Optimal. Leaf size=453 \[ -\frac{b x \sqrt{1-c^2 x^2} \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\right ) \sqrt{\frac{e x^2}{d}+1} \text{EllipticF}\left (\sin ^{-1}(c x),-\frac{e}{c^2 d}\right )}{225 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}+\frac{2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}-\frac{b c \sqrt{c^2 x^2-1} \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt{d+e x^2}}{225 d^2 \sqrt{c^2 x^2}}+\frac{b c^2 x \sqrt{1-c^2 x^2} \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{225 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}}-\frac{b c \sqrt{c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt{c^2 x^2}}-\frac{b c \sqrt{c^2 x^2-1} \left (12 c^2 d-e\right ) \sqrt{d+e x^2}}{225 d x^2 \sqrt{c^2 x^2}} \]
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Rubi [A] time = 0.620064, antiderivative size = 453, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 12, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.522, Rules used = {271, 264, 5239, 12, 580, 583, 524, 427, 426, 424, 421, 419} \[ \frac{2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}-\frac{b c \sqrt{c^2 x^2-1} \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt{d+e x^2}}{225 d^2 \sqrt{c^2 x^2}}-\frac{b x \sqrt{1-c^2 x^2} \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\right ) \sqrt{\frac{e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{225 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}+\frac{b c^2 x \sqrt{1-c^2 x^2} \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{225 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}}-\frac{b c \sqrt{c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt{c^2 x^2}}-\frac{b c \sqrt{c^2 x^2-1} \left (12 c^2 d-e\right ) \sqrt{d+e x^2}}{225 d x^2 \sqrt{c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 271
Rule 264
Rule 5239
Rule 12
Rule 580
Rule 583
Rule 524
Rule 427
Rule 426
Rule 424
Rule 421
Rule 419
Rubi steps
\begin{align*} \int \frac{\sqrt{d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx &=-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}+\frac{2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}+\frac{(b c x) \int \frac{\left (d+e x^2\right )^{3/2} \left (-3 d+2 e x^2\right )}{15 d^2 x^6 \sqrt{-1+c^2 x^2}} \, dx}{\sqrt{c^2 x^2}}\\ &=-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}+\frac{2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}+\frac{(b c x) \int \frac{\left (d+e x^2\right )^{3/2} \left (-3 d+2 e x^2\right )}{x^6 \sqrt{-1+c^2 x^2}} \, dx}{15 d^2 \sqrt{c^2 x^2}}\\ &=-\frac{b c \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt{c^2 x^2}}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}+\frac{2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}-\frac{(b c x) \int \frac{\sqrt{d+e x^2} \left (d \left (12 c^2 d-e\right )+\left (3 c^2 d-10 e\right ) e x^2\right )}{x^4 \sqrt{-1+c^2 x^2}} \, dx}{75 d^2 \sqrt{c^2 x^2}}\\ &=-\frac{b c \left (12 c^2 d-e\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{225 d 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 d x^4 \sqrt{c^2 x^2}}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}+\frac{2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}+\frac{(b c x) \int \frac{-d \left (24 c^4 d^2+19 c^2 d e-31 e^2\right )-2 e \left (6 c^4 d^2+4 c^2 d e-15 e^2\right ) x^2}{x^2 \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{225 d^2 \sqrt{c^2 x^2}}\\ &=-\frac{b c \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{225 d^2 \sqrt{c^2 x^2}}-\frac{b c \left (12 c^2 d-e\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{225 d 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 d x^4 \sqrt{c^2 x^2}}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}+\frac{2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}+\frac{(b c x) \int \frac{-2 d e \left (6 c^4 d^2+4 c^2 d e-15 e^2\right )+c^2 d e \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) x^2}{\sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{225 d^3 \sqrt{c^2 x^2}}\\ &=-\frac{b c \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{225 d^2 \sqrt{c^2 x^2}}-\frac{b c \left (12 c^2 d-e\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{225 d 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 d x^4 \sqrt{c^2 x^2}}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}+\frac{2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}+\frac{\left (b c^3 \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) x\right ) \int \frac{\sqrt{d+e x^2}}{\sqrt{-1+c^2 x^2}} \, dx}{225 d^2 \sqrt{c^2 x^2}}-\frac{\left (b c \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\right ) x\right ) \int \frac{1}{\sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{225 d^2 \sqrt{c^2 x^2}}\\ &=-\frac{b c \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{225 d^2 \sqrt{c^2 x^2}}-\frac{b c \left (12 c^2 d-e\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{225 d 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 d x^4 \sqrt{c^2 x^2}}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}+\frac{2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}+\frac{\left (b c^3 \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) x \sqrt{1-c^2 x^2}\right ) \int \frac{\sqrt{d+e x^2}}{\sqrt{1-c^2 x^2}} \, dx}{225 d^2 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2}}-\frac{\left (b c \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\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}{225 d^2 \sqrt{c^2 x^2} \sqrt{d+e x^2}}\\ &=-\frac{b c \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{225 d^2 \sqrt{c^2 x^2}}-\frac{b c \left (12 c^2 d-e\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{225 d 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 d x^4 \sqrt{c^2 x^2}}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}+\frac{2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}+\frac{\left (b c^3 \left (24 c^4 d^2+19 c^2 d e-31 e^2\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}{225 d^2 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{1+\frac{e x^2}{d}}}-\frac{\left (b c \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\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}{225 d^2 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}\\ &=-\frac{b c \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{225 d^2 \sqrt{c^2 x^2}}-\frac{b c \left (12 c^2 d-e\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{225 d 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 d x^4 \sqrt{c^2 x^2}}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}+\frac{2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}+\frac{b c^2 \left (24 c^4 d^2+19 c^2 d e-31 e^2\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 )}{225 d^2 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{1+\frac{e x^2}{d}}}-\frac{b \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\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 )}{225 d^2 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}\\ \end{align*}
Mathematica [C] time = 0.698121, size = 325, normalized size = 0.72 \[ -\frac{\sqrt{d+e x^2} \left (15 a \left (3 d^2+d e x^2-2 e^2 x^4\right )+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left (3 d^2 \left (8 c^4 x^4+4 c^2 x^2+3\right )+d e x^2 \left (19 c^2 x^2+8\right )-31 e^2 x^4\right )+15 b \csc ^{-1}(c x) \left (3 d^2+d e x^2-2 e^2 x^4\right )\right )}{225 d^2 x^5}+\frac{i b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{e x^2}{d}+1} \left (\left (-31 c^4 d^2 e-24 c^6 d^3+23 c^2 d e^2+30 e^3\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-c^2} x\right ),-\frac{e}{c^2 d}\right )+c^2 d \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) E\left (i \sinh ^{-1}\left (\sqrt{-c^2} x\right )|-\frac{e}{c^2 d}\right )\right )}{225 \sqrt{-c^2} d^2 \sqrt{1-c^2 x^2} \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 7.089, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\rm arccsc} \left (cx\right )}{{x}^{6}}\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{arccsc}\left (c x\right ) + a\right )}}{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{\sqrt{e x^{2} + d}{\left (b \operatorname{arccsc}\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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