Optimal. Leaf size=312 \[ d^4 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {4}{3} d^3 e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {6}{5} d^2 e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {4}{7} d e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} e^4 x^9 \left (a+b \sinh ^{-1}(c x)\right )-\frac {4 b e^3 \left (c^2 x^2+1\right )^{7/2} \left (9 c^2 d-7 e\right )}{441 c^9}-\frac {b e^4 \left (c^2 x^2+1\right )^{9/2}}{81 c^9}-\frac {2 b e^2 \left (c^2 x^2+1\right )^{5/2} \left (63 c^4 d^2-90 c^2 d e+35 e^2\right )}{525 c^9}-\frac {4 b e \left (c^2 x^2+1\right )^{3/2} \left (105 c^6 d^3-189 c^4 d^2 e+135 c^2 d e^2-35 e^3\right )}{945 c^9}-\frac {b \sqrt {c^2 x^2+1} \left (315 c^8 d^4-420 c^6 d^3 e+378 c^4 d^2 e^2-180 c^2 d e^3+35 e^4\right )}{315 c^9} \]
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Rubi [A] time = 0.35, antiderivative size = 312, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {194, 5704, 12, 1799, 1850} \[ \frac {6}{5} d^2 e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {4}{3} d^3 e x^3 \left (a+b \sinh ^{-1}(c x)\right )+d^4 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {4}{7} d e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} e^4 x^9 \left (a+b \sinh ^{-1}(c x)\right )-\frac {2 b e^2 \left (c^2 x^2+1\right )^{5/2} \left (63 c^4 d^2-90 c^2 d e+35 e^2\right )}{525 c^9}-\frac {4 b e \left (c^2 x^2+1\right )^{3/2} \left (-189 c^4 d^2 e+105 c^6 d^3+135 c^2 d e^2-35 e^3\right )}{945 c^9}-\frac {b \sqrt {c^2 x^2+1} \left (378 c^4 d^2 e^2-420 c^6 d^3 e+315 c^8 d^4-180 c^2 d e^3+35 e^4\right )}{315 c^9}-\frac {4 b e^3 \left (c^2 x^2+1\right )^{7/2} \left (9 c^2 d-7 e\right )}{441 c^9}-\frac {b e^4 \left (c^2 x^2+1\right )^{9/2}}{81 c^9} \]
Antiderivative was successfully verified.
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Rule 12
Rule 194
Rule 1799
Rule 1850
Rule 5704
Rubi steps
\begin {align*} \int \left (d+e x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=d^4 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {4}{3} d^3 e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {6}{5} d^2 e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {4}{7} d e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} e^4 x^9 \left (a+b \sinh ^{-1}(c x)\right )-(b c) \int \frac {x \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right )}{315 \sqrt {1+c^2 x^2}} \, dx\\ &=d^4 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {4}{3} d^3 e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {6}{5} d^2 e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {4}{7} d e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} e^4 x^9 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{315} (b c) \int \frac {x \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right )}{\sqrt {1+c^2 x^2}} \, dx\\ &=d^4 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {4}{3} d^3 e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {6}{5} d^2 e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {4}{7} d e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} e^4 x^9 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{630} (b c) \operatorname {Subst}\left (\int \frac {315 d^4+420 d^3 e x+378 d^2 e^2 x^2+180 d e^3 x^3+35 e^4 x^4}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )\\ &=d^4 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {4}{3} d^3 e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {6}{5} d^2 e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {4}{7} d e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} e^4 x^9 