Optimal. Leaf size=312 \[ -\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (c^2 d+e\right ) \left (2 c^2 d+3 e\right ) \sqrt{\frac{e x^2}{d}+1} \text{EllipticF}\left (\sin ^{-1}(c x),-\frac{e}{c^2 d}\right )}{9 c d \sqrt{d+e x^2}}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 d x^3}+\frac{2 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (c^2 d+2 e\right ) \sqrt{d+e x^2}}{9 d x}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{9 x^3}+\frac{2 b c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (c^2 d+2 e\right ) \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{9 d \sqrt{\frac{e x^2}{d}+1}} \]
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Rubi [A] time = 0.379898, antiderivative size = 312, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {264, 6301, 12, 474, 583, 524, 426, 424, 421, 419} \[ -\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 d x^3}+\frac{2 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (c^2 d+2 e\right ) \sqrt{d+e x^2}}{9 d x}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{9 x^3}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (c^2 d+e\right ) \left (2 c^2 d+3 e\right ) \sqrt{\frac{e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{9 c d \sqrt{d+e x^2}}+\frac{2 b c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (c^2 d+2 e\right ) \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{9 d \sqrt{\frac{e x^2}{d}+1}} \]
Antiderivative was successfully verified.
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Rule 264
Rule 6301
Rule 12
Rule 474
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^4} \, dx &=-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 d 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}}{3 d x^4 \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 )}{3 d 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}}{x^4 \sqrt{1-c^2 x^2}} \, dx}{3 d}\\ &=\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{9 x^3}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 d x^3}-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{2 d \left (c^2 d+2 e\right )+e \left (c^2 d+3 e\right ) x^2}{x^2 \sqrt{1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{9 d}\\ &=\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{9 x^3}+\frac{2 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}}{9 d x}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 d x^3}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{-d e \left (c^2 d+3 e\right )+2 c^2 d e \left (c^2 d+2 e\right ) x^2}{\sqrt{1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{9 d^2}\\ &=\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{9 x^3}+\frac{2 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}}{9 d x}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 d x^3}+\frac{\left (2 b c^2 \left (c^2 d+2 e\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}{9 d}-\frac{\left (b \left (c^2 d+e\right ) \left (2 c^2 d+3 e\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}{9 d}\\ &=\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{9 x^3}+\frac{2 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}}{9 d x}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 d x^3}+\frac{\left (2 b c^2 \left (c^2 d+2 e\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}{9 d \sqrt{1+\frac{e x^2}{d}}}-\frac{\left (b \left (c^2 d+e\right ) \left (2 c^2 d+3 e\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}{9 d \sqrt{d+e x^2}}\\ &=\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{9 x^3}+\frac{2 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}}{9 d x}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 d x^3}+\frac{2 b c \left (c^2 d+2 e\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 )}{9 d \sqrt{1+\frac{e x^2}{d}}}-\frac{b \left (c^2 d+e\right ) \left (2 c^2 d+3 e\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 )}{9 c d \sqrt{d+e x^2}}\\ \end{align*}
Mathematica [C] time = 3.96936, size = 576, normalized size = 1.85 \[ \frac{-\frac{2 i b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (c \sqrt{d}-i \sqrt{e}\right )^2 \sqrt{\frac{c \left (\sqrt{d}-i \sqrt{e} x\right )}{(c x+1) \left (c \sqrt{d}-i \sqrt{e}\right )}} \sqrt{\frac{c \left (\sqrt{d}+i \sqrt{e} x\right )}{(c x+1) \left (c \sqrt{d}+i \sqrt{e}\right )}} \left (\sqrt{e} \left (-3 \sqrt{e}+2 i c \sqrt{d}\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{(1-c x) \left (c^2 d+e\right )}{(c x+1) \left (c \sqrt{d}+i \sqrt{e}\right )^2}}\right ),\frac{\left (c \sqrt{d}+i \sqrt{e}\right )^2}{\left (c \sqrt{d}-i \sqrt{e}\right )^2}\right )+\left (c^2 d+2 e\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{\left (d c^2+e\right ) (1-c x)}{\left (\sqrt{d} c+i \sqrt{e}\right )^2 (c x+1)}}\right )|\frac{\left (\sqrt{d} c+i \sqrt{e}\right )^2}{\left (c \sqrt{d}-i \sqrt{e}\right )^2}\right )\right )}{c d \sqrt{-\frac{(c x-1) \left (c \sqrt{d}-i \sqrt{e}\right )}{(c x+1) \left (c \sqrt{d}+i \sqrt{e}\right )}}}-\frac{3 a \left (d+e x^2\right )^2}{d x^3}+\frac{2 b \sqrt{\frac{1-c x}{c x+1}} \left (c^2 d+2 e\right ) \left (d+e x^2\right )}{d x}+\frac{b c \sqrt{\frac{1-c x}{c x+1}} \left (d+e x^2\right )}{x^2}+\frac{b \sqrt{\frac{1-c x}{c x+1}} \left (d+e x^2\right )}{x^3}-\frac{3 b \text{sech}^{-1}(c x) \left (d+e x^2\right )^2}{d x^3}}{9 \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.318, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\rm arcsech} \left (cx\right )}{{x}^{4}}\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^{4}}, x\right ) \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{asech}{\left (c x \right )}\right ) \sqrt{d + e x^{2}}}{x^{4}}\, 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 \operatorname{arsech}\left (c x\right ) + a\right )}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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