Optimal. Leaf size=160 \[ -\frac {11}{5} b c d^3 \sqrt {1+c^2 x^2}-\frac {1}{5} b c d^3 \left (1+c^2 x^2\right )^{3/2}-\frac {1}{25} b c d^3 \left (1+c^2 x^2\right )^{5/2}-\frac {d^3 \left (a+b \sinh ^{-1}(c x)\right )}{x}+3 c^2 d^3 x \left (a+b \sinh ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} c^6 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )-b c d^3 \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right ) \]
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Rubi [A]
time = 0.14, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {276, 5803, 12,
1813, 1634, 65, 214} \begin {gather*} \frac {1}{5} c^6 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )+3 c^2 d^3 x \left (a+b \sinh ^{-1}(c x)\right )-\frac {d^3 \left (a+b \sinh ^{-1}(c x)\right )}{x}-\frac {1}{25} b c d^3 \left (c^2 x^2+1\right )^{5/2}-\frac {1}{5} b c d^3 \left (c^2 x^2+1\right )^{3/2}-\frac {11}{5} b c d^3 \sqrt {c^2 x^2+1}-b c d^3 \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 214
Rule 276
Rule 1634
Rule 1813
Rule 5803
Rubi steps
\begin {align*} \int \frac {\left (d+c^2 d x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )}{x^2} \, dx &=-\frac {d^3 \left (a+b \sinh ^{-1}(c x)\right )}{x}+3 c^2 d^3 x \left (a+b \sinh ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} c^6 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )-(b c) \int \frac {d^3 \left (-5+15 c^2 x^2+5 c^4 x^4+c^6 x^6\right )}{5 x \sqrt {1+c^2 x^2}} \, dx\\ &=-\frac {d^3 \left (a+b \sinh ^{-1}(c x)\right )}{x}+3 c^2 d^3 x \left (a+b \sinh ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} c^6 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{5} \left (b c d^3\right ) \int \frac {-5+15 c^2 x^2+5 c^4 x^4+c^6 x^6}{x \sqrt {1+c^2 x^2}} \, dx\\ &=-\frac {d^3 \left (a+b \sinh ^{-1}(c x)\right )}{x}+3 c^2 d^3 x \left (a+b \sinh ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} c^6 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{10} \left (b c d^3\right ) \text {Subst}\left (\int \frac {-5+15 c^2 x+5 c^4 x^2+c^6 x^3}{x \sqrt {1+c^2 x}} \, dx,x,x^2\right )\\ &=-\frac {d^3 \left (a+b \sinh ^{-1}(c x)\right )}{x}+3 c^2 d^3 x \left (a+b \sinh ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} c^6 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{10} \left (b c d^3\right ) \text {Subst}\left (\int \left (\frac {11 c^2}{\sqrt {1+c^2 x}}-\frac {5}{x \sqrt {1+c^2 x}}+3 c^2 \sqrt {1+c^2 x}+c^2 \left (1+c^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )\\ &=-\frac {11}{5} b c d^3 \sqrt {1+c^2 x^2}-\frac {1}{5} b c d^3 \left (1+c^2 x^2\right )^{3/2}-\frac {1}{25} b c d^3 \left (1+c^2 x^2\right )^{5/2}-\frac {d^3 \left (a+b \sinh ^{-1}(c x)\right )}{x}+3 c^2 d^3 x \left (a+b \sinh ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} c^6 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{2} \left (b c d^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+c^2 x}} \, dx,x,x^2\right )\\ &=-\frac {11}{5} b c d^3 \sqrt {1+c^2 x^2}-\frac {1}{5} b c d^3 \left (1+c^2 x^2\right )^{3/2}-\frac {1}{25} b c d^3 \left (1+c^2 x^2\right )^{5/2}-\frac {d^3 \left (a+b \sinh ^{-1}(c x)\right )}{x}+3 c^2 d^3 x \left (a+b \sinh ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} c^6 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {\left (b d^3\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {1+c^2 x^2}\right )}{c}\\ &=-\frac {11}{5} b c d^3 \sqrt {1+c^2 x^2}-\frac {1}{5} b c d^3 \left (1+c^2 x^2\right )^{3/2}-\frac {1}{25} b c d^3 \left (1+c^2 x^2\right )^{5/2}-\frac {d^3 \left (a+b \sinh ^{-1}(c x)\right )}{x}+3 c^2 d^3 x \left (a+b \sinh ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} c^6 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )-b c d^3 \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 163, normalized size = 1.02 \begin {gather*} \frac {d^3 \left (-25 a+75 a c^2 x^2+25 a c^4 x^4+5 a c^6 x^6-61 b c x \sqrt {1+c^2 x^2}-7 b c^3 x^3 \sqrt {1+c^2 x^2}-b c^5 x^5 \sqrt {1+c^2 x^2}+5 b \left (-5+15 c^2 x^2+5 c^4 x^4+c^6 x^6\right ) \sinh ^{-1}(c x)+25 b c x \log (x)-25 b c x \log \left (1+\sqrt {1+c^2 x^2}\right )\right )}{25 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.64, size = 151, normalized size = 0.94
