Integrand size = 33, antiderivative size = 520 \[ \int \frac {\left (e f+2 d h x+e h x^2\right ) (a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=-\frac {2 b^2 h x}{e}+\frac {2 a b h \sqrt {1-c^2 x^2}}{c e}+\frac {2 b^2 h \sqrt {1-c^2 x^2} \arcsin (c x)}{c e}+\frac {h x (a+b \arcsin (c x))^2}{e}-\frac {\left (f-\frac {d^2 h}{e^2}\right ) (a+b \arcsin (c x))^2}{d+e x}+\frac {2 a b c \left (e^2 f-d^2 h\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {2 i b^2 c \left (e^2 f-d^2 h\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {2 i b^2 c \left (e^2 f-d^2 h\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {2 b^2 c \left (e^2 f-d^2 h\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {2 b^2 c \left (e^2 f-d^2 h\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}} \]
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Time = 1.06 (sec) , antiderivative size = 520, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {697, 4841, 6874, 267, 739, 210, 4883, 1668, 12, 4881, 4767, 8, 4857, 3404, 2296, 2221, 2317, 2438} \[ \int \frac {\left (e f+2 d h x+e h x^2\right ) (a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=-\frac {\left (f-\frac {d^2 h}{e^2}\right ) (a+b \arcsin (c x))^2}{d+e x}+\frac {h x (a+b \arcsin (c x))^2}{e}+\frac {2 a b c \left (e^2 f-d^2 h\right ) \arctan \left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {2 a b h \sqrt {1-c^2 x^2}}{c e}-\frac {2 b^2 c \left (e^2 f-d^2 h\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {2 b^2 c \left (e^2 f-d^2 h\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {2 i b^2 c \arcsin (c x) \left (e^2 f-d^2 h\right ) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {2 i b^2 c \arcsin (c x) \left (e^2 f-d^2 h\right ) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {2 b^2 h \sqrt {1-c^2 x^2} \arcsin (c x)}{c e}-\frac {2 b^2 h x}{e} \]
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Rule 8
Rule 12
Rule 210
Rule 267
Rule 697
Rule 739
Rule 1668
Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 3404
Rule 4767
Rule 4841
Rule 4857
Rule 4881
Rule 4883
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {h x (a+b \arcsin (c x))^2}{e}-\frac {\left (f-\frac {d^2 h}{e^2}\right ) (a+b \arcsin (c x))^2}{d+e x}-(2 b c) \int \frac {\left (\frac {h x}{e}-\frac {f-\frac {d^2 h}{e^2}}{d+e x}\right ) (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}} \, dx \\ & = \frac {h x (a+b \arcsin (c x))^2}{e}-\frac {\left (f-\frac {d^2 h}{e^2}\right ) (a+b \arcsin (c x))^2}{d+e x}-(2 b c) \int \left (\frac {a \left (-e^2 f+d^2 h+d e h x+e^2 h x^2\right )}{e^2 (d+e x) \sqrt {1-c^2 x^2}}+\frac {b \left (-e^2 f+d^2 h+d e h x+e^2 h x^2\right ) \arcsin (c x)}{e^2 (d+e x) \sqrt {1-c^2 x^2}}\right ) \, dx \\ & = \frac {h x (a+b \arcsin (c x))^2}{e}-\frac {\left (f-\frac {d^2 h}{e^2}\right ) (a+b \arcsin (c x))^2}{d+e x}-\frac {(2 a b c) \int \frac {-e^2 f+d^2 h+d e h x+e^2 h x^2}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{e^2}-\frac {\left (2 b^2 c\right ) \int \frac {\left (-e^2 f+d^2 h+d e h x+e^2 h x^2\right ) \arcsin (c x)}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{e^2} \\ & = \frac {2 a b h \sqrt {1-c^2 x^2}}{c e}+\frac {h x (a+b \arcsin (c x))^2}{e}-\frac {\left (f-\frac {d^2 h}{e^2}\right ) (a+b \arcsin (c x))^2}{d+e x}+\frac {(2 a b) \int \frac {c^2 e^2 \left (e^2 f-d^2 h\right )}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{c e^4}-\frac {\left (2 b^2 c\right ) \int \left (\frac {e h x \arcsin (c