Optimal. Leaf size=593 \[ -\frac {\left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h-d e g+e^2 f\right )}{5 e^3 (d+e x)^5}-\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac {h \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac {b c \sqrt {1-c^2 x^2} \left (5 e^2 (e g-2 d h)-c^2 d \left (-3 d^2 h-2 d e g+7 e^2 f\right )\right )}{60 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^3}+\frac {b c \sqrt {1-c^2 x^2} \left (d^2 h-d e g+e^2 f\right )}{20 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^4}+\frac {b c \sqrt {1-c^2 x^2} \left (c^4 d^2 \left (-4 d^2 h-d e g+26 e^2 f\right )+c^2 e^2 \left (19 d^2 h-34 d e g+9 e^2 f\right )+20 e^4 h\right )}{120 e^2 \left (c^2 d^2-e^2\right )^3 (d+e x)^2}+\frac {b c^3 \sqrt {1-c^2 x^2} \left (c^4 d^3 (d g+10 e f)+c^2 d e \left (d^2 h-18 d e g+11 e^2 f\right )-4 e^3 (e g-5 d h)\right )}{24 e \left (c^2 d^2-e^2\right )^4 (d+e x)}+\frac {b c^3 \tan ^{-1}\left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right ) \left (2 c^6 d^4 \left (2 d^2 h+3 d e g+12 e^2 f\right )+3 c^4 d^2 e^2 \left (-6 d^2 h-19 d e g+24 e^2 f\right )+9 c^2 e^4 \left (11 d^2 h-6 d e g+e^2 f\right )+20 e^6 h\right )}{120 e^3 \left (c^2 d^2-e^2\right )^{9/2}} \]
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Rubi [A] time = 1.26, antiderivative size = 593, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {698, 4753, 12, 1651, 835, 807, 725, 204} \[ -\frac {\left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h-d e g+e^2 f\right )}{5 e^3 (d+e x)^5}-\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac {h \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}+\frac {b c^3 \sqrt {1-c^2 x^2} \left (c^2 d e \left (d^2 h-18 d e g+11 e^2 f\right )+c^4 d^3 (d g+10 e f)-4 e^3 (e g-5 d h)\right )}{24 e \left (c^2 d^2-e^2\right )^4 (d+e x)}+\frac {b c \sqrt {1-c^2 x^2} \left (c^4 d^2 \left (-4 d^2 h-d e g+26 e^2 f\right )+c^2 e^2 \left (19 d^2 h-34 d e g+9 e^2 f\right )+20 e^4 h\right )}{120 e^2 \left (c^2 d^2-e^2\right )^3 (d+e x)^2}-\frac {b c \sqrt {1-c^2 x^2} \left (5 e^2 (e g-2 d h)-c^2 d \left (-3 d^2 h-2 d e g+7 e^2 f\right )\right )}{60 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^3}+\frac {b c \sqrt {1-c^2 x^2} \left (d^2 h-d e g+e^2 f\right )}{20 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^4}+\frac {b c^3 \tan ^{-1}\left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right ) \left (2 c^6 d^4 \left (2 d^2 h+3 d e g+12 e^2 f\right )+3 c^4 d^2 e^2 \left (-6 d^2 h-19 d e g+24 e^2 f\right )+9 c^2 e^4 \left (11 d^2 h-6 d e g+e^2 f\right )+20 e^6 h\right )}{120 e^3 \left (c^2 d^2-e^2\right )^{9/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 204
Rule 698
Rule 725
Rule 807
Rule 835
Rule 1651
Rule 4753
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)^6} \, dx &=-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{5 e^3 (d+e x)^5}-\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac {h \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-(b c) \int \frac {-12 e^2 f-3 d e g-2 d^2 h-5 e (3 e g+2 d h) x-20 e^2 h x^2}{60 e^3 (d+e x)^5 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{5 e^3 (d+e x)^5}-\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac {h \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac {(b c) \int \frac {-12 e^2 f-3 d e g-2 d^2 h-5 e (3 e