Optimal. Leaf size=470 \[ -\frac {\left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h-d e g+e^2 f\right )}{4 e^3 (d+e x)^4}-\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac {h \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}-\frac {b c \sqrt {1-c^2 x^2} \left (4 e^2 (e g-2 d h)-c^2 d \left (-3 d^2 h-d e g+5 e^2 f\right )\right )}{24 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^2}+\frac {b c \sqrt {1-c^2 x^2} \left (d^2 h-d e g+e^2 f\right )}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^3}+\frac {b c \sqrt {1-c^2 x^2} \left (c^4 d^2 \left (d^2 (-h)+d e g+11 e^2 f\right )+4 c^2 e^2 \left (d^2 h-4 d e g+e^2 f\right )+12 e^4 h\right )}{24 e^2 \left (c^2 d^2-e^2\right )^3 (d+e x)}-\frac {b c^3 \tan ^{-1}\left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right ) \left (-2 c^4 d^3 \left (d^2 h+d e g+3 e^2 f\right )-c^2 d e^2 \left (-7 d^2 h-13 d e g+9 e^2 f\right )+4 e^4 (e g-5 d h)\right )}{24 e^3 \left (c^2 d^2-e^2\right )^{7/2}} \]
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Rubi [A] time = 0.94, antiderivative size = 470, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ -\frac {\left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h-d e g+e^2 f\right )}{4 e^3 (d+e x)^4}-\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac {h \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}+\frac {b c \sqrt {1-c^2 x^2} \left (c^4 d^2 \left (d^2 (-h)+d e g+11 e^2 f\right )+4 c^2 e^2 \left (d^2 h-4 d e g+e^2 f\right )+12 e^4 h\right )}{24 e^2 \left (c^2 d^2-e^2\right )^3 (d+e x)}-\frac {b c \sqrt {1-c^2 x^2} \left (4 e^2 (e g-2 d h)-c^2 d \left (-3 d^2 h-d e g+5 e^2 f\right )\right )}{24 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^2}+\frac {b c \sqrt {1-c^2 x^2} \left (d^2 h-d e g+e^2 f\right )}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac {b c^3 \tan ^{-1}\left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right ) \left (-2 c^4 d^3 \left (d^2 h+d e g+3 e^2 f\right )-c^2 d e^2 \left (-7 d^2 h-13 d e g+9 e^2 f\right )+4 e^4 (e g-5 d h)\right )}{24 e^3 \left (c^2 d^2-e^2\right )^{7/2}} \]
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)^5} \, dx &=-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac {h \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}-(b c) \int \frac {-3 e^2 f-d e g-d^2 h-4 e (e g+d h) x-6 e^2 h x^2}{12 e^3 (d+e x)^4 \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 )}{4 e^3 (d+e x)^4}-\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac {h \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}-\frac {(b c) \int \frac {-3 e^2 f-d e g-d^2 h-4 e (e g+d h) x-6 e^2 h x^2}{(d+e x)^4 \sqrt {1-c^2 x^2}} \, dx}{12 e^3}\\ &=\frac {b c \left (e^2 f-d e g+d^2 h\right ) \sqrt {1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac {h \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}-\frac {(b c) \int \frac {3 \left (2 e^2 (2 e g-d h)-c^2 d \left (3 e^2 f+d e g+d^2 h\right )\right )+6 e \left (3 e^2 h+c^2 \left (e^2 f-d e g-2 d^2 h\right )\right ) x}{(d+e x)^3 \sqrt {1-c^2 x^2}} \, dx}{36 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}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac {b c \left (4 e^2 (e g-2 d h)-c^2 d \left (5 e^2 f-d e g-3 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{24 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^2}-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac {h \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}-\frac {(b c) \int \frac {-6 \left (6 e^4 h+2 c^2 e^2 \left (e^2 f-3 d e g-d^2 h\right )+c^4 d^2 \left (3 e^2 f+d e g+d^2 h\right )\right )-3 c^2 e \left (4 e^2 (e g-2 d h)-c^2 d \left (5 e^2 f-d e g-3 d^2 h\right )\right ) x}{(d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{72 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}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac {b c \left (4 e^2 (e g-2 d h)-c^2 d \left (5 e^2 f-d e g-3 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{24 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^2}+\frac {b c \left (12 e^4 h+c^4 d^2 \left (11 e^2 f+d e g-d^2 h\right )+4 c^2 e^2 \left (e^2 f-4 d e g+d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{24 e^2 \left (c^2 d^2-e^2\right )^3 (d+e x)}-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac {h \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}-\frac {\left (b c^3 \left (4 e^4 (e g-5 d h)-c^2 d e^2 \left (9 e^2 f-13 d e g-7 d^2 h\right )-2 c^4 d^3 \left (3 e^2 f+d e g+d^2 h\right )\right )\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{24 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}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac {b c \left (4 e^2 (e g-2 d h)-c^2 d \left (5 e^2 f-d e g-3 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{24 