Optimal. Leaf size=482 \[ \frac {d x^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{35 c^2}-\frac {16 b c d x^5 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{175 \sqrt {c^2 x^2+1}}+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {3}{35} d x^4 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {2 b d x^3 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{105 c \sqrt {c^2 x^2+1}}-\frac {2 d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{35 c^4}+\frac {4 a b d x \sqrt {c^2 d x^2+d}}{35 c^3 \sqrt {c^2 x^2+1}}-\frac {2 b c^3 d x^7 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{49 \sqrt {c^2 x^2+1}}+\frac {2 b^2 d \left (c^2 x^2+1\right )^3 \sqrt {c^2 d x^2+d}}{343 c^4}-\frac {38 b^2 d \left (c^2 x^2+1\right )^2 \sqrt {c^2 d x^2+d}}{6125 c^4}-\frac {304 b^2 d \sqrt {c^2 d x^2+d}}{3675 c^4}-\frac {152 b^2 d \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d}}{11025 c^4}+\frac {4 b^2 d x \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{35 c^3 \sqrt {c^2 x^2+1}} \]
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Rubi [A] time = 0.81, antiderivative size = 482, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 14, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5744, 5742, 5758, 5717, 5653, 261, 5661, 266, 43, 14, 5730, 12, 446, 77} \[ \frac {4 a b d x \sqrt {c^2 d x^2+d}}{35 c^3 \sqrt {c^2 x^2+1}}-\frac {2 b c^3 d x^7 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{49 \sqrt {c^2 x^2+1}}-\frac {16 b c d x^5 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{175 \sqrt {c^2 x^2+1}}+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {3}{35} d x^4 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {2 b d x^3 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{105 c \sqrt {c^2 x^2+1}}+\frac {d x^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{35 c^2}-\frac {2 d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{35 c^4}+\frac {2 b^2 d \left (c^2 x^2+1\right )^3 \sqrt {c^2 d x^2+d}}{343 c^4}-\frac {38 b^2 d \left (c^2 x^2+1\right )^2 \sqrt {c^2 d x^2+d}}{6125 c^4}-\frac {304 b^2 d \sqrt {c^2 d x^2+d}}{3675 c^4}-\frac {152 b^2 d \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d}}{11025 c^4}+\frac {4 b^2 d x \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{35 c^3 \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 43
Rule 77
Rule 261
Rule 266
Rule 446
Rule 5653
Rule 5661
Rule 5717
Rule 5730
Rule 5742
Rule 5744
Rule 5758
Rubi steps
\begin {align*} \int x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac {1}{7} x^4 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{7} (3 d) \int x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac {\left (2 b c d \sqrt {d+c^2 d x^2}\right ) \int x^4 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{7 \sqrt {1+c^2 x^2}}\\ &=-\frac {2 b c d x^5 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{35 \sqrt {1+c^2 x^2}}-\frac {2 b c^3 d x^7 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{49 \sqrt {1+c^2 x^2}}+\frac {3}{35} d x^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{7} x^4 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {\left (3 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}} \, dx}{35 \sqrt {1+c^2 x^2}}-\frac {\left (6 b c d \sqrt {d+c^2 d x^2}\right ) \int x^4 \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{35 \sqrt {1+c^2 x^2}}+\frac {\left (2 b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^5 \left (7+5 c^2 x^2\right )}{35 \sqrt {1+c^2 x^2}} \, dx}{7 \sqrt {1+c^2 x^2}}\\ &=-\frac {16 b c d x^5 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{175 \sqrt {1+c^2 x^2}}-\frac {2 b c^3 d x^7 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{49 \sqrt {1+c^2 x^2}}+\frac {d x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{35 c^2}+\frac {3}{35} d x^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{7} x^4 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}} \, dx}{35 c^2 \sqrt {1+c^2 x^2}}-\frac {\left (2 b d \sqrt {d+c^2 d x^2}\right ) \int x^2 \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{35 c \sqrt {1+c^2 x^2}}+\frac {\left (2 b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^5 \left (7+5 c^2 x^2\right )}{\sqrt {1+c^2 x^2}} \, dx}{245 \sqrt {1+c^2 x^2}}+\frac {\left (6 b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^5}{\sqrt {1+c^2 x^2}} \, dx}{175 \sqrt {1+c^2 x^2}}\\ &=-\frac {2 b d x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{105 c \sqrt {1+c^2 x^2}}-\frac {16 b c d x^5 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{175 \sqrt {1+c^2 x^2}}-\frac {2 b c^3 d x^7 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{49 \sqrt {1+c^2 x^2}}-\frac {2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{35 c^4}+\frac {d x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{35 c^2}+\frac {3}{35} d x^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{7} x^4 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {\left (2 b^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^3}{\sqrt {1+c^2 x^2}} \, dx}{105 \sqrt {1+c^2 x^2}}+\frac {\left (4 b d \sqrt {d+c^2 d x^2}\right ) \int \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{35 c^3 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (7+5 c^2 x\right )}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )}{245 \sqrt {1+c^2 x^2}}+\frac {\left (3 b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )}{175 \sqrt {1+c^2 x^2}}\\ &=\frac {4 a b d x \sqrt {d+c^2 d x^2}}{35 c^3 \sqrt {1+c^2 x^2}}-\frac {2 b d x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{105 c \sqrt {1+c^2 x^2}}-\frac {16 b c d x^5 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{175 \sqrt {1+c^2 x^2}}-\frac {2 b c^3 d x^7 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{49 \sqrt {1+c^2 x^2}}-\frac {2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{35 c^4}+\frac {d x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{35 c^2}+\frac {3}{35} d x^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{7} x^4 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {\left (b^2 d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )}{105 \sqrt {1+c^2 x^2}}+\frac {\left (4 b^2 d \sqrt {d+c^2 d x^2}\right ) \int \sinh ^{-1}(c x) \, dx}{35 c^3 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {2}{c^4 \sqrt {1+c^2 x}}+\frac {\sqrt {1+c^2 x}}{c^4}-\frac {8 \left (1+c^2 x\right )^{3/2}}{c^4}+\frac {5 \left (1+c^2 x\right )^{5/2}}{c^4}\right ) \, dx,x,x^2\right )}{245 \sqrt {1+c^2 x^2}}+\frac {\left (3 b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{c^4 \sqrt {1+c^2 x}}-\frac {2 \sqrt {1+c^2 x}}{c^4}+\frac {\left (1+c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )}{175 \sqrt {1+c^2 x^2}}\\ &=\frac {62 b^2 d \sqrt {d+c^2 d x^2}}{1225 c^4}+\frac {4 a b d x \sqrt {d+c^2 d x^2}}{35 c^3 \sqrt {1+c^2 x^2}}-\frac {74 b^2 d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{3675 c^4}-\frac {38 b^2 d \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2}}{6125 c^4}+\frac {2 b^2 d \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2}}{343 c^4}+\frac {4 b^2 d x \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{35 c^3 \sqrt {1+c^2 x^2}}-\frac {2 b d x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{105 c \sqrt {1+c^2 x^2}}-\frac {16 b c d x^5 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{175 \sqrt {1+c^2 x^2}}-\frac {2 b c^3 d x^7 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{49 \sqrt {1+c^2 x^2}}-\frac {2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{35 c^4}+\frac {d x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{35 c^2}+\frac {3}{35} d x^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{7} x^4 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {\left (b^2 d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{c^2 \sqrt {1+c^2 x}}+\frac {\sqrt {1+c^2 x}}{c^2}\right ) \, dx,x,x^2\right )}{105 \sqrt {1+c^2 x^2}}-\frac {\left (4 b^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x}{\sqrt {1+c^2 x^2}} \, dx}{35 c^2 \sqrt {1+c^2 x^2}}\\ &=-\frac {304 b^2 d \sqrt {d+c^2 d x^2}}{3675 c^4}+\frac {4 a b d x \sqrt {d+c^2 d x^2}}{35 c^3 \sqrt {1+c^2 x^2}}-\frac {152 b^2 d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{11025 c^4}-\frac {38 b^2 d \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2}}{6125 