g+2 d h) x-20 e^2 h x^2}{(d+e x)^5 \sqrt {1-c^2 x^2}} \, dx}{60 e^3}\\ &=\frac {b c \left (e^2 f-d e g+d^2 h\right ) \sqrt {1-c^2 x^2}}{20 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{5 e^3 (d+e x)^5}-\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac {h \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac {(b c) \int \frac {4 \left (5 e^2 (3 e g-2 d h)-c^2 d \left (12 e^2 f+3 d e g+2 d^2 h\right )\right )+4 e \left (20 e^2 h+c^2 \left (9 e^2 f-9 d e g-11 d^2 h\right )\right ) x}{(d+e x)^4 \sqrt {1-c^2 x^2}} \, dx}{240 e^3 \left (c^2 d^2-e^2\right )}\\ &=\frac {b c \left (e^2 f-d e g+d^2 h\right ) \sqrt {1-c^2 x^2}}{20 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac {b c \left (5 e^2 (e g-2 d h)-c^2 d \left (7 e^2 f-2 d e g-3 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{60 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^3}-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{5 e^3 (d+e x)^5}-\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac {h \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac {(b c) \int \frac {-12 \left (20 e^4 h+c^2 e^2 \left (9 e^2 f-24 d e g-d^2 h\right )+c^4 d^2 \left (12 e^2 f+3 d e g+2 d^2 h\right )\right )-24 c^2 e \left (5 e^2 (e g-2 d h)-c^2 d \left (7 e^2 f-2 d e g-3 d^2 h\right )\right ) x}{(d+e x)^3 \sqrt {1-c^2 x^2}} \, dx}{720 e^3 \left (c^2 d^2-e^2\right )^2}\\ &=\frac {b c \left (e^2 f-d e g+d^2 h\right ) \sqrt {1-c^2 x^2}}{20 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac {b c \left (5 e^2 (e g-2 d h)-c^2 d \left (7 e^2 f-2 d e g-3 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{60 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^3}+\frac {b c \left (20 e^4 h+c^4 d^2 \left (26 e^2 f-d e g-4 d^2 h\right )+c^2 e^2 \left (9 e^2 f-34 d e g+19 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{120 e^2 \left (c^2 d^2-e^2\right )^3 (d+e x)^2}-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{5 e^3 (d+e x)^5}-\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac {h \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac {(b c) \int \frac {24 c^2 \left (10 e^4 (e g-4 d h)-c^2 d e^2 \left (23 e^2 f-28 d e g-7 d^2 h\right )-c^4 d^3 \left (12 e^2 f+3 d e g+2 d^2 h\right )\right )+12 c^2 e \left (20 e^4 h+c^4 d^2 \left (26 e^2 f-d e g-4 d^2 h\right )+c^2 e^2 \left (9 e^2 f-34 d e g+19 d^2 h\right )\right ) x}{(d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{1440 e^3 \left (c^2 d^2-e^2\right )^3}\\ &=\frac {b c \left (e^2 f-d e g+d^2 h\right ) \sqrt {1-c^2 x^2}}{20 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac {b c \left (5 e^2 (e g-2 d h)-c^2 d \left (7 e^2 f-2 d e g-3 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{60 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^3}+\frac {b c \left (20 e^4 h+c^4 d^2 \left (26 e^2 f-d e g-4 d^2 h\right )+c^2 e^2 \left (9 e^2 f-34 d e g+19 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{120 e^2 \left (c^2 d^2-e^2\right )^3 (d+e x)^2}+\frac {b c^3 \left (c^4 d^3 (10 e f+d g)-4 e^3 (e g-5 d h)+c^2 d e \left (11 e^2 f-18 d e g+d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{24 e \left (c^2 d^2-e^2\right )^4 (d+e x)}-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{5 e^3 (d+e x)^5}-\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac {h \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}+\frac {\left (b c^3 \left (20 e^6 h+3 c^4 d^2 e^2 \left (24 e^2 f-19 d e g-6 d^2 h\right )+2 c^6 d^4 \left (12 e^2 f+3 d e g+2 d^2 h\right )+9 c^2 e^4 \left (e^2 f-6 d e g+11 d^2 h\right )\right )\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{120 e^3 \left (c^2 d^2-e^2\right )^4}\\ &=\frac {b c \left (e^2 f-d e g+d^2 h\right ) \sqrt {1-c^2 x^2}}{20 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac {b c \left (5 e^2 (e g-2 d h)-c^2 d \left (7 e^2 f-2 d e g-3 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{60 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^3}+\frac {b c \left (20 e^4 h+c^4 d^2 \left (26 e^2 f-d e g-4 d^2 h\right )+c^2 e^2 \left (9 e^2 f-34 d e g+19 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{120 e^2 \left (c^2 d^2-e^2\right )^3 (d+e x)^2}+\frac {b c^3 \left (c^4 d^3 (10 e f+d g)-4 e^3 (e g-5 d h)+c^2 d e \left (11 e^2 f-18 d e g+d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{24 e \left (c^2 d^2-e^2\right )^4 (d+e x)}-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{5 e^3 (d+e x)^5}-\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac {h \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac {\left (b c^3 \left (20 e^6 h+3 c^4 d^2 e^2 \left (24 e^2 f-19 d e g-6 d^2 h\right )+2 c^6 d^4 \left (12 e^2 f+3 d e g+2 d^2 h\right )+9 c^2 e^4 \left (e^2 f-6 d e g+11 d^2 h\right )\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 )}{120 e^3 \left (c^2 d^2-e^2\right )^4}\\ &=\frac {b c \left (e^2 f-d e g+d^2 h\right ) \sqrt {1-c^2 x^2}}{20 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac {b c \left (5 e^2 (e g-2 d h)-c^2 d \left (7 e^2 f-2 d e g-3 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{60 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^3}+\frac {b c \left (20 e^4 h+c^4 d^2 \left (26 e^2 f-d e g-4 d^2 h\right )+c^2 e^2 \left (9 e^2 f-34 d e g+19 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{120 e^2 \left (c^2 d^2-e^2\right )^3 (d+e x)^2}+\frac {b c^3 \left (c^4 d^3 (10 e f+d g)-4 e^3 (e g-5 d h)+c^2 d e \left (11 e^2 f-18 d e g+d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{24 e \left (c^2 d^2-e^2\right )^4 (d+e x)}-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{5 e^3 (d+e x)^5}-\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac {h \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}+\frac {b c^3 \left (20 e^6 h+3 c^4 d^2 e^2 \left (24 e^2 f-19 d e g-6 d^2 h\right )+2 c^6 d^4 \left (12 e^2 f+3 d e g+2 d^2 h\right )+9 c^2 e^4 \left (e^2 f-6 d e g+11 d^2 h\right )\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{120 e^3 \left (c^2 d^2-e^2\right )^{9/2}}\\ \end {align*}
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Mathematica [A] time = 2.59, size = 682, normalized size = 1.15 \[ -\frac {\frac {24 a \left (d^2 h-d e g+e^2 f\right )}{(d+e x)^5}+\frac {30 a (e g-2 d h)}{(d+e x)^4}+\frac {40 a h}{(d+e x)^3}-\frac {b c e \sqrt {1-c^2 x^2} \left (-2 \left (e^2-c^2 d^2\right )^2 (d+e x) \left (c^2 d \left (3 d^2 h+2 d e g-7 e^2 f\right )+5 e^2 (e g-2 d h)\right )+6 \left (c^2 d^2-e^2\right )^3 \left (d^2 h-d e g+e^2 f\right )-\left (e^2-c^2 d^2\right ) (d+e x)^2 \left (c^4 \left (-d^2\right ) \left (4 d^2 h+d e g-26 e^2 f\right )+c^2 e^2 \left (19 d^2 h-34 d e g+9 e^2 f\right )+20 e^4 h\right )+5 c^2 e (d+e x)^3 \left (c^4 d^3 (d g+10 e f)+c^2 d e \left (d^2 h-18 d e g+11 e^2 f\right )-4 e^3 (e g-5 d h)\right )\right )}{\left (e^2-c^2 d^2\right )^4 (d+e x)^4}+\frac {b c^3 \log \left (\sqrt {1-c^2 x^2} \sqrt {e^2-c^2 d^2}+c^2 d x+e\right ) \left (2 c^6 d^4 \left (2 d^2 h+3 d e g+12 e^2 f\right )-3 c^4 d^2 e^2 \left (6 d^2 h+19 d e g-24 e^2 f\right )+9 c^2 e^4 \left (11 d^2 h-6 d e g+e^2 f\right )+20 e^6 h\right )}{(e-c d)^4 (c d+e)^4 \sqrt {e^2-c^2 d^2}}-\frac {b c^3 \log (d+e x) \left (2 c^6 d^4 \left (2 d^2 h+3 d e g+12 e^2 f\right )-3 c^4 d^2 e^2 \left (6 d^2 h+19 d e g-24 e^2 f\right )+9 c^2 e^4 \left (11 d^2 h-6 d e g+e^2 f\right )+20 e^6 h\right )}{(e-c d)^4 (c d+e)^4 \sqrt {e^2-c^2 d^2}}+\frac {2 b \sin ^{-1}(c x) \left (2 d^2 h+d e (3 g+10 h x)+e^2 (12 f+5 x (3 g+4 h x))\right )}{(d+e x)^5}}{120 e^3} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {{\left (h x^{2} + g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (e x + d\right )}^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 4077, normalized size = 6.88 \[ \text {output too large to display} \]
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 \[ -\frac {{\left (5 \, e x + d\right )} a g}{20 \, {\left (e^{7} x^{5} + 5 \, d e^{6} x^{4} + 10 \, d^{2} e^{5} x^{3} + 10 \, d^{3} e^{4} x^{2} + 5 \, d^{4} e^{3} x + d^{5} e^{2}\right )}} - \frac {{\left (10 \, e^{2} x^{2} + 5 \, d e x + d^{2}\right )} a h}{30 \, {\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} - \frac {a f}{5 \, {\left (e^{6} x^{5} + 5 \, d e^{5} x^{4} + 10 \, d^{2} e^{4} x^{3} + 10 \, d^{3} e^{3} x^{2} + 5 \, d^{4} e^{2} x + d^{5} e\right )}} - \frac {{\left (20 \, b e^{2} h x^{2} + 12 \, b e^{2} f + 3 \, b d e g + 2 \, b d^{2} h + 5 \, {\left (3 \, b e^{2} g + 2 \, b d e h\right )} x\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + {\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )} \int \frac {{\left (20 \, b c e^{2} h x^{2} + 12 \, b c e^{2} f + 3 \, b c d e g + 2 \, b c d^{2} h + 5 \, {\left (3 \, b c e^{2} g + 2 \, b c d e h\right )} x\right )} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (-c x + 1\right )\right )}}{c^{4} e^{8} x^{9} + 5 \, c^{4} d e^{7} x^{8} - 5 \, c^{2} d^{4} e^{4} x^{3} - c^{2} d^{5} e^{3} x^{2} + {\left (10 \, c^{4} d^{2} e^{6} - c^{2} e^{8}\right )} x^{7} + 5 \, {\left (2 \, c^{4} d^{3} e^{5} - c^{2} d e^{7}\right )} x^{6} + 5 \, {\left (c^{4} d^{4} e^{4} - 2 \, c^{2} d^{2} e^{6}\right )} x^{5} + {\left (c^{4} d^{5} e^{3} - 10 \, c^{2} d^{3} e^{5}\right )} x^{4} - {\left (c^{2} e^{8} x^{7} + 5 \, c^{2} d e^{7} x^{6} - 5 \, d^{4} e^{4} x - d^{5} e^{3} + {\left (10 \, c^{2} d^{2} e^{6} - e^{8}\right )} x^{5} + 5 \, {\left (2 \, c^{2} d^{3} e^{5} - d e^{7}\right )} x^{4} + 5 \, {\left (c^{2} d^{4} e^{4} - 2 \, d^{2} e^{6}\right )} x^{3} + {\left (c^{2} d^{5} e^{3} - 10 \, d^{3} e^{5}\right )} x^{2}\right )} {\left (c x + 1\right )} {\left (c x - 1\right )}}\,{d x}}{60 \, {\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \]
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 \[ \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (h\,x^2+g\,x+f\right )}{{\left (d+e\,x\right )}^6} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x + h x^{2}\right )}{\left (d + e x\right )^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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