Optimal. Leaf size=787 \[ \frac{3 i b \sqrt{-d} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 e^{5/2}}-\frac{3 i b \sqrt{-d} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 e^{5/2}}+\frac{3 i b \sqrt{-d} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 e^{5/2}}-\frac{3 i b \sqrt{-d} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 e^{5/2}}+\frac{3 \sqrt{-d} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 e^{5/2}}-\frac{3 \sqrt{-d} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 e^{5/2}}+\frac{3 \sqrt{-d} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 e^{5/2}}-\frac{3 \sqrt{-d} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 e^{5/2}}-\frac{d \left (a+b \sin ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{d \left (a+b \sin ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{a x}{e^2}+\frac{b c d \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right )}{4 e^{5/2} \sqrt{c^2 d+e}}+\frac{b c d \tanh ^{-1}\left (\frac{c^2 \sqrt{-d} x+\sqrt{e}}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right )}{4 e^{5/2} \sqrt{c^2 d+e}}+\frac{b \sqrt{1-c^2 x^2}}{c e^2}+\frac{b x \sin ^{-1}(c x)}{e^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 2.02723, antiderivative size = 787, normalized size of antiderivative = 1., number of steps used = 49, number of rules used = 12, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {4733, 4619, 261, 4667, 4743, 725, 206, 4741, 4521, 2190, 2279, 2391} \[ \frac{3 i b \sqrt{-d} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 e^{5/2}}-\frac{3 i b \sqrt{-d} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 e^{5/2}}+\frac{3 i b \sqrt{-d} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 e^{5/2}}-\frac{3 i b \sqrt{-d} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 e^{5/2}}+\frac{3 \sqrt{-d} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 e^{5/2}}-\frac{3 \sqrt{-d} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 e^{5/2}}+\frac{3 \sqrt{-d} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 e^{5/2}}-\frac{3 \sqrt{-d} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 e^{5/2}}-\frac{d \left (a+b \sin ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{d \left (a+b \sin ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{a x}{e^2}+\frac{b c d \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right )}{4 e^{5/2} \sqrt{c^2 d+e}}+\frac{b c d \tanh ^{-1}\left (\frac{c^2 \sqrt{-d} x+\sqrt{e}}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right )}{4 e^{5/2} \sqrt{c^2 d+e}}+\frac{b \sqrt{1-c^2 x^2}}{c e^2}+\frac{b x \sin ^{-1}(c x)}{e^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4733
Rule 4619
Rule 261
Rule 4667
Rule 4743
Rule 725
Rule 206
Rule 4741
Rule 4521
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{x^4 \left (a+b \sin ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx &=\int \left (\frac{a+b \sin ^{-1}(c x)}{e^2}+\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{e^2 \left (d+e x^2\right )^2}-\frac{2 d \left (a+b \sin ^{-1}(c x)\right )}{e^2 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac{\int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{e^2}-\frac{(2 d) \int \frac{a+b \sin ^{-1}(c x)}{d+e x^2} \, dx}{e^2}+\frac{d^2 \int \frac{a+b \sin ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx}{e^2}\\ &=\frac{a x}{e^2}+\frac{b \int \sin ^{-1}(c x) \, dx}{e^2}-\frac{(2 d) \int \left (\frac{\sqrt{-d} \left (a+b \sin ^{-1}(c x)\right )}{2 d \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\sqrt{-d} \left (a+b \sin ^{-1}(c x)\right )}{2 d \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{e^2}+\frac{d^2 \int \left (-\frac{e \left (a+b \sin ^{-1}(c x)\right )}{4 d \left (\sqrt{-d} \sqrt{e}-e x\right )^2}-\frac{e \left (a+b \sin ^{-1}(c x)\right )}{4 d \left (\sqrt{-d} \sqrt{e}+e x\right )^2}-\frac{e \left (a+b \sin ^{-1}(c x)\right )}{2 d \left (-d