c^4}+\frac {2 b^2 d \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2}}{343 c^4}+\frac {4 b^2 d x \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{35 c^3 \sqrt {1+c^2 x^2}}-\frac {2 b d x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{105 c \sqrt {1+c^2 x^2}}-\frac {16 b c d x^5 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{175 \sqrt {1+c^2 x^2}}-\frac {2 b c^3 d x^7 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{49 \sqrt {1+c^2 x^2}}-\frac {2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{35 c^4}+\frac {d x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{35 c^2}+\frac {3}{35} d x^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{7} x^4 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2\\ \end {align*}
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Mathematica [A] time = 0.39, size = 251, normalized size = 0.52 \[ \frac {d \sqrt {c^2 d x^2+d} \left (11025 a^2 \left (5 c^2 x^2-2\right ) \left (c^2 x^2+1\right )^3-210 a b c x \left (75 c^6 x^6+168 c^4 x^4+35 c^2 x^2-210\right ) \sqrt {c^2 x^2+1}-210 b \sinh ^{-1}(c x) \left (b c x \sqrt {c^2 x^2+1} \left (75 c^6 x^6+168 c^4 x^4+35 c^2 x^2-210\right )-105 a \left (c^2 x^2+1\right )^3 \left (5 c^2 x^2-2\right )\right )+11025 b^2 \left (5 c^2 x^2-2\right ) \left (c^2 x^2+1\right )^3 \sinh ^{-1}(c x)^2+2 b^2 \left (1125 c^8 x^8+3303 c^6 x^6+499 c^4 x^4-20371 c^2 x^2-18692\right )\right )}{385875 c^4 \left (c^2 x^2+1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 402, normalized size = 0.83 \[ \frac {11025 \, {\left (5 \, b^{2} c^{8} d x^{8} + 13 \, b^{2} c^{6} d x^{6} + 9 \, b^{2} c^{4} d x^{4} - b^{2} c^{2} d x^{2} - 2 \, b^{2} d\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 210 \, {\left (525 \, a b c^{8} d x^{8} + 1365 \, a b c^{6} d x^{6} + 945 \, a b c^{4} d x^{4} - 105 \, a b c^{2} d x^{2} - 210 \, a b d - {\left (75 \, b^{2} c^{7} d x^{7} + 168 \, b^{2} c^{5} d x^{5} + 35 \, b^{2} c^{3} d x^{3} - 210 \, b^{2} c d x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (1125 \, {\left (49 \, a^{2} + 2 \, b^{2}\right )} c^{8} d x^{8} + 9 \, {\left (15925 \, a^{2} + 734 \, b^{2}\right )} c^{6} d x^{6} + {\left (99225 \, a^{2} + 998 \, b^{2}\right )} c^{4} d x^{4} - {\left (11025 \, a^{2} + 40742 \, b^{2}\right )} c^{2} d x^{2} - 2 \, {\left (11025 \, a^{2} + 18692 \, b^{2}\right )} d - 210 \, {\left (75 \, a b c^{7} d x^{7} + 168 \, a b c^{5} d x^{5} + 35 \, a b c^{3} d x^{3} - 210 \, a b c d x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d}}{385875 \, {\left (c^{6} x^{2} + c^{4}\right )}} \]
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 [B] time = 0.54, size = 1766, normalized size = 3.66 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 346, normalized size = 0.72 \[ \frac {1}{35} \, {\left (\frac {5 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} - \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} b^{2} \operatorname {arsinh}\left (c x\right )^{2} + \frac {2}{35} \, {\left (\frac {5 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} - \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} a b \operatorname {arsinh}\left (c x\right ) + \frac {1}{35} \, {\left (\frac {5 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} - \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} a^{2} + \frac {2}{385875} \, b^{2} {\left (\frac {1125 \, \sqrt {c^{2} x^{2} + 1} c^{4} d^{\frac {3}{2}} x^{6} + 2178 \, \sqrt {c^{2} x^{2} + 1} c^{2} d^{\frac {3}{2}} x^{4} - 1679 \, \sqrt {c^{2} x^{2} + 1} d^{\frac {3}{2}} x^{2} - \frac {18692 \, \sqrt {c^{2} x^{2} + 1} d^{\frac {3}{2}}}{c^{2}}}{c^{2}} - \frac {105 \, {\left (75 \, c^{6} d^{\frac {3}{2}} x^{7} + 168 \, c^{4} d^{\frac {3}{2}} x^{5} + 35 \, c^{2} d^{\frac {3}{2}} x^{3} - 210 \, d^{\frac {3}{2}} x\right )} \operatorname {arsinh}\left (c x\right )}{c^{3}}\right )} - \frac {2 \, {\left (75 \, c^{6} d^{\frac {3}{2}} x^{7} + 168 \, c^{4} d^{\frac {3}{2}} x^{5} + 35 \, c^{2} d^{\frac {3}{2}} x^{3} - 210 \, d^{\frac {3}{2}} x\right )} a b}{3675 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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