Integrand size = 21, antiderivative size = 1092 \[ \int \frac {x^2 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\frac {b c \sqrt {1-c^2 x^2}}{16 \sqrt {-d} e \left (c^2 d+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {b c \sqrt {1-c^2 x^2}}{16 \sqrt {-d} e \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {a+b \arcsin (c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {a+b \arcsin (c x)}{16 d e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \arcsin (c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {a+b \arcsin (c x)}{16 d e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c^3 \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 e^{3/2} \left (c^2 d+e\right )^{3/2}}+\frac {b c \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d e^{3/2} \sqrt {c^2 d+e}}-\frac {b c^3 \text {arctanh}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 e^{3/2} \left (c^2 d+e\right )^{3/2}}+\frac {b c \text {arctanh}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d e^{3/2} \sqrt {c^2 d+e}}-\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 )}{16 (-d)^{3/2} e^{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 )}{16 (-d)^{3/2} e^{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 )}{16 (-d)^{3/2} e^{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 )}{16 (-d)^{3/2} e^{3/2}}-\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 )}{16 (-d)^{3/2} e^{3/2}}+\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 )}{16 (-d)^{3/2} e^{3/2}}-\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 )}{16 (-d)^{3/2} e^{3/2}}+\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 )}{16 (-d)^{3/2} e^{3/2}} \]
[Out]
Time = 1.73 (sec) , antiderivative size = 1092, normalized size of antiderivative = 1.00, number of steps used = 62, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {4817, 4757, 4827, 745, 739, 212, 4825, 4617, 2221, 2317, 2438} \[ \int \frac {x^2 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=-\frac {b \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right ) c^3}{16 e^{3/2} \left (d c^2+e\right )^{3/2}}-\frac {b \text {arctanh}\left (\frac {\sqrt {-d} x c^2+\sqrt {e}}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right ) c^3}{16 e^{3/2} \left (d c^2+e\right )^{3/2}}+\frac {b \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right ) c}{16 d e^{3/2} \sqrt {d c^2+e}}+\frac {b \text {arctanh}\left (\frac {\sqrt {-d} x c^2+\sqrt {e}}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right ) c}{16 d e^{3/2} \sqrt {d c^2+e}}+\frac {b \sqrt {1-c^2 x^2} c}{16 \sqrt {-d} e \left (d c^2+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {b \sqrt {1-c^2 x^2} c}{16 \sqrt {-d} e \left (d c^2+e\right ) \left (\sqrt {e} x+\sqrt {-d}\right )}-\frac {a+b \arcsin (c x)}{16 d e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \arcsin (c x)}{16 d e^{3/2} \left (\sqrt {e} x+\sqrt {-d}\right )}-\frac {a+b \arcsin (c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}+\frac {a+b \arcsin (c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {e} x+\sqrt {-d}\right )^2}-\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {(a+b \arcsin (c x)) \log \left (\frac {e^{i \arcsin (c x)} \sqrt {e}}{i c \sqrt {-d}-\sqrt {d c^2+e}}+1\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {(a+b \arcsin (c x)) \log \left (\frac {e^{i \arcsin (c x)} \sqrt {e}}{i \sqrt {-d} c+\sqrt {d c^2+e}}+1\right )}{16 (-d)^{3/2} e^{3/2}}-\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 )}{16 (-d)^{3/2} e^{3/2}}+\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 )}{16 (-d)^{3/2} e^{3/2}}-\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 )}{16 (-d)^{3/2} e^{3/2}}+\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 )}{16 (-d)^{3/2} e^{3/2}} \]
[In]
[Out]
Rule 212
Rule 739
Rule 745
Rule 2221
Rule 2317
Rule 2438
Rule 4617
Rule 4757
Rule 4817
Rule 4825
Rule 4827
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {d (a+b \arcsin (c x))}{e \left (d+e x^2\right )^3}+\frac {a+b \arcsin (c x)}{e \left (d+e x^2\right )^2}\right ) \, dx \\ & = \frac {\int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^2} \, dx}{e}-\frac {d \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^3} \, dx}{e} \\ & = \frac {\int \left (-\frac {e (a+b \arcsin (c x))}{4 d \left (\sqrt {-d} \sqrt {e}-e x\right )^2}-\frac {e (a+b \arcsin (c x))}{4 d \left (\sqrt {-d} \sqrt {e}+e x\right )^2}-\frac {e (a+b \arcsin (c x))}{2 d \left (-d e-e^2 x^2\right )}\right ) \, dx}{e}-\frac {d \int \left (-\frac {e^{3/2} (a+b \arcsin (c x))}{8 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}-e x\right )^3}-\frac {3 e (a+b \arcsin (c x))}{16 d^2 \left (\sqrt {-d} \sqrt {e}-e x\right )^2}-\frac {e^{3/2} (a+b \arcsin (c x))}{8 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}+e x\right )^3}-\frac {3 e (a+b \arcsin (c