Optimal. Leaf size=252 \[ -\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac{8 b c d^{3/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{3 e^3 \sqrt{c^2 x^2}}+\frac{b x \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}{6 c e^2 \sqrt{c^2 x^2}}-\frac{b x \left (9 c^2 d-e\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right )}{6 c^2 e^{5/2} \sqrt{c^2 x^2}} \]
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Rubi [A] time = 1.07354, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {266, 43, 5239, 12, 1615, 157, 63, 217, 206, 93, 204} \[ -\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac{8 b c d^{3/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{3 e^3 \sqrt{c^2 x^2}}+\frac{b x \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}{6 c e^2 \sqrt{c^2 x^2}}-\frac{b x \left (9 c^2 d-e\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right )}{6 c^2 e^{5/2} \sqrt{c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 5239
Rule 12
Rule 1615
Rule 157
Rule 63
Rule 217
Rule 206
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx &=-\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac{(b c x) \int \frac{-8 d^2-4 d e x^2+e^2 x^4}{3 e^3 x \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{\sqrt{c^2 x^2}}\\ &=-\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac{(b c x) \int \frac{-8 d^2-4 d e x^2+e^2 x^4}{x \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{3 e^3 \sqrt{c^2 x^2}}\\ &=-\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac{(b c x) \operatorname{Subst}\left (\int \frac{-8 d^2-4 d e x+e^2 x^2}{x \sqrt{-1+c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{6 e^3 \sqrt{c^2 x^2}}\\ &=\frac{b x \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{6 c e^2 \sqrt{c^2 x^2}}-\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac{(b x) \operatorname{Subst}\left (\int \frac{-8 c^2 d^2 e-\frac{1}{2} \left (9 c^2 d-e\right ) e^2 x}{x \sqrt{-1+c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{6 c e^4 \sqrt{c^2 x^2}}\\ &=\frac{b x \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{6 c e^2 \sqrt{c^2 x^2}}-\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}-\frac{\left (4 b c d^2 x\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{-1+c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{3 e^3 \sqrt{c^2 x^2}}-\frac{\left (b \left (9 c^2 d-e\right ) x\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{12 c e^2 \sqrt{c^2 x^2}}\\ &=\frac{b x \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{6 c e^2 \sqrt{c^2 x^2}}-\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}-\frac{\left (8 b c d^2 x\right ) \operatorname{Subst}\left (\int \frac{1}{-d-x^2} \, dx,x,\frac{\sqrt{d+e x^2}}{\sqrt{-1+c^2 x^2}}\right )}{3 e^3 \sqrt{c^2 x^2}}-\frac{\left (b \left (9 c^2 d-e\right ) x\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{d+\frac{e}{c^2}+\frac{e x^2}{c^2}}} \, dx,x,\sqrt{-1+c^2 x^2}\right )}{6 c^3 e^2 \sqrt{c^2 x^2}}\\ &=\frac{b x \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{6 c e^2 \sqrt{c^2 x^2}}-\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac{8 b c d^{3/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-1+c^2 x^2}}\right )}{3 e^3 \sqrt{c^2 x^2}}-\frac{\left (b \left (9 c^2 d-e\right ) x\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{e x^2}{c^2}} \, dx,x,\frac{\sqrt{-1+c^2 x^2}}{\sqrt{d+e x^2}}\right )}{6 c^3 e^2 \sqrt{c^2 x^2}}\\ &=\frac{b x \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{6 c e^2 \sqrt{c^2 x^2}}-\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac{8 b c d^{3/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-1+c^2 x^2}}\right )}{3 e^3 \sqrt{c^2 x^2}}-\frac{b \left (9 c^2 d-e\right ) x \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{-1+c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{6 c^2 e^{5/2} \sqrt{c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.607621, size = 302, normalized size = 1.2 \[ \frac{-2 a c \left (8 d^2+4 d e x^2-e^2 x^4\right )+b e x \sqrt{1-\frac{1}{c^2 x^2}} \left (d+e x^2\right )-2 b c \csc ^{-1}(c x) \left (8 d^2+4 d e x^2-e^2 x^4\right )}{6 c e^3 \sqrt{d+e x^2}}-\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left (16 c^5 d^{3/2} \sqrt{d+e x^2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right )+\sqrt{c^2} \sqrt{e} \left (9 c^2 d-e\right ) \sqrt{c^2 d+e} \sqrt{\frac{c^2 \left (d+e x^2\right )}{c^2 d+e}} \sinh ^{-1}\left (\frac{c \sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2} \sqrt{c^2 d+e}}\right )\right )}{6 c^4 e^3 \sqrt{c^2 x^2-1} \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 3.643, size = 0, normalized size = 0. \begin{align*} \int{{x}^{5} \left ( a+b{\rm arccsc} \left (cx\right ) \right ) \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 [A] time = 7.19572, size = 3167, normalized size = 12.57 \begin{align*} \text{result too large to display} \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 (b \operatorname{arccsc}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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