Optimal. Leaf size=597 \[ \frac{i b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}+\frac{i b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}+\frac{i b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}+\frac{i b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}-\frac{i b \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{2 d^2}-\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}-\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}-\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}-\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}+\frac{\log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^2}+\frac{a+b \sin ^{-1}(c x)}{2 d \left (d+e x^2\right )}-\frac{b c \tan ^{-1}\left (\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{2 d^{3/2} \sqrt{c^2 d+e}} \]
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Rubi [A] time = 1.00921, antiderivative size = 597, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 11, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.524, Rules used = {4733, 4625, 3717, 2190, 2279, 2391, 4729, 377, 205, 4741, 4521} \[ \frac{i b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}+\frac{i b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}+\frac{i b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}+\frac{i b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}-\frac{i b \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{2 d^2}-\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}-\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}-\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}-\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}+\frac{\log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^2}+\frac{a+b \sin ^{-1}(c x)}{2 d \left (d+e x^2\right )}-\frac{b c \tan ^{-1}\left (\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{2 d^{3/2} \sqrt{c^2 d+e}} \]
Antiderivative was successfully verified.
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Rule 4733
Rule 4625
Rule 3717
Rule 2190
Rule 2279
Rule 2391
Rule 4729
Rule 377
Rule 205
Rule 4741
Rule 4521
Rubi steps
\begin{align*} \int \frac{a+b \sin ^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx &=\int \left (\frac{a+b \sin ^{-1}(c x)}{d^2 x}-\frac{e x \left (a+b \sin ^{-1}(c x)\right )}{d \left (d+e x^2\right )^2}-\frac{e x \left (a+b \sin ^{-1}(c x)\right )}{d^2 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{a+b \sin ^{-1}(c x)}{x} \, dx}{d^2}-\frac{e \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{d+e x^2} \, dx}{d^2}-\frac{e \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx}{d}\\ &=\frac{a+b \sin ^{-1}(c x)}{2 d \left (d+e x^2\right )}+\frac{\operatorname{Subst}\left (\int (a+b x) \cot (x) \, dx,x,\sin ^{-1}(c x)\right )}{d^2}-\frac{(b c) \int \frac{1}{\sqrt{1-c^2 x^2} \left (d+e x^2\right )} \, dx}{2 d}-\frac{e \int \left (-\frac{a+b \sin ^{-1}(c x)}{2 \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sin ^{-1}(c x)}{2 \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{d^2}\\ &=\frac{a+b \sin ^{-1}(c x)}{2 d \left (d+e x^2\right )}-\frac{i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b d^2}-\frac{(2 i) \operatorname{Subst}\left (\int \frac{e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{d^2}-\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{d-\left (-c^2 d-e\right ) x^2} \, dx,x,\frac{x}{\sqrt{1-c^2 x^2}}\right )}{2 d}+\frac{\sqrt{e} \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{2 d^2}-\frac{\sqrt{e} \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{2 d^2}\\ &=\frac{a+b \sin ^{-1}(c x)}{2 d \left (d+e x^2\right )}-\frac{i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b d^2}-\frac{b c \tan ^{-1}\left (\frac{\sqrt{c^2 d+e} x}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{2 d^{3/2} \sqrt{c^2 d+e}}+\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{d^2}-\frac{b \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2}+\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{(a+b x) \cos (x)}{c \sqrt{-d}-\sqrt{e} \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2}-\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{(a+b x) \cos (x)}{c \sqrt{-d}+\sqrt{e} \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2}\\ &=\frac{a+b \sin ^{-1}(c x)}{2 d \left (d+e x^2\right )}-\frac{b c \tan ^{-1}\left (\frac{\sqrt{c^2 d+e} x}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{2 d^{3/2} \sqrt{c^2 d+e}}+\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{d^2}+\frac{(i b) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{2 d^2}+\frac{\left (i \sqrt{e}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{i c \sqrt{-d}-\sqrt{c^2 d+e}-\sqrt{e} e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2}+\frac{\left (i \sqrt{e}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{i c \sqrt{-d}+\sqrt{c^2 d+e}-\sqrt{e} e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2}-\frac{\left (i \sqrt{e}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{i c \sqrt{-d}-\sqrt{c^2 d+e}+\sqrt{e} e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2}-\frac{\left (i \sqrt{e}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{i c \sqrt{-d}+\sqrt{c^2 d+e}+\sqrt{e} e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2}\\ &=\frac{a+b \sin ^{-1}(c x)}{2 d \left (d+e x^2\right )}-\frac{b c \tan ^{-1}\left (\frac{\sqrt{c^2 