Optimal. Leaf size=492 \[ \frac{b e x \left (4 c^4 d^2-11 c^2 d e+15 e^2\right ) \sqrt{d+e x^2} \text{EllipticF}\left (\tan ^{-1}(c x),1-\frac{e}{c^2 d}\right )}{75 d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left (c^2 x^2+1\right )}}}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 d x^5}+\frac{b c^3 x^2 \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) \sqrt{d+e x^2}}{75 d \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1}}+\frac{b c \sqrt{-c^2 x^2-1} \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) \sqrt{d+e x^2}}{75 d \sqrt{-c^2 x^2}}-\frac{b c^2 x \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) \sqrt{d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{75 d \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left (c^2 x^2+1\right )}}}+\frac{b c \sqrt{-c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{25 x^4 \sqrt{-c^2 x^2}}-\frac{4 b c \sqrt{-c^2 x^2-1} \left (c^2 d-2 e\right ) \sqrt{d+e x^2}}{75 x^2 \sqrt{-c^2 x^2}} \]
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Rubi [A] time = 0.575472, antiderivative size = 492, 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, 6302, 12, 474, 580, 583, 531, 418, 492, 411} \[ -\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 d x^5}+\frac{b c^3 x^2 \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) \sqrt{d+e x^2}}{75 d \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1}}+\frac{b c \sqrt{-c^2 x^2-1} \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) \sqrt{d+e x^2}}{75 d \sqrt{-c^2 x^2}}+\frac{b e x \left (4 c^4 d^2-11 c^2 d e+15 e^2\right ) \sqrt{d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{75 d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left (c^2 x^2+1\right )}}}-\frac{b c^2 x \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) \sqrt{d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{75 d \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left (c^2 x^2+1\right )}}}+\frac{b c \sqrt{-c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{25 x^4 \sqrt{-c^2 x^2}}-\frac{4 b c \sqrt{-c^2 x^2-1} \left (c^2 d-2 e\right ) \sqrt{d+e x^2}}{75 x^2 \sqrt{-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 264
Rule 6302
Rule 12
Rule 474
Rule 580
Rule 583
Rule 531
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{x^6} \, dx &=-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 d x^5}-\frac{(b c x) \int -\frac{\left (d+e x^2\right )^{5/2}}{5 d x^6 \sqrt{-1-c^2 x^2}} \, dx}{\sqrt{-c^2 x^2}}\\ &=-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 d x^5}+\frac{(b c x) \int \frac{\left (d+e x^2\right )^{5/2}}{x^6 \sqrt{-1-c^2 x^2}} \, dx}{5 d \sqrt{-c^2 x^2}}\\ &=\frac{b c \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 x^4 \sqrt{-c^2 x^2}}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 d x^5}-\frac{(b c x) \int \frac{\sqrt{d+e x^2} \left (4 d \left (c^2 d-2 e\right )+\left (c^2 d-5 e\right ) e x^2\right )}{x^4 \sqrt{-1-c^2 x^2}} \, dx}{25 d \sqrt{-c^2 x^2}}\\ &=-\frac{4 b c \left (c^2 d-2 e\right ) \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{75 x^2 \sqrt{-c^2 x^2}}+\frac{b c \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 x^4 \sqrt{-c^2 x^2}}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 d x^5}+\frac{(b c x) \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 \sqrt{-c^2 x^2}}\\ &=\frac{b c \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{75 d \sqrt{-c^2 x^2}}-\frac{4 b c \left (c^2 d-2 e\right ) \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{75 x^2 \sqrt{-c^2 x^2}}+\frac{b c \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 x^4 \sqrt{-c^2 x^2}}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 d x^5}+\frac{(b c x) \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 \sqrt{-c^2 x^2}}\\ &=\frac{b c \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{75 d \sqrt{-c^2 x^2}}-\frac{4 b c \left (c^2 d-2 e\right ) \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{75 x^2 \sqrt{-c^2 x^2}}+\frac{b c \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 x^4 \sqrt{-c^2 x^2}}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 d x^5}+\frac{\left (b c e \left (4 c^4 d^2-11 c^2 d e+15 e^2\right ) x\right ) \int \frac{1}{\sqrt{-1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{75 d \sqrt{-c^2 x^2}}+\frac{\left (b c^3 e \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) x\right ) \int \frac{x^2}{\sqrt{-1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{75 d \sqrt{-c^2 x^2}}\\ &=\frac{b c^3 \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) x^2 \sqrt{d+e x^2}}{75 d \sqrt{-c^2 x^2} \sqrt{-1-c^2 x^2}}+\frac{b c \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{75 d \sqrt{-c^2 x^2}}-\frac{4 b c \left (c^2 d-2 e\right ) \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{75 x^2 \sqrt{-c^2 x^2}}+\frac{b c \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 x^4 \sqrt{-c^2 x^2}}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 d x^5}+\frac{b e \left (4 c^4 d^2-11 c^2 d e+15 e^2\right ) x \sqrt{d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{75 d^2 \sqrt{-c^2 x^2} \sqrt{-1-c^2 x^2} \sqrt{\frac{d+e x^2}{d \left (1+c^2 x^2\right )}}}+\frac{\left (b c^3 \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) x\right ) \int \frac{\sqrt{d+e x^2}}{\left (-1-c^2 x^2\right )^{3/2}} \, dx}{75 d \sqrt{-c^2 x^2}}\\ &=\frac{b c^3 \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) x^2 \sqrt{d+e x^2}}{75 d \sqrt{-c^2 x^2} \sqrt{-1-c^2 x^2}}+\frac{b c \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{75 d \sqrt{-c^2 x^2}}-\frac{4 b c \left (c^2 d-2 e\right ) \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{75 x^2 \sqrt{-c^2 x^2}}+\frac{b c \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 x^4 \sqrt{-c^2 x^2}}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 d x^5}-\frac{b c^2 \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) x \sqrt{d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{75 d \sqrt{-c^2 x^2} \sqrt{-1-c^2 x^2} \sqrt{\frac{d+e x^2}{d \left (1+c^2 x^2\right )}}}+\frac{b e \left (4 c^4 d^2-11 c^2 d e+15 e^2\right ) x \sqrt{d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{75 d^2 \sqrt{-c^2 x^2} \sqrt{-1-c^2 x^2} \sqrt{\frac{d+e x^2}{d \left (1+c^2 x^2\right )}}}\\ \end{align*}
Mathematica [C] time = 0.682093, size = 291, normalized size = 0.59 \[ \frac{\sqrt{d+e x^2} \left (-15 a \left (d+e x^2\right )^2+b c x \sqrt{\frac{1}{c^2 x^2}+1} \left (d^2 \left (8 c^4 x^4-4 c^2 x^2+3\right )+d e x^2 \left (11-23 c^2 x^2\right )+23 e^2 x^4\right )-15 b \text{csch}^{-1}(c x) \left (d+e x^2\right )^2\right )}{75 d x^5}+\frac{i b c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{e x^2}{d}+1} \left (\left (27 c^4 d^2 e-8 c^6 d^3-34 c^2 d e^2+15 e^3\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{c^2} x\right ),\frac{e}{c^2 d}\right )+c^2 d \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) E\left (i \sinh ^{-1}\left (\sqrt{c^2} x\right )|\frac{e}{c^2 d}\right )\right )}{75 \sqrt{c^2} d \sqrt{c^2 x^2+1} \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.453, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\rm arccsch} \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{arcsch}\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{arcsch}\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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