Optimal. Leaf size=178 \[ -\frac {\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}+\frac {b c \left (4 c^2 d-7 e\right ) \sqrt {d+e x^2}}{40 x^2}+\frac {b \left (c^2 d-e\right )^{5/2} \tanh ^{-1}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{5 d}-\frac {b c \left (8 c^4 d^2-20 c^2 d e+15 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{40 \sqrt {d}}-\frac {b c \left (d+e x^2\right )^{3/2}}{20 x^4} \]
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Rubi [A] time = 0.32, antiderivative size = 178, 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 = {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 12
Rule 63
Rule 98
Rule 149
Rule 156
Rule 208
Rule 264
Rule 446
Rule 4976
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*}
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Mathematica [C] time = 0.50, size = 334, normalized size = 1.88 \[ \frac {-\sqrt {d+e x^2} \left (8 a \left (d+e x^2\right )^2+b c d x \left (d \left (2-4 c^2 x^2\right )+9 e x^2\right )\right )+4 b x^5 \left (c^2 d-e\right )^{5/2} \log \left (-\frac {20 c d \left (\sqrt {c^2 d-e} \sqrt {d+e x^2}+c d-i e x\right )}{b (c x+i) \left (c^2 d-e\right )^{7/2}}\right )+4 b x^5 \left (c^2 d-e\right )^{5/2} \log \left (-\frac {20 c d \left (\sqrt {c^2 d-e} \sqrt {d+e x^2}+c d+i e x\right )}{b (c x-i) \left (c^2 d-e\right )^{7/2}}\right )+b c \sqrt {d} x^5 \log (x) \left (8 c^4 d^2-20 c^2 d e+15 e^2\right )-b c \sqrt {d} x^5 \left (8 c^4 d^2-20 c^2 d e+15 e^2\right ) \log \left (\sqrt {d} \sqrt {d+e x^2}+d\right )-8 b \tan ^{-1}(c x) \left (d+e x^2\right )^{5/2}}{40 d x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.27, size = 1145, normalized size = 6.43 \[ \text {result too large to display} \]
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 = 1.18, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arctan \left (c x \right )\right )}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^{3/2}}{x^6} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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