Optimal. Leaf size=199 \[ \frac {d^3 \left (c^2 x^2+1\right )^5 \left (a+b \sinh ^{-1}(c x)\right )}{10 c^4}-\frac {d^3 \left (c^2 x^2+1\right )^4 \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4}+\frac {49 b d^3 \sinh ^{-1}(c x)}{5120 c^4}-\frac {b d^3 x \left (c^2 x^2+1\right )^{9/2}}{100 c^3}+\frac {7 b d^3 x \left (c^2 x^2+1\right )^{7/2}}{1600 c^3}+\frac {49 b d^3 x \left (c^2 x^2+1\right )^{5/2}}{9600 c^3}+\frac {49 b d^3 x \left (c^2 x^2+1\right )^{3/2}}{7680 c^3}+\frac {49 b d^3 x \sqrt {c^2 x^2+1}}{5120 c^3} \]
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Rubi [A] time = 0.17, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {266, 43, 5730, 12, 388, 195, 215} \[ \frac {d^3 \left (c^2 x^2+1\right )^5 \left (a+b \sinh ^{-1}(c x)\right )}{10 c^4}-\frac {d^3 \left (c^2 x^2+1\right )^4 \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4}-\frac {b d^3 x \left (c^2 x^2+1\right )^{9/2}}{100 c^3}+\frac {7 b d^3 x \left (c^2 x^2+1\right )^{7/2}}{1600 c^3}+\frac {49 b d^3 x \left (c^2 x^2+1\right )^{5/2}}{9600 c^3}+\frac {49 b d^3 x \left (c^2 x^2+1\right )^{3/2}}{7680 c^3}+\frac {49 b d^3 x \sqrt {c^2 x^2+1}}{5120 c^3}+\frac {49 b d^3 \sinh ^{-1}(c x)}{5120 c^4} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 195
Rule 215
Rule 266
Rule 388
Rule 5730
Rubi steps
\begin {align*} \int x^3 \left (d+c^2 d x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=-\frac {d^3 \left (1+c^2 x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4}+\frac {d^3 \left (1+c^2 x^2\right )^5 \left (a+b \sinh ^{-1}(c x)\right )}{10 c^4}-(b c) \int \frac {d^3 \left (1+c^2 x^2\right )^{7/2} \left (-1+4 c^2 x^2\right )}{40 c^4} \, dx\\ &=-\frac {d^3 \left (1+c^2 x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4}+\frac {d^3 \left (1+c^2 x^2\right )^5 \left (a+b \sinh ^{-1}(c x)\right )}{10 c^4}-\frac {\left (b d^3\right ) \int \left (1+c^2 x^2\right )^{7/2} \left (-1+4 c^2 x^2\right ) \, dx}{40 c^3}\\ &=-\frac {b d^3 x \left (1+c^2 x^2\right )^{9/2}}{100 c^3}-\frac {d^3 \left (1+c^2 x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4}+\frac {d^3 \left (1+c^2 x^2\right )^5 \left (a+b \sinh ^{-1}(c x)\right )}{10 c^4}+\frac {\left (7 b d^3\right ) \int \left (1+c^2 x^2\right )^{7/2} \, dx}{200 c^3}\\ &=\frac {7 b d^3 x \left (1+c^2 x^2\right )^{7/2}}{1600 c^3}-\frac {b d^3 x \left (1+c^2 x^2\right )^{9/2}}{100 c^3}-\frac {d^3 \left (1+c^2 x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4}+\frac {d^3 \left (1+c^2 x^2\right )^5 \left (a+b \sinh ^{-1}(c x)\right )}{10 c^4}+\frac {\left (49 b d^3\right ) \int \left (1+c^2 x^2\right )^{5/2} \, dx}{1600 c^3}\\ &=\frac {49 b d^3 x \left (1+c^2 x^2\right )^{5/2}}{9600 c^3}+\frac {7 b d^3 x \left (1+c^2 x^2\right )^{7/2}}{1600 c^3}-\frac {b d^3 x \left (1+c^2 x^2\right )^{9/2}}{100 c^3}-\frac {d^3 \left (1+c^2 x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4}+\frac {d^3 \left (1+c^2 x^2\right )^5 \left (a+b \sinh ^{-1}(c x)\right )}{10 c^4}+\frac {\left (49 b d^3\right ) \int \left (1+c^2 x^2\right )^{3/2} \, dx}{1920 c^3}\\ &=\frac {49 b d^3 x \left (1+c^2 x^2\right )^{3/2}}{7680 c^3}+\frac {49 b d^3 x \left (1+c^2 x^2\right )^{5/2}}{9600 c^3}+\frac {7 b d^3 x \left (1+c^2 x^2\right )^{7/2}}{1600 c^3}-\frac {b d^3 x \left (1+c^2 x^2\right )^{9/2}}{100 c^3}-\frac {d^3 \left (1+c^2 x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4}+\frac {d^3 \left (1+c^2 x^2\right )^5 \left (a+b \sinh ^{-1}(c x)\right )}{10 c^4}+\frac {\left (49 b d^3\right ) \int \sqrt {1+c^2 x^2} \, dx}{2560 c^3}\\ &=\frac {49 b d^3 x \sqrt {1+c^2 x^2}}{5120 c^3}+\frac {49 b d^3 x \left (1+c^2 x^2\right )^{3/2}}{7680 c^3}+\frac {49 b d^3 x \left (1+c^2 x^2\right )^{5/2}}{9600 c^3}+\frac {7 b d^3 x \left (1+c^2 x^2\right )^{7/2}}{1600 c^3}-\frac {b d^3 x \left (1+c^2 x^2\right )^{9/2}}{100 c^3}-\frac {d^3 \left (1+c^2 x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4}+\frac {d^3 \left (1+c^2 x^2\right )^5 \left (a+b \sinh ^{-1}(c x)\right )}{10 c^4}+\frac {\left (49 b d^3\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{5120 c^3}\\ &=\frac {49 b d^3 x \sqrt {1+c^2 x^2}}{5120 c^3}+\frac {49 b d^3 x \left (1+c^2 x^2\right )^{3/2}}{7680 c^3}+\frac {49 b d^3 x \left (1+c^2 x^2\right )^{5/2}}{9600 c^3}+\frac {7 b d^3 x \left (1+c^2 x^2\right )^{7/2}}{1600 c^3}-\frac {b d^3 x \left (1+c^2 x^2\right )^{9/2}}{100 c^3}+\frac {49 b d^3 \sinh ^{-1}(c x)}{5120 c^4}-\frac {d^3 \left (1+c^2 x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4}+\frac {d^3 \left (1+c^2 x^2\right )^5 \left (a+b \sinh ^{-1}(c x)\right )}{10 c^4}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 139, normalized size = 0.70 \[ \frac {d^3 \left (1920 a c^4 x^4 \left (4 c^6 x^6+15 c^4 x^4+20 c^2 x^2+10\right )+15 b \left (512 c^{10} x^{10}+1920 c^8 x^8+2560 c^6 x^6+1280 c^4 x^4-79\right ) \sinh ^{-1}(c x)-b c x \sqrt {c^2 x^2+1} \left (768 c^8 x^8+2736 c^6 x^6+3208 c^4 x^4+790 c^2 x^2-1185\right )\right )}{76800 c^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 197, normalized size = 0.99 \[ \frac {7680 \, a c^{10} d^{3} x^{10} + 28800 \, a c^{8} d^{3} x^{8} + 38400 \, a c^{6} d^{3} x^{6} + 19200 \, a c^{4} d^{3} x^{4} + 15 \, {\left (512 \, b c^{10} d^{3} x^{10} + 1920 \, b c^{8} d^{3} x^{8} + 2560 \, b c^{6} d^{3} x^{6} + 1280 \, b c^{4} d^{3} x^{4} - 79 \, b d^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (768 \, b c^{9} d^{3} x^{9} + 2736 \, b c^{7} d^{3} x^{7} + 3208 \, b c^{5} d^{3} x^{5} + 790 \, b c^{3} d^{3} x^{3} - 1185 \, b c d^{3} x\right )} \sqrt {c^{2} x^{2} + 1}}{76800 \, c^{4}} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 195, normalized size = 0.98 \[ \frac {d^{3} a \left (\frac {1}{10} c^{10} x^{10}+\frac {3}{8} c^{8} x^{8}+\frac {1}{2} c^{6} x^{6}+\frac {1}{4} c^{4} x^{4}\right )+d^{3} b \left (\frac {\arcsinh \left (c x \right ) c^{10} x^{10}}{10}+\frac {3 \arcsinh \left (c x \right ) c^{8} x^{8}}{8}+\frac {\arcsinh \left (c x \right ) c^{6} x^{6}}{2}+\frac {\arcsinh \left (c x \right ) c^{4} x^{4}}{4}-\frac {c^{9} x^{9} \sqrt {c^{2} x^{2}+1}}{100}-\frac {57 c^{7} x^{7} \sqrt {c^{2} x^{2}+1}}{1600}-\frac {401 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}}{9600}-\frac {79 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{7680}+\frac {79 c x \sqrt {c^{2} x^{2}+1}}{5120}-\frac {79 \arcsinh \left (c x \right )}{5120}\right )}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 429, normalized size = 2.16 \[ \frac {1}{10} \, a c^{6} d^{3} x^{10} + \frac {3}{8} \, a c^{4} d^{3} x^{8} + \frac {1}{2} \, a c^{2} d^{3} x^{6} + \frac {1}{12800} \, {\left (1280 \, x^{10} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {128 \, \sqrt {c^{2} x^{2} + 1} x^{9}}{c^{2}} - \frac {144 \, \sqrt {c^{2} x^{2} + 1} x^{7}}{c^{4}} + \frac {168 \, \sqrt {c^{2} x^{2} + 1} x^{5}}{c^{6}} - \frac {210 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{8}} + \frac {315 \, \sqrt {c^{2} x^{2} + 1} x}{c^{10}} - \frac {315 \, \operatorname {arsinh}\left (c x\right )}{c^{11}}\right )} c\right )} b c^{6} d^{3} + \frac {1}{1024} \, {\left (384 \, x^{8} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {48 \, \sqrt {c^{2} x^{2} + 1} x^{7}}{c^{2}} - \frac {56 \, \sqrt {c^{2} x^{2} + 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{6}} - \frac {105 \, \sqrt {c^{2} x^{2} + 1} x}{c^{8}} + \frac {105 \, \operatorname {arsinh}\left (c x\right )}{c^{9}}\right )} c\right )} b c^{4} d^{3} + \frac {1}{4} \, a d^{3} x^{4} + \frac {1}{96} \, {\left (48 \, x^{6} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{5}}{c^{2}} - \frac {10 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \operatorname {arsinh}\left (c x\right )}{c^{7}}\right )} c\right )} b c^{2} d^{3} + \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c\right )} b d^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 28.24, size = 280, normalized size = 1.41 \[ \begin {cases} \frac {a c^{6} d^{3} x^{10}}{10} + \frac {3 a c^{4} d^{3} x^{8}}{8} + \frac {a c^{2} d^{3} x^{6}}{2} + \frac {a d^{3} x^{4}}{4} + \frac {b c^{6} d^{3} x^{10} \operatorname {asinh}{\left (c x \right )}}{10} - \frac {b c^{5} d^{3} x^{9} \sqrt {c^{2} x^{2} + 1}}{100} + \frac {3 b c^{4} d^{3} x^{8} \operatorname {asinh}{\left (c x \right )}}{8} - \frac {57 b c^{3} d^{3} x^{7} \sqrt {c^{2} x^{2} + 1}}{1600} + \frac {b c^{2} d^{3} x^{6} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {401 b c d^{3} x^{5} \sqrt {c^{2} x^{2} + 1}}{9600} + \frac {b d^{3} x^{4} \operatorname {asinh}{\left (c x \right )}}{4} - \frac {79 b d^{3} x^{3} \sqrt {c^{2} x^{2} + 1}}{7680 c} + \frac {79 b d^{3} x \sqrt {c^{2} x^{2} + 1}}{5120 c^{3}} - \frac {79 b d^{3} \operatorname {asinh}{\left (c x \right )}}{5120 c^{4}} & \text {for}\: c \neq 0 \\\frac {a d^{3} x^{4}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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