Optimal. Leaf size=362 \[ \frac{2 b x \sqrt{1-c^2 x^2} \left (c^2 d-3 e\right ) \left (c^2 d+e\right ) \sqrt{\frac{e x^2}{d}+1} \text{EllipticF}\left (\sin ^{-1}(c x),-\frac{e}{c^2 d}\right )}{9 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}+\frac{2 e \sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}-\frac{\sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}+\frac{b c \sqrt{c^2 x^2-1} \left (2 c^2 d-5 e\right ) \sqrt{d+e x^2}}{9 d^2 \sqrt{c^2 x^2}}-\frac{b c^2 x \sqrt{1-c^2 x^2} \left (2 c^2 d-5 e\right ) \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{9 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}}+\frac{b c \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}{9 d x^2 \sqrt{c^2 x^2}} \]
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Rubi [A] time = 0.446629, antiderivative size = 362, normalized size of antiderivative = 1., number of steps used = 11, 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} \[ \frac{2 e \sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}-\frac{\sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}+\frac{b c \sqrt{c^2 x^2-1} \left (2 c^2 d-5 e\right ) \sqrt{d+e x^2}}{9 d^2 \sqrt{c^2 x^2}}+\frac{2 b x \sqrt{1-c^2 x^2} \left (c^2 d-3 e\right ) \left (c^2 d+e\right ) \sqrt{\frac{e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{9 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}-\frac{b c^2 x \sqrt{1-c^2 x^2} \left (2 c^2 d-5 e\right ) \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{9 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}}+\frac{b c \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}{9 d x^2 \sqrt{c^2 x^2}} \]
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{a+b \sec ^{-1}(c x)}{x^4 \sqrt{d+e x^2}} \, dx &=-\frac{\sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}+\frac{2 e \sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}-\frac{(b c x) \int \frac{\sqrt{d+e x^2} \left (-d+2 e x^2\right )}{3 d^2 x^4 \sqrt{-1+c^2 x^2}} \, dx}{\sqrt{c^2 x^2}}\\ &=-\frac{\sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}+\frac{2 e \sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}-\frac{(b c x) \int \frac{\sqrt{d+e x^2} \left (-d+2 e x^2\right )}{x^4 \sqrt{-1+c^2 x^2}} \, dx}{3 d^2 \sqrt{c^2 x^2}}\\ &=\frac{b c \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 d x^2 \sqrt{c^2 x^2}}-\frac{\sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}+\frac{2 e \sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}+\frac{(b c x) \int \frac{d \left (2 c^2 d-5 e\right )+\left (c^2 d-6 e\right ) e x^2}{x^2 \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{9 d^2 \sqrt{c^2 x^2}}\\ &=\frac{b c \left (2 c^2 d-5 e\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 d^2 \sqrt{c^2 x^2}}+\frac{b c \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 d x^2 \sqrt{c^2 x^2}}-\frac{\sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}+\frac{2 e \sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}+\frac{(b c x) \int \frac{d \left (c^2 d-6 e\right ) e-c^2 d \left (2 c^2 d-5 e\right ) e x^2}{\sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{9 d^3 \sqrt{c^2 x^2}}\\ &=\frac{b c \left (2 c^2 d-5 e\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 d^2 \sqrt{c^2 x^2}}+\frac{b c \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 d x^2 \sqrt{c^2 x^2}}-\frac{\sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}+\frac{2 e \sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}-\frac{\left (b c^3 \left (2 c^2 d-5 e\right ) x\right ) \int \frac{\sqrt{d+e x^2}}{\sqrt{-1+c^2 x^2}} \, dx}{9 d^2 \sqrt{c^2 x^2}}+\frac{\left (2 b c \left (c^2 d-3 e\right ) \left (c^2 d+e\right ) x\right ) \int \frac{1}{\sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{9 d^2 \sqrt{c^2 x^2}}\\ &=\frac{b c \left (2 c^2 d-5 e\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 d^2 \sqrt{c^2 x^2}}+\frac{b c \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 d x^2 \sqrt{c^2 x^2}}-\frac{\sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}+\frac{2 e \sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}-\frac{\left (b c^3 \left (2 c^2 d-5 e\right ) x \sqrt{1-c^2 x^2}\right ) \int \frac{\sqrt{d+e x^2}}{\sqrt{1-c^2 x^2}} \, dx}{9 d^2 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2}}+\frac{\left (2 b c \left (c^2 d-3 e\right ) \left (c^2 d+e\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}{9 d^2 \sqrt{c^2 x^2} \sqrt{d+e x^2}}\\ &=\frac{b c \left (2 c^2 d-5 e\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 d^2 \sqrt{c^2 x^2}}+\frac{b c \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 d x^2 \sqrt{c^2 x^2}}-\frac{\sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}+\frac{2 e \sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}-\frac{\left (b c^3 \left (2 c^2 d-5 e\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}{9 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-3 e\right ) \left (c^2 d+e\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}{9 d^2 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}\\ &=\frac{b c \left (2 c^2 d-5 e\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 d^2 \sqrt{c^2 x^2}}+\frac{b c \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 d x^2 \sqrt{c^2 x^2}}-\frac{\sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}+\frac{2 e \sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}-\frac{b c^2 \left (2 c^2 d-5 e\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 )}{9 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-3 e\right ) \left (c^2 d+e\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 )}{9 d^2 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}\\ \end{align*}
Mathematica [C] time = 0.635467, size = 249, normalized size = 0.69 \[ \frac{\sqrt{d+e x^2} \left (-3 a \left (d-2 e x^2\right )+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left (2 c^2 d x^2+d-5 e x^2\right )-3 b \sec ^{-1}(c x) \left (d-2 e x^2\right )\right )}{9 d^2 x^3}-\frac{i b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{e x^2}{d}+1} \left (2 \left (c^4 \left (-d^2\right )+2 c^2 d e+3 e^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-c^2} x\right ),-\frac{e}{c^2 d}\right )+c^2 d \left (2 c^2 d-5 e\right ) E\left (i \sinh ^{-1}\left (\sqrt{-c^2} x\right )|-\frac{e}{c^2 d}\right )\right )}{9 \sqrt{-c^2} d^2 \sqrt{1-c^2 x^2} \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.717, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\rm arcsec} \left (cx\right )}{{x}^{4}}{\frac{1}{\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{arcsec}\left (c x\right ) + a\right )}}{e x^{6} + d x^{4}}, 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{b \operatorname{arcsec}\left (c x\right ) + a}{\sqrt{e x^{2} + d} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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