Optimal. Leaf size=191 \[ -\frac{a+b \sin ^{-1}(c x)}{3 e (d+e x)^3}+\frac{b c^3 d \sqrt{1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac{b c \sqrt{1-c^2 x^2}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac{b c^3 \left (2 c^2 d^2+e^2\right ) \tan ^{-1}\left (\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right )}{6 e \left (c^2 d^2-e^2\right )^{5/2}} \]
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Rubi [A] time = 0.140214, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {4743, 745, 807, 725, 204} \[ -\frac{a+b \sin ^{-1}(c x)}{3 e (d+e x)^3}+\frac{b c^3 d \sqrt{1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac{b c \sqrt{1-c^2 x^2}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac{b c^3 \left (2 c^2 d^2+e^2\right ) \tan ^{-1}\left (\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right )}{6 e \left (c^2 d^2-e^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 4743
Rule 745
Rule 807
Rule 725
Rule 204
Rubi steps
\begin{align*} \int \frac{a+b \sin ^{-1}(c x)}{(d+e x)^4} \, dx &=-\frac{a+b \sin ^{-1}(c x)}{3 e (d+e x)^3}+\frac{(b c) \int \frac{1}{(d+e x)^3 \sqrt{1-c^2 x^2}} \, dx}{3 e}\\ &=\frac{b c \sqrt{1-c^2 x^2}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac{a+b \sin ^{-1}(c x)}{3 e (d+e x)^3}-\frac{\left (b c^3\right ) \int \frac{-2 d+e x}{(d+e x)^2 \sqrt{1-c^2 x^2}} \, dx}{6 e \left (c^2 d^2-e^2\right )}\\ &=\frac{b c \sqrt{1-c^2 x^2}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac{b c^3 d \sqrt{1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac{a+b \sin ^{-1}(c x)}{3 e (d+e x)^3}+\frac{\left (b c^3 \left (2 c^2 d^2+e^2\right )\right ) \int \frac{1}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{6 e \left (c^2 d^2-e^2\right )^2}\\ &=\frac{b c \sqrt{1-c^2 x^2}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac{b c^3 d \sqrt{1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac{a+b \sin ^{-1}(c x)}{3 e (d+e x)^3}-\frac{\left (b c^3 \left (2 c^2 d^2+e^2\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 \left (c^2 d^2-e^2\right )^2}\\ &=\frac{b c \sqrt{1-c^2 x^2}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac{b c^3 d \sqrt{1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac{a+b \sin ^{-1}(c x)}{3 e (d+e x)^3}+\frac{b c^3 \left (2 c^2 d^2+e^2\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 \left (c^2 d^2-e^2\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.521047, size = 241, normalized size = 1.26 \[ \frac{1}{6} \left (-\frac{2 a}{e (d+e x)^3}+\frac{b \sqrt{1-c^2 x^2} \left (c^3 d (4 d+3 e x)-c e^2\right )}{\left (e^2-c^2 d^2\right )^2 (d+e x)^2}-\frac{b c^3 \left (2 c^2 d^2+e^2\right ) \log \left (\sqrt{1-c^2 x^2} \sqrt{e^2-c^2 d^2}+c^2 d x+e\right )}{e (e-c d)^2 (c d+e)^2 \sqrt{e^2-c^2 d^2}}+\frac{b c^3 \left (2 c^2 d^2+e^2\right ) \log (d+e x)}{e (e-c d)^2 (c d+e)^2 \sqrt{e^2-c^2 d^2}}-\frac{2 b \sin ^{-1}(c x)}{e (d+e x)^3}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.008, size = 560, normalized size = 2.9 \begin{align*} -{\frac{a{c}^{3}}{3\, \left ( ecx+dc \right ) ^{3}e}}-{\frac{{c}^{3}b\arcsin \left ( cx \right ) }{3\, \left ( ecx+dc \right ) ^{3}e}}+{\frac{{c}^{3}b}{6\,{e}^{2} \left ({c}^{2}{d}^{2}-{e}^{2} \right ) }\sqrt{- \left ( cx+{\frac{dc}{e}} \right ) ^{2}+2\,{\frac{dc}{e} \left ( cx+{\frac{dc}{e}} \right ) }-{\frac{{c}^{2}{d}^{2}-{e}^{2}}{{e}^{2}}}} \left ( cx+{\frac{dc}{e}} \right ) ^{-2}}+{\frac{b{c}^{4}d}{2\,e \left ({c}^{2}{d}^{2}-{e}^{2} \right ) ^{2}}\sqrt{- \left ( cx+{\frac{dc}{e}} \right ) ^{2}+2\,{\frac{dc}{e} \left ( cx+{\frac{dc}{e}} \right ) }-{\frac{{c}^{2}{d}^{2}-{e}^{2}}{{e}^{2}}}} \left ( cx+{\frac{dc}{e}} \right ) ^{-1}}-{\frac{{c}^{5}b{d}^{2}}{2\,{e}^{2} \left ({c}^{2}{d}^{2}-{e}^{2} \right ) ^{2}}\ln \left ({ \left ( -2\,{\frac{{c}^{2}{d}^{2}-{e}^{2}}{{e}^{2}}}+2\,{\frac{dc}{e} \left ( cx+{\frac{dc}{e}} \right ) }+2\,\sqrt{-{\frac{{c}^{2}{d}^{2}-{e}^{2}}{{e}^{2}}}}\sqrt{- \left ( cx+{\frac{dc}{e}} \right ) ^{2}+2\,{\frac{dc}{e} \left ( cx+{\frac{dc}{e}} \right ) }-{\frac{{c}^{2}{d}^{2}-{e}^{2}}{{e}^{2}}}} \right ) \left ( cx+{\frac{dc}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{{c}^{2}{d}^{2}-{e}^{2}}{{e}^{2}}}}}}}+{\frac{{c}^{3}b}{6\,{e}^{2} \left ({c}^{2}{d}^{2}-{e}^{2} \right ) }\ln \left ({ \left ( -2\,{\frac{{c}^{2}{d}^{2}-{e}^{2}}{{e}^{2}}}+2\,{\frac{dc}{e} \left ( cx+{\frac{dc}{e}} \right ) }+2\,\sqrt{-{\frac{{c}^{2}{d}^{2}-{e}^{2}}{{e}^{2}}}}\sqrt{- \left ( cx+{\frac{dc}{e}} \right ) ^{2}+2\,{\frac{dc}{e} \left ( cx+{\frac{dc}{e}} \right ) }-{\frac{{c}^{2}{d}^{2}-{e}^{2}}{{e}^{2}}}} \right ) \left ( cx+{\frac{dc}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{{c}^{2}{d}^{2}-{e}^{2}}{{e}^{2}}}}}}} \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 ({\left (c e^{4} x^{3} + 3 \, c d e^{3} x^{2} + 3 \, c d^{2} e^{2} x + c d^{3} e\right )} \int \frac{e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (-c x + 1\right )\right )}}{c^{4} e^{4} x^{7} + 3 \, c^{4} d e^{3} x^{6} - 3 \, c^{2} d^{2} e^{2} x^{3} - c^{2} d^{3} e x^{2} +{\left (3 \, c^{4} d^{2} e^{2} - c^{2} e^{4}\right )} x^{5} +{\left (c^{4} d^{3} e - 3 \, c^{2} d e^{3}\right )} x^{4} -{\left (c^{2} e^{4} x^{5} + 3 \, c^{2} d e^{3} x^{4} - 3 \, d^{2} e^{2} x - d^{3} e +{\left (3 \, c^{2} d^{2} e^{2} - e^{4}\right )} x^{3} +{\left (c^{2} d^{3} e - 3 \, d e^{3}\right )} x^{2}\right )}{\left (c x + 1\right )}{\left (c x - 1\right )}}\,{d x} + \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )\right )} b}{3 \,{\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} - \frac{a}{3 \,{\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 12.2468, size = 2233, normalized size = 11.69 \begin{align*} \text{result too large to display} \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{a + b \operatorname{asin}{\left (c x \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{b \arcsin \left (c x\right ) + a}{{\left (e x + d\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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