Optimal. Leaf size=283 \[ \frac {1}{7} d x^5 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {4 b d x^4 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{175 c}-\frac {2 b d \left (c^2 x^2+1\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c^5}+\frac {4 b d \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{35 c^5}-\frac {2 b d \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{21 c^5}-\frac {32 b d \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{525 c^5}+\frac {16 b d x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{525 c^3}+\frac {2}{35} d x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {304 b^2 d x}{3675 c^4}+\frac {2}{343} b^2 c^2 d x^7-\frac {152 b^2 d x^3}{11025 c^2}+\frac {38 b^2 d x^5}{6125} \]
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Rubi [A] time = 0.48, antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5744, 5661, 5758, 5717, 8, 30, 266, 43, 5732, 12} \[ \frac {1}{7} d x^5 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {4 b d x^4 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{175 c}+\frac {16 b d x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{525 c^3}-\frac {2 b d \left (c^2 x^2+1\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c^5}+\frac {4 b d \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{35 c^5}-\frac {2 b d \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{21 c^5}-\frac {32 b d \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{525 c^5}+\frac {2}{35} d x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{343} b^2 c^2 d x^7-\frac {152 b^2 d x^3}{11025 c^2}+\frac {304 b^2 d x}{3675 c^4}+\frac {38 b^2 d x^5}{6125} \]
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 30
Rule 43
Rule 266
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 ) \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac {1}{7} d x^5 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{7} (2 d) \int x^4 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac {1}{7} (2 b c d) \int x^5 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=-\frac {2 b d \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{21 c^5}+\frac {4 b d \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{35 c^5}-\frac {2 b d \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c^5}+\frac {2}{35} d x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{7} d x^5 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {1}{35} (4 b c d) \int \frac {x^5 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx+\frac {1}{7} \left (2 b^2 c^2 d\right ) \int \frac {8-4 c^2 x^2+3 c^4 x^4+15 c^6 x^6}{105 c^6} \, dx\\ &=-\frac {4 b d x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{175 c}-\frac {2 b d \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{21 c^5}+\frac {4 b d \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{35 c^5}-\frac {2 b d \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c^5}+\frac {2}{35} d x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{7} d x^5 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{175} \left (4 b^2 d\right ) \int x^4 \, dx+\frac {\left (2 b^2 d\right ) \int \left (8-4 c^2 x^2+3 c^4 x^4+15 c^6 x^6\right ) \, dx}{735 c^4}+\frac {(16 b d) \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{175 c}\\ &=\frac {16 b^2 d x}{735 c^4}-\frac {8 b^2 d x^3}{2205 c^2}+\frac {38 b^2 d x^5}{6125}+\frac {2}{343} b^2 c^2 d x^7+\frac {16 b d x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{525 c^3}-\frac {4 b d x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{175 c}-\frac {2 b d \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{21 c^5}+\frac {4 b d \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{35 c^5}-\frac {2 b d \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c^5}+\frac {2}{35} d x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{7} d x^5 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {(32 b d) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{525 c^3}-\frac {\left (16 b^2 d\right ) \int x^2 \, dx}{525 c^2}\\ &=\frac {16 b^2 d x}{735 c^4}-\frac {152 b^2 d x^3}{11025 c^2}+\frac {38 b^2 d x^5}{6125}+\frac {2}{343} b^2 c^2 d x^7-\frac {32 b d \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{525 c^5}+\frac {16 b d x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{525 c^3}-\frac {4 b d x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{175 c}-\frac {2 b d \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{21 c^5}+\frac {4 b d \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{35 c^5}-\frac {2 b d \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c^5}+\frac {2}{35} d x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{7} d x^5 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {\left (32 b^2 d\right ) \int 1 \, dx}{525 c^4}\\ &=\frac {304 b^2 d x}{3675 c^4}-\frac {152 b^2 d x^3}{11025 c^2}+\frac {38 b^2 d x^5}{6125}+\frac {2}{343} b^2 c^2 d x^7-\frac {32 b d \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{525 c^5}+\frac {16 b d x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{525 c^3}-\frac {4 b d x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{175 c}-\frac {2 b d \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{21 c^5}+\frac {4 b d \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{35 c^5}-\frac {2 b d \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c^5}+\frac {2}{35} d x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{7} d x^5 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2\\ \end {align*}
