Optimal. Leaf size=124 \[ -\frac {2 b d \sqrt {1+c^2 x^2}}{35 c^5}-\frac {b d \left (1+c^2 x^2\right )^{3/2}}{105 c^5}+\frac {8 b d \left (1+c^2 x^2\right )^{5/2}}{175 c^5}-\frac {b d \left (1+c^2 x^2\right )^{7/2}}{49 c^5}+\frac {1}{5} d x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{7} c^2 d x^7 \left (a+b \sinh ^{-1}(c x)\right ) \]
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Rubi [A]
time = 0.07, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {14, 5803, 12,
457, 78} \begin {gather*} \frac {1}{7} c^2 d x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} d x^5 \left (a+b \sinh ^{-1}(c x)\right )-\frac {b d \left (c^2 x^2+1\right )^{7/2}}{49 c^5}+\frac {8 b d \left (c^2 x^2+1\right )^{5/2}}{175 c^5}-\frac {b d \left (c^2 x^2+1\right )^{3/2}}{105 c^5}-\frac {2 b d \sqrt {c^2 x^2+1}}{35 c^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 78
Rule 457
Rule 5803
Rubi steps
\begin {align*} \int x^4 \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac {1}{5} d x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{7} c^2 d x^7 \left (a+b \sinh ^{-1}(c x)\right )-(b c) \int \frac {d x^5 \left (7+5 c^2 x^2\right )}{35 \sqrt {1+c^2 x^2}} \, dx\\ &=\frac {1}{5} d x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{7} c^2 d x^7 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{35} (b c d) \int \frac {x^5 \left (7+5 c^2 x^2\right )}{\sqrt {1+c^2 x^2}} \, dx\\ &=\frac {1}{5} d x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{7} c^2 d x^7 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{70} (b c d) \text {Subst}\left (\int \frac {x^2 \left (7+5 c^2 x\right )}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )\\ &=\frac {1}{5} d x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{7} c^2 d x^7 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{70} (b c d) \text {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 )\\ &=-\frac {2 b d \sqrt {1+c^2 x^2}}{35 c^5}-\frac {b d \left (1+c^2 x^2\right )^{3/2}}{105 c^5}+\frac {8 b d \left (1+c^2 x^2\right )^{5/2}}{175 c^5}-\frac {b d \left (1+c^2 x^2\right )^{7/2}}{49 c^5}+\frac {1}{5} d x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{7} c^2 d x^7 \left (a+b \sinh ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 87, normalized size = 0.70 \begin {gather*} \frac {d \left (105 a x^5 \left (7+5 c^2 x^2\right )-\frac {b \sqrt {1+c^2 x^2} \left (152-76 c^2 x^2+57 c^4 x^4+75 c^6 x^6\right )}{c^5}+105 b x^5 \left (7+5 c^2 x^2\right ) \sinh ^{-1}(c x)\right )}{3675} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.96, size = 124, normalized size = 1.00
method | result | size |
derivativedivides | \(\frac {d a \left (\frac {1}{7} c^{7} x^{7}+\frac {1}{5} c^{5} x^{5}\right )+b d \left (\frac {\arcsinh \left (c x \right ) c^{7} x^{7}}{7}+\frac {\arcsinh \left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{49}-\frac {19 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{1225}+\frac {76 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3675}-\frac {152 \sqrt {c^{2} x^{2}+1}}{3675}\right )}{c^{5}}\) | \(124\) |
default | \(\frac {d a \left (\frac {1}{7} c^{7} x^{7}+\frac {1}{5} c^{5} x^{5}\right )+b d \left (\frac {\arcsinh \left (c x \right ) c^{7} x^{7}}{7}+\frac {\arcsinh \left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{49}-\frac {19 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{1225}+\frac {76 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3675}-\frac {152 \sqrt {c^{2} x^{2}+1}}{3675}\right )}{c^{5}}\) | \(124\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 184, normalized size = 1.48 \begin {gather*} \frac {1}{7} \, a c^{2} d x^{7} + \frac {1}{5} \, a d x^{5} + \frac {1}{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 c^{2} d + \frac {1}{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 )} b d \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 113, normalized size = 0.91 \begin {gather*} \frac {525 \, a c^{7} d x^{7} + 735 \, a c^{5} d x^{5} + 105 \, {\left (5 \, b c^{7} d x^{7} + 7 \, b c^{5} d x^{5}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (75 \, b c^{6} d x^{6} + 57 \, b c^{4} d x^{4} - 76 \, b c^{2} d x^{2} + 152 \, b d\right )} \sqrt {c^{2} x^{2} + 1}}{3675 \, c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.76, size = 151, normalized size = 1.22 \begin {gather*} \begin {cases} \frac {a c^{2} d x^{7}}{7} + \frac {a d x^{5}}{5} + \frac {b c^{2} d x^{7} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {b c d x^{6} \sqrt {c^{2} x^{2} + 1}}{49} + \frac {b d x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {19 b d x^{4} \sqrt {c^{2} x^{2} + 1}}{1225 c} + \frac {76 b d x^{2} \sqrt {c^{2} x^{2} + 1}}{3675 c^{3}} - \frac {152 b d \sqrt {c^{2} x^{2} + 1}}{3675 c^{5}} & \text {for}\: c \neq 0 \\\frac {a d x^{5}}{5} & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
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 \begin {gather*} \int x^4\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\left (d\,c^2\,x^2+d\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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