Optimal. Leaf size=554 \[ \frac{2 b x \sqrt{1-c^2 x^2} \left (c^2 d+e\right ) \left (204 c^4 d^2 e+120 c^6 d^3+17 c^2 d e^2-105 e^3\right ) \sqrt{\frac{e x^2}{d}+1} \text{EllipticF}\left (\sin ^{-1}(c x),-\frac{e}{c^2 d}\right )}{3675 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{35 d^2 x^5}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{7 d x^7}+\frac{b c \sqrt{c^2 x^2-1} \left (528 c^4 d^2 e+240 c^6 d^3+193 c^2 d e^2-247 e^3\right ) \sqrt{d+e x^2}}{3675 d^2 \sqrt{c^2 x^2}}+\frac{b c \sqrt{c^2 x^2-1} \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt{d+e x^2}}{3675 d x^2 \sqrt{c^2 x^2}}-\frac{b c^2 x \sqrt{1-c^2 x^2} \left (528 c^4 d^2 e+240 c^6 d^3+193 c^2 d e^2-247 e^3\right ) \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3675 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}}+\frac{b c \sqrt{c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt{c^2 x^2}}+\frac{b c \sqrt{c^2 x^2-1} \left (30 c^2 d+11 e\right ) \left (d+e x^2\right )^{3/2}}{1225 d x^4 \sqrt{c^2 x^2}} \]
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Rubi [A] time = 0.777962, antiderivative size = 554, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 12, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.522, Rules used = {271, 264, 5238, 12, 580, 583, 524, 427, 426, 424, 421, 419} \[ \frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{35 d^2 x^5}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{7 d x^7}+\frac{b c \sqrt{c^2 x^2-1} \left (528 c^4 d^2 e+240 c^6 d^3+193 c^2 d e^2-247 e^3\right ) \sqrt{d+e x^2}}{3675 d^2 \sqrt{c^2 x^2}}+\frac{b c \sqrt{c^2 x^2-1} \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt{d+e x^2}}{3675 d x^2 \sqrt{c^2 x^2}}+\frac{2 b x \sqrt{1-c^2 x^2} \left (c^2 d+e\right ) \left (204 c^4 d^2 e+120 c^6 d^3+17 c^2 d e^2-105 e^3\right ) \sqrt{\frac{e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3675 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}-\frac{b c^2 x \sqrt{1-c^2 x^2} \left (528 c^4 d^2 e+240 c^6 d^3+193 c^2 d e^2-247 e^3\right ) \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3675 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}}+\frac{b c \sqrt{c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt{c^2 x^2}}+\frac{b c \sqrt{c^2 x^2-1} \left (30 c^2 d+11 e\right ) \left (d+e x^2\right )^{3/2}}{1225 d x^4 \sqrt{c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 271
Rule 264
Rule 5238
Rule 12
Rule 580
Rule 583
Rule 524
Rule 427
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 \sec ^{-1}(c x)\right )}{x^8} \, dx &=-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{35 d^2 x^5}-\frac{(b c x) \int \frac{\left (d+e x^2\right )^{5/2} \left (-5 d+2 e x^2\right )}{35 d^2 x^8 \sqrt{-1+c^2 x^2}} \, dx}{\sqrt{c^2 x^2}}\\ &=-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{35 d^2 x^5}-\frac{(b c x) \int \frac{\left (d+e x^2\right )^{5/2} \left (-5 d+2 e x^2\right )}{x^8 \sqrt{-1+c^2 x^2}} \, dx}{35 d^2 \sqrt{c^2 x^2}}\\ &=\frac{b c \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt{c^2 x^2}}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{35 d^2 x^5}+\frac{(b c x) \int \frac{\left (d+e x^2\right )^{3/2} \left (d \left (30 c^2 d+11 e\right )+\left (5 c^2 d-14 e\right ) e x^2\right )}{x^6 \sqrt{-1+c^2 x^2}} \, dx}{245 d^2 \sqrt{c^2 x^2}}\\ &=\frac{b c \left (30 c^2 d+11 e\right ) \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{1225 d x^4 \sqrt{c^2 x^2}}+\frac{b c \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt{c^2 x^2}}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{35 d^2 x^5}-\frac{(b c x) \int \frac{\sqrt{d+e x^2} \left (-d \left (120 c^4 d^2+159 c^2 d e-37 e^2\right )-2 e \left (15 c^4 d^2+18 c^2 d e-35 e^2\right ) x^2\right )}{x^4 \sqrt{-1+c^2 x^2}} \, dx}{1225 d^2 \sqrt{c^2 x^2}}\\ &=\frac{b c \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{3675 d x^2 \sqrt{c^2 x^2}}+\frac{b c \left (30 c^2 d+11 e\right ) \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{1225 d x^4 \sqrt{c^2 x^2}}+\frac{b c \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt{c^2 x^2}}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{35 d^2 x^5}+\frac{(b c x) \int \frac{d \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right )+e \left (120 c^6 d^3+249 c^4 d^2 e+71 c^2 d e^2-210 e^3\right ) x^2}{x^2 \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{3675 d^2 \sqrt{c^2 x^2}}\\ &=\frac{b c \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{3675 d^2 \sqrt{c^2 x^2}}+\frac{b c \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{3675 d x^2 \sqrt{c^2 x^2}}+\frac{b c \left (30 c^2 d+11 e\right ) \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{1225 d x^4 \sqrt{c^2 x^2}}+\frac{b c \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt{c^2 x^2}}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{35 d^2 x^5}+\frac{(b c x) \int \frac{d e \left (120 c^6 d^3+249 c^4 d^2 e+71 c^2 d e^2-210 e^3\right )-c^2 d e \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) x^2}{\sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{3675 d^3 \sqrt{c^2 x^2}}\\ &=\frac{b c \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{3675 d^2 \sqrt{c^2 x^2}}+\frac{b c \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{3675 d x^2 \sqrt{c^2 x^2}}+\frac{b c \left (30 c^2 d+11 e\right ) \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{1225 d x^4 \sqrt{c^2 x^2}}+\frac{b c \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt{c^2 x^2}}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{35 d^2 x^5}-\frac{\left (b c^3 \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) x\right ) \int \frac{\sqrt{d+e x^2}}{\sqrt{-1+c^2 x^2}} \, dx}{3675 d^2 \sqrt{c^2 x^2}}+\frac{\left (2 b c \left (c^2 d+e\right ) \left (120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 e^3\right ) x\right ) \int \frac{1}{\sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{3675 d^2 \sqrt{c^2 x^2}}\\ &=\frac{b c \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{3675 d^2 \sqrt{c^2 x^2}}+\frac{b c \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{3675 d x^2 \sqrt{c^2 x^2}}+\frac{b c \left (30 c^2 d+11 e\right ) \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{1225 d x^4 \sqrt{c^2 x^2}}+\frac{b c \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt{c^2 x^2}}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{35 d^2 x^5}-\frac{\left (b c^3 \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) x \sqrt{1-c^2 x^2}\right ) \int \frac{\sqrt{d+e x^2}}{\sqrt{1-c^2 x^2}} \, dx}{3675 d^2 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2}}+\frac{\left (2 b c \left (c^2 d+e\right ) \left (120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 e^3\right ) 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}{3675 d^2 \sqrt{c^2 x^2} \sqrt{d+e x^2}}\\ &=\frac{b c \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{3675 d^2 \sqrt{c^2 x^2}}+\frac{b c \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{3675 d x^2 \sqrt{c^2 x^2}}+\frac{b c \left (30 c^2 d+11 e\right ) \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{1225 d x^4 \sqrt{c^2 x^2}}+\frac{b c \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt{c^2 x^2}}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{35 d^2 x^5}-\frac{\left (b c^3 \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) x \sqrt{1-c^2 x^2} \sqrt{d+e x^2}\right ) \int \frac{\sqrt{1+\frac{e x^2}{d}}}{\sqrt{1-c^2 x^2}} \, dx}{3675 d^2 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{1+\frac{e x^2}{d}}}+\frac{\left (2 b c \left (c^2 d+e\right ) \left (120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 e^3\right ) x \sqrt{1-c^2 x^2} \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}{3675 d^2 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}\\ &=\frac{b c \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{3675 d^2 \sqrt{c^2 x^2}}+\frac{b c \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{3675 d x^2 \sqrt{c^2 x^2}}+\frac{b c \left (30 c^2 d+11 e\right ) \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{1225 d x^4 \sqrt{c^2 x^2}}+\frac{b c \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt{c^2 x^2}}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{35 d^2 x^5}-\frac{b c^2 \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) x \sqrt{1-c^2 x^2} \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3675 d^2 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{1+\frac{e x^2}{d}}}+\frac{2 b \left (c^2 d+e\right ) \left (120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 e^3\right ) x \sqrt{1-c^2 x^2} \sqrt{1+\frac{e x^2}{d}} F\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3675 d^2 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}\\ \end{align*}
Mathematica [C] time = 0.834749, size = 383, normalized size = 0.69 \[ \frac{\sqrt{d+e x^2} \left (-105 a \left (5 d-2 e x^2\right ) \left (d+e x^2\right )^2+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left (3 d^2 e x^2 \left (176 c^4 x^4+83 c^2 x^2+61\right )+15 d^3 \left (16 c^6 x^6+8 c^4 x^4+6 c^2 x^2+5\right )+d e^2 x^4 \left (193 c^2 x^2+71\right )-247 e^3 x^6\right )-105 b \sec ^{-1}(c x) \left (5 d-2 e x^2\right ) \left (d+e x^2\right )^2\right )}{3675 d^2 x^7}-\frac{i b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{e x^2}{d}+1} \left (c^2 d \left (528 c^4 d^2 e+240 c^6 d^3+193 c^2 d e^2-247 e^3\right ) E\left (i \sinh ^{-1}\left (\sqrt{-c^2} x\right )|-\frac{e}{c^2 d}\right )-2 \left (221 c^4 d^2 e^2+324 c^6 d^3 e+120 c^8 d^4-88 c^2 d e^3-105 e^4\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-c^2} x\right ),-\frac{e}{c^2 d}\right )\right )}{3675 \sqrt{-c^2} d^2 \sqrt{1-c^2 x^2} \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 2.136, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\rm arcsec} \left (cx\right )}{{x}^{8}} \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{arcsec}\left (c x\right )\right )} \sqrt{e x^{2} + d}}{x^{8}}, 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{arcsec}\left (c x\right ) + a\right )}}{x^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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