Optimal. Leaf size=147 \[ -\frac {b \left (15 c^4 d^2-10 c^2 d e+3 e^2\right ) \sqrt {1+c^2 x^2}}{15 c^5}-\frac {2 b \left (5 c^2 d-3 e\right ) e \left (1+c^2 x^2\right )^{3/2}}{45 c^5}-\frac {b e^2 \left (1+c^2 x^2\right )^{5/2}}{25 c^5}+d^2 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right ) \]
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Rubi [A]
time = 0.10, antiderivative size = 147, 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 = {200, 5792, 12,
1261, 712} \begin {gather*} d^2 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )-\frac {2 b e \left (c^2 x^2+1\right )^{3/2} \left (5 c^2 d-3 e\right )}{45 c^5}-\frac {b e^2 \left (c^2 x^2+1\right )^{5/2}}{25 c^5}-\frac {b \sqrt {c^2 x^2+1} \left (15 c^4 d^2-10 c^2 d e+3 e^2\right )}{15 c^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 200
Rule 712
Rule 1261
Rule 5792
Rubi steps
\begin {align*} \int \left (d+e x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=d^2 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )-(b c) \int \frac {x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )}{15 \sqrt {1+c^2 x^2}} \, dx\\ &=d^2 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{15} (b c) \int \frac {x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )}{\sqrt {1+c^2 x^2}} \, dx\\ &=d^2 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{30} (b c) \text {Subst}\left (\int \frac {15 d^2+10 d e x+3 e^2 x^2}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )\\ &=d^2 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{30} (b c) \text {Subst}\left (\int \left (\frac {15 c^4 d^2-10 c^2 d e+3 e^2}{c^4 \sqrt {1+c^2 x}}+\frac {2 \left (5 c^2 d-3 e\right ) e \sqrt {1+c^2 x}}{c^4}+\frac {3 e^2 \left (1+c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )\\ &=-\frac {b \left (15 c^4 d^2-10 c^2 d e+3 e^2\right ) \sqrt {1+c^2 x^2}}{15 c^5}-\frac {2 b \left (5 c^2 d-3 e\right ) e \left (1+c^2 x^2\right )^{3/2}}{45 c^5}-\frac {b e^2 \left (1+c^2 x^2\right )^{5/2}}{25 c^5}+d^2 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 125, normalized size = 0.85 \begin {gather*} \frac {1}{225} \left (15 a x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )-\frac {b \sqrt {1+c^2 x^2} \left (24 e^2-4 c^2 e \left (25 d+3 e x^2\right )+c^4 \left (225 d^2+50 d e x^2+9 e^2 x^4\right )\right )}{c^5}+15 b x \left (15 d^2+10 d e x^2+3 e^2 x^4\right ) \sinh ^{-1}(c x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.64, size = 204, normalized size = 1.39
method | result | size |
derivativedivides | \(\frac {\frac {a \left (d^{2} c^{5} x +\frac {2}{3} d \,c^{5} e \,x^{3}+\frac {1}{5} e^{2} c^{5} x^{5}\right )}{c^{4}}+\frac {b \left (\arcsinh \left (c x \right ) d^{2} c^{5} x +\frac {2 \arcsinh \left (c x \right ) d \,c^{5} e \,x^{3}}{3}+\frac {\arcsinh \left (c x \right ) e^{2} c^{5} x^{5}}{5}-d^{2} c^{4} \sqrt {c^{2} x^{2}+1}-\frac {2 d \,c^{2} e \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{3}-\frac {e^{2} \left (\frac {\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}}{5}-\frac {4 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{15}+\frac {8 \sqrt {c^{2} x^{2}+1}}{15}\right )}{5}\right )}{c^{4}}}{c}\) | \(204\) |
default | \(\frac {\frac {a \left (d^{2} c^{5} x +\frac {2}{3} d \,c^{5} e \,x^{3}+\frac {1}{5} e^{2} c^{5} x^{5}\right )}{c^{4}}+\frac {b \left (\arcsinh \left (c x \right ) d^{2} c^{5} x +\frac {2 \arcsinh \left (c x \right ) d \,c^{5} e \,x^{3}}{3}+\frac {\arcsinh \left (c x \right ) e^{2} c^{5} x^{5}}{5}-d^{2} c^{4} \sqrt {c^{2} x^{2}+1}-\frac {2 d \,c^{2} e \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{3}-\frac {e^{2} \left (\frac {\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}}{5}-\frac {4 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{15}+\frac {8 \sqrt {c^{2} x^{2}+1}}{15}\right )}{5}\right )}{c^{4}}}{c}\) | \(204\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 180, normalized size = 1.22 \begin {gather*} \frac {1}{5} \, a x^{5} e^{2} + \frac {2}{3} \, a d x^{3} e + a d^{2} x + \frac {2}{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 e + \frac {{\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b d^{2}}{c} + \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 e^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 321 vs.