d+e} x}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{2 d^{3/2} \sqrt{c^2 d+e}}-\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 d^2}-\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 d^2}-\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 d^2}-\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 d^2}+\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{d^2}-\frac{i b \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{2 d^2}+\frac{b \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{e} e^{i x}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2}+\frac{b \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{e} e^{i x}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2}+\frac{b \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{e} e^{i x}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2}+\frac{b \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{e} e^{i x}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2}\\ &=\frac{a+b \sin ^{-1}(c x)}{2 d \left (d+e x^2\right )}-\frac{b c \tan ^{-1}\left (\frac{\sqrt{c^2 d+e} x}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{2 d^{3/2} \sqrt{c^2 d+e}}-\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 d^2}-\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 d^2}-\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 d^2}-\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 d^2}+\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{d^2}-\frac{i b \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{2 d^2}-\frac{(i b) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{e} x}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 d^2}-\frac{(i b) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{e} x}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 d^2}-\frac{(i b) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{e} x}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 d^2}-\frac{(i b) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{e} x}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 d^2}\\ &=\frac{a+b \sin ^{-1}(c x)}{2 d \left (d+e x^2\right )}-\frac{b c \tan ^{-1}\left (\frac{\sqrt{c^2 d+e} x}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{2 d^{3/2} \sqrt{c^2 d+e}}-\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 d^2}-\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 d^2}-\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 d^2}-\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 d^2}+\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{d^2}+\frac{i b \text{Li}_2\left (-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 d^2}+\frac{i b \text{Li}_2\left (\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 d^2}+\frac{i b \text{Li}_2\left (-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 d^2}+\frac{i b \text{Li}_2\left (\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 d^2}-\frac{i b \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{2 d^2}\\ \end{align*}
Mathematica [F] time = 3.6245, size = 0, normalized size = 0. \[ \int \frac{a+b \sin ^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx \]
Verification is Not applicable to the result.
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Maple [C] time = 0.234, size = 491, normalized size = 0.8 \begin{align*}{\frac{a{c}^{2}}{2\,d \left ({c}^{2}e{x}^{2}+{c}^{2}d \right ) }}-{\frac{a\ln \left ({c}^{2}e{x}^{2}+{c}^{2}d \right ) }{2\,{d}^{2}}}+{\frac{a\ln \left ( cx \right ) }{{d}^{2}}}+{\frac{b{c}^{2}\arcsin \left ( cx \right ) }{2\,d \left ({c}^{2}e{x}^{2}+{c}^{2}d \right ) }}+{\frac{{\frac{i}{2}}b}{{d}^{2} \left ({c}^{2}d+e \right ) }\sqrt{{c}^{2}d \left ({c}^{2}d+e \right ) }{\it Artanh} \left ({\frac{1}{4} \left ( 2\, \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) ^{2}e-4\,{c}^{2}d-2\,e \right ){\frac{1}{\sqrt{{d}^{2}{c}^{4}+{c}^{2}ed}}}} \right ) }+{\frac{ib}{{d}^{2}}{\it dilog} \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) }+{\frac{b\arcsin \left ( cx \right ) }{{d}^{2}}\ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) }-{\frac{ib}{{d}^{2}}{\it dilog} \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) }+{\frac{{\frac{i}{4}}b}{{d}^{2}}\sum _{{\it \_R1}={\it RootOf} \left ( e{{\it \_Z}}^{4}+ \left ( -4\,{c}^{2}d-2\,e \right ){{\it \_Z}}^{2}+e \right ) }{\frac{{{\it \_R1}}^{2}e-4\,{c}^{2}d-e}{{{\it \_R1}}^{2}e-2\,{c}^{2}d-e} \left ( i\arcsin \left ( cx \right ) \ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \right ) \right ) }}+{\frac{{\frac{i}{4}}be}{{d}^{2}}\sum _{{\it \_R1}={\it RootOf} \left ( e{{\it \_Z}}^{4}+ \left ( -4\,{c}^{2}d-2\,e \right ){{\it \_Z}}^{2}+e \right ) }{\frac{{{\it \_R1}}^{2}-1}{{{\it \_R1}}^{2}e-2\,{c}^{2}d-e} \left ( i\arcsin \left ( cx \right ) \ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \right ) \right ) }} \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}{2} \, a{\left (\frac{1}{d e x^{2} + d^{2}} - \frac{\log \left (e x^{2} + d\right )}{d^{2}} + \frac{2 \, \log \left (x\right )}{d^{2}}\right )} + b \int \frac{\arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{e^{2} x^{5} + 2 \, d e x^{3} + d^{2} x}\,{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{b \arcsin \left (c x\right ) + a}{e^{2} x^{5} + 2 \, d e x^{3} + d^{2} x}, 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 \arcsin \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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