Optimal. Leaf size=224 \[ \frac{2 e \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{15 d^2 x^3}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}-\frac{b c \left (24 c^4 d^2-20 c^2 d e-15 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{120 d^{3/2}}+\frac{b \left (3 c^2 d+2 e\right ) \left (c^2 d-e\right )^{3/2} \tanh ^{-1}\left (\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right )}{15 d^2}+\frac{b c \left (12 c^2 d-e\right ) \sqrt{d+e x^2}}{120 d x^2}-\frac{b c \left (d+e x^2\right )^{3/2}}{20 d x^4} \]
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Rubi [A] time = 0.352288, antiderivative size = 224, normalized size of antiderivative = 1., 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 271
Rule 264
Rule 4976
Rule 12
Rule 573
Rule 149
Rule 156
Rule 63
Rule 208
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*}
Mathematica [C] time = 0.508839, size = 413, normalized size = 1.84 \[ \frac{-\sqrt{d+e x^2} \left (8 a \left (3 d^2+d e x^2-2 e^2 x^4\right )+b c d x \left (d \left (6-12 c^2 x^2\right )+7 e x^2\right )\right )+b c \sqrt{d} x^5 \log (x) \left (24 c^4 d^2-20 c^2 d e-15 e^2\right )-b c \sqrt{d} x^5 \left (24 c^4 d^2-20 c^2 d e-15 e^2\right ) \log \left (\sqrt{d} \sqrt{d+e x^2}+d\right )+4 b x^5 \left (c^2 d-e\right )^{3/2} \left (3 c^2 d+2 e\right ) \log \left (-\frac{60 c d^2 \left (\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d-i e x\right )}{b (c x+i) \left (c^2 d-e\right )^{5/2} \left (3 c^2 d+2 e\right )}\right )+4 b x^5 \left (c^2 d-e\right )^{3/2} \left (3 c^2 d+2 e\right ) \log \left (-\frac{60 c d^2 \left (\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d+i e x\right )}{b (c x-i) \left (c^2 d-e\right )^{5/2} \left (3 c^2 d+2 e\right )}\right )-8 b \tan ^{-1}(c x) \sqrt{d+e x^2} \left (3 d^2+d e x^2-2 e^2 x^4\right )}{120 d^2 x^5} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.881, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\arctan \left ( cx \right ) }{{x}^{6}}\sqrt{e{x}^{2}+d}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 6.67949, size = 2593, normalized size = 11.58 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{atan}{\left (c x \right )}\right ) \sqrt{d + e x^{2}}}{x^{6}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e x^{2} + d}{\left (b \arctan \left (c x\right ) + a\right )}}{x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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