Optimal. Leaf size=276 \[ \frac{b x \sqrt{1-c^2 x^2} \sqrt{\frac{e x^2}{d}+1} \text{EllipticF}\left (\sin ^{-1}(c x),-\frac{e}{c^2 d}\right )}{3 d e \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}+\frac{x^3 \left (a+b \csc ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{b c x^2 \sqrt{c^2 x^2-1}}{3 d \sqrt{c^2 x^2} \left (c^2 d+e\right ) \sqrt{d+e x^2}}-\frac{b c^2 x \sqrt{1-c^2 x^2} \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3 d e \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \left (c^2 d+e\right ) \sqrt{\frac{e x^2}{d}+1}} \]
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Rubi [A] time = 0.283373, antiderivative size = 276, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {264, 5239, 12, 471, 423, 427, 426, 424, 421, 419} \[ \frac{x^3 \left (a+b \csc ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{b c x^2 \sqrt{c^2 x^2-1}}{3 d \sqrt{c^2 x^2} \left (c^2 d+e\right ) \sqrt{d+e x^2}}+\frac{b x \sqrt{1-c^2 x^2} \sqrt{\frac{e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3 d e \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}-\frac{b c^2 x \sqrt{1-c^2 x^2} \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3 d e \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \left (c^2 d+e\right ) \sqrt{\frac{e x^2}{d}+1}} \]
Antiderivative was successfully verified.
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Rule 264
Rule 5239
Rule 12
Rule 471
Rule 423
Rule 427
Rule 426
Rule 424
Rule 421
Rule 419
Rubi steps
\begin{align*} \int \frac{x^2 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=\frac{x^3 \left (a+b \csc ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{(b c x) \int \frac{x^2}{3 d \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{\sqrt{c^2 x^2}}\\ &=\frac{x^3 \left (a+b \csc ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{(b c x) \int \frac{x^2}{\sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{3 d \sqrt{c^2 x^2}}\\ &=\frac{b c x^2 \sqrt{-1+c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt{c^2 x^2} \sqrt{d+e x^2}}+\frac{x^3 \left (a+b \csc ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac{(b c x) \int \frac{\sqrt{-1+c^2 x^2}}{\sqrt{d+e x^2}} \, dx}{3 d \left (c^2 d+e\right ) \sqrt{c^2 x^2}}\\ &=\frac{b c x^2 \sqrt{-1+c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt{c^2 x^2} \sqrt{d+e x^2}}+\frac{x^3 \left (a+b \csc ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{(b c x) \int \frac{1}{\sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{3 d e \sqrt{c^2 x^2}}-\frac{\left (b c^3 x\right ) \int \frac{\sqrt{d+e x^2}}{\sqrt{-1+c^2 x^2}} \, dx}{3 d e \left (c^2 d+e\right ) \sqrt{c^2 x^2}}\\ &=\frac{b c x^2 \sqrt{-1+c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt{c^2 x^2} \sqrt{d+e x^2}}+\frac{x^3 \left (a+b \csc ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac{\left (b c^3 x \sqrt{1-c^2 x^2}\right ) \int \frac{\sqrt{d+e x^2}}{\sqrt{1-c^2 x^2}} \, dx}{3 d e \left (c^2 d+e\right ) \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2}}+\frac{\left (b c 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}{3 d e \sqrt{c^2 x^2} \sqrt{d+e x^2}}\\ &=\frac{b c x^2 \sqrt{-1+c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt{c^2 x^2} \sqrt{d+e x^2}}+\frac{x^3 \left (a+b \csc ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac{\left (b c^3 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}{3 d e \left (c^2 d+e\right ) \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{1+\frac{e x^2}{d}}}+\frac{\left (b c 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}{3 d e \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}\\ &=\frac{b c x^2 \sqrt{-1+c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt{c^2 x^2} \sqrt{d+e x^2}}+\frac{x^3 \left (a+b \csc ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac{b c^2 x \sqrt{1-c^2 x^2} \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3 d e \left (c^2 d+e\right ) \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{1+\frac{e x^2}{d}}}+\frac{b 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 )}{3 d e \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}\\ \end{align*}
Mathematica [A] time = 0.27801, size = 185, normalized size = 0.67 \[ \frac{x^2 \left (a x \left (c^2 d+e\right )+b c \sqrt{1-\frac{1}{c^2 x^2}} \left (d+e x^2\right )+b x \left (c^2 d+e\right ) \csc ^{-1}(c x)\right )}{3 d \left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}-\frac{b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{e x^2}{d}+1} E\left (\sin ^{-1}\left (\sqrt{-\frac{e}{d}} x\right )|-\frac{c^2 d}{e}\right )}{3 d \sqrt{1-c^2 x^2} \sqrt{-\frac{e}{d}} \left (c^2 d+e\right ) \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 2.079, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ( a+b{\rm arccsc} \left (cx\right ) \right ) \left ( e{x}^{2}+d \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{3} \, a{\left (\frac{x}{{\left (e x^{2} + d\right )}^{\frac{3}{2}} e} - \frac{x}{\sqrt{e x^{2} + d} d e}\right )} + b \int \frac{x^{2} \arctan \left (1, \sqrt{c x + 1} \sqrt{c x - 1}\right )}{{\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \sqrt{e x^{2} + d}}\,{d x} \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 (b x^{2} \operatorname{arccsc}\left (c x\right ) + a x^{2}\right )} \sqrt{e x^{2} + d}}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}, 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 (b \operatorname{arccsc}\left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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