Optimal. Leaf size=349 \[ -\frac{\left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h-d e g+e^2 f\right )}{3 e^3 (d+e x)^3}-\frac{(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}-\frac{h \left (a+b \sin ^{-1}(c x)\right )}{e^3 (d+e x)}+\frac{b c \sqrt{1-c^2 x^2} \left (d^2 h-d e g+e^2 f\right )}{6 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac{b c \sqrt{1-c^2 x^2} \left (e^2 (e g-2 d h)-c^2 \left (d e^2 f-d^3 h\right )\right )}{2 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac{b c \tan ^{-1}\left (\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right ) \left (c^4 d^2 \left (2 d^2 h+d e g+2 e^2 f\right )+c^2 e^2 \left (-5 d^2 h-4 d e g+e^2 f\right )+6 e^4 h\right )}{6 e^3 \left (c^2 d^2-e^2\right )^{5/2}} \]
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Rubi [A] time = 0.617012, antiderivative size = 349, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {698, 4753, 12, 1651, 807, 725, 204} \[ -\frac{\left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h-d e g+e^2 f\right )}{3 e^3 (d+e x)^3}-\frac{(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}-\frac{h \left (a+b \sin ^{-1}(c x)\right )}{e^3 (d+e x)}+\frac{b c \sqrt{1-c^2 x^2} \left (d^2 h-d e g+e^2 f\right )}{6 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac{b c \sqrt{1-c^2 x^2} \left (e^2 (e g-2 d h)-c^2 \left (d e^2 f-d^3 h\right )\right )}{2 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac{b c \tan ^{-1}\left (\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right ) \left (c^4 d^2 \left (2 d^2 h+d e g+2 e^2 f\right )+c^2 e^2 \left (-5 d^2 h-4 d e g+e^2 f\right )+6 e^4 h\right )}{6 e^3 \left (c^2 d^2-e^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 698
Rule 4753
Rule 12
Rule 1651
Rule 807
Rule 725
Rule 204
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)^4} \, dx &=-\frac{\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac{(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}-\frac{h \left (a+b \sin ^{-1}(c x)\right )}{e^3 (d+e x)}-(b c) \int \frac{-2 e^2 f-d e g-2 d^2 h-3 e (e g+2 d h) x-6 e^2 h x^2}{6 e^3 (d+e x)^3 \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 )}{3 e^3 (d+e x)^3}-\frac{(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}-\frac{h \left (a+b \sin ^{-1}(c x)\right )}{e^3 (d+e x)}-\frac{(b c) \int \frac{-2 e^2 f-d e g-2 d^2 h-3 e (e g+2 d h) x-6 e^2 h x^2}{(d+e x)^3 \sqrt{1-c^2 x^2}} \, dx}{6 e^3}\\ &=\frac{b c \left (e^2 f-d e g+d^2 h\right ) \sqrt{1-c^2 x^2}}{6 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac{\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac{(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}-\frac{h \left (a+b \sin ^{-1}(c x)\right )}{e^3 (d+e x)}-\frac{(b c) \int \frac{2 \left (3 e^3 g-c^2 d \left (2 e^2 f+d e g+2 d^2 h\right )\right )+2 e \left (6 e^2 h+c^2 \left (e^2 f-d e g-5 d^2 h\right )\right ) x}{(d+e x)^2 \sqrt{1-c^2 x^2}} \, dx}{12 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}}{6 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac{b c \left (e^2 (e g-2 d h)-c^2 \left (d e^2 f-d^3 h\right )\right ) \sqrt{1-c^2 x^2}}{2 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac{\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac{(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}-\frac{h \left (a+b \sin ^{-1}(c x)\right )}{e^3 (d+e x)}+\frac{\left (b c \left (6 e^4 h+c^2 e^2 \left (e^2 f-4 d e g-5 d^2 h\right )+c^4 d^2 \left (2 e^2 f+d e g+2 d^2 h\right )\right )\right ) \int \frac{1}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{6 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}}{6 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac{b c \left (e^2 (e g-2 d h)-c^2 \left (d e^2 f-d^3 h\right )\right ) \sqrt{1-c^2 x^2}}{2 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac{\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac{(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}-\frac{h \left (a+b \sin ^{-1}(c x)\right )}{e^3 (d+e x)}-\frac{\left (b c \left (6 e^4 h+c^2 e^2 \left (e^2 f-4 d e g-5 d^2 h\right )+c^4 d^2 \left (2 e^2 f+d e g+2 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 )}{6 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}}{6 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac{b c \left (e^2 (e g-2 d h)-c^2 \left (d e^2 f-d^3 h\right )\right ) \sqrt{1-c^2 x^2}}{2 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac{\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac{(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}-\frac{h \left (a+b \sin ^{-1}(c x)\right )}{e^3 (d+e x)}+\frac{b c \left (6 e^4 h+c^2 e^2 \left (e^2 f-4 d e g-5 d^2 h\right )+c^4 d^2 \left (2 e^2 f+d e g+2 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 )}{6 e^3 \left (c^2 d^2-e^2\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 1.82451, size = 442, normalized size = 1.27 \[ -\frac{\frac{2 a \left (d^2 h-d e g+e^2 f\right )}{(d+e x)^3}+\frac{3 a (e g-2 d h)}{(d+e x)^2}+\frac{6 a h}{d+e x}+\frac{b c e \sqrt{1-c^2 x^2} \left (c^2 d \left (d^2 e (g+3 h x)+2 d^3 h-4 d e^2 f-3 e^3 f x\right )+e^2 \left (-5 d^2 h+2 d e (g-3 h x)+e^2 (f+3 g x)\right )\right )}{\left (e^2-c^2 d^2\right )^2 (d+e x)^2}+\frac{b c \log \left (\sqrt{1-c^2 x^2} \sqrt{e^2-c^2 d^2}+c^2 d x+e\right ) \left (c^4 d^2 \left (2 d^2 h+d e g+2 e^2 f\right )+c^2 e^2 \left (-5 d^2 h-4 d e g+e^2 f\right )+6 e^4 h\right )}{(e-c d)^2 (c d+e)^2 \sqrt{e^2-c^2 d^2}}-\frac{b c \log (d+e x) \left (c^4 d^2 \left (2 d^2 h+d e g+2 e^2 f\right )+c^2 e^2 \left (-5 d^2 h-4 d e g+e^2 f\right )+6 e^4 h\right )}{(e-c d)^2 (c d+e)^2 \sqrt{e^2-c^2 d^2}}+\frac{b \sin ^{-1}(c x) \left (2 d^2 h+d e (g+6 h x)+e^2 (2 f+3 x (g+2 h x))\right )}{(d+e x)^3}}{6 e^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.015, size = 2173, normalized size = 6.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (3 \, e x + d\right )} a g}{6 \,{\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )}} - \frac{{\left (3 \, e^{2} x^{2} + 3 \, d e x + d^{2}\right )} a h}{3 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} - \frac{a f}{3 \,{\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} - \frac{{\left (6 \, b e^{2} h x^{2} + 2 \, b e^{2} f + b d e g + 2 \, b d^{2} h + 3 \,{\left (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^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )} \int \frac{{\left (6 \, b c e^{2} h x^{2} + 2 \, b c e^{2} f + b c d e g + 2 \, b c d^{2} h + 3 \,{\left (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^{6} x^{7} + 3 \, c^{4} d e^{5} x^{6} - 3 \, c^{2} d^{2} e^{4} x^{3} - c^{2} d^{3} e^{3} x^{2} +{\left (3 \, c^{4} d^{2} e^{4} - c^{2} e^{6}\right )} x^{5} +{\left (c^{4} d^{3} e^{3} - 3 \, c^{2} d e^{5}\right )} x^{4} -{\left (c^{2} e^{6} x^{5} + 3 \, c^{2} d e^{5} x^{4} - 3 \, d^{2} e^{4} x - d^{3} e^{3} +{\left (3 \, c^{2} d^{2} e^{4} - e^{6}\right )} x^{3} +{\left (c^{2} d^{3} e^{3} - 3 \, d e^{5}\right )} x^{2}\right )}{\left (c x + 1\right )}{\left (c x - 1\right )}}\,{d x}}{6 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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 )^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (h x^{2} + g x + f\right )}{\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (e x + d\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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