Integrand size = 23, antiderivative size = 270 \[ \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^2} \, dx=-\frac {(a+b \arcsin (c+d x))^4}{d e^2 (c+d x)}-\frac {8 b (a+b \arcsin (c+d x))^3 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right )}{d e^2}+\frac {12 i b^2 (a+b \arcsin (c+d x))^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )}{d e^2}-\frac {12 i b^2 (a+b \arcsin (c+d x))^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )}{d e^2}-\frac {24 b^3 (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (3,-e^{i \arcsin (c+d x)}\right )}{d e^2}+\frac {24 b^3 (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (3,e^{i \arcsin (c+d x)}\right )}{d e^2}-\frac {24 i b^4 \operatorname {PolyLog}\left (4,-e^{i \arcsin (c+d x)}\right )}{d e^2}+\frac {24 i b^4 \operatorname {PolyLog}\left (4,e^{i \arcsin (c+d x)}\right )}{d e^2} \]
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Time = 0.22 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {4889, 12, 4723, 4803, 4268, 2611, 6744, 2320, 6724} \[ \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^2} \, dx=-\frac {8 b \text {arctanh}\left (e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^3}{d e^2}-\frac {24 b^3 \operatorname {PolyLog}\left (3,-e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))}{d e^2}+\frac {24 b^3 \operatorname {PolyLog}\left (3,e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))}{d e^2}+\frac {12 i b^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^2}{d e^2}-\frac {12 i b^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^2}{d e^2}-\frac {(a+b \arcsin (c+d x))^4}{d e^2 (c+d x)}-\frac {24 i b^4 \operatorname {PolyLog}\left (4,-e^{i \arcsin (c+d x)}\right )}{d e^2}+\frac {24 i b^4 \operatorname {PolyLog}\left (4,e^{i \arcsin (c+d x)}\right )}{d e^2} \]
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Rule 12
Rule 2320
Rule 2611
Rule 4268
Rule 4723
Rule 4803
Rule 4889
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+b \arcsin (x))^4}{e^2 x^2} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {(a+b \arcsin (x))^4}{x^2} \, dx,x,c+d x\right )}{d e^2} \\ & = -\frac {(a+b \arcsin (c+d x))^4}{d e^2 (c+d x)}+\frac {(4 b) \text {Subst}\left (\int \frac {(a+b \arcsin (x))^3}{x \sqrt {1-x^2}} \, dx,x,c+d x\right )}{d e^2} \\ & = -\frac {(a+b \arcsin (c+d x))^4}{d e^2 (c+d x)}+\frac {(4 b) \text {Subst}\left (\int (a+b x)^3 \csc (x) \, dx,x,\arcsin (c+d x)\right )}{d e^2} \\ & = -\frac {(a+b \arcsin (c+d x))^4}{d e^2 (c+d x)}-\frac {8 b (a+b \arcsin (c+d x))^3 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right )}{d e^2}-\frac {\left (12 b^2\right ) \text {Subst}\left (\int (a+b x)^2 \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (c+d x)\right )}{d e^2}+\frac {\left (12 b^2\right ) \text {Subst}\left (\int (a+b x)^2 \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (c+d x)\right )}{d e^2} \\ & = -\frac {(a+b \arcsin (c+d x))^4}{d e^2 (c+d x)}-\frac {8 b (a+b \arcsin (c+d x))^3 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right )}{d e^2}+\frac {12 i b^2 (a+b \arcsin (c+d x))^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )}{d e^2}-\frac {12 i b^2 (a+b \arcsin (c+d x))^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )}{d e^2}-\frac {\left (24 i b^3\right ) \text {Subst}\left (\int (a+b x) \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx,x,\arcsin (c+d x)\right )}{d e^2}+\frac {\left (24 i b^3\right ) \text {Subst}\left (\int (a+b x) \operatorname {PolyLog}\left (2,e^{i x}\right ) \, dx,x,\arcsin (c+d x)\right )}{d e^2} \\ & = -\frac {(a+b \arcsin (c+d x))^4}{d e^2 (c+d x)}-\frac {8 b (a+b \arcsin (c+d x))^3 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right )}{d e^2}+\frac {12 i b^2 (a+b \arcsin (c+d x))^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )}{d e^2}-\frac {12 i b^2 (a+b \arcsin (c+d x))^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )}{d e^2}-\frac {24 b^3 (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (3,-e^{i \arcsin (c+d x)}\right )}{d e^2}+\frac {24 b^3 (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (3,e^{i \arcsin (c+d x)}\right )}{d e^2}+\frac {\left (24 b^4\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-e^{i x}\right ) \, dx,x,\arcsin (c+d x)\right )}{d e^2}-\frac {\left (24 b^4\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,e^{i x}\right ) \, dx,x,\arcsin (c+d x)\right )}{d e^2} \\ & = -\frac {(a+b \arcsin (c+d x))^4}{d e^2 (c+d x)}-\frac {8 b (a+b \arcsin (c+d x))^3 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right )}{d e^2}+\frac {12 i b^2 (a+b \arcsin (c+d x))^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )}{d e^2}-\frac {12 i b^2 (a+b \arcsin (c+d x))^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )}{d e^2}-\frac {24 b^3 (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (3,-e^{i \arcsin (c+d x)}\right )}{d e^2}+\frac {24 b^3 (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (3,e^{i \arcsin (c+d x)}\right )}{d e^2}-\frac {\left (24 i b^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{i \arcsin (c+d x)}\right )}{d e^2}+\frac {\left (24 i b^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{i \arcsin (c+d x)}\right )}{d e^2} \\ & = -\frac {(a+b \arcsin (c+d x))^4}{d e^2 (c+d x)}-\frac {8 b (a+b \arcsin (c+d x))^3 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right )}{d e^2}+\frac {12 i b^2 (a+b \arcsin (c+d x))^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )}{d e^2}-\frac {12 i b^2 (a+b \arcsin (c+d x))^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )}{d e^2}-\frac {24 b^3 (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (3,-e^{i \arcsin (c+d x)}\right )}{d e^2}+\frac {24 b^3 (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (3,e^{i \arcsin (c+d x)}\right )}{d e^2}-\frac {24 i b^4 \operatorname {PolyLog}\left (4,-e^{i \arcsin (c+d x)}\right )}{d e^2}+\frac {24 i b^4 \operatorname {PolyLog}\left (4,e^{i \arcsin (c+d x)}\right )}{d e^2} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(575\) vs. \(2(270)=540\).
Time = 2.32 (sec) , antiderivative size = 575, normalized size of antiderivative = 2.13 \[ \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^2} \, dx=\frac {-\frac {a^4}{c+d x}-4 a^3 b \left (\frac {\arcsin (c+d x)}{c+d x}+\log \left (\frac {1}{2} (c+d x) \csc \left (\frac {1}{2} \arcsin (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} \arcsin (c+d x)\right )\right )\right )+6 a^2 b^2 \left (\arcsin (c+d x) \left (-\frac {\arcsin (c+d x)}{c+d x}+2 \log \left (1-e^{i \arcsin (c+d x)}\right )-2 \log \left (1+e^{i \arcsin (c+d x)}\right )\right )+2 i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )-2 i \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )\right )+4 a b^3 \left (-\frac {\arcsin (c+d x)^3}{c+d x}+3 \arcsin (c+d x)^2 \log \left (1-e^{i \arcsin (c+d x)}\right )-3 \arcsin (c+d x)^2 \log \left (1+e^{i \arcsin (c+d x)}\right )+6 i \arcsin (c+d x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )-6 i \arcsin (c+d x) \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )-6 \operatorname {PolyLog}\left (3,-e^{i \arcsin (c+d x)}\right )+6 \operatorname {PolyLog}\left (3,e^{i \arcsin (c+d x)}\right )\right )+b^4 \left (-\frac {i \pi ^4}{2}+i \arcsin (c+d x)^4-\frac {\arcsin (c+d x)^4}{c+d x}+4 \arcsin (c+d x)^3 \log \left (1-e^{-i \arcsin (c+d x)}\right )-4 \arcsin (c+d x)^3 \log \left (1+e^{i \arcsin (c+d x)}\right )+12 i \arcsin (c+d x)^2 \operatorname {PolyLog}\left (2,e^{-i \arcsin (c+d x)}\right )+12 i \arcsin (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )+24 \arcsin (c+d x) \operatorname {PolyLog}\left (3,e^{-i \arcsin (c+d x)}\right )-24 \arcsin (c+d x) \operatorname {PolyLog}\left (3,-e^{i \arcsin (c+d x)}\right )-24 i \operatorname {PolyLog}\left (4,e^{-i \arcsin (c+d x)}\right )-24 i \operatorname {PolyLog}\left (4,-e^{i \arcsin (c+d x)}\right )\right )}{d e^2} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 720 vs. \(2 (336 ) = 672\).
