Optimal. Leaf size=459 \[ \frac {\log (d+e x) \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h-d e g+e^2 f\right )}{e^3}+\frac {x (e g-d h) \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac {h x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e}-\frac {i b \left (d^2 h-d e g+e^2 f\right ) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {i b \left (d^2 h-d e g+e^2 f\right ) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {b \sin ^{-1}(c x) \left (d^2 h-d e g+e^2 f\right ) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {b \sin ^{-1}(c x) \left (d^2 h-d e g+e^2 f\right ) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e^3}+\frac {b \sqrt {1-c^2 x^2} (4 (e g-d h)+e h x)}{4 c e^2}-\frac {b h \sin ^{-1}(c x)}{4 c^2 e}-\frac {i b \sin ^{-1}(c x)^2 \left (d^2 h-d e g+e^2 f\right )}{2 e^3}-\frac {b \sin ^{-1}(c x) \log (d+e x) \left (d^2 h-d e g+e^2 f\right )}{e^3} \]
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Rubi [A] time = 0.79, antiderivative size = 459, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {698, 4753, 12, 6742, 780, 216, 2404, 4741, 4519, 2190, 2279, 2391} \[ -\frac {i b \left (d^2 h-d e g+e^2 f\right ) \text {PolyLog}\left (2,\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {i b \left (d^2 h-d e g+e^2 f\right ) \text {PolyLog}\left (2,\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e^3}+\frac {\log (d+e x) \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h-d e g+e^2 f\right )}{e^3}+\frac {x (e g-d h) \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac {h x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e}+\frac {b \sin ^{-1}(c x) \left (d^2 h-d e g+e^2 f\right ) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {b \sin ^{-1}(c x) \left (d^2 h-d e g+e^2 f\right ) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e^3}+\frac {b \sqrt {1-c^2 x^2} (4 (e g-d h)+e h x)}{4 c e^2}-\frac {b h \sin ^{-1}(c x)}{4 c^2 e}-\frac {i b \sin ^{-1}(c x)^2 \left (d^2 h-d e g+e^2 f\right )}{2 e^3}-\frac {b \sin ^{-1}(c x) \log (d+e x) \left (d^2 h-d e g+e^2 f\right )}{e^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 216
Rule 698
Rule 780
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} \, dx &=\frac {(e g-d h) x \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac {h x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e}+\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^3}-(b c) \int \frac {e x (2 e g-2 d h+e h x)+2 \left (e^2 f-d e g+d^2 h\right ) \log (d+e x)}{2 e^3 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {(e g-d h) x \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac {h x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e}+\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^3}-\frac {(b c) \int \frac {e x (2 e g-2 d h+e h x)+2 \left (e^2 f-d e g+d^2 h\right ) \log (d+e x)}{\sqrt {1-c^2 x^2}} \, dx}{2 e^3}\\ &=\frac {(e g-d h) x \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac {h x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e}+\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^3}-\frac {(b c) \int \left (\frac {e x (2 e g-2 d h+e h x)}{\sqrt {1-c^2 x^2}}+\frac {2 \left (e^2 f-d e g+d^2 h\right ) \log (d+e x)}{\sqrt {1-c^2 x^2}}\right ) \, dx}{2 e^3}\\ &=\frac {(e g-d h) x \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac {h x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e}+\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^3}-\frac {(b c) \int \frac {x (2 e g-2 d h+e h x)}{\sqrt {1-c^2 x^2}} \, dx}{2 e^2}-\frac {\left (b c \left (e^2 f-d e g+d^2 h\right )\right ) \int \frac {\log (d+e x)}{\sqrt {1-c^2 x^2}} \, dx}{e^3}\\ &=\frac {b (4 (e g-d h)+e h x) \sqrt {1-c^2 x^2}}{4 c e^2}+\frac {(e g-d h) x \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac {h x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e}-\frac {b \left (e^2 f-d e g+d^2 h\right ) \sin ^{-1}(c x) \log (d+e x)}{e^3}+\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^3}-\frac {(b h) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{4 c e}+\frac {\left (b c \left (e^2 f-d e g+d^2 h\right )\right ) \int \frac {\sin ^{-1}(c x)}{c d+c e x} \, dx}{e^2}\\ &=\frac {b (4 (e g-d h)+e h x) \sqrt {1-c^2 x^2}}{4 c e^2}-\frac {b h \sin ^{-1}(c x)}{4 c^2 e}+\frac {(e g-d h) x \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac {h x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e}-\frac {b \left (e^2 f-d e g+d^2 h\right ) \sin ^{-1}(c x) \log (d+e x)}{e^3}+\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^3}+\frac {\left (b c \left (e^2 f-d e g+d^2 h\right )\right ) \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 (4 (e g-d h)+e h x) \sqrt {1-c^2 x^2}}{4 c e^2}-\frac {b h \sin ^{-1}(c x)}{4 c^2 e}-\frac {i b \left (e^2 f-d e g+d^2 h\right ) \sin ^{-1}(c x)^2}{2 e^3}+\frac {(e g-d h) x \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac {h x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e}-\frac {b \left (e^2 f-d e g+d^2 h\right ) \sin ^{-1}(c x) \log (d+e x)}{e^3}+\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^3}+\frac {\left (b c \left (e^2 f-d e g+d^2 h\right )\right ) \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 {\left (b c \left (e^2 f-d e g+d^2 h\right )\right ) \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 (4 (e g-d h)+e h x) \sqrt {1-c^2 x^2}}{4 c e^2}-\frac {b h \sin ^{-1}(c x)}{4 c^2 e}-\frac {i b \left (e^2 f-d e g+d^2 h\right ) \sin ^{-1}(c x)^2}{2 e^3}+\frac {(e g-d h) x \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac {h x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e}+\frac {b \left (e^2 f-d e g+d^2 h\right ) \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 \left (e^2 f-d e g+d^2 h\right ) \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 \left (e^2 f-d e g+d^2 h\right ) \sin ^{-1}(c x) \log (d+e x)}{e^3}+\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^3}-\frac {\left (b \left (e^2 f-d e g+d^2 h\right )\right ) \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 {\left (b \left (e^2 f-d e g+d^2 h\right )\right ) \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 (4 (e g-d h)+e h x) \sqrt {1-c^2 x^2}}{4 c e^2}-\frac {b h \sin ^{-1}(c x)}{4 c^2 e}-\frac {i b \left (e^2 f-d e g+d^2 h\right ) \sin ^{-1}(c x)^2}{2 e^3}+\frac {(e g-d h) x \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac {h x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e}+\frac {b \left (e^2 f-d e g+d^2 h\right ) \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 \left (e^2 f-d e g+d^2 h\right ) \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 \left (e^2 f-d e g+d^2 h\right ) \sin ^{-1}(c x) \log (d+e x)}{e^3}+\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^3}+\frac {\left (i b \left (e^2 f-d e g+d^2 h\right )\right ) \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 {\left (i b \left (e^2 f-d e g+d^2 h\right )\right ) \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 (4 (e g-d h)+e h x) \sqrt {1-c^2 x^2}}{4 c e^2}-\frac {b h \sin ^{-1}(c x)}{4 c^2 e}-\frac {i b \left (e^2 f-d e g+d^2 h\right ) \sin ^{-1}(c x)^2}{2 e^3}+\frac {(e g-d h) x \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac {h x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e}+\frac {b \left (e^2 f-d e g+d^2 h\right ) \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 \left (e^2 f-d e g+d^2 h\right ) \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 \left (e^2 f-d e g+d^2 h\right ) \sin ^{-1}(c x) \log (d+e x)}{e^3}+\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^3}-\frac {i b \left (e^2 f-d e g+d^2 h\right ) \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 \left (e^2 f-d e g+d^2 h\right ) \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 = 0.65, size = 381, normalized size = 0.83 \[ \frac {2 \log (d+e x) \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h-d e g+e^2 f\right )+2 e x (e g-d h) \left (a+b \sin ^{-1}(c x)\right )+e^2 h x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \left (-2 i c^2 \left (d^2 h-d e g+e^2 f\right ) \left (2 \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )+2 \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )+\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+2 i \left (\log \left (1+\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}-c d}\right )+\log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )\right )\right )\right )-4 c^2 \sin ^{-1}(c x) \log (d+e x) \left (d^2 h-d e g+e^2 f\right )+4 c e \sqrt {1-c^2 x^2} (e g-d h)+c e^2 h x \sqrt {1-c^2 x^2}-e^2 h \sin ^{-1}(c x)\right )}{2 c^2}}{2 e^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a h x^{2} + a g x + a f + {\left (b h x^{2} + b g x + b f\right )} \arcsin \left (c x\right )}{e x + d}, x\right ) \]
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 \[ \int \frac {{\left (h x^{2} + g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.74, size = 2446, normalized size = 5.33 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ a g {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} + \frac {1}{2} \, a h {\left (\frac {2 \, d^{2} \log \left (e x + d\right )}{e^{3}} + \frac {e x^{2} - 2 \, d x}{e^{2}}\right )} + \frac {a f \log \left (e x + d\right )}{e} + \int \frac {{\left (b h x^{2} + b g x + b f\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{e x + d}\,{d x} \]
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 \[ \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (h\,x^2+g\,x+f\right )}{d+e\,x} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x + h x^{2}\right )}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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