Optimal. Leaf size=398 \[ -\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}+\frac {\sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c^5 d^2 \sqrt {c^2 d x^2+d}}-\frac {4 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^5 d^2 \sqrt {c^2 d x^2+d}}+\frac {8 b \sqrt {c^2 x^2+1} \log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {c^2 d x^2+d}}-\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{c^4 d^2 \sqrt {c^2 d x^2+d}}-\frac {b x^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}}+\frac {4 b^2 \sqrt {c^2 x^2+1} \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{3 c^5 d^2 \sqrt {c^2 d x^2+d}}+\frac {b^2 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)}{3 c^5 d^2 \sqrt {c^2 d x^2+d}}-\frac {b^2 x}{3 c^4 d^2 \sqrt {c^2 d x^2+d}} \]
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Rubi [A] time = 0.76, antiderivative size = 398, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5751, 5677, 5675, 5714, 3718, 2190, 2279, 2391, 288, 215} \[ \frac {4 b^2 \sqrt {c^2 x^2+1} \text {PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{3 c^5 d^2 \sqrt {c^2 d x^2+d}}-\frac {b x^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}}-\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{c^4 d^2 \sqrt {c^2 d x^2+d}}+\frac {\sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c^5 d^2 \sqrt {c^2 d x^2+d}}-\frac {4 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^5 d^2 \sqrt {c^2 d x^2+d}}+\frac {8 b \sqrt {c^2 x^2+1} \log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {c^2 d x^2+d}}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}-\frac {b^2 x}{3 c^4 d^2 \sqrt {c^2 d x^2+d}}+\frac {b^2 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)}{3 c^5 d^2 \sqrt {c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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Rule 215
Rule 288
Rule 2190
Rule 2279
Rule 2391
Rule 3718
Rule 5675
Rule 5677
Rule 5714
Rule 5751
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 )^{5/2}} \, dx &=-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {\int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^{3/2}} \, 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 )}{\left (1+c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt {d+c^2 d x^2}}\\ &=-\frac {b x^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {\int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+c^2 d x^2}} \, dx}{c^4 d^2}+\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}{3 c^3 d^2 \sqrt {d+c^2 d x^2}}+\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^2 \sqrt {d+c^2 d x^2}}+\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x^2}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{3 c^2 d^2 \sqrt {d+c^2 d x^2}}\\ &=-\frac {b^2 x}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{c^4 d^2 \sqrt {d+c^2 d x^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 )}{3 c^5 d^2 \sqrt {d+c^2 d x^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^2 \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}} \, dx}{c^4 d^2 \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}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}\\ &=-\frac {b^2 x}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {b^2 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{c^4 d^2 \sqrt {d+c^2 d x^2}}-\frac {4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c^5 d^2 \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 )}{3 c^5 d^2 \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^2 \sqrt {d+c^2 d x^2}}\\ &=-\frac {b^2 x}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {b^2 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{c^4 d^2 \sqrt {d+c^2 d x^2}}-\frac {4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {8 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 )}{3 c^5 d^2 \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 )}{3 c^5 d^2 \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^2 \sqrt {d+c^2 d x^2}}\\ &=-\frac {b^2 x}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {b^2 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{c^4 d^2 \sqrt {d+c^2 d x^2}}-\frac {4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {8 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 )}{3 c^5 d^2 \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 )}{3 c^5 d^2 \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^2 \sqrt {d+c^2 d x^2}}\\ &=-\frac {b^2 x}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {b^2 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{c^4 d^2 \sqrt {d+c^2 d x^2}}-\frac {4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {8 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 )}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {4 b^2 \sqrt {1+c^2 x^2} \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 1.25, size = 359, normalized size = 0.90 \[ \frac {a^2 (-c) \sqrt {d} x \left (4 c^2 x^2+3\right )+3 a^2 \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} \log \left (\sqrt {d} \sqrt {c^2 d x^2+d}+c d x\right )+a b \sqrt {d} \left (\sqrt {c^2 x^2+1}-8 c x \left (c^2 x^2+1\right ) \sinh ^{-1}(c x)+\left (c^2 x^2+1\right )^{3/2} \left (4 \log \left (c^2 x^2+1\right )+3 \sinh ^{-1}(c x)^2\right )+2 c x \sinh ^{-1}(c x)\right )-b^2 \sqrt {d} \left (c^3 x^3+4 c^3 x^3 \sinh ^{-1}(c x)^2+4 \left (c^2 x^2+1\right )^{3/2} \text {Li}_2\left (-e^{-2 \sinh ^{-1}(c x)}\right )-\left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x)^3-4 \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x)^2-\sqrt {c^2 x^2+1} \sinh ^{-1}(c x)-8 \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x) \log \left (e^{-2 \sinh ^{-1}(c x)}+1\right )+c x+3 c x \sinh ^{-1}(c x)^2\right )}{3 c^5 d^{5/2} \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{4} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, a b x^{4} \operatorname {arsinh}\left (c x\right ) + a^{2} x^{4}\right )} \sqrt {c^{2} d x^{2} + d}}{c^{6} d^{3} x^{6} + 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} + d^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{4}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.56, size = 3705, normalized size = 9.31 \[ \text {output too large to display} \]
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 \[ -\frac {1}{3} \, {\left (x {\left (\frac {3 \, x^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d} + \frac {2}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{4} d}\right )} + \frac {x}{\sqrt {c^{2} d x^{2} + d} c^{4} d^{2}} - \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5} d^{\frac {5}{2}}}\right )} a^{2} + \int \frac {b^{2} x^{4} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} + \frac {2 \, a b x^{4} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {x^4\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d\,c^2\,x^2+d\right )}^{5/2}} \,d x \]
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 \[ \int \frac {x^{4} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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