Optimal. Leaf size=355 \[ \frac {5}{2} c^2 d^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )-\frac {5 c^2 d^2 \sqrt {c^2 d x^2+d} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 x^2+1}}+\frac {5}{6} c^2 d \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (c^2 d x^2+d\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}-\frac {5 b c^2 d^2 \sqrt {c^2 d x^2+d} \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {5 b c^2 d^2 \sqrt {c^2 d x^2+d} \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt {c^2 x^2+1}}-\frac {b c d^2 \sqrt {c^2 d x^2+d}}{2 x \sqrt {c^2 x^2+1}}-\frac {b c^5 d^2 x^3 \sqrt {c^2 d x^2+d}}{9 \sqrt {c^2 x^2+1}}-\frac {7 b c^3 d^2 x \sqrt {c^2 d x^2+d}}{3 \sqrt {c^2 x^2+1}} \]
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Rubi [A] time = 0.45, antiderivative size = 355, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {5739, 5744, 5742, 5760, 4182, 2279, 2391, 8, 270} \[ -\frac {5 b c^2 d^2 \sqrt {c^2 d x^2+d} \text {PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {5 b c^2 d^2 \sqrt {c^2 d x^2+d} \text {PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {5}{2} c^2 d^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )-\frac {5 c^2 d^2 \sqrt {c^2 d x^2+d} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 x^2+1}}+\frac {5}{6} c^2 d \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (c^2 d x^2+d\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}-\frac {b c^5 d^2 x^3 \sqrt {c^2 d x^2+d}}{9 \sqrt {c^2 x^2+1}}-\frac {7 b c^3 d^2 x \sqrt {c^2 d x^2+d}}{3 \sqrt {c^2 x^2+1}}-\frac {b c d^2 \sqrt {c^2 d x^2+d}}{2 x \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 270
Rule 2279
Rule 2391
Rule 4182
Rule 5739
Rule 5742
Rule 5744
Rule 5760
Rubi steps
\begin {align*} \int \frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{x^3} \, dx &=-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} \left (5 c^2 d\right ) \int \frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx+\frac {\left (b c d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (1+c^2 x^2\right )^2}{x^2} \, dx}{2 \sqrt {1+c^2 x^2}}\\ &=\frac {5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} \left (5 c^2 d^2\right ) \int \frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx+\frac {\left (b c d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (2 c^2+\frac {1}{x^2}+c^4 x^2\right ) \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right ) \, dx}{6 \sqrt {1+c^2 x^2}}\\ &=-\frac {b c d^2 \sqrt {d+c^2 d x^2}}{2 x \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 x \sqrt {d+c^2 d x^2}}{6 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^3 \sqrt {d+c^2 d x^2}}{9 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^2 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}+\frac {\left (5 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x \sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \int 1 \, dx}{2 \sqrt {1+c^2 x^2}}\\ &=-\frac {b c d^2 \sqrt {d+c^2 d x^2}}{2 x \sqrt {1+c^2 x^2}}-\frac {7 b c^3 d^2 x \sqrt {d+c^2 d x^2}}{3 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^3 \sqrt {d+c^2 d x^2}}{9 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^2 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}+\frac {\left (5 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}\\ &=-\frac {b c d^2 \sqrt {d+c^2 d x^2}}{2 x \sqrt {1+c^2 x^2}}-\frac {7 b c^3 d^2 x \sqrt {d+c^2 d x^2}}{3 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^3 \sqrt {d+c^2 d x^2}}{9 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^2 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}-\frac {5 c^2 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (5 b c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+\frac {\left (5 b c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}\\ &=-\frac {b c d^2 \sqrt {d+c^2 d x^2}}{2 x \sqrt {1+c^2 x^2}}-\frac {7 b c^3 d^2 x \sqrt {d+c^2 d x^2}}{3 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^3 \sqrt {d+c^2 d x^2}}{9 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^2 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}-\frac {5 c^2 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (5 b c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt {1+c^2 x^2}}+\frac {\left (5 b c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt {1+c^2 x^2}}\\ &=-\frac {b c d^2 \sqrt {d+c^2 d x^2}}{2 x \sqrt {1+c^2 x^2}}-\frac {7 b c^3 d^2 x \sqrt {d+c^2 d x^2}}{3 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^3 \sqrt {d+c^2 d x^2}}{9 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^2 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}-\frac {5 c^2 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {5 b c^2 d^2 \sqrt {d+c^2 d x^2} \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt {1+c^2 x^2}}+\frac {5 b c^2 d^2 \sqrt {d+c^2 d x^2} \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 6.74, size = 467, normalized size = 1.32 \[ -\frac {5}{2} a c^2 d^{5/2} \log \left (\sqrt {d} \sqrt {d \left (c^2 x^2+1\right )}+d\right )+\frac {5}{2} a c^2 d^{5/2} \log (x)+\sqrt {d \left (c^2 x^2+1\right )} \left (\frac {1}{3} a c^4 d^2 x^2+\frac {7}{3} a c^2 d^2-\frac {a d^2}{2 x^2}\right )+\frac {2 b c^2 d^2 \sqrt {d \left (c^2 x^2+1\right )} \left (\sqrt {c^2 x^2+1} \sinh ^{-1}(c x)+\text {Li}_2\left (-e^{-\sinh ^{-1}(c x)}\right )-\text {Li}_2\left (e^{-\sinh ^{-1}(c x)}\right )-c x+\sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-\sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )\right )}{\sqrt {c^2 x^2+1}}+\frac {b c^2 d^2 \sqrt {d \left (c^2 x^2+1\right )} \left (4 \text {Li}_2\left (-e^{-\sinh ^{-1}(c x)}\right )-4 \text {Li}_2\left (e^{-\sinh ^{-1}(c x)}\right )+4 \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-4 \sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )+2 \tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )-2 \coth \left (\frac {1}{2} \sinh ^{-1}(c x)\right )-\sinh ^{-1}(c x) \text {csch}^2\left (\frac {1}{2} \sinh ^{-1}(c x)\right )-\sinh ^{-1}(c x) \text {sech}^2\left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )}{8 \sqrt {c^2 x^2+1}}+b c^2 d^2 \left (\frac {1}{3} \left (c^2 x^2+1\right ) \sqrt {d \left (c^2 x^2+1\right )} \sinh ^{-1}(c x)-\frac {c x \left (c^2 x^2+3\right ) \sqrt {d \left (c^2 x^2+1\right )}}{9 \sqrt {c^2 x^2+1}}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a c^{4} d^{2} x^{4} + 2 \, a c^{2} d^{2} x^{2} + a d^{2} + {\left (b c^{4} d^{2} x^{4} + 2 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \operatorname {arsinh}\left (c x\right )\right )} \sqrt {c^{2} d x^{2} + d}}{x^{3}}, x\right ) \]
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 = 0.30, size = 588, normalized size = 1.66 \[ -\frac {a \left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{2 d \,x^{2}}+\frac {a \,c^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{2}+\frac {5 a \,c^{2} d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{6}-\frac {5 a \,c^{2} d^{\frac {5}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )}{2}+\frac {5 a \,c^{2} \sqrt {c^{2} d \,x^{2}+d}\, d^{2}}{2}+\frac {5 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) c^{2} d^{2}}{2 \sqrt {c^{2} x^{2}+1}}-\frac {5 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) c^{2} d^{2}}{2 \sqrt {c^{2} x^{2}+1}}+\frac {5 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) c^{2} d^{2}}{2 \sqrt {c^{2} x^{2}+1}}-\frac {5 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) c^{2} d^{2}}{2 \sqrt {c^{2} x^{2}+1}}+\frac {11 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{2} d^{2} \arcsinh \left (c x \right )}{6 \left (c^{2} x^{2}+1\right )}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d^{2} \arcsinh \left (c x \right )}{2 x^{2} \left (c^{2} x^{2}+1\right )}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{6} d^{2} \arcsinh \left (c x \right ) x^{4}}{3 c^{2} x^{2}+3}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{5} d^{2} x^{3}}{9 \sqrt {c^{2} x^{2}+1}}+\frac {8 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{4} d^{2} \arcsinh \left (c x \right ) x^{2}}{3 \left (c^{2} x^{2}+1\right )}-\frac {7 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{3} d^{2} x}{3 \sqrt {c^{2} x^{2}+1}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d^{2} c}{2 x \sqrt {c^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{6} \, {\left (15 \, c^{2} d^{\frac {5}{2}} \operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right ) - 3 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2} - 5 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d - 15 \, \sqrt {c^{2} d x^{2} + d} c^{2} d^{2} + \frac {3 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{d x^{2}}\right )} a + b \int \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{5/2}}{x^3} \,d x \]
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 \[ \int \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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