Optimal. Leaf size=224 \[ \frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}+\frac {b \left (3 c^2 d+2 e\right ) \left (c^2 d-e\right )^{3/2} \tanh ^{-1}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{15 d^2}+\frac {b c \left (12 c^2 d-e\right ) \sqrt {d+e x^2}}{120 d x^2}-\frac {b c \left (24 c^4 d^2-20 c^2 d e-15 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{120 d^{3/2}}-\frac {b c \left (d+e x^2\right )^{3/2}}{20 d x^4} \]
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Rubi [A] time = 0.35, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {271, 264, 4976, 12, 573, 149, 156, 63, 208} \[ \frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}-\frac {b c \left (24 c^4 d^2-20 c^2 d e-15 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{120 d^{3/2}}+\frac {b \left (3 c^2 d+2 e\right ) \left (c^2 d-e\right )^{3/2} \tanh ^{-1}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{15 d^2}+\frac {b c \left (12 c^2 d-e\right ) \sqrt {d+e x^2}}{120 d x^2}-\frac {b c \left (d+e x^2\right )^{3/2}}{20 d x^4} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 149
Rule 156
Rule 208
Rule 264
Rule 271
Rule 573
Rule 4976
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{x^6} \, dx &=-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{15 d^2 x^3}-(b c) \int \frac {\left (d+e x^2\right )^{3/2} \left (-3 d+2 e x^2\right )}{15 d^2 x^5 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {(b c) \int \frac {\left (d+e x^2\right )^{3/2} \left (-3 d+2 e x^2\right )}{x^5 \left (1+c^2 x^2\right )} \, dx}{15 d^2}\\ &=-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {(b c) \operatorname {Subst}\left (\int \frac {(d+e x)^{3/2} (-3 d+2 e x)}{x^3 \left (1+c^2 x\right )} \, dx,x,x^2\right )}{30 d^2}\\ &=-\frac {b c \left (d+e x^2\right )^{3/2}}{20 d x^4}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {(b c) \operatorname {Subst}\left (\int \frac {\sqrt {d+e x} \left (\frac {1}{2} d \left (12 c^2 d-e\right )+\frac {1}{2} e \left (3 c^2 d+8 e\right ) x\right )}{x^2 \left (1+c^2 x\right )} \, dx,x,x^2\right )}{60 d^2}\\ &=\frac {b c \left (12 c^2 d-e\right ) \sqrt {d+e x^2}}{120 d x^2}-\frac {b c \left (d+e x^2\right )^{3/2}}{20 d x^4}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {(b c) \operatorname {Subst}\left (\int \frac {-\frac {1}{4} d \left (24 c^4 d^2-20 c^2 d e-15 e^2\right )-\frac {1}{4} e \left (12 c^4 d^2-7 c^2 d e-16 e^2\right ) x}{x \left (1+c^2 x\right ) \sqrt {d+e x}} \, dx,x,x^2\right )}{60 d^2}\\ &=\frac {b c \left (12 c^2 d-e\right ) \sqrt {d+e x^2}}{120 d x^2}-\frac {b c \left (d+e x^2\right )^{3/2}}{20 d x^4}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (b c \left (c^2 d-e\right )^2 \left (3 c^2 d+2 e\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+c^2 x\right ) \sqrt {d+e x}} \, dx,x,x^2\right )}{30 d^2}+\frac {\left (b c \left (24 c^4 d^2-20 c^2 d e-15 e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )}{240 d}\\ &=\frac {b c \left (12 c^2 d-e\right ) \sqrt {d+e x^2}}{120 d x^2}-\frac {b c \left (d+e x^2\right )^{3/2}}{20 d x^4}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (b c \left (c^2 d-e\right )^2 \left (3 c^2 d+2 e\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {c^2 d}{e}+\frac {c^2 x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{15 d^2 e}+\frac {\left (b c \left (24 c^4 d^2-20 c^2 d e-15 e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{120 d e}\\ &=\frac {b c \left (12 c^2 d-e\right ) \sqrt {d+e x^2}}{120 d x^2}-\frac {b c \left (d+e x^2\right )^{3/2}}{20 d x^4}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {b c \left (24 c^4 d^2-20 c^2 d e-15 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{120 d^{3/2}}+\frac {b \left (c^2 d-e\right )^{3/2} \left (3 c^2 d+2 e\right ) \tanh ^{-1}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{15 d^2}\\ \end {align*}
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Mathematica [C] time = 0.55, size = 413, normalized size = 1.84 \[ \frac {-\sqrt {d+e x^2} \left (8 a \left (3 d^2+d e x^2-2 e^2 x^4\right )+b c d x \left (d \left (6-12 c^2 x^2\right )+7 e x^2\right )\right )+4 b x^5 \left (c^2 d-e\right )^{3/2} \left (3 c^2 d+2 e\right ) \log \left (-\frac {60 c d^2 \left (\sqrt {c^2 d-e} \sqrt {d+e x^2}+c d-i e x\right )}{b (c x+i) \left (c^2 d-e\right )^{5/2} \left (3 c^2 d+2 e\right )}\right )+4 b x^5 \left (c^2 d-e\right )^{3/2} \left (3 c^2 d+2 e\right ) \log \left (-\frac {60 c d^2 \left (\sqrt {c^2 d-e} \sqrt {d+e x^2}+c d+i e x\right )}{b (c x-i) \left (c^2 d-e\right )^{5/2} \left (3 c^2 d+2 e\right )}\right )+b c \sqrt {d} x^5 \log (x) \left (24 c^4 d^2-20 c^2 d e-15 e^2\right )-b c \sqrt {d} x^5 \left (24 c^4 d^2-20 c^2 d e-15 e^2\right ) \log \left (\sqrt {d} \sqrt {d+e x^2}+d\right )-8 b \tan ^{-1}(c x) \sqrt {d+e x^2} \left (3 d^2+d e x^2-2 e^2 x^4\right )}{120 d^2 x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.09, size = 1156, normalized size = 5.16 \[ \text {result too large to display} \]
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 [F] time = 1.23, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e \,x^{2}+d}\, \left (a +b \arctan \left (c x \right )\right )}{x^{6}}\, dx \]
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 \[ \frac {1}{15} \, a {\left (\frac {2 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} e}{d^{2} x^{3}} - \frac {3 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}}}{d x^{5}}\right )} + b \int \frac {\sqrt {e x^{2} + d} \arctan \left (c x\right )}{x^{6}}\,{d x} \]
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 \[ \int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,\sqrt {e\,x^2+d}}{x^6} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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