Optimal. Leaf size=95 \[ b \text {Int}\left (\frac {\tan ^{-1}(c x) \left (d+e x^2\right )^{3/2}}{x^5},x\right )-\frac {3 a e^2 \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{8 \sqrt {d}}-\frac {3 a e \sqrt {d+e x^2}}{8 x^2}-\frac {a \left (d+e x^2\right )^{3/2}}{4 x^4} \]
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Rubi [A] time = 0.20, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{x^5} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{x^5} \, dx &=a \int \frac {\left (d+e x^2\right )^{3/2}}{x^5} \, dx+b \int \frac {\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^5} \, dx\\ &=\frac {1}{2} a \operatorname {Subst}\left (\int \frac {(d+e x)^{3/2}}{x^3} \, dx,x,x^2\right )+b \int \frac {\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^5} \, dx\\ &=-\frac {a \left (d+e x^2\right )^{3/2}}{4 x^4}+b \int \frac {\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^5} \, dx+\frac {1}{8} (3 a e) \operatorname {Subst}\left (\int \frac {\sqrt {d+e x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {3 a e \sqrt {d+e x^2}}{8 x^2}-\frac {a \left (d+e x^2\right )^{3/2}}{4 x^4}+b \int \frac {\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^5} \, dx+\frac {1}{16} \left (3 a e^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )\\ &=-\frac {3 a e \sqrt {d+e x^2}}{8 x^2}-\frac {a \left (d+e x^2\right )^{3/2}}{4 x^4}+b \int \frac {\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^5} \, dx+\frac {1}{8} (3 a e) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )\\ &=-\frac {3 a e \sqrt {d+e x^2}}{8 x^2}-\frac {a \left (d+e x^2\right )^{3/2}}{4 x^4}-\frac {3 a e^2 \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{8 \sqrt {d}}+b \int \frac {\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^5} \, dx\\ \end {align*}
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Mathematica [A] time = 57.17, size = 0, normalized size = 0.00 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{x^5} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a e x^{2} + a d + {\left (b e x^{2} + b d\right )} \arctan \left (c x\right )\right )} \sqrt {e x^{2} + d}}{x^{5}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.06, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arctan \left (c x \right )\right )}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{8} \, {\left (\frac {3 \, e^{2} \operatorname {arsinh}\left (\frac {d}{\sqrt {d e} {\left | x \right |}}\right )}{\sqrt {d}} - \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} e^{2}}{d^{2}} - \frac {3 \, \sqrt {e x^{2} + d} e^{2}}{d} + \frac {{\left (e x^{2} + d\right )}^{\frac {5}{2}} e}{d^{2} x^{2}} + \frac {2 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}}}{d x^{4}}\right )} a + \frac {1}{2} \, b \int \frac {2 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} \arctan \left (c x\right )}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^{3/2}}{x^5} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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