} \, dx}{c^3 (5+m) (7+m)}\\ &=-\frac{b c d^3 \left (2271+1329 m+284 m^2+27 m^3+m^4\right ) x^{2+m} \sqrt{1+c^2 x^2}}{(3+m)^2 (5+m)^2 (7+m)^2}-\frac{b c^3 d^3 (9+m) (13+2 m) x^{4+m} \sqrt{1+c^2 x^2}}{(5+m)^2 (7+m)^2}-\frac{b c^5 d^3 x^{6+m} \sqrt{1+c^2 x^2}}{(7+m)^2}+\frac{d^3 x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{1+m}+\frac{3 c^2 d^3 x^{3+m} \left (a+b \sinh ^{-1}(c x)\right )}{3+m}+\frac{3 c^4 d^3 x^{5+m} \left (a+b \sinh ^{-1}(c x)\right )}{5+m}+\frac{c^6 d^3 x^{7+m} \left (a+b \sinh ^{-1}(c x)\right )}{7+m}-\frac{\left (3 b c d^3 \left (2161+1813 m+455 m^2+35 m^3\right )\right ) \int \frac{x^{1+m}}{\sqrt{1+c^2 x^2}} \, dx}{(1+m) (3+m)^2 (5+m)^2 (7+m)^2}\\ &=-\frac{b c d^3 \left (2271+1329 m+284 m^2+27 m^3+m^4\right ) x^{2+m} \sqrt{1+c^2 x^2}}{(3+m)^2 (5+m)^2 (7+m)^2}-\frac{b c^3 d^3 (9+m) (13+2 m) x^{4+m} \sqrt{1+c^2 x^2}}{(5+m)^2 (7+m)^2}-\frac{b c^5 d^3 x^{6+m} \sqrt{1+c^2 x^2}}{(7+m)^2}+\frac{d^3 x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{1+m}+\frac{3 c^2 d^3 x^{3+m} \left (a+b \sinh ^{-1}(c x)\right )}{3+m}+\frac{3 c^4 d^3 x^{5+m} \left (a+b \sinh ^{-1}(c x)\right )}{5+m}+\frac{c^6 d^3 x^{7+m} \left (a+b \sinh ^{-1}(c x)\right )}{7+m}-\frac{3 b c d^3 \left (2161+1813 m+455 m^2+35 m^3\right ) x^{2+m} \, _2F_1\left (\frac{1}{2},\frac{2+m}{2};\frac{4+m}{2};-c^2 x^2\right )}{(1+m) (2+m) (3+m)^2 (5+m)^2 (7+m)^2}\\ \end{align*}
Mathematica [A] time = 0.51656, size = 257, normalized size = 0.82 \[ \frac{x^{m+1} \left (\frac{6 d \left (\frac{4 d^2 \left (-b c (m+1) x \text{Hypergeometric2F1}\left (-\frac{1}{2},\frac{m}{2}+1,\frac{m}{2}+2,-c^2 x^2\right )-2 b c x \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m}{2}+1,\frac{m}{2}+2,-c^2 x^2\right )+(m+2) \left (c^2 m x^2+c^2 x^2+m+3\right ) \left (a+b \sinh ^{-1}(c x)\right )\right )}{(m+1) (m+2) (m+3)}-\frac{b c d^2 x \text{Hypergeometric2F1}\left (-\frac{3}{2},\frac{m}{2}+1,\frac{m}{2}+2,-c^2 x^2\right )}{m+2}+\left (c^2 d x^2+d\right )^2 \left (a+b \sinh ^{-1}(c x)\right )\right )}{m+5}-\frac{b c d^3 x \text{Hypergeometric2F1}\left (-\frac{5}{2},\frac{m}{2}+1,\frac{m}{2}+2,-c^2 x^2\right )}{m+2}+\left (c^2 d x^2+d\right )^3 \left (a+b \sinh ^{-1}(c x)\right )\right )}{m+7} \]
Antiderivative was successfully verified.
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Maple [F] time = 2.484, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ({c}^{2}d{x}^{2}+d \right ) ^{3} \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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^{6} d^{3} x^{6} + 3 \, a c^{4} d^{3} x^{4} + 3 \, a c^{2} d^{3} x^{2} + a d^{3} +{\left (b c^{6} d^{3} x^{6} + 3 \, b c^{4} d^{3} x^{4} + 3 \, b c^{2} d^{3} x^{2} + b d^{3}\right )} \operatorname{arsinh}\left (c x\right )\right )} 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] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c^{2} d x^{2} + d\right )}^{3}{\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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