Optimal. Leaf size=382 \[ \frac{1}{9} d^3 x^3 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{2}{21} d^3 x^3 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{8}{105} d^3 x^3 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{32 b d^3 x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{945 c}-\frac{2 b d^3 \left (c^2 x^2+1\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^3}+\frac{2 b d^3 \left (c^2 x^2+1\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{441 c^3}+\frac{4 b d^3 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{525 c^3}+\frac{16 b d^3 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c^3}+\frac{64 b d^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{945 c^3}+\frac{16}{315} d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{2}{729} b^2 c^6 d^3 x^9+\frac{374 b^2 c^4 d^3 x^7}{27783}+\frac{4198 b^2 c^2 d^3 x^5}{165375}-\frac{10516 b^2 d^3 x}{99225 c^2}+\frac{5258 b^2 d^3 x^3}{297675} \]
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Rubi [A] time = 0.861356, antiderivative size = 382, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 11, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.423, Rules used = {5744, 5661, 5758, 5717, 8, 30, 266, 43, 5732, 12, 373} \[ \frac{1}{9} d^3 x^3 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{2}{21} d^3 x^3 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{8}{105} d^3 x^3 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{32 b d^3 x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{945 c}-\frac{2 b d^3 \left (c^2 x^2+1\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^3}+\frac{2 b d^3 \left (c^2 x^2+1\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{441 c^3}+\frac{4 b d^3 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{525 c^3}+\frac{16 b d^3 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c^3}+\frac{64 b d^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{945 c^3}+\frac{16}{315} d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{2}{729} b^2 c^6 d^3 x^9+\frac{374 b^2 c^4 d^3 x^7}{27783}+\frac{4198 b^2 c^2 d^3 x^5}{165375}-\frac{10516 b^2 d^3 x}{99225 c^2}+\frac{5258 b^2 d^3 x^3}{297675} \]
Antiderivative was successfully verified.
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Rule 5744
Rule 5661
Rule 5758
Rule 5717
Rule 8
Rule 30
Rule 266
Rule 43
Rule 5732
Rule 12
Rule 373
Rubi steps
\begin{align*} \int x^2 \left (d+c^2 d x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac{1}{9} d^3 x^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{3} (2 d) \int x^2 \left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac{1}{9} \left (2 b c d^3\right ) \int x^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=\frac{2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{63 c^3}-\frac{2 b d^3 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^3}+\frac{2}{21} d^3 x^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{9} d^3 x^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{21} \left (8 d^2\right ) \int x^2 \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac{1}{21} \left (4 b c d^3\right ) \int x^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx+\frac{1}{9} \left (2 b^2 c^2 d^3\right ) \int \frac{\left (1+c^2 x^2\right )^3 \left (-2+7 c^2 x^2\right )}{63 c^4} \, dx\\ &=\frac{4 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{105 c^3}+\frac{2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{441 c^3}-\frac{2 b d^3 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^3}+\frac{8}{105} d^3 x^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{2}{21} d^3 x^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{9} d^3 x^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{105} \left (16 d^3\right ) \int x^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+\frac{\left (2 b^2 d^3\right ) \int \left (1+c^2 x^2\right )^3 \left (-2+7 c^2 x^2\right ) \, dx}{567 c^2}-\frac{1}{105} \left (16 b c d^3\right ) \int x^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx+\frac{1}{21} \left (4 