Optimal. Leaf size=360 \[ -\frac{(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{4 e^2 (d+e x)^4}-\frac{g \left (a+b \sin ^{-1}(c x)\right )}{3 e^2 (d+e x)^3}+\frac{b c^3 \sqrt{1-c^2 x^2} \left (c^2 d^2 (d g+11 e f)+4 e^2 (e f-4 d g)\right )}{24 e \left (c^2 d^2-e^2\right )^3 (d+e x)}-\frac{b c \sqrt{1-c^2 x^2} \left (4 e^2 g-c^2 d (5 e f-d g)\right )}{24 e \left (c^2 d^2-e^2\right )^2 (d+e x)^2}+\frac{b c \sqrt{1-c^2 x^2} (e f-d g)}{12 e \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac{b c^3 \left (-2 c^4 d^3 (d g+3 e f)-c^2 d e^2 (9 e f-13 d g)+4 e^4 g\right ) \tan ^{-1}\left (\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right )}{24 e^2 \left (c^2 d^2-e^2\right )^{7/2}} \]
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Rubi [A] time = 0.695172, antiderivative size = 360, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {43, 4753, 12, 835, 807, 725, 204} \[ -\frac{(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{4 e^2 (d+e x)^4}-\frac{g \left (a+b \sin ^{-1}(c x)\right )}{3 e^2 (d+e x)^3}+\frac{b c^3 \sqrt{1-c^2 x^2} \left (c^2 d^2 (d g+11 e f)+4 e^2 (e f-4 d g)\right )}{24 e \left (c^2 d^2-e^2\right )^3 (d+e x)}-\frac{b c \sqrt{1-c^2 x^2} \left (4 e^2 g-c^2 d (5 e f-d g)\right )}{24 e \left (c^2 d^2-e^2\right )^2 (d+e x)^2}+\frac{b c \sqrt{1-c^2 x^2} (e f-d g)}{12 e \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac{b c^3 \left (-2 c^4 d^3 (d g+3 e f)-c^2 d e^2 (9 e f-13 d g)+4 e^4 g\right ) \tan ^{-1}\left (\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right )}{24 e^2 \left (c^2 d^2-e^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 4753
Rule 12
Rule 835
Rule 807
Rule 725
Rule 204
Rubi steps
\begin{align*} \int \frac{(f+g x) \left (a+b \sin ^{-1}(c x)\right )}{(d+e x)^5} \, dx &=-\frac{(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{4 e^2 (d+e x)^4}-\frac{g \left (a+b \sin ^{-1}(c x)\right )}{3 e^2 (d+e x)^3}-(b c) \int \frac{-3 e f-d g-4 e g x}{12 e^2 (d+e x)^4 \sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{4 e^2 (d+e x)^4}-\frac{g \left (a+b \sin ^{-1}(c x)\right )}{3 e^2 (d+e x)^3}-\frac{(b c) \int \frac{-3 e f-d g-4 e g x}{(d+e x)^4 \sqrt{1-c^2 x^2}} \, dx}{12 e^2}\\ &=\frac{b c (e f-d g) \sqrt{1-c^2 x^2}}{12 e \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac{(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{4 e^2 (d+e x)^4}-\frac{g \left (a+b \sin ^{-1}(c x)\right )}{3 e^2 (d+e x)^3}-\frac{(b c) \int \frac{3 \left (4 e^2 g-c^2 d (3 e f+d g)\right )+6 c^2 e (e f-d g) x}{(d+e x)^3 \sqrt{1-c^2 x^2}} \, dx}{36 e^2 \left (c^2 d^2-e^2\right )}\\ &=\frac{b c (e f-d g) \sqrt{1-c^2 x^2}}{12 e \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac{b c \left (4 e^2 g-c^2 d (5 e f-d g)\right ) \sqrt{1-c^2 x^2}}{24 e \left (c^2 d^2-e^2\right )^2 (d+e x)^2}-\frac{(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{4 e^2 (d+e x)^4}-\frac{g \left (a+b \sin ^{-1}(c x)\right )}{3 e^2 (d+e x)^3}-\frac{(b c) \int \frac{-6 c^2 \left (2 e^2 (e f-3 d g)+c^2 d^2 (3 e f+d g)\right )-3 c^2 e \left (4 e^2 g-c^2 d (5 e f-d g)\right ) x}{(d+e x)^2 \sqrt{1-c^2 x^2}} \, dx}{72 e^2 \left (c^2 d^2-e^2\right )^2}\\ &=\frac{b c (e f-d g) \sqrt{1-c^2 x^2}}{12 e \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac{b c \left (4 e^2 g-c^2 d (5 e f-d g)\right ) \sqrt{1-c^2 x^2}}{24 e \left (c^2 d^2-e^2\right )^2 (d+e x)^2}+\frac{b c^3 \left (4 e^2 (e f-4 d g)+c^2 d^2 (11 e f+d g)\right ) \sqrt{1-c^2 x^2}}{24 e \left (c^2 d^2-e^2\right )^3 (d+e x)}-\frac{(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{4 e^2 (d+e x)^4}-\frac{g \left (a+b \sin ^{-1}(c x)\right )}{3 e^2 (d+e x)^3}-\frac{\left (b c^3 \left (4 e^4 g-c^2 d e^2 (9 e f-13 d g)-2 c^4 d^3 (3 e f+d g)\right )\right ) \int \frac{1}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{24 e^2 \left (c^2 d^2-e^2\right )^3}\\ &=\frac{b c (e f-d g) \sqrt{1-c^2 x^2}}{12 e \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac{b c \left (4 e^2 g-c^2 d (5 e f-d g)\right ) \sqrt{1-c^2 x^2}}{24 e \left (c^2 d^2-e^2\right )^2 (d+e x)^2}+\frac{b c^3 \left (4 e^2 (e f-4 d g)+c^2 d^2 (11 e f+d g)\right ) \sqrt{1-c^2 x^2}}{24 e \left (c^2 d^2-e^2\right )^3 (d+e x)}-\frac{(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{4 e^2 (d+e x)^4}-\frac{g \left (a+b \sin ^{-1}(c x)\right )}{3 e^2 (d+e x)^3}+\frac{\left (b c^3 \left (4 e^4 g-c^2 d e^2 (9 e f-13 d g)-2 c^4 d^3 (3 e f+d g)\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^2 \left (c^2 d^2-e^2\right )^3}\\ &=\frac{b c (e f-d g) \sqrt{1-c^2 x^2}}{12 e \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac{b c \left (4 e^2 g-c^2 d (5 e f-d g)\right ) \sqrt{1-c^2 x^2}}{24 e \left (c^2 d^2-e^2\right )^2 (d+e x)^2}+\frac{b c^3 \left (4 e^2 (e f-4 d g)+c^2 d^2 (11 e f+d g)\right ) \sqrt{1-c^2 x^2}}{24 e \left (c^2 d^2-e^2\right )^3 (d+e x)}-\frac{(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{4 e^2 (d+e x)^4}-\frac{g \left (a+b \sin ^{-1}(c x)\right )}{3 e^2 (d+e x)^3}-\frac{b c^3 \left (4 e^4 g-c^2 d e^2 (9 e f-13 d g)-2 c^4 d^3 (3 e f+d g)\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^2 \left (c^2 d^2-e^2\right )^{7/2}}\\ \end{align*}
Mathematica [A] time = 1.27819, size = 418, normalized size = 1.16 \[ \frac{\frac{a (6 d g-6 e f)}{(d+e x)^4}-\frac{8 a g}{(d+e x)^3}-\frac{b e \sqrt{1-c^2 x^2} \left (c^5 d^2 \left (d^2 e (18 f+g x)-2 d^3 g+d e^2 x (27 f+g x)+11 e^3 f x^2\right )-c^3 e^2 \left (5 d^2 e (f+7 g x)+15 d^3 g+d e^2 x (16 g x-3 f)-4 e^3 f x^2\right )+2 c e^4 (d g+e (f+2 g x))\right )}{\left (e^2-c^2 d^2\right )^3 (d+e x)^3}+\frac{b c^3 \left (2 c^4 d^3 (d g+3 e f)+c^2 d e^2 (9 e f-13 d g)-4 e^4 g\right ) \log \left (\sqrt{1-c^2 x^2} \sqrt{e^2-c^2 d^2}+c^2 d x+e\right )}{(e-c d)^3 (c d+e)^3 \sqrt{e^2-c^2 d^2}}+\frac{b c^3 \log (d+e x) \left (-2 c^4 d^3 (d g+3 e f)+c^2 d e^2 (13 d g-9 e f)+4 e^4 g\right )}{(e-c d)^3 (c d+e)^3 \sqrt{e^2-c^2 d^2}}-\frac{2 b \sin ^{-1}(c x) (d g+3 e f+4 e g x)}{(d+e x)^4}}{24 e^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.019, size = 1804, normalized size = 5. \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 (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{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 (4 \, b e g x + 3 \, b e f + b d g\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) +{\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 )} \int \frac{{\left (4 \, b c e g x + 3 \, b c e f + b c d g\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^{8} + 4 \, c^{4} d e^{5} x^{7} - 4 \, c^{2} d^{3} e^{3} x^{3} - c^{2} d^{4} e^{2} x^{2} +{\left (6 \, c^{4} d^{2} e^{4} - c^{2} e^{6}\right )} x^{6} + 4 \,{\left (c^{4} d^{3} e^{3} - c^{2} d e^{5}\right )} x^{5} +{\left (c^{4} d^{4} e^{2} - 6 \, c^{2} d^{2} e^{4}\right )} x^{4} -{\left (c^{2} e^{6} x^{6} + 4 \, c^{2} d e^{5} x^{5} - 4 \, d^{3} e^{3} x - d^{4} e^{2} +{\left (6 \, c^{2} d^{2} e^{4} - e^{6}\right )} x^{4} + 4 \,{\left (c^{2} d^{3} e^{3} - d e^{5}\right )} x^{3} +{\left (c^{2} d^{4} e^{2} - 6 \, d^{2} e^{4}\right )} x^{2}\right )}{\left (c x + 1\right )}{\left (c x - 1\right )}}\,{d x}}{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 )}} \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\right )}{\left (d + e x\right )^{5}}\, 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 (g x + f\right )}{\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (e x + d\right )}^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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