method | result | size |
derivativedivides | \(c \left (a \,d^{3} \left (\frac {c^{5} x^{5}}{5}+c^{3} x^{3}+3 c x -\frac {1}{c x}\right )+d^{3} b \left (\frac {\arcsinh \left (c x \right ) c^{5} x^{5}}{5}+\arcsinh \left (c x \right ) c^{3} x^{3}+3 \arcsinh \left (c x \right ) c x -\frac {\arcsinh \left (c x \right )}{c x}-\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{25}-\frac {7 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{25}-\frac {61 \sqrt {c^{2} x^{2}+1}}{25}-\arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )\right )\right )\) | \(151\) |
default | \(c \left (a \,d^{3} \left (\frac {c^{5} x^{5}}{5}+c^{3} x^{3}+3 c x -\frac {1}{c x}\right )+d^{3} b \left (\frac {\arcsinh \left (c x \right ) c^{5} x^{5}}{5}+\arcsinh \left (c x \right ) c^{3} x^{3}+3 \arcsinh \left (c x \right ) c x -\frac {\arcsinh \left (c x \right )}{c x}-\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{25}-\frac {7 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{25}-\frac {61 \sqrt {c^{2} x^{2}+1}}{25}-\arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )\right )\right )\) | \(151\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 231, normalized size = 1.44 \begin {gather*} \frac {1}{5} \, a c^{6} d^{3} x^{5} + \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 c^{6} d^{3} + a c^{4} d^{3} x^{3} + \frac {1}{3} \, {\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 c^{4} d^{3} + 3 \, a c^{2} d^{3} x + 3 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b c d^{3} - {\left (c \operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arsinh}\left (c x\right )}{x}\right )} b d^{3} - \frac {a d^{3}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 276, normalized size = 1.72 \begin {gather*} \frac {5 \, a c^{6} d^{3} x^{6} + 25 \, a c^{4} d^{3} x^{4} + 75 \, a c^{2} d^{3} x^{2} - 25 \, b c d^{3} x \log \left (-c x + \sqrt {c^{2} x^{2} + 1} + 1\right ) + 25 \, b c d^{3} x \log \left (-c x + \sqrt {c^{2} x^{2} + 1} - 1\right ) - 5 \, {\left (b c^{6} + 5 \, b c^{4} + 15 \, b c^{2} - 5 \, b\right )} d^{3} x \log \left (-c x + \sqrt {c^{2} x^{2} + 1}\right ) - 25 \, a d^{3} + 5 \, {\left (b c^{6} d^{3} x^{6} + 5 \, b c^{4} d^{3} x^{4} + 15 \, b c^{2} d^{3} x^{2} - {\left (b c^{6} + 5 \, b c^{4} + 15 \, b c^{2} - 5 \, b\right )} d^{3} x - 5 \, b d^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (b c^{5} d^{3} x^{5} + 7 \, b c^{3} d^{3} x^{3} + 61 \, b c d^{3} x\right )} \sqrt {c^{2} x^{2} + 1}}{25 \, x} \end {gather*}
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 \begin {gather*} d^{3} \left (\int 3 a c^{2}\, dx + \int \frac {a}{x^{2}}\, dx + \int 3 a c^{4} x^{2}\, dx + \int a c^{6} x^{4}\, dx + \int 3 b c^{2} \operatorname {asinh}{\left (c x \right )}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{x^{2}}\, dx + \int 3 b c^{4} x^{2} \operatorname {asinh}{\left (c x \right )}\, dx + \int b c^{6} x^{4} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \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: TypeError} \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 \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^3}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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