Time = 0.91 (sec) , antiderivative size = 721, normalized size of antiderivative = 2.67
method | result | size |
derivativedivides | \(\frac {-\frac {a^{4}}{e^{2} \left (d x +c \right )}+\frac {b^{4} \left (-\frac {\arcsin \left (d x +c \right )^{4}}{d x +c}-4 \arcsin \left (d x +c \right )^{3} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+4 \arcsin \left (d x +c \right )^{3} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-24 \arcsin \left (d x +c \right ) \operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+24 \arcsin \left (d x +c \right ) \operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+12 i \arcsin \left (d x +c \right )^{2} \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-12 i \arcsin \left (d x +c \right )^{2} \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-24 i \operatorname {polylog}\left (4, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+24 i \operatorname {polylog}\left (4, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e^{2}}+\frac {4 a \,b^{3} \left (-\frac {\arcsin \left (d x +c \right )^{3}}{d x +c}+3 \arcsin \left (d x +c \right )^{2} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-6 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+6 \operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-3 \arcsin \left (d x +c \right )^{2} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+6 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-6 \operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e^{2}}+\frac {6 a^{2} b^{2} \left (-\frac {\arcsin \left (d x +c \right )^{2}}{d x +c}+2 \arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-2 \arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+2 i \operatorname {dilog}\left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-2 i \operatorname {dilog}\left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e^{2}}+\frac {4 a^{3} b \left (-\frac {\arcsin \left (d x +c \right )}{d x +c}-\operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )\right )}{e^{2}}}{d}\) | \(721\) |
default | \(\frac {-\frac {a^{4}}{e^{2} \left (d x +c \right )}+\frac {b^{4} \left (-\frac {\arcsin \left (d x +c \right )^{4}}{d x +c}-4 \arcsin \left (d x +c \right )^{3} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+4 \arcsin \left (d x +c \right )^{3} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-24 \arcsin \left (d x +c \right ) \operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+24 \arcsin \left (d x +c \right ) \operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+12 i \arcsin \left (d x +c \right )^{2} \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-12 i \arcsin \left (d x +c \right )^{2} \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-24 i \operatorname {polylog}\left (4, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+24 i \operatorname {polylog}\left (4, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e^{2}}+\frac {4 a \,b^{3} \left (-\frac {\arcsin \left (d x +c \right )^{3}}{d x +c}+3 \arcsin \left (d x +c \right )^{2} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-6 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+6 \operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-3 \arcsin \left (d x +c \right )^{2} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+6 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-6 \operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e^{2}}+\frac {6 a^{2} b^{2} \left (-\frac {\arcsin \left (d x +c \right )^{2}}{d x +c}+2 \arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-2 \arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+2 i \operatorname {dilog}\left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-2 i \operatorname {dilog}\left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e^{2}}+\frac {4 a^{3} b \left (-\frac {\arcsin \left (d x +c \right )}{d x +c}-\operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )\right )}{e^{2}}}{d}\) | \(721\) |
parts | \(-\frac {a^{4}}{e^{2} \left (d x +c \right ) d}+\frac {b^{4} \left (-\frac {\arcsin \left (d x +c \right )^{4}}{d x +c}-4 \arcsin \left (d x +c \right )^{3} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+4 \arcsin \left (d x +c \right )^{3} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-24 \arcsin \left (d x +c \right ) \operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+24 \arcsin \left (d x +c \right ) \operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+12 i \arcsin \left (d x +c \right )^{2} \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-12 i \arcsin \left (d x +c \right )^{2} \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-24 i \operatorname {polylog}\left (4, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+24 i \operatorname {polylog}\left (4, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e^{2} d}+\frac {4 a \,b^{3} \left (-\frac {\arcsin \left (d x +c \right )^{3}}{d x +c}+3 \arcsin \left (d x +c \right )^{2} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-6 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+6 \operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-3 \arcsin \left (d x +c \right )^{2} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+6 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-6 \operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e^{2} d}+\frac {6 a^{2} b^{2} \left (-\frac {\arcsin \left (d x +c \right )^{2}}{d x +c}+2 \arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-2 \arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+2 i \operatorname {dilog}\left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-2 i \operatorname {dilog}\left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e^{2} d}+\frac {4 a^{3} b \left (-\frac {\arcsin \left (d x +c \right )}{d x +c}-\operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )\right )}{e^{2} d}\) | \(732\) |
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\[ \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^2} \, dx=\int { \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{2}} \,d x } \]
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\[ \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^2} \, dx=\frac {\int \frac {a^{4}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {b^{4} \operatorname {asin}^{4}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {4 a b^{3} \operatorname {asin}^{3}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {6 a^{2} b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {4 a^{3} b \operatorname {asin}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx}{e^{2}} \]
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Exception generated. \[ \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^2} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^2} \, dx=\int { \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^4}{{\left (c\,e+d\,e\,x\right )}^2} \,d x \]
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