Optimal. Leaf size=386 \[ \frac {1}{9} d^2 x^5 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {4}{63} d^2 x^5 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {16 b d^2 x^4 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{1575 c}-\frac {2 b d^2 \left (c^2 x^2+1\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^5}+\frac {20 b d^2 \left (c^2 x^2+1\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{441 c^5}+\frac {2 b d^2 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c^5}-\frac {8 b d^2 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{189 c^5}-\frac {128 b d^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{4725 c^5}+\frac {64 b d^2 x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{4725 c^3}+\frac {8}{315} d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{729} b^2 c^4 d^2 x^9+\frac {4208 b^2 d^2 x}{99225 c^4}+\frac {212 b^2 c^2 d^2 x^7}{27783}-\frac {2104 b^2 d^2 x^3}{297675 c^2}+\frac {526 b^2 d^2 x^5}{165375} \]
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Rubi [A] time = 0.74, antiderivative size = 386, normalized size of antiderivative = 1.00, number of steps used = 16, 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, 1153} \[ \frac {1}{9} d^2 x^5 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {4}{63} d^2 x^5 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {16 b d^2 x^4 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{1575 c}+\frac {64 b d^2 x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{4725 c^3}-\frac {2 b d^2 \left (c^2 x^2+1\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^5}+\frac {20 b d^2 \left (c^2 x^2+1\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{441 c^5}+\frac {2 b d^2 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c^5}-\frac {8 b d^2 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{189 c^5}-\frac {128 b d^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{4725 c^5}+\frac {8}{315} d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{729} b^2 c^4 d^2 x^9+\frac {212 b^2 c^2 d^2 x^7}{27783}-\frac {2104 b^2 d^2 x^3}{297675 c^2}+\frac {4208 b^2 d^2 x}{99225 c^4}+\frac {526 b^2 d^2 x^5}{165375} \]
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 30
Rule 43
Rule 266
Rule 1153
Rule 5661
Rule 5717
Rule 5732
Rule 5744
Rule 5758
Rubi steps
\begin {align*} \int x^4 \left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac {1}{9} d^2 x^5 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{9} (4 d) \int x^4 \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac {1}{9} \left (2 b c d^2\right ) \int x^5 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=-\frac {2 b d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{45 c^5}+\frac {4 b d^2 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{63 c^5}-\frac {2 b d^2 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^5}+\frac {4}{63} d^2 x^5 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{9} d^2 x^5 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{63} \left (8 d^2\right ) \int x^4 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac {1}{63} \left (8 b c d^2\right ) \int x^5 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx+\frac {1}{9} \left (2 b^2 c^2 d^2\right ) \int \frac {\left (1+c^2 x^2\right )^2 \left (8-20 c^2 x^2+35 c^4 x^4\right )}{315 c^6} \, dx\\ &=-\frac {8 b d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{189 c^5}+\frac {2 b d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c^5}+\frac {20 b d^2 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{441 c^5}-\frac {2 b d^2 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^5}+\frac {8}{315} d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {4}{63} d^2 x^5 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{9} d^2 x^5 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {\left (2 b^2 d^2\right ) \int \left (1+c^2 x^2\right )^2 \left (8-20 c^2 x^2+35 c^4 x^4\right ) \, dx}{2835 c^4}-\frac {1}{315} \left (16 b c d^2\right ) \int \frac {x^5 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx+\frac {1}{63} \left (8 b^2 c^2 d^2\right ) \int \frac {8-4 c^2 x^2+3 c^4 x^4+15 c^6 x^6}{105 c^6} \, dx\\ &=-\frac {16 b d^2 x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{1575 c}-\frac {8 b d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{189 c^5}+\frac {2 b d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c^5}+\frac {20 b d^2 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{441 c^5}-\frac {2 b d^2 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^5}+\frac {8}{315} d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {4}{63} d^2 x^5 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{9} d^2 x^5 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {\left (16 b^2 d^2\right ) \int x^4 \, dx}{1575}+\frac {\left (2 b^2 d^2\right ) \int \left (8-4 c^2 x^2+3 c^4 x^4+50 c^6 x^6+35 c^8 x^8\right ) \, dx}{2835 c^4}+\frac {\left (8 b^2 d^2\right ) \int \left (8-4 c^2 x^2+3 c^4 x^4+15 c^6 x^6\right ) \, dx}{6615 c^4}+\frac {\left (64 b d^2\right ) \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{1575 c}\\ &=\frac {304 b^2 d^2 x}{19845 c^4}-\frac {152 b^2 d^2 x^3}{59535 c^2}+\frac {526 b^2 d^2 x^5}{165375}+\frac {212 b^2 c^2 d^2 x^7}{27783}+\frac {2}{729} b^2 c^4 d^2 x^9+\frac {64 b d^2 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4725 c^3}-\frac {16 b d^2 x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{1575 c}-\frac {8 b d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{189 c^5}+\frac {2 b d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c^5}+\frac {20 b d^2 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{441 c^5}-\frac {2 b d^2 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^5}+\frac {8}{315} d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {4}{63} d^2 x^5 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{9} d^2 x^5 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (128 b d^2\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{4725 c^3}-\frac {\left (64 b^2 d^2\right ) \int x^2 \, dx}{4725 c^2}\\ &=\frac {304 b^2 d^2 x}{19845 c^4}-\frac {2104 b^2 d^2 x^3}{297675 c^2}+\frac {526 b^2 d^2 x^5}{165375}+\frac {212 b^2 c^2 d^2 x^7}{27783}+\frac {2}{729} b^2 c^4 d^2 x^9-\frac {128 b d^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4725 c^5}+\frac {64 b d^2 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4725 c^3}-\frac {16 b d^2 x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{1575 c}-\frac {8 b d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{189 c^5}+\frac {2 b d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c^5}+\frac {20 b d^2 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{441 c^5}-\frac {2 b d^2 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^5}+\frac {8}{315} d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {4}{63} d^2 x^5 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{9} d^2 x^5 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {\left (128 b^2 d^2\right ) \int 1 \, dx}{4725 c^4}\\ &=\frac {4208 b^2 d^2 x}{99225 c^4}-\frac {2104 b^2 d^2 x^3}{297675 c^2}+\frac {526 b^2 d^2 x^5}{165375}+\frac {212 b^2 c^2 d^2 x^7}{27783}+\frac {2}{729} b^2 c^4 d^2 x^9-\frac {128 b d^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4725 c^5}+\frac {64 b d^2 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4725 c^3}-\frac {16 b d^2 x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{1575 c}-\frac {8 b d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{189 c^5}+\frac {2 b d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{315 c^5}+\frac {20 b d^2 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{441 c^5}-\frac {2 b d^2 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{81 c^5}+\frac {8}{315} d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {4}{63} d^2 x^5 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{9} d^2 x^5 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2\\ \end {align*}