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Mathematica [A] time = 0.29, size = 201, normalized size = 0.71 \[ \frac {d \left (11025 a^2 c^5 x^5 \left (5 c^2 x^2+7\right )-210 a b \sqrt {c^2 x^2+1} \left (75 c^6 x^6+57 c^4 x^4-76 c^2 x^2+152\right )-210 b \sinh ^{-1}(c x) \left (b \sqrt {c^2 x^2+1} \left (75 c^6 x^6+57 c^4 x^4-76 c^2 x^2+152\right )-105 a c^5 x^5 \left (5 c^2 x^2+7\right )\right )+11025 b^2 c^5 x^5 \left (5 c^2 x^2+7\right ) \sinh ^{-1}(c x)^2+b^2 \left (2250 c^7 x^7+2394 c^5 x^5-5320 c^3 x^3+31920 c x\right )\right )}{385875 c^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 260, normalized size = 0.92 \[ \frac {1125 \, {\left (49 \, a^{2} + 2 \, b^{2}\right )} c^{7} d x^{7} + 63 \, {\left (1225 \, a^{2} + 38 \, b^{2}\right )} c^{5} d x^{5} - 5320 \, b^{2} c^{3} d x^{3} + 31920 \, b^{2} c d x + 11025 \, {\left (5 \, b^{2} c^{7} d x^{7} + 7 \, b^{2} c^{5} d x^{5}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 210 \, {\left (525 \, a b c^{7} d x^{7} + 735 \, a b c^{5} d x^{5} - {\left (75 \, b^{2} c^{6} d x^{6} + 57 \, b^{2} c^{4} d x^{4} - 76 \, b^{2} c^{2} d x^{2} + 152 \, b^{2} d\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 210 \, {\left (75 \, a b c^{6} d x^{6} + 57 \, a b c^{4} d x^{4} - 76 \, a b c^{2} d x^{2} + 152 \, a b d\right )} \sqrt {c^{2} x^{2} + 1}}{385875 \, c^{5}} \]
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 = 330, normalized size = 1.17 \[ \frac {d \,a^{2} \left (\frac {1}{7} c^{7} x^{7}+\frac {1}{5} c^{5} x^{5}\right )+d \,b^{2} \left (\frac {\arcsinh \left (c x \right )^{2} c^{3} x^{3} \left (c^{2} x^{2}+1\right )^{2}}{7}-\frac {3 \arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{35}+\frac {2 \arcsinh \left (c x \right )^{2} c x}{35}+\frac {\arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{35}-\frac {2 \arcsinh \left (c x \right ) c^{2} x^{2} \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{49}+\frac {62 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{1225}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{3}}{343}+\frac {37384 c x}{385875}-\frac {484 c x \left (c^{2} x^{2}+1\right )^{2}}{42875}-\frac {3358 c x \left (c^{2} x^{2}+1\right )}{385875}-\frac {4 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{35}-\frac {2 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{105}\right )+2 d a b \left (\frac {\arcsinh \left (c x \right ) c^{7} x^{7}}{7}+\frac {\arcsinh \left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{49}-\frac {19 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{1225}+\frac {76 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3675}-\frac {152 \sqrt {c^{2} x^{2}+1}}{3675}\right )}{c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 441, normalized size = 1.56 \[ \frac {1}{7} \, b^{2} c^{2} d x^{7} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{7} \, a^{2} c^{2} d x^{7} + \frac {1}{5} \, b^{2} d x^{5} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{5} \, a^{2} d x^{5} + \frac {2}{245} \, {\left (35 \, x^{7} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac {6 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac {16 \, \sqrt {c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} a b c^{2} d - \frac {2}{25725} \, {\left (105 \, {\left (\frac {5 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac {6 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac {16 \, \sqrt {c^{2} x^{2} + 1}}{c^{8}}\right )} c \operatorname {arsinh}\left (c x\right ) - \frac {75 \, c^{6} x^{7} - 126 \, c^{4} x^{5} + 280 \, c^{2} x^{3} - 1680 \, x}{c^{6}}\right )} b^{2} c^{2} d + \frac {2}{75} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b d - \frac {2}{1125} \, {\left (15 \, {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c \operatorname {arsinh}\left (c x\right ) - \frac {9 \, c^{4} x^{5} - 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} d \]
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 \[ \int x^4\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\left (d\,c^2\,x^2+d\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.40, size = 388, normalized size = 1.37 \[ \begin {cases} \frac {a^{2} c^{2} d x^{7}}{7} + \frac {a^{2} d x^{5}}{5} + \frac {2 a b c^{2} d x^{7} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {2 a b c d x^{6} \sqrt {c^{2} x^{2} + 1}}{49} + \frac {2 a b d x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {38 a b d x^{4} \sqrt {c^{2} x^{2} + 1}}{1225 c} + \frac {152 a b d x^{2} \sqrt {c^{2} x^{2} + 1}}{3675 c^{3}} - \frac {304 a b d \sqrt {c^{2} x^{2} + 1}}{3675 c^{5}} + \frac {b^{2} c^{2} d x^{7} \operatorname {asinh}^{2}{\left (c x \right )}}{7} + \frac {2 b^{2} c^{2} d x^{7}}{343} - \frac {2 b^{2} c d x^{6} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{49} + \frac {b^{2} d x^{5} \operatorname {asinh}^{2}{\left (c x \right )}}{5} + \frac {38 b^{2} d x^{5}}{6125} - \frac {38 b^{2} d x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{1225 c} - \frac {152 b^{2} d x^{3}}{11025 c^{2}} + \frac {152 b^{2} d x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3675 c^{3}} + \frac {304 b^{2} d x}{3675 c^{4}} - \frac {304 b^{2} d \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3675 c^{5}} & \text {for}\: c \neq 0 \\\frac {a^{2} d x^{5}}{5} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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