\(2 (132) = 264\).
time = 0.40, size = 321, normalized size = 2.18 \begin {gather*} \frac {45 \, a c^{5} x^{5} \cosh \left (1\right )^{2} + 45 \, a c^{5} x^{5} \sinh \left (1\right )^{2} + 150 \, a c^{5} d x^{3} \cosh \left (1\right ) + 225 \, a c^{5} d^{2} x + 15 \, {\left (3 \, b c^{5} x^{5} \cosh \left (1\right )^{2} + 3 \, b c^{5} x^{5} \sinh \left (1\right )^{2} + 10 \, b c^{5} d x^{3} \cosh \left (1\right ) + 15 \, b c^{5} d^{2} x + 2 \, {\left (3 \, b c^{5} x^{5} \cosh \left (1\right ) + 5 \, b c^{5} d x^{3}\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 30 \, {\left (3 \, a c^{5} x^{5} \cosh \left (1\right ) + 5 \, a c^{5} d x^{3}\right )} \sinh \left (1\right ) - {\left (225 \, b c^{4} d^{2} + 3 \, {\left (3 \, b c^{4} x^{4} - 4 \, b c^{2} x^{2} + 8 \, b\right )} \cosh \left (1\right )^{2} + 3 \, {\left (3 \, b c^{4} x^{4} - 4 \, b c^{2} x^{2} + 8 \, b\right )} \sinh \left (1\right )^{2} + 50 \, {\left (b c^{4} d x^{2} - 2 \, b c^{2} d\right )} \cosh \left (1\right ) + 2 \, {\left (25 \, b c^{4} d x^{2} - 50 \, b c^{2} d + 3 \, {\left (3 \, b c^{4} x^{4} - 4 \, b c^{2} x^{2} + 8 \, b\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} + 1}}{225 \, c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.36, size = 240, normalized size = 1.63 \begin {gather*} \begin {cases} a d^{2} x + \frac {2 a d e x^{3}}{3} + \frac {a e^{2} x^{5}}{5} + b d^{2} x \operatorname {asinh}{\left (c x \right )} + \frac {2 b d e x^{3} \operatorname {asinh}{\left (c x \right )}}{3} + \frac {b e^{2} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {b d^{2} \sqrt {c^{2} x^{2} + 1}}{c} - \frac {2 b d e x^{2} \sqrt {c^{2} x^{2} + 1}}{9 c} - \frac {b e^{2} x^{4} \sqrt {c^{2} x^{2} + 1}}{25 c} + \frac {4 b d e \sqrt {c^{2} x^{2} + 1}}{9 c^{3}} + \frac {4 b e^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{75 c^{3}} - \frac {8 b e^{2} \sqrt {c^{2} x^{2} + 1}}{75 c^{5}} & \text {for}\: c \neq 0 \\a \left (d^{2} x + \frac {2 d e x^{3}}{3} + \frac {e^{2} x^{5}}{5}\right ) & \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 \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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