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Mathematica [A] time = 0.41, size = 251, normalized size = 0.65 \[ \frac {d^2 \left (99225 a^2 c^5 x^5 \left (35 c^4 x^4+90 c^2 x^2+63\right )-630 a b \sqrt {c^2 x^2+1} \left (1225 c^8 x^8+2650 c^6 x^6+789 c^4 x^4-1052 c^2 x^2+2104\right )-630 b \sinh ^{-1}(c x) \left (b \sqrt {c^2 x^2+1} \left (1225 c^8 x^8+2650 c^6 x^6+789 c^4 x^4-1052 c^2 x^2+2104\right )-315 a c^5 x^5 \left (35 c^4 x^4+90 c^2 x^2+63\right )\right )+99225 b^2 c^5 x^5 \left (35 c^4 x^4+90 c^2 x^2+63\right ) \sinh ^{-1}(c x)^2+2 b^2 c x \left (42875 c^8 x^8+119250 c^6 x^6+49707 c^4 x^4-110460 c^2 x^2+662760\right )\right )}{31255875 c^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 368, normalized size = 0.95 \[ \frac {42875 \, {\left (81 \, a^{2} + 2 \, b^{2}\right )} c^{9} d^{2} x^{9} + 2250 \, {\left (3969 \, a^{2} + 106 \, b^{2}\right )} c^{7} d^{2} x^{7} + 189 \, {\left (33075 \, a^{2} + 526 \, b^{2}\right )} c^{5} d^{2} x^{5} - 220920 \, b^{2} c^{3} d^{2} x^{3} + 1325520 \, b^{2} c d^{2} x + 99225 \, {\left (35 \, b^{2} c^{9} d^{2} x^{9} + 90 \, b^{2} c^{7} d^{2} x^{7} + 63 \, b^{2} c^{5} d^{2} x^{5}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 630 \, {\left (11025 \, a b c^{9} d^{2} x^{9} + 28350 \, a b c^{7} d^{2} x^{7} + 19845 \, a b c^{5} d^{2} x^{5} - {\left (1225 \, b^{2} c^{8} d^{2} x^{8} + 2650 \, b^{2} c^{6} d^{2} x^{6} + 789 \, b^{2} c^{4} d^{2} x^{4} - 1052 \, b^{2} c^{2} d^{2} x^{2} + 2104 \, b^{2} d^{2}\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 630 \, {\left (1225 \, a b c^{8} d^{2} x^{8} + 2650 \, a b c^{6} d^{2} x^{6} + 789 \, a b c^{4} d^{2} x^{4} - 1052 \, a b c^{2} d^{2} x^{2} + 2104 \, a b d^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{31255875 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 428, normalized size = 1.11 \[ \frac {d^{2} a^{2} \left (\frac {1}{9} c^{9} x^{9}+\frac {2}{7} c^{7} x^{7}+\frac {1}{5} c^{5} x^{5}\right )+d^{2} b^{2} \left (\frac {\arcsinh \left (c x \right )^{2} c^{3} x^{3} \left (c^{2} x^{2}+1\right )^{3}}{9}-\frac {\arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{3}}{21}+\frac {8 \arcsinh \left (c x \right )^{2} c x}{315}+\frac {\arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{105}+\frac {4 \arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{315}-\frac {2 \arcsinh \left (c x \right ) c^{2} x^{2} \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{81}+\frac {82 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{3969}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{4}}{729}+\frac {1493104 c x}{31255875}-\frac {836 c x \left (c^{2} x^{2}+1\right )^{3}}{250047}-\frac {33862 c x \left (c^{2} x^{2}+1\right )^{2}}{10418625}-\frac {47248 c x \left (c^{2} x^{2}+1\right )}{31255875}-\frac {16 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{315}-\frac {2 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{525}-\frac {8 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{945}\right )+2 d^{2} a b \left (\frac {\arcsinh \left (c x \right ) c^{9} x^{9}}{9}+\frac {2 \arcsinh \left (c x \right ) c^{7} x^{7}}{7}+\frac {\arcsinh \left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{8} x^{8} \sqrt {c^{2} x^{2}+1}}{81}-\frac {106 c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{3969}-\frac {263 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{33075}+\frac {1052 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{99225}-\frac {2104 \sqrt {c^{2} x^{2}+1}}{99225}\right )}{c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 760, normalized size = 1.97 \[ \frac {1}{9} \, b^{2} c^{4} d^{2} x^{9} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{9} \, a^{2} c^{4} d^{2} x^{9} + \frac {2}{7} \, b^{2} c^{2} d^{2} x^{7} \operatorname {arsinh}\left (c x\right )^{2} + \frac {2}{7} \, a^{2} c^{2} d^{2} x^{7} + \frac {1}{5} \, b^{2} d^{2} x^{5} \operatorname {arsinh}\left (c x\right )^{2} + \frac {2}{2835} \, {\left (315 \, x^{9} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {35 \, \sqrt {c^{2} x^{2} + 1} x^{8}}{c^{2}} - \frac {40 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{6}} - \frac {64 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} + 1}}{c^{10}}\right )} c\right )} a b c^{4} d^{2} - \frac {2}{893025} \, {\left (315 \, {\left (\frac {35 \, \sqrt {c^{2} x^{2} + 1} x^{8}}{c^{2}} - \frac {40 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{6}} - \frac {64 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} + 1}}{c^{10}}\right )} c \operatorname {arsinh}\left (c x\right ) - \frac {1225 \, c^{8} x^{9} - 1800 \, c^{6} x^{7} + 3024 \, c^{4} x^{5} - 6720 \, c^{2} x^{3} + 40320 \, x}{c^{8}}\right )} b^{2} c^{4} d^{2} + \frac {1}{5} \, a^{2} d^{2} x^{5} + \frac {4}{245} \, {\left (35 \, x^{7} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac {6 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac {16 \, \sqrt {c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} a b c^{2} d^{2} - \frac {4}{25725} \, {\left (105 \, {\left (\frac {5 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac {6 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac {16 \, \sqrt {c^{2} x^{2} + 1}}{c^{8}}\right )} c \operatorname {arsinh}\left (c x\right ) - \frac {75 \, c^{6} x^{7} - 126 \, c^{4} x^{5} + 280 \, c^{2} x^{3} - 1680 \, x}{c^{6}}\right )} b^{2} c^{2} d^{2} + \frac {2}{75} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b d^{2} - \frac {2}{1125} \, {\left (15 \, {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c \operatorname {arsinh}\left (c x\right ) - \frac {9 \, c^{4} x^{5} - 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} d^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^4\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 30.82, size = 563, normalized size = 1.46 \[ \begin {cases} \frac {a^{2} c^{4} d^{2} x^{9}}{9} + \frac {2 a^{2} c^{2} d^{2} x^{7}}{7} + \frac {a^{2} d^{2} x^{5}}{5} + \frac {2 a b c^{4} d^{2} x^{9} \operatorname {asinh}{\left (c x \right )}}{9} - \frac {2 a b c^{3} d^{2} x^{8} \sqrt {c^{2} x^{2} + 1}}{81} + \frac {4 a b c^{2} d^{2} x^{7} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {212 a b c d^{2} x^{6} \sqrt {c^{2} x^{2} + 1}}{3969} + \frac {2 a b d^{2} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {526 a b d^{2} x^{4} \sqrt {c^{2} x^{2} + 1}}{33075 c} + \frac {2104 a b d^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{99225 c^{3}} - \frac {4208 a b d^{2} \sqrt {c^{2} x^{2} + 1}}{99225 c^{5}} + \frac {b^{2} c^{4} d^{2} x^{9} \operatorname {asinh}^{2}{\left (c x \right )}}{9} + \frac {2 b^{2} c^{4} d^{2} x^{9}}{729} - \frac {2 b^{2} c^{3} d^{2} x^{8} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{81} + \frac {2 b^{2} c^{2} d^{2} x^{7} \operatorname {asinh}^{2}{\left (c x \right )}}{7} + \frac {212 b^{2} c^{2} d^{2} x^{7}}{27783} - \frac {212 b^{2} c d^{2} x^{6} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3969} + \frac {b^{2} d^{2} x^{5} \operatorname {asinh}^{2}{\left (c x \right )}}{5} + \frac {526 b^{2} d^{2} x^{5}}{165375} - \frac {526 b^{2} d^{2} x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{33075 c} - \frac {2104 b^{2} d^{2} x^{3}}{297675 c^{2}} + \frac {2104 b^{2} d^{2} x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{99225 c^{3}} + \frac {4208 b^{2} d^{2} x}{99225 c^{4}} - \frac {4208 b^{2} d^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{99225 c^{5}} & \text {for}\: c \neq 0 \\\frac {a^{2} d^{2} x^{5}}{5} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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