b^2 c^2 d^3\right ) \int \frac{\left (1+c^2 x^2\right )^2 \left (-2+5 c^2 x^2\right )}{35 c^4} \, dx\\ &=\frac{16 b d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c^3}+\frac{4 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{525 c^3}+\frac{2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{441 c^3}-\frac{2 b d^3 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^3}+\frac{16}{315} d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{8}{105} d^3 x^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{2}{21} d^3 x^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{9} d^3 x^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\left (2 b^2 d^3\right ) \int \left (-2+c^2 x^2+15 c^4 x^4+19 c^6 x^6+7 c^8 x^8\right ) \, dx}{567 c^2}+\frac{\left (4 b^2 d^3\right ) \int \left (1+c^2 x^2\right )^2 \left (-2+5 c^2 x^2\right ) \, dx}{735 c^2}-\frac{1}{315} \left (32 b c d^3\right ) \int \frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx+\frac{1}{105} \left (16 b^2 c^2 d^3\right ) \int \frac{-2+c^2 x^2+3 c^4 x^4}{15 c^4} \, dx\\ &=-\frac{4 b^2 d^3 x}{567 c^2}+\frac{2 b^2 d^3 x^3}{1701}+\frac{2}{189} b^2 c^2 d^3 x^5+\frac{38 b^2 c^4 d^3 x^7}{3969}+\frac{2}{729} b^2 c^6 d^3 x^9-\frac{32 b d^3 x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{945 c}+\frac{16 b d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c^3}+\frac{4 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{525 c^3}+\frac{2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{441 c^3}-\frac{2 b d^3 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^3}+\frac{16}{315} d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{8}{105} d^3 x^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{2}{21} d^3 x^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{9} d^3 x^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{945} \left (32 b^2 d^3\right ) \int x^2 \, dx+\frac{\left (4 b^2 d^3\right ) \int \left (-2+c^2 x^2+8 c^4 x^4+5 c^6 x^6\right ) \, dx}{735 c^2}+\frac{\left (16 b^2 d^3\right ) \int \left (-2+c^2 x^2+3 c^4 x^4\right ) \, dx}{1575 c^2}+\frac{\left (64 b d^3\right ) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx}{945 c}\\ &=-\frac{3796 b^2 d^3 x}{99225 c^2}+\frac{5258 b^2 d^3 x^3}{297675}+\frac{4198 b^2 c^2 d^3 x^5}{165375}+\frac{374 b^2 c^4 d^3 x^7}{27783}+\frac{2}{729} b^2 c^6 d^3 x^9+\frac{64 b d^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{945 c^3}-\frac{32 b d^3 x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{945 c}+\frac{16 b d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c^3}+\frac{4 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{525 c^3}+\frac{2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{441 c^3}-\frac{2 b d^3 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^3}+\frac{16}{315} d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{8}{105} d^3 x^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{2}{21} d^3 x^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{9} d^3 x^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{\left (64 b^2 d^3\right ) \int 1 \, dx}{945 c^2}\\ &=-\frac{10516 b^2 d^3 x}{99225 c^2}+\frac{5258 b^2 d^3 x^3}{297675}+\frac{4198 b^2 c^2 d^3 x^5}{165375}+\frac{374 b^2 c^4 d^3 x^7}{27783}+\frac{2}{729} b^2 c^6 d^3 x^9+\frac{64 b d^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{945 c^3}-\frac{32 b d^3 x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{945 c}+\frac{16 b d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c^3}+\frac{4 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{525 c^3}+\frac{2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{441 c^3}-\frac{2 b d^3 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^3}+\frac{16}{315} d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{8}{105} d^3 x^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{2}{21} d^3 x^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{9} d^3 x^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2\\ \end{align*}
