Optimal. Leaf size=409 \[ -\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (c^2 d+e\right ) \left (8 c^4 d^2+19 c^2 d e+15 e^2\right ) \sqrt{\frac{e x^2}{d}+1} \text{EllipticF}\left (\sin ^{-1}(c x),-\frac{e}{c^2 d}\right )}{75 c d \sqrt{d+e x^2}}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 d x^5}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt{d+e x^2}}{75 d x}+\frac{b c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{75 d \sqrt{\frac{e x^2}{d}+1}}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 x^5}+\frac{4 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (c^2 d+2 e\right ) \sqrt{d+e x^2}}{75 x^3} \]
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Rubi [A] time = 0.532908, antiderivative size = 409, 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 = {264, 6301, 12, 474, 580, 583, 524, 426, 424, 421, 419} \[ -\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 d x^5}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt{d+e x^2}}{75 d x}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (c^2 d+e\right ) \left (8 c^4 d^2+19 c^2 d e+15 e^2\right ) \sqrt{\frac{e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{75 c d \sqrt{d+e x^2}}+\frac{b c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{75 d \sqrt{\frac{e x^2}{d}+1}}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 x^5}+\frac{4 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (c^2 d+2 e\right ) \sqrt{d+e x^2}}{75 x^3} \]
Antiderivative was successfully verified.
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Rule 264
Rule 6301
Rule 12
Rule 474
Rule 580
Rule 583
Rule 524
Rule 426
Rule 424
Rule 421
Rule 419
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{x^6} \, dx &=-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 d x^5}+\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int -\frac{\left (d+e x^2\right )^{5/2}}{5 d x^6 \sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 d x^5}-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{\left (d+e x^2\right )^{5/2}}{x^6 \sqrt{1-c^2 x^2}} \, dx}{5 d}\\ &=\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 x^5}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 d x^5}-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{\sqrt{d+e x^2} \left (4 d \left (c^2 d+2 e\right )+e \left (c^2 d+5 e\right ) x^2\right )}{x^4 \sqrt{1-c^2 x^2}} \, dx}{25 d}\\ &=\frac{4 b \left (c^2 d+2 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{75 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 x^5}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 d x^5}-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{d \left (8 c^4 d^2+23 c^2 d e+23 e^2\right )+e \left (4 c^4 d^2+11 c^2 d e+15 e^2\right ) x^2}{x^2 \sqrt{1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{75 d}\\ &=\frac{4 b \left (c^2 d+2 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{75 x^3}+\frac{b \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{75 d 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 x^5}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 d x^5}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{-d e \left (4 c^4 d^2+11 c^2 d e+15 e^2\right )+c^2 d e \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) x^2}{\sqrt{1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{75 d^2}\\ &=\frac{4 b \left (c^2 d+2 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{75 x^3}+\frac{b \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{75 d 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 x^5}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 d x^5}-\frac{\left (b \left (c^2 d+e\right ) \left (8 c^4 d^2+19 c^2 d e+15 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}{75 d}+\frac{\left (b c^2 \left (8 c^4 d^2+23 c^2 d e+23 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}{75 d}\\ &=\frac{4 b \left (c^2 d+2 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{75 x^3}+\frac{b \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{75 d 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 x^5}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 d x^5}+\frac{\left (b c^2 \left (8 c^4 d^2+23 c^2 d e+23 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}{75 d \sqrt{1+\frac{e x^2}{d}}}-\frac{\left (b \left (c^2 d+e\right ) \left (8 c^4 d^2+19 c^2 d e+15 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}{75 d \sqrt{d+e x^2}}\\ &=\frac{4 b \left (c^2 d+2 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{75 x^3}+\frac{b \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{75 d 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 x^5}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 d x^5}+\frac{b c \left (8 c^4 d^2+23 c^2 d e+23 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 )}{75 d \sqrt{1+\frac{e x^2}{d}}}-\frac{b \left (c^2 d+e\right ) \left (8 c^4 d^2+19 c^2 d e+15 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 )}{75 c d \sqrt{d+e x^2}}\\ \end{align*}
Mathematica [C] time = 6.27375, size = 620, normalized size = 1.52 \[ \frac{\frac{b \sqrt{\frac{1-c x}{c x+1}} \left (c^2 \left (-\left (8 c^4 d^2+23 c^2 d e+23 e^2\right )\right ) \left (d+e x^2\right )-\frac{i (c x+1) \left (c \sqrt{d}-i \sqrt{e}\right )^2 \sqrt{\frac{c \left (\sqrt{d}-i \sqrt{e} x\right )}{(c x+1) \left (c \sqrt{d}-i \sqrt{e}\right )}} \sqrt{\frac{c \left (\sqrt{d}+i \sqrt{e} x\right )}{(c x+1) \left (c \sqrt{d}+i \sqrt{e}\right )}} \left (2 \sqrt{e} \left (8 i c^3 d^{3/2}-12 c^2 d \sqrt{e}+7 i c \sqrt{d} e-15 e^{3/2}\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{(1-c x) \left (c^2 d+e\right )}{(c x+1) \left (c \sqrt{d}+i \sqrt{e}\right )^2}}\right ),\frac{\left (c \sqrt{d}+i \sqrt{e}\right )^2}{\left (c \sqrt{d}-i \sqrt{e}\right )^2}\right )+\left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{\left (d c^2+e\right ) (1-c x)}{\left (\sqrt{d} c+i \sqrt{e}\right )^2 (c x+1)}}\right )|\frac{\left (\sqrt{d} c+i \sqrt{e}\right )^2}{\left (c \sqrt{d}-i \sqrt{e}\right )^2}\right )\right )}{\sqrt{-\frac{(c x-1) \left (c \sqrt{d}-i \sqrt{e}\right )}{(c x+1) \left (c \sqrt{d}+i \sqrt{e}\right )}}}\right )}{c}-\frac{15 a \left (d+e x^2\right )^3}{x^5}+\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (d+e x^2\right ) \left (d^2 \left (8 c^4 x^4+4 c^2 x^2+3\right )+d e x^2 \left (23 c^2 x^2+11\right )+23 e^2 x^4\right )}{x^5}-\frac{15 b \text{sech}^{-1}(c x) \left (d+e x^2\right )^3}{x^5}}{75 d \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.224, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\rm arcsech} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a e x^{2} + a d +{\left (b e x^{2} + b d\right )} \operatorname{arsech}\left (c x\right )\right )} \sqrt{e x^{2} + d}}{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{{\left (e x^{2} + d\right )}^{\frac{3}{2}}{\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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