Optimal. Leaf size=120 \[ \frac {1}{3} c^4 d^2 x^3 \left (a+b \sinh ^{-1}(c x)\right )+2 c^2 d^2 x \left (a+b \sinh ^{-1}(c x)\right )-\frac {d^2 \left (a+b \sinh ^{-1}(c x)\right )}{x}-\frac {1}{9} b c d^2 \left (c^2 x^2+1\right )^{3/2}-\frac {5}{3} b c d^2 \sqrt {c^2 x^2+1}-b c d^2 \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right ) \]
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Rubi [A] time = 0.16, antiderivative size = 120, 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 = {270, 5730, 12, 1251, 897, 1153, 208} \[ \frac {1}{3} c^4 d^2 x^3 \left (a+b \sinh ^{-1}(c x)\right )+2 c^2 d^2 x \left (a+b \sinh ^{-1}(c x)\right )-\frac {d^2 \left (a+b \sinh ^{-1}(c x)\right )}{x}-\frac {1}{9} b c d^2 \left (c^2 x^2+1\right )^{3/2}-\frac {5}{3} b c d^2 \sqrt {c^2 x^2+1}-b c d^2 \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 208
Rule 270
Rule 897
Rule 1153
Rule 1251
Rule 5730
Rubi steps
\begin {align*} \int \frac {\left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{x^2} \, dx &=-\frac {d^2 \left (a+b \sinh ^{-1}(c x)\right )}{x}+2 c^2 d^2 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^2 x^3 \left (a+b \sinh ^{-1}(c x)\right )-(b c) \int \frac {d^2 \left (-3+6 c^2 x^2+c^4 x^4\right )}{3 x \sqrt {1+c^2 x^2}} \, dx\\ &=-\frac {d^2 \left (a+b \sinh ^{-1}(c x)\right )}{x}+2 c^2 d^2 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^2 x^3 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{3} \left (b c d^2\right ) \int \frac {-3+6 c^2 x^2+c^4 x^4}{x \sqrt {1+c^2 x^2}} \, dx\\ &=-\frac {d^2 \left (a+b \sinh ^{-1}(c x)\right )}{x}+2 c^2 d^2 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^2 x^3 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{6} \left (b c d^2\right ) \operatorname {Subst}\left (\int \frac {-3+6 c^2 x+c^4 x^2}{x \sqrt {1+c^2 x}} \, dx,x,x^2\right )\\ &=-\frac {d^2 \left (a+b \sinh ^{-1}(c x)\right )}{x}+2 c^2 d^2 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^2 x^3 \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (b d^2\right ) \operatorname {Subst}\left (\int \frac {-8+4 x^2+x^4}{-\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {1+c^2 x^2}\right )}{3 c}\\ &=-\frac {d^2 \left (a+b \sinh ^{-1}(c x)\right )}{x}+2 c^2 d^2 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^2 x^3 \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (b d^2\right ) \operatorname {Subst}\left (\int \left (5 c^2+c^2 x^2-\frac {3}{-\frac {1}{c^2}+\frac {x^2}{c^2}}\right ) \, dx,x,\sqrt {1+c^2 x^2}\right )}{3 c}\\ &=-\frac {5}{3} b c d^2 \sqrt {1+c^2 x^2}-\frac {1}{9} b c d^2 \left (1+c^2 x^2\right )^{3/2}-\frac {d^2 \left (a+b \sinh ^{-1}(c x)\right )}{x}+2 c^2 d^2 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^2 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {\left (b d^2\right ) \operatorname {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 {5}{3} b c d^2 \sqrt {1+c^2 x^2}-\frac {1}{9} b c d^2 \left (1+c^2 x^2\right )^{3/2}-\frac {d^2 \left (a+b \sinh ^{-1}(c x)\right )}{x}+2 c^2 d^2 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^2 x^3 \left (a+b \sinh ^{-1}(c x)\right )-b c d^2 \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.12, size = 124, normalized size = 1.03 \[ \frac {d^2 \left (3 a c^4 x^4+18 a c^2 x^2-9 a-16 b c x \sqrt {c^2 x^2+1}-9 b c x \log \left (\sqrt {c^2 x^2+1}+1\right )+3 b \left (c^4 x^4+6 c^2 x^2-3\right ) \sinh ^{-1}(c x)-b c^3 x^3 \sqrt {c^2 x^2+1}+9 b c x \log (x)\right )}{9 x} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 228, normalized size = 1.90 \[ \frac {3 \, a c^{4} d^{2} x^{4} + 18 \, a c^{2} d^{2} x^{2} - 9 \, b c d^{2} x \log \left (-c x + \sqrt {c^{2} x^{2} + 1} + 1\right ) + 9 \, b c d^{2} x \log \left (-c x + \sqrt {c^{2} x^{2} + 1} - 1\right ) - 3 \, {\left (b c^{4} + 6 \, b c^{2} - 3 \, b\right )} d^{2} x \log \left (-c x + \sqrt {c^{2} x^{2} + 1}\right ) - 9 \, a d^{2} + 3 \, {\left (b c^{4} d^{2} x^{4} + 6 \, b c^{2} d^{2} x^{2} - {\left (b c^{4} + 6 \, b c^{2} - 3 \, b\right )} d^{2} x - 3 \, b d^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (b c^{3} d^{2} x^{3} + 16 \, b c d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}}{9 \, x} \]
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 [A] time = 0.01, size = 114, normalized size = 0.95 \[ c \left (d^{2} a \left (\frac {c^{3} x^{3}}{3}+2 c x -\frac {1}{c x}\right )+d^{2} b \left (\frac {\arcsinh \left (c x \right ) c^{3} x^{3}}{3}+2 \arcsinh \left (c x \right ) c x -\frac {\arcsinh \left (c x \right )}{c x}-\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {16 \sqrt {c^{2} x^{2}+1}}{9}-\arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 143, normalized size = 1.19 \[ \frac {1}{3} \, a c^{4} d^{2} x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b c^{4} d^{2} + 2 \, a c^{2} d^{2} x + 2 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b c d^{2} - {\left (c \operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arsinh}\left (c x\right )}{x}\right )} b d^{2} - \frac {a d^{2}}{x} \]
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 \[ \int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^2}{x^2} \,d x \]
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 \[ d^{2} \left (\int 2 a c^{2}\, dx + \int \frac {a}{x^{2}}\, dx + \int a c^{4} x^{2}\, dx + \int 2 b c^{2} \operatorname {asinh}{\left (c x \right )}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{x^{2}}\, dx + \int b c^{4} x^{2} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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