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^2}+\frac {b c \left (12 e^4 h+c^4 d^2 \left (11 e^2 f+d e g-d^2 h\right )+4 c^2 e^2 \left (e^2 f-4 d e g+d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{24 e^2 \left (c^2 d^2-e^2\right )^3 (d+e x)}-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac {h \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}+\frac {\left (b c^3 \left (4 e^4 (e g-5 d h)-c^2 d e^2 \left (9 e^2 f-13 d e g-7 d^2 h\right )-2 c^4 d^3 \left (3 e^2 f+d e g+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 )}{24 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}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac {b c \left (4 e^2 (e g-2 d h)-c^2 d \left (5 e^2 f-d e g-3 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{24 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^2}+\frac {b c \left (12 e^4 h+c^4 d^2 \left (11 e^2 f+d e g-d^2 h\right )+4 c^2 e^2 \left (e^2 f-4 d e g+d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{24 e^2 \left (c^2 d^2-e^2\right )^3 (d+e x)}-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac {h \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}-\frac {b c^3 \left (4 e^4 (e g-5 d h)-c^2 d e^2 \left (9 e^2 f-13 d e g-7 d^2 h\right )-2 c^4 d^3 \left (3 e^2 f+d e g+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 )}{24 e^3 \left (c^2 d^2-e^2\right )^{7/2}}\\ \end {align*}
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Mathematica [A] time = 2.92, size = 575, normalized size = 1.22 \[ -\frac {\frac {6 a \left (d^2 h-d e g+e^2 f\right )}{(d+e x)^4}+\frac {8 a (e g-2 d h)}{(d+e x)^3}+\frac {12 a h}{(d+e x)^2}+\frac {b c e \sqrt {1-c^2 x^2} \left (c^4 d^2 \left (-2 d^4 h-d^3 e (2 g+5 h x)+d^2 e^2 (18 f+x (g-h x))+d e^3 x (27 f+g x)+11 e^4 f x^2\right )+c^2 e^2 \left (11 d^4 h+d^3 e (19 h x-15 g)+d^2 e^2 (x (4 h x-35 g)-5 f)+d e^3 x (3 f-16 g x)+4 e^4 f x^2\right )+2 e^4 \left (3 d^2 h+d e (g+8 h x)+e^2 (f+2 x (g+3 h x))\right )\right )}{\left (e^2-c^2 d^2\right )^3 (d+e x)^3}+\frac {b c^3 \log \left (\sqrt {1-c^2 x^2} \sqrt {e^2-c^2 d^2}+c^2 d x+e\right ) \left (2 c^4 d^3 \left (d^2 h+d e g+3 e^2 f\right )+c^2 d e^2 \left (-7 d^2 h-13 d e g+9 e^2 f\right )-4 e^4 (e g-5 d h)\right )}{(c d-e)^3 (c d+e)^3 \sqrt {e^2-c^2 d^2}}-\frac {b c^3 \log (d+e x) \left (2 c^4 d^3 \left (d^2 h+d e g+3 e^2 f\right )+c^2 d e^2 \left (-7 d^2 h-13 d e g+9 e^2 f\right )-4 e^4 (e g-5 d h)\right )}{(c d-e)^3 (c d+e)^3 \sqrt {e^2-c^2 d^2}}+\frac {2 b \sin ^{-1}(c x) \left (d^2 h+d e (g+4 h x)+e^2 \left (3 f+4 g x+6 h x^2\right )\right )}{(d+e x)^4}}{24 e^3} \]
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(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 3005, normalized size = 6.39 \[ \text {output too large to display} \]
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 \[ -\frac {{\left (4 \, e x + d\right )} a g}{12 \, {\left (e^{6} x^{4} + 4 \, d e^{5} x^{3} + 6 \, d^{2} e^{4} x^{2} + 4 \, d^{3} e^{3} x + d^{4} e^{2}\right )}} - \frac {{\left (6 \, e^{2} x^{2} + 4 \, d e x + d^{2}\right )} a h}{12 \, {\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} - \frac {a f}{4 \, {\left (e^{5} x^{4} + 4 \, d e^{4} x^{3} + 6 \, d^{2} e^{3} x^{2} + 4 \, d^{3} e^{2} x + d^{4} e\right )}} - \frac {{\left (6 \, b e^{2} h x^{2} + 3 \, b e^{2} f + b d e g + b d^{2} h + 4 \, {\left (b e^{2} g + b d e h\right )} x\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + {\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )} \int \frac {{\left (6 \, b c e^{2} h x^{2} + 3 \, b c e^{2} f + b c d e g + b c d^{2} h + 4 \, {\left (b c e^{2} g + b c d e h\right )} x\right )} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (-c x + 1\right )\right )}}{c^{4} e^{7} x^{8} + 4 \, c^{4} d e^{6} x^{7} - 4 \, c^{2} d^{3} e^{4} x^{3} - c^{2} d^{4} e^{3} x^{2} + {\left (6 \, c^{4} d^{2} e^{5} - c^{2} e^{7}\right )} x^{6} + 4 \, {\left (c^{4} d^{3} e^{4} - c^{2} d e^{6}\right )} x^{5} + {\left (c^{4} d^{4} e^{3} - 6 \, c^{2} d^{2} e^{5}\right )} x^{4} - {\left (c^{2} e^{7} x^{6} + 4 \, c^{2} d e^{6} x^{5} - 4 \, d^{3} e^{4} x - d^{4} e^{3} + {\left (6 \, c^{2} d^{2} e^{5} - e^{7}\right )} x^{4} + 4 \, {\left (c^{2} d^{3} e^{4} - d e^{6}\right )} x^{3} + {\left (c^{2} d^{4} e^{3} - 6 \, d^{2} e^{5}\right )} x^{2}\right )} {\left (c x + 1\right )} {\left (c x - 1\right )}}\,{d x}}{12 \, {\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \]
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 \[ \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (h\,x^2+g\,x+f\right )}{{\left (d+e\,x\right )}^5} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x + h x^{2}\right )}{\left (d + e x\right )^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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