Optimal. Leaf size=178 \[ -\frac{\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}-\frac{b c \left (8 c^4 d^2-20 c^2 d e+15 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{40 \sqrt{d}}+\frac{b c \left (4 c^2 d-7 e\right ) \sqrt{d+e x^2}}{40 x^2}+\frac{b \left (c^2 d-e\right )^{5/2} \tanh ^{-1}\left (\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right )}{5 d}-\frac{b c \left (d+e x^2\right )^{3/2}}{20 x^4} \]
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Rubi [A] time = 0.3246, antiderivative size = 178, 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 = {264, 4976, 12, 446, 98, 149, 156, 63, 208} \[ -\frac{\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}-\frac{b c \left (8 c^4 d^2-20 c^2 d e+15 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{40 \sqrt{d}}+\frac{b c \left (4 c^2 d-7 e\right ) \sqrt{d+e x^2}}{40 x^2}+\frac{b \left (c^2 d-e\right )^{5/2} \tanh ^{-1}\left (\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right )}{5 d}-\frac{b c \left (d+e x^2\right )^{3/2}}{20 x^4} \]
Antiderivative was successfully verified.
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Rule 264
Rule 4976
Rule 12
Rule 446
Rule 98
Rule 149
Rule 156
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{x^6} \, dx &=-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}-(b c) \int \frac{\left (d+e x^2\right )^{5/2}}{5 x^5 \left (-d-c^2 d x^2\right )} \, dx\\ &=-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}-\frac{1}{5} (b c) \int \frac{\left (d+e x^2\right )^{5/2}}{x^5 \left (-d-c^2 d x^2\right )} \, dx\\ &=-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}-\frac{1}{10} (b c) \operatorname{Subst}\left (\int \frac{(d+e x)^{5/2}}{x^3 \left (-d-c^2 d x\right )} \, dx,x,x^2\right )\\ &=-\frac{b c \left (d+e x^2\right )^{3/2}}{20 x^4}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}-\frac{(b c) \operatorname{Subst}\left (\int \frac{\sqrt{d+e x} \left (-\frac{1}{2} d^2 \left (4 c^2 d-7 e\right )-\frac{1}{2} d \left (c^2 d-4 e\right ) e x\right )}{x^2 \left (-d-c^2 d x\right )} \, dx,x,x^2\right )}{20 d}\\ &=\frac{b c \left (4 c^2 d-7 e\right ) \sqrt{d+e x^2}}{40 x^2}-\frac{b c \left (d+e x^2\right )^{3/2}}{20 x^4}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}+\frac{(b c) \operatorname{Subst}\left (\int \frac{-\frac{1}{4} d^3 \left (8 c^4 d^2-20 c^2 d e+15 e^2\right )-\frac{1}{4} d^2 e \left (4 c^4 d^2-9 c^2 d e+8 e^2\right ) x}{x \left (-d-c^2 d x\right ) \sqrt{d+e x}} \, dx,x,x^2\right )}{20 d^2}\\ &=\frac{b c \left (4 c^2 d-7 e\right ) \sqrt{d+e x^2}}{40 x^2}-\frac{b c \left (d+e x^2\right )^{3/2}}{20 x^4}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}+\frac{1}{10} \left (b c \left (c^2 d-e\right )^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (-d-c^2 d x\right ) \sqrt{d+e x}} \, dx,x,x^2\right )+\frac{1}{80} \left (b c \left (8 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 )\\ &=\frac{b c \left (4 c^2 d-7 e\right ) \sqrt{d+e x^2}}{40 x^2}-\frac{b c \left (d+e x^2\right )^{3/2}}{20 x^4}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}+\frac{\left (b c \left (c^2 d-e\right )^3\right ) \operatorname{Subst}\left (\int \frac{1}{-d+\frac{c^2 d^2}{e}-\frac{c^2 d x^2}{e}} \, dx,x,\sqrt{d+e x^2}\right )}{5 e}+\frac{\left (b c \left (8 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 )}{40 e}\\ &=\frac{b c \left (4 c^2 d-7 e\right ) \sqrt{d+e x^2}}{40 x^2}-\frac{b c \left (d+e x^2\right )^{3/2}}{20 x^4}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}-\frac{b c \left (8 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 )}{40 \sqrt{d}}+\frac{b \left (c^2 d-e\right )^{5/2} \tanh ^{-1}\left (\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right )}{5 d}\\ \end{align*}
Mathematica [C] time = 0.463967, size = 334, normalized size = 1.88 \[ \frac{-\sqrt{d+e x^2} \left (8 a \left (d+e x^2\right )^2+b c d x \left (d \left (2-4 c^2 x^2\right )+9 e x^2\right )\right )+b c \sqrt{d} x^5 \log (x) \left (8 c^4 d^2-20 c^2 d e+15 e^2\right )-b c \sqrt{d} x^5 \left (8 c^4 d^2-20 c^2 d e+15 e^2\right ) \log \left (\sqrt{d} \sqrt{d+e x^2}+d\right )+4 b x^5 \left (c^2 d-e\right )^{5/2} \log \left (-\frac{20 c d \left (\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d-i e x\right )}{b (c x+i) \left (c^2 d-e\right )^{7/2}}\right )+4 b x^5 \left (c^2 d-e\right )^{5/2} \log \left (-\frac{20 c d \left (\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d+i e x\right )}{b (c x-i) \left (c^2 d-e\right )^{7/2}}\right )-8 b \tan ^{-1}(c x) \left (d+e x^2\right )^{5/2}}{40 d x^5} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.682, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\arctan \left ( cx \right ) }{{x}^{6}} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}\, 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 = 6.77062, size = 2538, normalized size = 14.26 \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{{\left (e x^{2} + d\right )}^{\frac{3}{2}}{\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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