x)}{\sqrt {1-c^2 x^2}}+\frac {\left (-e^2 f+d^2 h\right ) \arcsin (c x)}{(d+e x) \sqrt {1-c^2 x^2}}\right ) \, dx}{e^2} \\ & = \frac {2 a b h \sqrt {1-c^2 x^2}}{c e}+\frac {h x (a+b \arcsin (c x))^2}{e}-\frac {\left (f-\frac {d^2 h}{e^2}\right ) (a+b \arcsin (c x))^2}{d+e x}-\frac {\left (2 b^2 c h\right ) \int \frac {x \arcsin (c x)}{\sqrt {1-c^2 x^2}} \, dx}{e}+\frac {\left (2 a b c \left (e^2 f-d^2 h\right )\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{e^2}-\frac {\left (2 b^2 c \left (-e^2 f+d^2 h\right )\right ) \int \frac {\arcsin (c x)}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{e^2} \\ & = \frac {2 a b h \sqrt {1-c^2 x^2}}{c e}+\frac {2 b^2 h \sqrt {1-c^2 x^2} \arcsin (c x)}{c e}+\frac {h x (a+b \arcsin (c x))^2}{e}-\frac {\left (f-\frac {d^2 h}{e^2}\right ) (a+b \arcsin (c x))^2}{d+e x}-\frac {\left (2 b^2 h\right ) \int 1 \, dx}{e}-\frac {\left (2 a b c \left (e^2 f-d^2 h\right )\right ) \text {Subst}\left (\int \frac {1}{-c^2 d^2+e^2-x^2} \, dx,x,\frac {e+c^2 d x}{\sqrt {1-c^2 x^2}}\right )}{e^2}-\frac {\left (2 b^2 c \left (-e^2 f+d^2 h\right )\right ) \text {Subst}\left (\int \frac {x}{c d+e \sin (x)} \, dx,x,\arcsin (c x)\right )}{e^2} \\ & = -\frac {2 b^2 h x}{e}+\frac {2 a b h \sqrt {1-c^2 x^2}}{c e}+\frac {2 b^2 h \sqrt {1-c^2 x^2} \arcsin (c x)}{c e}+\frac {h x (a+b \arcsin (c x))^2}{e}-\frac {\left (f-\frac {d^2 h}{e^2}\right ) (a+b \arcsin (c x))^2}{d+e x}+\frac {2 a b c \left (e^2 f-d^2 h\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {\left (4 b^2 c \left (-e^2 f+d^2 h\right )\right ) \text {Subst}\left (\int \frac {e^{i x} x}{i e+2 c d e^{i x}-i e e^{2 i x}} \, dx,x,\arcsin (c x)\right )}{e^2} \\ & = -\frac {2 b^2 h x}{e}+\frac {2 a b h \sqrt {1-c^2 x^2}}{c e}+\frac {2 b^2 h \sqrt {1-c^2 x^2} \arcsin (c x)}{c e}+\frac {h x (a+b \arcsin (c x))^2}{e}-\frac {\left (f-\frac {d^2 h}{e^2}\right ) (a+b \arcsin (c x))^2}{d+e x}+\frac {2 a b c \left (e^2 f-d^2 h\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {\left (4 i b^2 c \left (e^2 f-d^2 h\right )\right ) \text {Subst}\left (\int \frac {e^{i x} x}{2 c d-2 \sqrt {c^2 d^2-e^2}-2 i e e^{i x}} \, dx,x,\arcsin (c x)\right )}{e \sqrt {c^2 d^2-e^2}}+\frac {\left (4 i b^2 c \left (e^2 f-d^2 h\right )\right ) \text {Subst}\left (\int \frac {e^{i x} x}{2 c d+2 \sqrt {c^2 d^2-e^2}-2 i e e^{i x}} \, dx,x,\arcsin (c x)\right )}{e \sqrt {c^2 d^2-e^2}} \\ & = -\frac {2 b^2 h x}{e}+\frac {2 a b h \sqrt {1-c^2 x^2}}{c e}+\frac {2 b^2 h \sqrt {1-c^2 x^2} \arcsin (c x)}{c e}+\frac {h x (a+b \arcsin (c x))^2}{e}-\frac {\left (f-\frac {d^2 h}{e^2}\right ) (a+b \arcsin (c x))^2}{d+e x}+\frac {2 a b c \left (e^2 f-d^2 h\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {2 i b^2 c \left (e^2 f-d^2 h\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {2 i b^2 c \left (e^2 f-d^2 h\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {\left (2 i b^2 c \left (e^2 f-d^2 h\right )\right ) \text {Subst}\left (\int \log \left (1-\frac {2 i e e^{i x}}{2 c d-2 \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\arcsin (c x)\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {\left (2 i b^2 c \left (e^2 f-d^2 h\right )\right ) \text {Subst}\left (\int \log \left (1-\frac {2 i e e^{i x}}{2 c d+2 \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\arcsin (c x)\right )}{e^2 \sqrt {c^2 d^2-e^2}} \\ & = -\frac {2 b^2 h x}{e}+\frac {2 a b h \sqrt {1-c^2 x^2}}{c e}+\frac {2 b^2 h \sqrt {1-c^2 x^2} \arcsin (c x)}{c e}+\frac {h x (a+b \arcsin (c x))^2}{e}-\frac {\left (f-\frac {d^2 h}{e^2}\right ) (a+b \arcsin (c x))^2}{d+e x}+\frac {2 a b c \left (e^2 f-d^2 h\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {2 i b^2 c \left (e^2 f-d^2 h\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {2 i b^2 c \left (e^2 f-d^2 h\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {\left (2 b^2 c \left (e^2 f-d^2 h\right )\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i e x}{2 c d-2 \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {\left (2 b^2 c \left (e^2 f-d^2 h\right )\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i e x}{2 c d+2 \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{e^2 \sqrt {c^2 d^2-e^2}} \\ & = -\frac {2 b^2 h x}{e}+\frac {2 a b h \sqrt {1-c^2 x^2}}{c e}+\frac {2 b^2 h \sqrt {1-c^2 x^2} \arcsin (c x)}{c e}+\frac {h x (a+b \arcsin (c x))^2}{e}-\frac {\left (f-\frac {d^2 h}{e^2}\right ) (a+b \arcsin (c x))^2}{d+e x}+\frac {2 a b c \left (e^2 f-d^2 h\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {2 i b^2 c \left (e^2 f-d^2 h\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {2 i b^2 c \left (e^2 f-d^2 h\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {2 b^2 c \left (e^2 f-d^2 h\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {2 b^2 c \left (e^2 f-d^2 h\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}} \\ \end{align*}
Time = 0.52 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.59 \[ \int \frac {\left (e f+2 d h x+e h x^2\right ) (a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=\frac {h x (a+b \arcsin (c x))^2}{e}-\frac {\left (f-\frac {d^2 h}{e^2}\right ) (a+b \arcsin (c x))^2}{d+e x}-\frac {2 b h \left (b x-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}\right )}{e}+\frac {2 b c \left (e^2 f-d^2 h\right ) \left (-i (a+b \arcsin (c x)) \left (\log \left (1+\frac {i e e^{i \arcsin (c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )-\log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )-b \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )+b \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )}{e^2 \sqrt {c^2 d^2-e^2}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1250 vs. \(2 (520 ) = 1040\).
Time = 3.86 (sec) , antiderivative size = 1251, normalized size of antiderivative = 2.41
method | result | size |
parts | \(\text {Expression too large to display}\) | \(1251\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1277\) |
default | \(\text {Expression too large to display}\) | \(1277\) |
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\[ \int \frac {\left (e f+2 d h x+e h x^2\right ) (a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=\int { \frac {{\left (e h x^{2} + 2 \, d h x + e f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (e f+2 d h x+e h x^2\right ) (a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \cdot \left (2 d h x + e f + e h x^{2}\right )}{\left (d + e x\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {\left (e f+2 d h x+e h x^2\right ) (a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {\left (e f+2 d h x+e h x^2\right ) (a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=\int { \frac {{\left (e h x^{2} + 2 \, d h x + e f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (e f+2 d h x+e h x^2\right ) (a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\left (e\,h\,x^2+2\,d\,h\,x+e\,f\right )}{{\left (d+e\,x\right )}^2} \,d x \]
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