x))}{16 d^2 \left (\sqrt {-d} \sqrt {e}+e x\right )^2}-\frac {3 e (a+b \arcsin (c x))}{8 d^2 \left (-d e-e^2 x^2\right )}\right ) \, dx}{e} \\ & = \frac {3 \int \frac {a+b \arcsin (c x)}{\left (\sqrt {-d} \sqrt {e}-e x\right )^2} \, dx}{16 d}+\frac {3 \int \frac {a+b \arcsin (c x)}{\left (\sqrt {-d} \sqrt {e}+e x\right )^2} \, dx}{16 d}-\frac {\int \frac {a+b \arcsin (c x)}{\left (\sqrt {-d} \sqrt {e}-e x\right )^2} \, dx}{4 d}-\frac {\int \frac {a+b \arcsin (c x)}{\left (\sqrt {-d} \sqrt {e}+e x\right )^2} \, dx}{4 d}+\frac {3 \int \frac {a+b \arcsin (c x)}{-d e-e^2 x^2} \, dx}{8 d}-\frac {\int \frac {a+b \arcsin (c x)}{-d e-e^2 x^2} \, dx}{2 d}-\frac {\sqrt {e} \int \frac {a+b \arcsin (c x)}{\left (\sqrt {-d} \sqrt {e}-e x\right )^3} \, dx}{8 \sqrt {-d}}-\frac {\sqrt {e} \int \frac {a+b \arcsin (c x)}{\left (\sqrt {-d} \sqrt {e}+e x\right )^3} \, dx}{8 \sqrt {-d}} \\ & = -\frac {a+b \arcsin (c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {a+b \arcsin (c x)}{16 d e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \arcsin (c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {a+b \arcsin (c x)}{16 d e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {3 \int \left (-\frac {\sqrt {-d} (a+b \arcsin (c x))}{2 d e \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {-d} (a+b \arcsin (c x))}{2 d e \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{8 d}-\frac {\int \left (-\frac {\sqrt {-d} (a+b \arcsin (c x))}{2 d e \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {-d} (a+b \arcsin (c x))}{2 d e \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{2 d}-\frac {(3 b c) \int \frac {1}{\left (\sqrt {-d} \sqrt {e}-e x\right ) \sqrt {1-c^2 x^2}} \, dx}{16 d e}+\frac {(3 b c) \int \frac {1}{\left (\sqrt {-d} \sqrt {e}+e x\right ) \sqrt {1-c^2 x^2}} \, dx}{16 d e}+\frac {(b c) \int \frac {1}{\left (\sqrt {-d} \sqrt {e}-e x\right ) \sqrt {1-c^2 x^2}} \, dx}{4 d e}-\frac {(b c) \int \frac {1}{\left (\sqrt {-d} \sqrt {e}+e x\right ) \sqrt {1-c^2 x^2}} \, dx}{4 d e}+\frac {(b c) \int \frac {1}{\left (\sqrt {-d} \sqrt {e}-e x\right )^2 \sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {-d} \sqrt {e}}-\frac {(b c) \int \frac {1}{\left (\sqrt {-d} \sqrt {e}+e x\right )^2 \sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {-d} \sqrt {e}} \\ & = \frac {b c \sqrt {1-c^2 x^2}}{16 \sqrt {-d} e \left (c^2 d+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {b c \sqrt {1-c^2 x^2}}{16 \sqrt {-d} e \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {a+b \arcsin (c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {a+b \arcsin (c x)}{16 d e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \arcsin (c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {a+b \arcsin (c x)}{16 d e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {3 \int \frac {a+b \arcsin (c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{16 (-d)^{3/2} e}-\frac {3 \int \frac {a+b \arcsin (c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{16 (-d)^{3/2} e}+\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{4 (-d)^{3/2} e}+\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{4 (-d)^{3/2} e}+\frac {(3 b c) \text {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 )}{16 d e}-\frac {(3 b c) \text {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 )}{16 d e}-\frac {(b c) \text {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 d e}+\frac {(b c) \text {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 d e}-\frac {\left (b c^3\right ) \int \frac {1}{\left (\sqrt {-d} \sqrt {e}-e x\right ) \sqrt {1-c^2 x^2}} \, dx}{16 e \left (c^2 d+e\right )}+\frac {\left (b c^3\right ) \int \frac {1}{\left (\sqrt {-d} \sqrt {e}+e x\right ) \sqrt {1-c^2 x^2}} \, dx}{16 e \left (c^2 d+e\right )} \\ & = \frac {b c \sqrt {1-c^2 x^2}}{16 \sqrt {-d} e \left (c^2 d+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {b c \sqrt {1-c^2 x^2}}{16 \sqrt {-d} e \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {a+b \arcsin (c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {a+b \arcsin (c x)}{16 d e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \arcsin (c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {a+b \arcsin (c x)}{16 d e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d e^{3/2} \sqrt {c^2 d+e}}+\frac {b c \text {arctanh}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d e^{3/2} \sqrt {c^2 d+e}}-\frac {3 \text {Subst}\left (\int \frac {(a+b x) \cos (x)}{c \sqrt {-d}-\sqrt {e} \sin (x)} \, dx,x,\arcsin (c x)\right )}{16 (-d)^{3/2} e}-\frac {3 \text {Subst}\left (\int \frac {(a+b x) \cos (x)}{c \sqrt {-d}+\sqrt {e} \sin (x)} \, dx,x,\arcsin (c x)\right )}{16 (-d)^{3/2} 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 )}{4 (-d)^{3/2} 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 )}{4 (-d)^{3/2} e}+\frac {\left (b c^3\right ) \text {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 )}{16 e \left (c^2 d+e\right )}-\frac {\left (b c^3\right ) \text {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 )}{16 e \left (c^2 d+e\right )} \\ & = \frac {b c \sqrt {1-c^2 x^2}}{16 \sqrt {-d} e \left (c^2 d+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {b c \sqrt {1-c^2 x^2}}{16 \sqrt {-d} e \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {a+b \arcsin (c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {a+b \arcsin (c x)}{16 d e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \arcsin (c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {a+b \arcsin (c x)}{16 d e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c^3 \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 e^{3/2} \left (c^2 d+e\right )^{3/2}}+\frac {b c \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d e^{3/2} \sqrt {c^2 d+e}}-\frac {b c^3 \text {arctanh}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 e^{3/2} \left (c^2 d+e\right )^{3/2}}+\frac {b c \text {arctanh}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d e^{3/2} \sqrt {c^2 d+e}}-\frac {(3 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 )}{16 (-d)^{3/2} e}-\frac {(3 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 )}{16 (-d)^{3/2} e}-\frac {(3 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 )}{16 (-d)^{3/2} e}-\frac {(3 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 )}{16 (-d)^{3/2} e}+\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 )}{4 (-d)^{3/2} e}+\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 )}{4 (-d)^{3/2} e}+\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 )}{4 (-d)^{3/2} e}+\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 )}{4 (-d)^{3/2} e} \\ & = \frac {b c \sqrt {1-c^2 x^2}}{16 \sqrt {-d} e \left (c^2 d+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {b c \sqrt {1-c^2 x^2}}{16 \sqrt {-d} e \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {a+b \arcsin (c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {a+b \arcsin (c x)}{16 d e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \arcsin (c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {a+b \arcsin (c x)}{16 d e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c^3 \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 e^{3/2} \left (c^2 d+e\right )^{3/2}}+\frac {b c \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d e^{3/2} \sqrt {c^2 d+e}}-\frac {b c^3 \text {arctanh}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 e^{3/2} \left (c^2 d+e\right )^{3/2}}+\frac {b c \text {arctanh}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d e^{3/2} \sqrt {c^2 d+e}}-\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 )}{16 (-d)^{3/2} e^{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 )}{16 (-d)^{3/2} e^{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 )}{16 (-d)^{3/2} e^{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 )}{16 (-d)^{3/2} e^{3/2}}-\frac {(3 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 )}{16 (-d)^{3/2} e^{3/2}}+\frac {(3 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 )}{16 (-d)^{3/2} e^{3/2}}-\frac {(3 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 )}{16 (-d)^{3/2} e^{3/2}}+\frac {(3 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 )}{16 (-d)^{3/2} e^{3/2}}+\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 )}{4 (-d)^{3/2} e^{3/2}}-\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 )}{4 (-d)^{3/2} e^{3/2}}+\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 )}{4 (-d)^{3/2} e^{3/2}}-\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 )}{4 (-d)^{3/2} e^{3/2}} \\ & = \frac {b c \sqrt {1-c^2 x^2}}{16 \sqrt {-d} e \left (c^2 d+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {b c \sqrt {1-c^2 x^2}}{16 \sqrt {-d} e \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {a+b \arcsin (c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {a+b \arcsin (c x)}{16 d e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \arcsin (c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {a+b \arcsin (c x)}{16 d e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c^3 \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 e^{3/2} \left (c^2 d+e\right )^{3/2}}+\frac {b c \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d e^{3/2} \sqrt {c^2 d+e}}-\frac {b c^3 \text {arctanh}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 e^{3/2} \left (c^2 d+e\right )^{3/2}}+\frac {b c \text {arctanh}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d e^{3/2} \sqrt {c^2 d+e}}-\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 )}{16 (-d)^{3/2} e^{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 )}{16 (-d)^{3/2} e^{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 )}{16 (-d)^{3/2} e^{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 )}{16 (-d)^{3/2} e^{3/2}}+\frac {(3 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 )}{16 (-d)^{3/2} e^{3/2}}-\frac {(3 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 )}{16 (-d)^{3/2} e^{3/2}}+\frac {(3 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 )}{16 (-d)^{3/2} e^{3/2}}-\frac {(3 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 )}{16 (-d)^{3/2} e^{3/2}}-\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 )}{4 (-d)^{3/2} e^{3/2}}+\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 )}{4 (-d)^{3/2} e^{3/2}}-\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 )}{4 (-d)^{3/2} e^{3/2}}+\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 )}{4 (-d)^{3/2} e^{3/2}} \\ & = \frac {b c \sqrt {1-c^2 x^2}}{16 \sqrt {-d} e \left (c^2 d+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {b c \sqrt {1-c^2 x^2}}{16 \sqrt {-d} e \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {a+b \arcsin (c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {a+b \arcsin (c x)}{16 d e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \arcsin (c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {a+b \arcsin (c x)}{16 d e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c^3 \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 e^{3/2} \left (c^2 d+e\right )^{3/2}}+\frac {b c \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d e^{3/2} \sqrt {c^2 d+e}}-\frac {b c^3 \text {arctanh}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 e^{3/2} \left (c^2 d+e\right )^{3/2}}+\frac {b c \text {arctanh}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d e^{3/2} \sqrt {c^2 d+e}}-\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 )}{16 (-d)^{3/2} e^{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 )}{16 (-d)^{3/2} e^{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 )}{16 (-d)^{3/2} e^{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 )}{16 (-d)^{3/2} e^{3/2}}-\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 )}{16 (-d)^{3/2} e^{3/2}}+\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 )}{16 (-d)^{3/2} e^{3/2}}-\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 )}{16 (-d)^{3/2} e^{3/2}}+\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 )}{16 (-d)^{3/2} e^{3/2}} \\ \end{align*}
Time = 4.64 (sec) , antiderivative size = 1014, normalized size of antiderivative = 0.93 \[ \int \frac {x^2 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\frac {\frac {2 i b c \sqrt {e} \sqrt {1-c^2 x^2}}{\sqrt {d} \left (c^2 d+e\right ) \left (-i \sqrt {d}+\sqrt {e} x\right )}-\frac {2 i b c \sqrt {e} \sqrt {1-c^2 x^2}}{\sqrt {d} \left (c^2 d+e\right ) \left (i \sqrt {d}+\sqrt {e} x\right )}-\frac {8 a \sqrt {e} x}{\left (d+e x^2\right )^2}+\frac {4 a \sqrt {e} x}{d^2+d e x^2}+\frac {2 i b \arcsin (c x)}{\sqrt {d} \left (\sqrt {d}-i \sqrt {e} x\right )^2}-\frac {2 i b \arcsin (c x)}{\sqrt {d} \left (\sqrt {d}+i \sqrt {e} x\right )^2}+\frac {4 a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2}}+\frac {2 i b \left (\frac {\arcsin (c x)}{\sqrt {d}+i \sqrt {e} x}-\frac {c \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}}\right )}{d}+\frac {2 b \left (\frac {\arcsin (c x)}{i \sqrt {d}+\sqrt {e} x}+\frac {c \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}}\right )}{d}-\frac {2 b c^3 \left (\log (4)+\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 )\right )}{\left (c^2 d+e\right )^{3/2}}-\frac {2 b c^3 \left (\log (4)+\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 )\right )}{\left (c^2 d+e\right )^{3/2}}-\frac {b \left (\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )}{d^{3/2}}+\frac {b \left (\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+\log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )}{d^{3/2}}}{32 e^{3/2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 5.11 (sec) , antiderivative size = 1224, normalized size of antiderivative = 1.12
method | result | size |
parts | \(\text {Expression too large to display}\) | \(1224\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1231\) |
default | \(\text {Expression too large to display}\) | \(1231\) |
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\[ \int \frac {x^2 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
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\[ \int \frac {x^2 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^{2} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{3}}\, dx \]
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Exception generated. \[ \int \frac {x^2 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {x^2 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {x^2 (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]
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