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{630} (b c) \operatorname {Subst}\left (\int \left (\frac {315 c^8 d^4-420 c^6 d^3 e+378 c^4 d^2 e^2-180 c^2 d e^3+35 e^4}{c^8 \sqrt {1+c^2 x}}+\frac {4 e \left (105 c^6 d^3-189 c^4 d^2 e+135 c^2 d e^2-35 e^3\right ) \sqrt {1+c^2 x}}{c^8}+\frac {6 e^2 \left (63 c^4 d^2-90 c^2 d e+35 e^2\right ) \left (1+c^2 x\right )^{3/2}}{c^8}+\frac {20 \left (9 c^2 d-7 e\right ) e^3 \left (1+c^2 x\right )^{5/2}}{c^8}+\frac {35 e^4 \left (1+c^2 x\right )^{7/2}}{c^8}\right ) \, dx,x,x^2\right )\\ &=-\frac {b \left (315 c^8 d^4-420 c^6 d^3 e+378 c^4 d^2 e^2-180 c^2 d e^3+35 e^4\right ) \sqrt {1+c^2 x^2}}{315 c^9}-\frac {4 b e \left (105 c^6 d^3-189 c^4 d^2 e+135 c^2 d e^2-35 e^3\right ) \left (1+c^2 x^2\right )^{3/2}}{945 c^9}-\frac {2 b e^2 \left (63 c^4 d^2-90 c^2 d e+35 e^2\right ) \left (1+c^2 x^2\right )^{5/2}}{525 c^9}-\frac {4 b \left (9 c^2 d-7 e\right ) e^3 \left (1+c^2 x^2\right )^{7/2}}{441 c^9}-\frac {b e^4 \left (1+c^2 x^2\right )^{9/2}}{81 c^9}+d^4 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {4}{3} d^3 e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {6}{5} d^2 e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {4}{7} d e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} e^4 x^9 \left (a+b \sinh ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.36, size = 260, normalized size = 0.83 \[ \frac {315 a x \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right )-\frac {b \sqrt {c^2 x^2+1} \left (c^8 \left (99225 d^4+44100 d^3 e x^2+23814 d^2 e^2 x^4+8100 d e^3 x^6+1225 e^4 x^8\right )-8 c^6 e \left (11025 d^3+3969 d^2 e x^2+1215 d e^2 x^4+175 e^3 x^6\right )+48 c^4 e^2 \left (1323 d^2+270 d e x^2+35 e^2 x^4\right )-320 c^2 e^3 \left (81 d+7 e x^2\right )+4480 e^4\right )}{c^9}+315 b x \sinh ^{-1}(c x) \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right )}{99225} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 333, normalized size = 1.07 \[ \frac {11025 \, a c^{9} e^{4} x^{9} + 56700 \, a c^{9} d e^{3} x^{7} + 119070 \, a c^{9} d^{2} e^{2} x^{5} + 132300 \, a c^{9} d^{3} e x^{3} + 99225 \, a c^{9} d^{4} x + 315 \, {\left (35 \, b c^{9} e^{4} x^{9} + 180 \, b c^{9} d e^{3} x^{7} + 378 \, b c^{9} d^{2} e^{2} x^{5} + 420 \, b c^{9} d^{3} e x^{3} + 315 \, b c^{9} d^{4} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (1225 \, b c^{8} e^{4} x^{8} + 99225 \, b c^{8} d^{4} - 88200 \, b c^{6} d^{3} e + 63504 \, b c^{4} d^{2} e^{2} - 25920 \, b c^{2} d e^{3} + 100 \, {\left (81 \, b c^{8} d e^{3} - 14 \, b c^{6} e^{4}\right )} x^{6} + 4480 \, b e^{4} + 6 \, {\left (3969 \, b c^{8} d^{2} e^{2} - 1620 \, b c^{6} d e^{3} + 280 \, b c^{4} e^{4}\right )} x^{4} + 4 \, {\left (11025 \, b c^{8} d^{3} e - 7938 \, b c^{6} d^{2} e^{2} + 3240 \, b c^{4} d e^{3} - 560 \, b c^{2} e^{4}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{99225 \, c^{9}} \]
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 = 0.02, size = 451, normalized size = 1.45 \[ \frac {\frac {a \left (\frac {1}{9} e^{4} c^{9} x^{9}+\frac {4}{7} c^{9} d \,e^{3} x^{7}+\frac {6}{5} c^{9} d^{2} e^{2} x^{5}+\frac {4}{3} c^{9} d^{3} e \,x^{3}+c^{9} d^{4} x \right )}{c^{8}}+\frac {b \left (\frac {\arcsinh \left (c x \right ) e^{4} c^{9} x^{9}}{9}+\frac {4 \arcsinh \left (c x \right ) c^{9} d \,e^{3} x^{7}}{7}+\frac {6 \arcsinh \left (c x \right ) c^{9} d^{2} e^{2} x^{5}}{5}+\frac {4 \arcsinh \left (c x \right ) c^{9} d^{3} e \,x^{3}}{3}+\arcsinh \left (c x \right ) c^{9} d^{4} x -\frac {e^{4} \left (\frac {c^{8} x^{8} \sqrt {c^{2} x^{2}+1}}{9}-\frac {8 c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{63}+\frac {16 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{105}-\frac {64 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{315}+\frac {128 \sqrt {c^{2} x^{2}+1}}{315}\right )}{9}-\frac {4 c^{2} d \,e^{3} \left (\frac {c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{35}+\frac {8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {c^{2} x^{2}+1}}{35}\right )}{7}-\frac {6 c^{4} d^{2} e^{2} \left (\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{15}+\frac {8 \sqrt {c^{2} x^{2}+1}}{15}\right )}{5}-\frac {4 c^{6} d^{3} e \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{3}-c^{8} d^{4} \sqrt {c^{2} x^{2}+1}\right )}{c^{8}}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 415, normalized size = 1.33 \[ \frac {1}{9} \, a e^{4} x^{9} + \frac {4}{7} \, a d e^{3} x^{7} + \frac {6}{5} \, a d^{2} e^{2} x^{5} + \frac {4}{3} \, a d^{3} e x^{3} + \frac {4}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d^{3} e + \frac {2}{25} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d^{2} e^{2} + \frac {4}{245} \, {\left (35 \, x^{7} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac {6 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac {16 \, \sqrt {c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b d e^{3} + \frac {1}{2835} \, {\left (315 \, x^{9} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {35 \, \sqrt {c^{2} x^{2} + 1} x^{8}}{c^{2}} - \frac {40 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{6}} - \frac {64 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} + 1}}{c^{10}}\right )} c\right )} b e^{4} + a d^{4} x + \frac {{\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b d^{4}}{c} \]
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 \[ \int \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^4 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 18.00, size = 593, normalized size = 1.90 \[ \begin {cases} a d^{4} x + \frac {4 a d^{3} e x^{3}}{3} + \frac {6 a d^{2} e^{2} x^{5}}{5} + \frac {4 a d e^{3} x^{7}}{7} + \frac {a e^{4} x^{9}}{9} + b d^{4} x \operatorname {asinh}{\left (c x \right )} + \frac {4 b d^{3} e x^{3} \operatorname {asinh}{\left (c x \right )}}{3} + \frac {6 b d^{2} e^{2} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} + \frac {4 b d e^{3} x^{7} \operatorname {asinh}{\left (c x \right )}}{7} + \frac {b e^{4} x^{9} \operatorname {asinh}{\left (c x \right )}}{9} - \frac {b d^{4} \sqrt {c^{2} x^{2} + 1}}{c} - \frac {4 b d^{3} e x^{2} \sqrt {c^{2} x^{2} + 1}}{9 c} - \frac {6 b d^{2} e^{2} x^{4} \sqrt {c^{2} x^{2} + 1}}{25 c} - \frac {4 b d e^{3} x^{6} \sqrt {c^{2} x^{2} + 1}}{49 c} - \frac {b e^{4} x^{8} \sqrt {c^{2} x^{2} + 1}}{81 c} + \frac {8 b d^{3} e \sqrt {c^{2} x^{2} + 1}}{9 c^{3}} + \frac {8 b d^{2} e^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{25 c^{3}} + \frac {24 b d e^{3} x^{4} \sqrt {c^{2} x^{2} + 1}}{245 c^{3}} + \frac {8 b e^{4} x^{6} \sqrt {c^{2} x^{2} + 1}}{567 c^{3}} - \frac {16 b d^{2} e^{2} \sqrt {c^{2} x^{2} + 1}}{25 c^{5}} - \frac {32 b d e^{3} x^{2} \sqrt {c^{2} x^{2} + 1}}{245 c^{5}} - \frac {16 b e^{4} x^{4} \sqrt {c^{2} x^{2} + 1}}{945 c^{5}} + \frac {64 b d e^{3} \sqrt {c^{2} x^{2} + 1}}{245 c^{7}} + \frac {64 b e^{4} x^{2} \sqrt {c^{2} x^{2} + 1}}{2835 c^{7}} - \frac {128 b e^{4} \sqrt {c^{2} x^{2} + 1}}{2835 c^{9}} & \text {for}\: c \neq 0 \\a \left (d^{4} x + \frac {4 d^{3} e x^{3}}{3} + \frac {6 d^{2} e^{2} x^{5}}{5} + \frac {4 d e^{3} x^{7}}{7} + \frac {e^{4} x^{9}}{9}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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