Mathematica [A] time = 0.484018, size = 275, normalized size = 0.72 \[ \frac{d^3 \left (99225 a^2 c^3 x^3 \left (35 c^6 x^6+135 c^4 x^4+189 c^2 x^2+105\right )-630 a b \sqrt{c^2 x^2+1} \left (1225 c^8 x^8+4675 c^6 x^6+6297 c^4 x^4+2629 c^2 x^2-5258\right )-630 b \sinh ^{-1}(c x) \left (b \sqrt{c^2 x^2+1} \left (1225 c^8 x^8+4675 c^6 x^6+6297 c^4 x^4+2629 c^2 x^2-5258\right )-315 a c^3 x^3 \left (35 c^6 x^6+135 c^4 x^4+189 c^2 x^2+105\right )\right )+b^2 \left (85750 c^9 x^9+420750 c^7 x^7+793422 c^5 x^5+552090 c^3 x^3-3312540 c x\right )+99225 b^2 c^3 x^3 \left (35 c^6 x^6+135 c^4 x^4+189 c^2 x^2+105\right ) \sinh ^{-1}(c x)^2\right )}{31255875 c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 462, normalized size = 1.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.33379, size = 1245, normalized size = 3.26 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.75059, size = 963, normalized size = 2.52 \begin{align*} \frac{42875 \,{\left (81 \, a^{2} + 2 \, b^{2}\right )} c^{9} d^{3} x^{9} + 1125 \,{\left (11907 \, a^{2} + 374 \, b^{2}\right )} c^{7} d^{3} x^{7} + 189 \,{\left (99225 \, a^{2} + 4198 \, b^{2}\right )} c^{5} d^{3} x^{5} + 105 \,{\left (99225 \, a^{2} + 5258 \, b^{2}\right )} c^{3} d^{3} x^{3} - 3312540 \, b^{2} c d^{3} x + 99225 \,{\left (35 \, b^{2} c^{9} d^{3} x^{9} + 135 \, b^{2} c^{7} d^{3} x^{7} + 189 \, b^{2} c^{5} d^{3} x^{5} + 105 \, b^{2} c^{3} d^{3} x^{3}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 630 \,{\left (11025 \, a b c^{9} d^{3} x^{9} + 42525 \, a b c^{7} d^{3} x^{7} + 59535 \, a b c^{5} d^{3} x^{5} + 33075 \, a b c^{3} d^{3} x^{3} -{\left (1225 \, b^{2} c^{8} d^{3} x^{8} + 4675 \, b^{2} c^{6} d^{3} x^{6} + 6297 \, b^{2} c^{4} d^{3} x^{4} + 2629 \, b^{2} c^{2} d^{3} x^{2} - 5258 \, b^{2} d^{3}\right )} \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - 630 \,{\left (1225 \, a b c^{8} d^{3} x^{8} + 4675 \, a b c^{6} d^{3} x^{6} + 6297 \, a b c^{4} d^{3} x^{4} + 2629 \, a b c^{2} d^{3} x^{2} - 5258 \, a b d^{3}\right )} \sqrt{c^{2} x^{2} + 1}}{31255875 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 49.5627, size = 626, normalized size = 1.64 \begin{align*} \begin{cases} \frac{a^{2} c^{6} d^{3} x^{9}}{9} + \frac{3 a^{2} c^{4} d^{3} x^{7}}{7} + \frac{3 a^{2} c^{2} d^{3} x^{5}}{5} + \frac{a^{2} d^{3} x^{3}}{3} + \frac{2 a b c^{6} d^{3} x^{9} \operatorname{asinh}{\left (c x \right )}}{9} - \frac{2 a b c^{5} d^{3} x^{8} \sqrt{c^{2} x^{2} + 1}}{81} + \frac{6 a b c^{4} d^{3} x^{7} \operatorname{asinh}{\left (c x \right )}}{7} - \frac{374 a b c^{3} d^{3} x^{6} \sqrt{c^{2} x^{2} + 1}}{3969} + \frac{6 a b c^{2} d^{3} x^{5} \operatorname{asinh}{\left (c x \right )}}{5} - \frac{4198 a b c d^{3} x^{4} \sqrt{c^{2} x^{2} + 1}}{33075} + \frac{2 a b d^{3} x^{3} \operatorname{asinh}{\left (c x \right )}}{3} - \frac{5258 a b d^{3} x^{2} \sqrt{c^{2} x^{2} + 1}}{99225 c} + \frac{10516 a b d^{3} \sqrt{c^{2} x^{2} + 1}}{99225 c^{3}} + \frac{b^{2} c^{6} d^{3} x^{9} \operatorname{asinh}^{2}{\left (c x \right )}}{9} + \frac{2 b^{2} c^{6} d^{3} x^{9}}{729} - \frac{2 b^{2} c^{5} d^{3} x^{8} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{81} + \frac{3 b^{2} c^{4} d^{3} x^{7} \operatorname{asinh}^{2}{\left (c x \right )}}{7} + \frac{374 b^{2} c^{4} d^{3} x^{7}}{27783} - \frac{374 b^{2} c^{3} d^{3} x^{6} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{3969} + \frac{3 b^{2} c^{2} d^{3} x^{5} \operatorname{asinh}^{2}{\left (c x \right )}}{5} + \frac{4198 b^{2} c^{2} d^{3} x^{5}}{165375} - \frac{4198 b^{2} c d^{3} x^{4} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{33075} + \frac{b^{2} d^{3} x^{3} \operatorname{asinh}^{2}{\left (c x \right )}}{3} + \frac{5258 b^{2} d^{3} x^{3}}{297675} - \frac{5258 b^{2} d^{3} x^{2} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{99225 c} - \frac{10516 b^{2} d^{3} x}{99225 c^{2}} + \frac{10516 b^{2} d^{3} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{99225 c^{3}} & \text{for}\: c \neq 0 \\\frac{a^{2} d^{3} x^{3}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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