Optimal. Leaf size=446 \[ -\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\right ) \sqrt{\frac{e x^2}{d}+1} \text{EllipticF}\left (\sin ^{-1}(c x),-\frac{e}{c^2 d}\right )}{225 c d^2 \sqrt{d+e x^2}}+\frac{2 e \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{15 d^2 x^3}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 d x^5}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt{d+e x^2}}{225 d^2 x}+\frac{b c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{225 d^2 \sqrt{\frac{e x^2}{d}+1}}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^5}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (12 c^2 d-e\right ) \sqrt{d+e x^2}}{225 d x^3} \]
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Rubi [A] time = 0.560598, antiderivative size = 446, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {271, 264, 6301, 12, 580, 583, 524, 426, 424, 421, 419} \[ \frac{2 e \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{15 d^2 x^3}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 d x^5}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt{d+e x^2}}{225 d^2 x}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\right ) \sqrt{\frac{e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{225 c d^2 \sqrt{d+e x^2}}+\frac{b c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{225 d^2 \sqrt{\frac{e x^2}{d}+1}}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^5}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (12 c^2 d-e\right ) \sqrt{d+e x^2}}{225 d x^3} \]
Antiderivative was successfully verified.
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Rule 271
Rule 264
Rule 6301
Rule 12
Rule 580
Rule 583
Rule 524
Rule 426
Rule 424
Rule 421
Rule 419
Rubi steps
\begin{align*} \int \frac{\sqrt{d+e x^2} \left (a+b \text{sech}^{-1}(c x)\right )}{x^6} \, dx &=-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 d x^5}+\frac{2 e \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{15 d^2 x^3}+\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{\left (d+e x^2\right )^{3/2} \left (-3 d+2 e x^2\right )}{15 d^2 x^6 \sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 d x^5}+\frac{2 e \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{15 d^2 x^3}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{\left (d+e x^2\right )^{3/2} \left (-3 d+2 e x^2\right )}{x^6 \sqrt{1-c^2 x^2}} \, dx}{15 d^2}\\ &=\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^5}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 d x^5}+\frac{2 e \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{15 d^2 x^3}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{\sqrt{d+e x^2} \left (-d \left (12 c^2 d-e\right )-\left (3 c^2 d-10 e\right ) e x^2\right )}{x^4 \sqrt{1-c^2 x^2}} \, dx}{75 d^2}\\ &=\frac{b \left (12 c^2 d-e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{225 d x^3}+\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^5}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 d x^5}+\frac{2 e \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{15 d^2 x^3}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{-d \left (24 c^4 d^2+19 c^2 d e-31 e^2\right )-2 e \left (6 c^4 d^2+4 c^2 d e-15 e^2\right ) x^2}{x^2 \sqrt{1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{225 d^2}\\ &=\frac{b \left (12 c^2 d-e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{225 d x^3}+\frac{b \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{225 d^2 x}+\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^5}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 d x^5}+\frac{2 e \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{15 d^2 x^3}-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{2 d e \left (6 c^4 d^2+4 c^2 d e-15 e^2\right )-c^2 d e \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) x^2}{\sqrt{1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{225 d^3}\\ &=\frac{b \left (12 c^2 d-e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{225 d x^3}+\frac{b \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{225 d^2 x}+\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^5}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 d x^5}+\frac{2 e \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{15 d^2 x^3}+\frac{\left (b c^2 \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{\sqrt{d+e x^2}}{\sqrt{1-c^2 x^2}} \, dx}{225 d^2}-\frac{\left (b \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{\sqrt{1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{225 d^2}\\ &=\frac{b \left (12 c^2 d-e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{225 d x^3}+\frac{b \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{225 d^2 x}+\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^5}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 d x^5}+\frac{2 e \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{15 d^2 x^3}+\frac{\left (b c^2 \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{d+e x^2}\right ) \int \frac{\sqrt{1+\frac{e x^2}{d}}}{\sqrt{1-c^2 x^2}} \, dx}{225 d^2 \sqrt{1+\frac{e x^2}{d}}}-\frac{\left (b \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1+\frac{e x^2}{d}}\right ) \int \frac{1}{\sqrt{1-c^2 x^2} \sqrt{1+\frac{e x^2}{d}}} \, dx}{225 d^2 \sqrt{d+e x^2}}\\ &=\frac{b \left (12 c^2 d-e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{225 d x^3}+\frac{b \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{225 d^2 x}+\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^5}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 d x^5}+\frac{2 e \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{15 d^2 x^3}+\frac{b c \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{225 d^2 \sqrt{1+\frac{e x^2}{d}}}-\frac{b \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1+\frac{e x^2}{d}} F\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{225 c d^2 \sqrt{d+e x^2}}\\ \end{align*}
Mathematica [C] time = 6.12054, size = 641, normalized size = 1.44 \[ \frac{\frac{b \sqrt{\frac{1-c x}{c x+1}} \left (c^2 \left (-\left (24 c^4 d^2+19 c^2 d e-31 e^2\right )\right ) \left (d+e x^2\right )-\frac{i (c x+1) \left (c \sqrt{d}-i \sqrt{e}\right )^2 \sqrt{\frac{c \left (\sqrt{d}-i \sqrt{e} x\right )}{(c x+1) \left (c \sqrt{d}-i \sqrt{e}\right )}} \sqrt{\frac{c \left (\sqrt{d}+i \sqrt{e} x\right )}{(c x+1) \left (c \sqrt{d}+i \sqrt{e}\right )}} \left (2 \sqrt{e} \left (24 i c^3 d^{3/2}-36 c^2 d \sqrt{e}-29 i c \sqrt{d} e+30 e^{3/2}\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{(1-c x) \left (c^2 d+e\right )}{(c x+1) \left (c \sqrt{d}+i \sqrt{e}\right )^2}}\right ),\frac{\left (c \sqrt{d}+i \sqrt{e}\right )^2}{\left (c \sqrt{d}-i \sqrt{e}\right )^2}\right )+\left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{\left (d c^2+e\right ) (1-c x)}{\left (\sqrt{d} c+i \sqrt{e}\right )^2 (c x+1)}}\right )|\frac{\left (\sqrt{d} c+i \sqrt{e}\right )^2}{\left (c \sqrt{d}-i \sqrt{e}\right )^2}\right )\right )}{\sqrt{-\frac{(c x-1) \left (c \sqrt{d}-i \sqrt{e}\right )}{(c x+1) \left (c \sqrt{d}+i \sqrt{e}\right )}}}\right )}{c}+\frac{15 a \left (2 e x^2-3 d\right ) \left (d+e x^2\right )^2}{x^5}+\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (d+e x^2\right ) \left (3 d^2 \left (8 c^4 x^4+4 c^2 x^2+3\right )+d e x^2 \left (19 c^2 x^2+8\right )-31 e^2 x^4\right )}{x^5}+\frac{15 b \text{sech}^{-1}(c x) \left (2 e x^2-3 d\right ) \left (d+e x^2\right )^2}{x^5}}{225 d^2 \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.605, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\rm arcsech} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{e x^{2} + d}{\left (b \operatorname{arsech}\left (c x\right ) + a\right )}}{x^{6}}, x\right ) \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{\sqrt{e x^{2} + d}{\left (b \operatorname{arsech}\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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