Optimal. Leaf size=400 \[ -\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt {c^2 d x^2+d}}-\frac {\sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c^5 d \sqrt {c^2 d x^2+d}}+\frac {\sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{c^5 d \sqrt {c^2 d x^2+d}}-\frac {2 b \sqrt {c^2 x^2+1} \log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d \sqrt {c^2 d x^2+d}}+\frac {3 x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac {b x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^3 d \sqrt {c^2 d x^2+d}}-\frac {b^2 \sqrt {c^2 x^2+1} \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{c^5 d \sqrt {c^2 d x^2+d}}-\frac {b^2 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)}{4 c^5 d \sqrt {c^2 d x^2+d}}+\frac {b^2 x \left (c^2 x^2+1\right )}{4 c^4 d \sqrt {c^2 d x^2+d}} \]
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Rubi [A] time = 0.66, antiderivative size = 400, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 13, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {5751, 5758, 5677, 5675, 5661, 321, 215, 5767, 5714, 3718, 2190, 2279, 2391} \[ -\frac {b^2 \sqrt {c^2 x^2+1} \text {PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{c^5 d \sqrt {c^2 d x^2+d}}+\frac {3 x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt {c^2 d x^2+d}}-\frac {b x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^3 d \sqrt {c^2 d x^2+d}}-\frac {\sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c^5 d \sqrt {c^2 d x^2+d}}+\frac {\sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{c^5 d \sqrt {c^2 d x^2+d}}-\frac {2 b \sqrt {c^2 x^2+1} \log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d \sqrt {c^2 d x^2+d}}+\frac {b^2 x \left (c^2 x^2+1\right )}{4 c^4 d \sqrt {c^2 d x^2+d}}-\frac {b^2 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)}{4 c^5 d \sqrt {c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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Rule 215
Rule 321
Rule 2190
Rule 2279
Rule 2391
Rule 3718
Rule 5661
Rule 5675
Rule 5677
Rule 5714
Rule 5751
Rule 5758
Rule 5767
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx &=-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {3 \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+c^2 d x^2}} \, dx}{c^2 d}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{c d \sqrt {d+c^2 d x^2}}\\ &=\frac {b x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {3 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac {3 \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+c^2 d x^2}} \, dx}{2 c^4 d}-\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{c^3 d \sqrt {d+c^2 d x^2}}-\frac {\left (3 b \sqrt {1+c^2 x^2}\right ) \int x \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{c^3 d \sqrt {d+c^2 d x^2}}-\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{c^2 d \sqrt {d+c^2 d x^2}}\\ &=-\frac {b^2 x \left (1+c^2 x^2\right )}{2 c^4 d \sqrt {d+c^2 d x^2}}-\frac {b x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {3 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{c^5 d \sqrt {d+c^2 d x^2}}-\frac {\left (3 \sqrt {1+c^2 x^2}\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}} \, dx}{2 c^4 d \sqrt {d+c^2 d x^2}}+\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{2 c^4 d \sqrt {d+c^2 d x^2}}+\frac {\left (3 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{2 c^2 d \sqrt {d+c^2 d x^2}}\\ &=\frac {b^2 x \left (1+c^2 x^2\right )}{4 c^4 d \sqrt {d+c^2 d x^2}}+\frac {b^2 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{2 c^5 d \sqrt {d+c^2 d x^2}}-\frac {b x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c^5 d \sqrt {d+c^2 d x^2}}+\frac {3 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c^5 d \sqrt {d+c^2 d x^2}}-\frac {\left (4 b \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{c^5 d \sqrt {d+c^2 d x^2}}-\frac {\left (3 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{4 c^4 d \sqrt {d+c^2 d x^2}}\\ &=\frac {b^2 x \left (1+c^2 x^2\right )}{4 c^4 d \sqrt {d+c^2 d x^2}}-\frac {b^2 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{4 c^5 d \sqrt {d+c^2 d x^2}}-\frac {b x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c^5 d \sqrt {d+c^2 d x^2}}+\frac {3 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c^5 d \sqrt {d+c^2 d x^2}}-\frac {2 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c^5 d \sqrt {d+c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^5 d \sqrt {d+c^2 d x^2}}\\ &=\frac {b^2 x \left (1+c^2 x^2\right )}{4 c^4 d \sqrt {d+c^2 d x^2}}-\frac {b^2 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{4 c^5 d \sqrt {d+c^2 d x^2}}-\frac {b x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c^5 d \sqrt {d+c^2 d x^2}}+\frac {3 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c^5 d \sqrt {d+c^2 d x^2}}-\frac {2 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c^5 d \sqrt {d+c^2 d x^2}}+\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{c^5 d \sqrt {d+c^2 d x^2}}\\ &=\frac {b^2 x \left (1+c^2 x^2\right )}{4 c^4 d \sqrt {d+c^2 d x^2}}-\frac {b^2 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{4 c^5 d \sqrt {d+c^2 d x^2}}-\frac {b x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c^5 d \sqrt {d+c^2 d x^2}}+\frac {3 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c^5 d \sqrt {d+c^2 d x^2}}-\frac {2 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c^5 d \sqrt {d+c^2 d x^2}}-\frac {b^2 \sqrt {1+c^2 x^2} \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{c^5 d \sqrt {d+c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 1.90, size = 288, normalized size = 0.72 \[ \frac {4 a^2 c \sqrt {d} x \left (c^2 x^2+3\right )-12 a^2 \sqrt {c^2 d x^2+d} \log \left (\sqrt {d} \sqrt {c^2 d x^2+d}+c d x\right )+2 a b \sqrt {d} \left (8 c x \sinh ^{-1}(c x)-\sqrt {c^2 x^2+1} \left (4 \log \left (c^2 x^2+1\right )+6 \sinh ^{-1}(c x)^2-2 \sinh \left (2 \sinh ^{-1}(c x)\right ) \sinh ^{-1}(c x)+\cosh \left (2 \sinh ^{-1}(c x)\right )\right )\right )+b^2 \sqrt {d} \left (8 \sqrt {c^2 x^2+1} \text {Li}_2\left (-e^{-2 \sinh ^{-1}(c x)}\right )+\sqrt {c^2 x^2+1} \left (-4 \sinh ^{-1}(c x)^3+2 \left (\sinh \left (2 \sinh ^{-1}(c x)\right )-4\right ) \sinh ^{-1}(c x)^2+\sinh \left (2 \sinh ^{-1}(c x)\right )-2 \sinh ^{-1}(c x) \left (8 \log \left (e^{-2 \sinh ^{-1}(c x)}+1\right )+\cosh \left (2 \sinh ^{-1}(c x)\right )\right )\right )+8 c x \sinh ^{-1}(c x)^2\right )}{8 c^5 d^{3/2} \sqrt {c^2 d x^2+d}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{4} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, a b x^{4} \operatorname {arsinh}\left (c x\right ) + a^{2} x^{4}\right )} \sqrt {c^{2} d x^{2} + d}}{c^{4} d^{2} x^{4} + 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \]
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 [B] time = 0.54, size = 816, normalized size = 2.04 \[ \frac {a^{2} x^{3}}{2 c^{2} d \sqrt {c^{2} d \,x^{2}+d}}+\frac {3 a^{2} x}{2 c^{4} d \sqrt {c^{2} d \,x^{2}+d}}-\frac {3 a^{2} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 c^{4} d \sqrt {c^{2} d}}-\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, c^{5} d^{2}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2} x^{3}}{2 c^{2} d^{2} \left (c^{2} x^{2}+1\right )}+\frac {3 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2} x}{2 c^{4} d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2}}{c^{5} d^{2} \sqrt {c^{2} x^{2}+1}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )}{4 c^{5} d^{2} \sqrt {c^{2} x^{2}+1}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{3}}{2 \sqrt {c^{2} x^{2}+1}\, c^{5} d^{2}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{3}}{4 c^{2} d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, x}{4 c^{4} d^{2} \left (c^{2} x^{2}+1\right )}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, c^{5} d^{2}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x^{2}}{2 c^{3} d^{2} \sqrt {c^{2} x^{2}+1}}-\frac {3 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2}}{2 \sqrt {c^{2} x^{2}+1}\, c^{5} d^{2}}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x^{3}}{c^{2} d^{2} \left (c^{2} x^{2}+1\right )}-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{2}}{2 c^{3} d^{2} \sqrt {c^{2} x^{2}+1}}+\frac {3 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x}{c^{4} d^{2} \left (c^{2} x^{2}+1\right )}+\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )}{c^{5} d^{2} \sqrt {c^{2} x^{2}+1}}-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}}{4 c^{5} d^{2} \sqrt {c^{2} x^{2}+1}}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, c^{5} d^{2}} \]
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 \[ \frac {1}{2} \, a^{2} {\left (\frac {x^{3}}{\sqrt {c^{2} d x^{2} + d} c^{2} d} + \frac {3 \, x}{\sqrt {c^{2} d x^{2} + d} c^{4} d} - \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5} d^{\frac {3}{2}}}\right )} + \int \frac {b^{2} x^{4} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} + \frac {2 \, a b x^{4} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^4\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d\,c^2\,x^2+d\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{4} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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