Integrand size = 21, antiderivative size = 705 \[ \int \frac {x^5 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\frac {b c d x \sqrt {1-c^2 x^2}}{8 e^2 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {d^2 (a+b \arcsin (c x))}{4 e^3 \left (d+e x^2\right )^2}+\frac {d (a+b \arcsin (c x))}{e^3 \left (d+e x^2\right )}-\frac {i (a+b \arcsin (c x))^2}{2 b e^3}-\frac {b c \sqrt {d} \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{e^3 \sqrt {c^2 d+e}}+\frac {b c \sqrt {d} \left (2 c^2 d+e\right ) \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 e^3 \left (c^2 d+e\right )^{3/2}}+\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^3}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^3}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^3}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^3}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^3} \]
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Time = 0.77 (sec) , antiderivative size = 705, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {4817, 4813, 390, 385, 211, 4825, 4617, 2221, 2317, 2438} \[ \int \frac {x^5 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^3}+\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^3}+\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^3}+\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 e^3}-\frac {d^2 (a+b \arcsin (c x))}{4 e^3 \left (d+e x^2\right )^2}+\frac {d (a+b \arcsin (c x))}{e^3 \left (d+e x^2\right )}-\frac {i (a+b \arcsin (c x))^2}{2 b e^3}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{2 e^3}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{2 e^3}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{2 e^3}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{2 e^3}+\frac {b c \sqrt {d} \left (2 c^2 d+e\right ) \arctan \left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 e^3 \left (c^2 d+e\right )^{3/2}}-\frac {b c \sqrt {d} \arctan \left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{e^3 \sqrt {c^2 d+e}}+\frac {b c d x \sqrt {1-c^2 x^2}}{8 e^2 \left (c^2 d+e\right ) \left (d+e x^2\right )} \]
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Rule 211
Rule 385
Rule 390
Rule 2221
Rule 2317
Rule 2438
Rule 4617
Rule 4813
Rule 4817
Rule 4825
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^2 x (a+b \arcsin (c x))}{e^2 \left (d+e x^2\right )^3}-\frac {2 d x (a+b \arcsin (c x))}{e^2 \left (d+e x^2\right )^2}+\frac {x (a+b \arcsin (c x))}{e^2 \left (d+e x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {x (a+b \arcsin (c x))}{d+e x^2} \, dx}{e^2}-\frac {(2 d) \int \frac {x (a+b \arcsin (c x))}{\left (d+e x^2\right )^2} \, dx}{e^2}+\frac {d^2 \int \frac {x (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx}{e^2} \\ & = -\frac {d^2 (a+b \arcsin (c x))}{4 e^3 \left (d+e x^2\right )^2}+\frac {d (a+b \arcsin (c x))}{e^3 \left (d+e x^2\right )}-\frac {(b c d) \int \frac {1}{\sqrt {1-c^2 x^2} \left (d+e x^2\right )} \, dx}{e^3}+\frac {\left (b c d^2\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \left (d+e x^2\right )^2} \, dx}{4 e^3}+\frac {\int \left (-\frac {a+b \arcsin (c x)}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \arcsin (c x)}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{e^2} \\ & = \frac {b c d x \sqrt {1-c^2 x^2}}{8 e^2 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {d^2 (a+b \arcsin (c x))}{4 e^3 \left (d+e x^2\right )^2}+\frac {d (a+b \arcsin (c x))}{e^3 \left (d+e x^2\right )}-\frac {(b c d) \text {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 )}{e^3}-\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 e^{5/2}}+\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 e^{5/2}}+\frac {\left (b c d \left (2 c^2 d+e\right )\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \left (d+e x^2\right )} \, dx}{8 e^3 \left (c^2 d+e\right )} \\ & = \frac {b c d x \sqrt {1-c^2 x^2}}{8 e^2 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {d^2 (a+b \arcsin (c x))}{4 e^3 \left (d+e x^2\right )^2}+\frac {d (a+b \arcsin (c x))}{e^3 \left (d+e x^2\right )}-\frac {b c \sqrt {d} \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{e^3 \sqrt {c^2 d+e}}-\frac {\text {Subst}\left (\int \frac {(a+b x) \cos (x)}{c \sqrt {-d}-\sqrt {e} \sin (x)} \, dx,x,\arcsin (c x)\right )}{2 e^{5/2}}+\frac {\text {Subst}\left (\int \frac {(a+b x) \cos (x)}{c \sqrt {-d}+\sqrt {e} \sin (x)} \, dx,x,\arcsin (c x)\right )}{2 e^{5/2}}+\frac {\left (b c d \left (2 c^2 d+e\right )\right ) \text {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 )}{8 e^3 \left (c^2 d+e\right )} \\ & = \frac {b c d x \sqrt {1-c^2 x^2}}{8 e^2 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {d^2 (a+b \arcsin (c x))}{4 e^3 \left (d+e x^2\right )^2}+\frac {d (a+b \arcsin (c x))}{e^3 \left (d+e x^2\right )}-\frac {i (a+b \arcsin (c x))^2}{2 b e^3}-\frac {b c \sqrt {d} \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{e^3 \sqrt {c^2 d+e}}+\frac {b c \sqrt {d} \left (2 c^2 d+e\right ) \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 e^3 \left (c^2 d+e\right )^{3/2}}-\frac {i \text {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,\arcsin (c x)\right )}{2 e^{5/2}}-\frac {i \text {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,\arcsin (c x)\right )}{2 e^{5/2}}+\frac {i \text {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,\arcsin (c x)\right )}{2 e^{5/2}}+\frac {i \text {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,\arcsin (c x)\right )}{2 e^{5/2}} \\ & = \frac {b c d x \sqrt {1-c^2 x^2}}{8 e^2 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {d^2 (a+b \arcsin (c x))}{4 e^3 \left (d+e x^2\right )^2}+\frac {d (a+b \arcsin (c x))}{e^3 \left (d+e x^2\right )}-\frac {i (a+b \arcsin (c x))^2}{2 b e^3}-\frac {b c \sqrt {d} \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{e^3 \sqrt {c^2 d+e}}+\frac {b c \sqrt {d} \left (2 c^2 d+e\right ) \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 e^3 \left (c^2 d+e\right )^{3/2}}+\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^3}-\frac {b \text {Subst}\left (\int \log \left (1-\frac {\sqrt {e} e^{i x}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right ) \, dx,x,\arcsin (c x)\right )}{2 e^3}-\frac {b \text {Subst}\left (\int \log \left (1+\frac {\sqrt {e} e^{i x}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right ) \, dx,x,\arcsin (c x)\right )}{2 e^3}-\frac {b \text {Subst}\left (\int \log \left (1-\frac {\sqrt {e} e^{i x}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right ) \, dx,x,\arcsin (c x)\right )}{2 e^3}-\frac {b \text {Subst}\left (\int \log \left (1+\frac {\sqrt {e} e^{i x}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right ) \, dx,x,\arcsin (c x)\right )}{2 e^3} \\ & = \frac {b c d x \sqrt {1-c^2 x^2}}{8 e^2 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {d^2 (a+b \arcsin (c x))}{4 e^3 \left (d+e x^2\right )^2}+\frac {d (a+b \arcsin (c x))}{e^3 \left (d+e x^2\right )}-\frac {i (a+b \arcsin (c x))^2}{2 b e^3}-\frac {b c \sqrt {d} \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{e^3 \sqrt {c^2 d+e}}+\frac {b c \sqrt {d} \left (2 c^2 d+e\right ) \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 e^3 \left (c^2 d+e\right )^{3/2}}+\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {(i b) \text {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 \arcsin (c x)}\right )}{2 e^3}+\frac {(i b) \text {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 \arcsin (c x)}\right )}{2 e^3}+\frac {(i b) \text {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 \arcsin (c x)}\right )}{2 e^3}+\frac {(i b) \text {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 \arcsin (c x)}\right )}{2 e^3} \\ & = \frac {b c d x \sqrt {1-c^2 x^2}}{8 e^2 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {d^2 (a+b \arcsin (c x))}{4 e^3 \left (d+e x^2\right )^2}+\frac {d (a+b \arcsin (c x))}{e^3 \left (d+e x^2\right )}-\frac {i (a+b \arcsin (c x))^2}{2 b e^3}-\frac {b c \sqrt {d} \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{e^3 \sqrt {c^2 d+e}}+\frac {b c \sqrt {d} \left (2 c^2 d+e\right ) \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 e^3 \left (c^2 d+e\right )^{3/2}}+\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^3}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^3}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 e^3}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^3}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 e^3} \\ \end{align*}
Time = 5.02 (sec) , antiderivative size = 973, normalized size of antiderivative = 1.38 \[ \int \frac {x^5 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\frac {-\frac {4 a d^2}{\left (d+e x^2\right )^2}+\frac {16 a d}{d+e x^2}+8 a \log \left (d+e x^2\right )+b \left (\frac {c d \sqrt {e} \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \left (-i \sqrt {d}+\sqrt {e} x\right )}+\frac {c d \sqrt {e} \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \left (i \sqrt {d}+\sqrt {e} x\right )}+\frac {7 \sqrt {d} \arcsin (c x)}{\sqrt {d}-i \sqrt {e} x}-\frac {d \arcsin (c x)}{\left (\sqrt {d}+i \sqrt {e} x\right )^2}+\frac {7 \sqrt {d} \arcsin (c x)}{\sqrt {d}+i \sqrt {e} x}+\frac {d \arcsin (c x)}{\left (i \sqrt {d}+\sqrt {e} x\right )^2}-8 i \arcsin (c x)^2-\frac {7 c \sqrt {d} \arctan \left (\frac {i \sqrt {e}+c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}+\frac {7 i c \sqrt {d} \text {arctanh}\left (\frac {\sqrt {e}+i c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}+8 \arcsin (c x) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+8 \arcsin (c x) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+8 \arcsin (c x) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )+8 \arcsin (c x) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )+\frac {i c^3 d^{3/2} \log \left (\frac {e \sqrt {c^2 d+e} \left (\sqrt {e}-i c^2 \sqrt {d} x+\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}\right )}{c^3 \left (d+i \sqrt {d} \sqrt {e} x\right )}\right )}{\left (c^2 d+e\right )^{3/2}}-\frac {i c^3 d^{3/2} \log \left (\frac {e \sqrt {c^2 d+e} \left (\sqrt {e}+i c^2 \sqrt {d} x+\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}\right )}{c^3 \left (d-i \sqrt {d} \sqrt {e} x\right )}\right )}{\left (c^2 d+e\right )^{3/2}}-8 i \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )-8 i \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )-8 i \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )-8 i \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )}{16 e^3} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.99 (sec) , antiderivative size = 3508, normalized size of antiderivative = 4.98
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(3508\) |
default | \(\text {Expression too large to display}\) | \(3508\) |
parts | \(\text {Expression too large to display}\) | \(3513\) |
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\[ \int \frac {x^5 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
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\[ \int \frac {x^5 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^{5} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{3}}\, dx \]
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\[ \int \frac {x^5 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
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\[ \int \frac {x^5 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {x^5 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^5\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]
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