e-e^2 x^2\right )}\right ) \, dx}{e^2}\\ &=\frac{a x}{e^2}+\frac{b x \sin ^{-1}(c x)}{e^2}-\frac{(b c) \int \frac{x}{\sqrt{1-c^2 x^2}} \, dx}{e^2}-\frac{\sqrt{-d} \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{e^2}-\frac{\sqrt{-d} \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{e^2}-\frac{d \int \frac{a+b \sin ^{-1}(c x)}{\left (\sqrt{-d} \sqrt{e}-e x\right )^2} \, dx}{4 e}-\frac{d \int \frac{a+b \sin ^{-1}(c x)}{\left (\sqrt{-d} \sqrt{e}+e x\right )^2} \, dx}{4 e}-\frac{d \int \frac{a+b \sin ^{-1}(c x)}{-d e-e^2 x^2} \, dx}{2 e}\\ &=\frac{a x}{e^2}+\frac{b \sqrt{1-c^2 x^2}}{c e^2}+\frac{b x \sin ^{-1}(c x)}{e^2}-\frac{d \left (a+b \sin ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{d \left (a+b \sin ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{\sqrt{-d} \operatorname{Subst}\left (\int \frac{(a+b x) \cos (x)}{c \sqrt{-d}-\sqrt{e} \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e^2}-\frac{\sqrt{-d} \operatorname{Subst}\left (\int \frac{(a+b x) \cos (x)}{c \sqrt{-d}+\sqrt{e} \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e^2}+\frac{(b c d) \int \frac{1}{\left (\sqrt{-d} \sqrt{e}-e x\right ) \sqrt{1-c^2 x^2}} \, dx}{4 e^2}-\frac{(b c d) \int \frac{1}{\left (\sqrt{-d} \sqrt{e}+e x\right ) \sqrt{1-c^2 x^2}} \, dx}{4 e^2}-\frac{d \int \left (-\frac{\sqrt{-d} \left (a+b \sin ^{-1}(c x)\right )}{2 d e \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{-d} \left (a+b \sin ^{-1}(c x)\right )}{2 d e \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{2 e}\\ &=\frac{a x}{e^2}+\frac{b \sqrt{1-c^2 x^2}}{c e^2}+\frac{b x \sin ^{-1}(c x)}{e^2}-\frac{d \left (a+b \sin ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{d \left (a+b \sin ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{\left (i \sqrt{-d}\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 )}{e^2}-\frac{\left (i \sqrt{-d}\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 )}{e^2}-\frac{\left (i \sqrt{-d}\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 )}{e^2}-\frac{\left (i \sqrt{-d}\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 )}{e^2}+\frac{\sqrt{-d} \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{4 e^2}+\frac{\sqrt{-d} \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{4 e^2}-\frac{(b c d) \operatorname{Subst}\left (\int \frac{1}{c^2 d e+e^2-x^2} \, dx,x,\frac{-e+c^2 \sqrt{-d} \sqrt{e} x}{\sqrt{1-c^2 x^2}}\right )}{4 e^2}+\frac{(b c d) \operatorname{Subst}\left (\int \frac{1}{c^2 d e+e^2-x^2} \, dx,x,\frac{e+c^2 \sqrt{-d} \sqrt{e} x}{\sqrt{1-c^2 x^2}}\right )}{4 e^2}\\ &=\frac{a x}{e^2}+\frac{b \sqrt{1-c^2 x^2}}{c e^2}+\frac{b x \sin ^{-1}(c x)}{e^2}-\frac{d \left (a+b \sin ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{d \left (a+b \sin ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b c d \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{4 e^{5/2} \sqrt{c^2 d+e}}+\frac{b c d \tanh ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{4 e^{5/2} \sqrt{c^2 d+e}}+\frac{\sqrt{-d} \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 )}{e^{5/2}}-\frac{\sqrt{-d} \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 )}{e^{5/2}}+\frac{\sqrt{-d} \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 )}{e^{5/2}}-\frac{\sqrt{-d} \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 )}{e^{5/2}}-\frac{\left (b \sqrt{-d}\right ) \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 )}{e^{5/2}}+\frac{\left (b \sqrt{-d}\right ) \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 )}{e^{5/2}}-\frac{\left (b \sqrt{-d}\right ) \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 )}{e^{5/2}}+\frac{\left (b \sqrt{-d}\right ) \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 )}{e^{5/2}}+\frac{\sqrt{-d} \operatorname{Subst}\left (\int \frac{(a+b x) \cos (x)}{c \sqrt{-d}-\sqrt{e} \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{4 e^2}+\frac{\sqrt{-d} \operatorname{Subst}\left (\int \frac{(a+b x) \cos (x)}{c \sqrt{-d}+\sqrt{e} \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{4 e^2}\\ &=\frac{a x}{e^2}+\frac{b \sqrt{1-c^2 x^2}}{c e^2}+\frac{b x \sin ^{-1}(c x)}{e^2}-\frac{d \left (a+b \sin ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{d \left (a+b \sin ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b c d \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{4 e^{5/2} \sqrt{c^2 d+e}}+\frac{b c d \tanh ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{4 e^{5/2} \sqrt{c^2 d+e}}+\frac{\sqrt{-d} \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 )}{e^{5/2}}-\frac{\sqrt{-d} \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 )}{e^{5/2}}+\frac{\sqrt{-d} \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 )}{e^{5/2}}-\frac{\sqrt{-d} \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 )}{e^{5/2}}+\frac{\left (i b \sqrt{-d}\right ) \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 )}{e^{5/2}}-\frac{\left (i b \sqrt{-d}\right ) \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 )}{e^{5/2}}+\frac{\left (i b \sqrt{-d}\right ) \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 )}{e^{5/2}}-\frac{\left (i b \sqrt{-d}\right ) \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 )}{e^{5/2}}+\frac{\left (i \sqrt{-d}\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 )}{4 e^2}+\frac{\left (i \sqrt{-d}\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 )}{4 e^2}+\frac{\left (i \sqrt{-d}\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 )}{4 e^2}+\frac{\left (i \sqrt{-d}\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 )}{4 e^2}\\ &=\frac{a x}{e^2}+\frac{b \sqrt{1-c^2 x^2}}{c e^2}+\frac{b x \sin ^{-1}(c x)}{e^2}-\frac{d \left (a+b \sin ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{d \left (a+b \sin ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b c d \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{4 e^{5/2} \sqrt{c^2 d+e}}+\frac{b c d \tanh ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{4 e^{5/2} \sqrt{c^2 d+e}}+\frac{3 \sqrt{-d} \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 )}{4 e^{5/2}}-\frac{3 \sqrt{-d} \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 )}{4 e^{5/2}}+\frac{3 \sqrt{-d} \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 )}{4 e^{5/2}}-\frac{3 \sqrt{-d} \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 )}{4 e^{5/2}}+\frac{i b \sqrt{-d} \text{Li}_2\left (-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{e^{5/2}}-\frac{i b \sqrt{-d} \text{Li}_2\left (\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{e^{5/2}}+\frac{i b \sqrt{-d} \text{Li}_2\left (-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{e^{5/2}}-\frac{i b \sqrt{-d} \text{Li}_2\left (\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{e^{5/2}}+\frac{\left (b \sqrt{-d}\right ) \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 )}{4 e^{5/2}}-\frac{\left (b \sqrt{-d}\right ) \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 )}{4 e^{5/2}}+\frac{\left (b \sqrt{-d}\right ) \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 )}{4 e^{5/2}}-\frac{\left (b \sqrt{-d}\right ) \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 )}{4 e^{5/2}}\\ &=\frac{a x}{e^2}+\frac{b \sqrt{1-c^2 x^2}}{c e^2}+\frac{b x \sin ^{-1}(c x)}{e^2}-\frac{d \left (a+b \sin ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{d \left (a+b \sin ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b c d \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{4 e^{5/2} \sqrt{c^2 d+e}}+\frac{b c d \tanh ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{4 e^{5/2} \sqrt{c^2 d+e}}+\frac{3 \sqrt{-d} \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 )}{4 e^{5/2}}-\frac{3 \sqrt{-d} \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 )}{4 e^{5/2}}+\frac{3 \sqrt{-d} \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 )}{4 e^{5/2}}-\frac{3 \sqrt{-d} \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 )}{4 e^{5/2}}+\frac{i b \sqrt{-d} \text{Li}_2\left (-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{e^{5/2}}-\frac{i b \sqrt{-d} \text{Li}_2\left (\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{e^{5/2}}+\frac{i b \sqrt{-d} \text{Li}_2\left (-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{e^{5/2}}-\frac{i b \sqrt{-d} \text{Li}_2\left (\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{e^{5/2}}-\frac{\left (i b \sqrt{-d}\right ) \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 )}{4 e^{5/2}}+\frac{\left (i b \sqrt{-d}\right ) \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 )}{4 e^{5/2}}-\frac{\left (i b \sqrt{-d}\right ) \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 )}{4 e^{5/2}}+\frac{\left (i b \sqrt{-d}\right ) \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 )}{4 e^{5/2}}\\ &=\frac{a x}{e^2}+\frac{b \sqrt{1-c^2 x^2}}{c e^2}+\frac{b x \sin ^{-1}(c x)}{e^2}-\frac{d \left (a+b \sin ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{d \left (a+b \sin ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b c d \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{4 e^{5/2} \sqrt{c^2 d+e}}+\frac{b c d \tanh ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{4 e^{5/2} \sqrt{c^2 d+e}}+\frac{3 \sqrt{-d} \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 )}{4 e^{5/2}}-\frac{3 \sqrt{-d} \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 )}{4 e^{5/2}}+\frac{3 \sqrt{-d} \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 )}{4 e^{5/2}}-\frac{3 \sqrt{-d} \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 )}{4 e^{5/2}}+\frac{3 i b \sqrt{-d} \text{Li}_2\left (-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{4 e^{5/2}}-\frac{3 i b \sqrt{-d} \text{Li}_2\left (\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{4 e^{5/2}}+\frac{3 i b \sqrt{-d} \text{Li}_2\left (-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{4 e^{5/2}}-\frac{3 i b \sqrt{-d} \text{Li}_2\left (\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{4 e^{5/2}}\\ \end{align*}
Mathematica [A] time = 1.50994, size = 649, normalized size = 0.82 \[ \frac{b \left (3 \sqrt{d} \left (2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right )+2 \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )+\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+2 i \left (\log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right )+\log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )\right )\right )\right )-3 \sqrt{d} \left (2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )+\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+2 i \left (\log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right )+\log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )\right )\right )\right )+2 i d \left (\frac{\sin ^{-1}(c x)}{\sqrt{d}+i \sqrt{e} x}-\frac{c \tan ^{-1}\left (\frac{c^2 \sqrt{d} x+i \sqrt{e}}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right )}{\sqrt{c^2 d+e}}\right )+2 d \left (\frac{c \tanh ^{-1}\left (\frac{\sqrt{e}+i c^2 \sqrt{d} x}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right )}{\sqrt{c^2 d+e}}+\frac{\sin ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}\right )+\frac{8 \sqrt{e} \left (\sqrt{1-c^2 x^2}+c x \sin ^{-1}(c x)\right )}{c}\right )+\frac{4 a d \sqrt{e} x}{d+e x^2}-12 a \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )+8 a \sqrt{e} x}{8 e^{5/2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 1.484, size = 1738, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b x^{4} \arcsin \left (c x\right ) + a x^{4}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \left (a + b \operatorname{asin}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arcsin \left (c x\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]