Optimal. Leaf size=460 \[ -\frac {\left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h-d e g+e^2 f\right )}{e^3 (d+e x)}+\frac {(e g-2 d h) \log (d+e x) \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac {h x \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac {b c \tan ^{-1}\left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right ) \left (d^2 h-d e g+e^2 f\right )}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {i b (e g-2 d h) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {i b (e g-2 d h) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {b \sin ^{-1}(c x) (e g-2 d h) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {b \sin ^{-1}(c x) (e g-2 d h) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e^3}+\frac {b h \sqrt {1-c^2 x^2}}{c e^2}-\frac {i b \sin ^{-1}(c x)^2 (e g-2 d h)}{2 e^3}-\frac {b \sin ^{-1}(c x) (e g-2 d h) \log (d+e x)}{e^3} \]
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Rubi [A] time = 0.85, antiderivative size = 460, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 14, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {698, 4753, 12, 6742, 261, 725, 204, 216, 2404, 4741, 4519, 2190, 2279, 2391} \[ -\frac {i b (e g-2 d h) \text {PolyLog}\left (2,\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {i b (e g-2 d h) \text {PolyLog}\left (2,\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e^3}-\frac {\left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h-d e g+e^2 f\right )}{e^3 (d+e x)}+\frac {(e g-2 d h) \log (d+e x) \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac {h x \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac {b c \tan ^{-1}\left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right ) \left (d^2 h-d e g+e^2 f\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {b \sin ^{-1}(c x) (e g-2 d h) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {b \sin ^{-1}(c x) (e g-2 d h) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e^3}+\frac {b h \sqrt {1-c^2 x^2}}{c e^2}-\frac {i b \sin ^{-1}(c x)^2 (e g-2 d h)}{2 e^3}-\frac {b \sin ^{-1}(c x) (e g-2 d h) \log (d+e x)}{e^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 204
Rule 216
Rule 261
Rule 698
Rule 725
Rule 2190
Rule 2279
Rule 2391
Rule 2404
Rule 4519
Rule 4741
Rule 4753
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (f+g x+h x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{(d+e x)^2} \, dx &=\frac {h x \left (a+b \sin ^{-1}(c x)\right )}{e^2}-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^3 (d+e x)}+\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^3}-(b c) \int \frac {e h x-\frac {e^2 f-d e g+d^2 h}{d+e x}+(e g-2 d h) \log (d+e x)}{e^3 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {h x \left (a+b \sin ^{-1}(c x)\right )}{e^2}-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^3 (d+e x)}+\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^3}-\frac {(b c) \int \frac {e h x-\frac {e^2 f-d e g+d^2 h}{d+e x}+(e g-2 d h) \log (d+e x)}{\sqrt {1-c^2 x^2}} \, dx}{e^3}\\ &=\frac {h x \left (a+b \sin ^{-1}(c x)\right )}{e^2}-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^3 (d+e x)}+\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^3}-\frac {(b c) \int \left (\frac {e h x}{\sqrt {1-c^2 x^2}}+\frac {-e^2 f+d e g-d^2 h}{(d+e x) \sqrt {1-c^2 x^2}}+\frac {(e g-2 d h) \log (d+e x)}{\sqrt {1-c^2 x^2}}\right ) \, dx}{e^3}\\ &=\frac {h x \left (a+b \sin ^{-1}(c x)\right )}{e^2}-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^3 (d+e x)}+\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^3}-\frac {(b c h) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{e^2}-\frac {(b c (e g-2 d h)) \int \frac {\log (d+e x)}{\sqrt {1-c^2 x^2}} \, dx}{e^3}+\frac {\left (b c \left (e^2 f-d e g+d^2 h\right )\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{e^3}\\ &=\frac {b h \sqrt {1-c^2 x^2}}{c e^2}+\frac {h x \left (a+b \sin ^{-1}(c x)\right )}{e^2}-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^3 (d+e x)}-\frac {b (e g-2 d h) \sin ^{-1}(c x) \log (d+e x)}{e^3}+\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^3}+\frac {(b c (e g-2 d h)) \int \frac {\sin ^{-1}(c x)}{c d+c e x} \, dx}{e^2}-\frac {\left (b c \left (e^2 f-d e g+d^2 h\right )\right ) \operatorname {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^3}\\ &=\frac {b h \sqrt {1-c^2 x^2}}{c e^2}+\frac {h x \left (a+b \sin ^{-1}(c x)\right )}{e^2}-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^3 (d+e x)}+\frac {b c \left (e^2 f-d e g+d^2 h\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {b (e g-2 d h) \sin ^{-1}(c x) \log (d+e x)}{e^3}+\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^3}+\frac {(b c (e g-2 d h)) \operatorname {Subst}\left (\int \frac {x \cos (x)}{c^2 d+c e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e^2}\\ &=\frac {b h \sqrt {1-c^2 x^2}}{c e^2}-\frac {i b (e g-2 d h) \sin ^{-1}(c x)^2}{2 e^3}+\frac {h x \left (a+b \sin ^{-1}(c x)\right )}{e^2}-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^3 (d+e x)}+\frac {b c \left (e^2 f-d e g+d^2 h\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {b (e g-2 d h) \sin ^{-1}(c x) \log (d+e x)}{e^3}+\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^3}+\frac {(b c (e g-2 d h)) \operatorname {Subst}\left (\int \frac {e^{i x} x}{c^2 d-c \sqrt {c^2 d^2-e^2}-i c e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^2}+\frac {(b c (e g-2 d h)) \operatorname {Subst}\left (\int \frac {e^{i x} x}{c^2 d+c \sqrt {c^2 d^2-e^2}-i c e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^2}\\ &=\frac {b h \sqrt {1-c^2 x^2}}{c e^2}-\frac {i b (e g-2 d h) \sin ^{-1}(c x)^2}{2 e^3}+\frac {h x \left (a+b \sin ^{-1}(c x)\right )}{e^2}-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^3 (d+e x)}+\frac {b c \left (e^2 f-d e g+d^2 h\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {b (e g-2 d h) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {b (e g-2 d h) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {b (e g-2 d h) \sin ^{-1}(c x) \log (d+e x)}{e^3}+\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^3}-\frac {(b (e g-2 d h)) \operatorname {Subst}\left (\int \log \left (1-\frac {i c e e^{i x}}{c^2 d-c \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3}-\frac {(b (e g-2 d h)) \operatorname {Subst}\left (\int \log \left (1-\frac {i c e e^{i x}}{c^2 d+c \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3}\\ &=\frac {b h \sqrt {1-c^2 x^2}}{c e^2}-\frac {i b (e g-2 d h) \sin ^{-1}(c x)^2}{2 e^3}+\frac {h x \left (a+b \sin ^{-1}(c x)\right )}{e^2}-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^3 (d+e x)}+\frac {b c \left (e^2 f-d e g+d^2 h\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {b (e g-2 d h) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {b (e g-2 d h) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {b (e g-2 d h) \sin ^{-1}(c x) \log (d+e x)}{e^3}+\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^3}+\frac {(i b (e g-2 d h)) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {i c e x}{c^2 d-c \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^3}+\frac {(i b (e g-2 d h)) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {i c e x}{c^2 d+c \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^3}\\ &=\frac {b h \sqrt {1-c^2 x^2}}{c e^2}-\frac {i b (e g-2 d h) \sin ^{-1}(c x)^2}{2 e^3}+\frac {h x \left (a+b \sin ^{-1}(c x)\right )}{e^2}-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^3 (d+e x)}+\frac {b c \left (e^2 f-d e g+d^2 h\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {b (e g-2 d h) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {b (e g-2 d h) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {b (e g-2 d h) \sin ^{-1}(c x) \log (d+e x)}{e^3}+\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^3}-\frac {i b (e g-2 d h) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {i b (e g-2 d h) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}\\ \end {align*}
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Mathematica [A] time = 0.97, size = 392, normalized size = 0.85 \[ \frac {-\frac {\left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h-d e g+e^2 f\right )}{d+e x}+(e g-2 d h) \log (d+e x) \left (a+b \sin ^{-1}(c x)\right )+e h x \left (a+b \sin ^{-1}(c x)\right )+\frac {b c \tan ^{-1}\left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right ) \left (d^2 h-d e g+e^2 f\right )}{\sqrt {c^2 d^2-e^2}}-\frac {1}{2} i b (e g-2 d h) \left (2 \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )+2 \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )+\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+2 i \left (\log \left (1+\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}-c d}\right )+\log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )\right )\right )\right )+\frac {b e h \sqrt {1-c^2 x^2}}{c}-b \sin ^{-1}(c x) (e g-2 d h) \log (d+e x)}{e^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a h x^{2} + a g x + a f + {\left (b h x^{2} + b g x + b f\right )} \arcsin \left (c x\right )}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (h x^{2} + g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.49, size = 1910, normalized size = 4.15 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (h\,x^2+g\,x+f\right )}{{\left (d+e\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x + h x^{2}\right )}{\left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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