Optimal. Leaf size=183 \[ -\frac{a+b \sinh ^{-1}(c x)}{3 e (d+e x)^3}-\frac{b c^3 d \sqrt{c^2 x^2+1}}{2 \left (c^2 d^2+e^2\right )^2 (d+e x)}-\frac{b c \sqrt{c^2 x^2+1}}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac{b c^3 \left (2 c^2 d^2-e^2\right ) \tanh ^{-1}\left (\frac{e-c^2 d x}{\sqrt{c^2 x^2+1} \sqrt{c^2 d^2+e^2}}\right )}{6 e \left (c^2 d^2+e^2\right )^{5/2}} \]
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Rubi [A] time = 0.138058, antiderivative size = 183, 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 = {5801, 745, 807, 725, 206} \[ -\frac{a+b \sinh ^{-1}(c x)}{3 e (d+e x)^3}-\frac{b c^3 d \sqrt{c^2 x^2+1}}{2 \left (c^2 d^2+e^2\right )^2 (d+e x)}-\frac{b c \sqrt{c^2 x^2+1}}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac{b c^3 \left (2 c^2 d^2-e^2\right ) \tanh ^{-1}\left (\frac{e-c^2 d x}{\sqrt{c^2 x^2+1} \sqrt{c^2 d^2+e^2}}\right )}{6 e \left (c^2 d^2+e^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 5801
Rule 745
Rule 807
Rule 725
Rule 206
Rubi steps
\begin{align*} \int \frac{a+b \sinh ^{-1}(c x)}{(d+e x)^4} \, dx &=-\frac{a+b \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-1}(c x)}{3 e (d+e x)^3}-\frac{b c^3 \left (2 c^2 d^2-e^2\right ) \tanh ^{-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 = 0.388277, size = 205, normalized size = 1.12 \[ \frac{1}{6} \left (-\frac{2 a}{e (d+e x)^3}-\frac{b c \sqrt{c^2 x^2+1} \left (c^2 d (4 d+3 e x)+e^2\right )}{\left (c^2 d^2+e^2\right )^2 (d+e x)^2}+\frac{b c^3 \left (e^2-2 c^2 d^2\right ) \log \left (\sqrt{c^2 x^2+1} \sqrt{c^2 d^2+e^2}+c^2 (-d) x+e\right )}{e \left (c^2 d^2+e^2\right )^{5/2}}-\frac{b c^3 \left (e^2-2 c^2 d^2\right ) \log (d+e x)}{e \left (c^2 d^2+e^2\right )^{5/2}}-\frac{2 b \sinh ^{-1}(c x)}{e (d+e x)^3}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.012, size = 516, normalized size = 2.8 \begin{align*} -{\frac{{c}^{3}a}{3\, \left ( cex+cd \right ) ^{3}e}}-{\frac{{c}^{3}b{\it Arcsinh} \left ( cx \right ) }{3\, \left ( cex+cd \right ) ^{3}e}}-{\frac{{c}^{3}b}{6\,{e}^{2} \left ({c}^{2}{d}^{2}+{e}^{2} \right ) }\sqrt{ \left ( cx+{\frac{cd}{e}} \right ) ^{2}-2\,{\frac{cd}{e} \left ( cx+{\frac{cd}{e}} \right ) }+{\frac{{c}^{2}{d}^{2}+{e}^{2}}{{e}^{2}}}} \left ( cx+{\frac{cd}{e}} \right ) ^{-2}}-{\frac{b{c}^{4}d}{2\,e \left ({c}^{2}{d}^{2}+{e}^{2} \right ) ^{2}}\sqrt{ \left ( cx+{\frac{cd}{e}} \right ) ^{2}-2\,{\frac{cd}{e} \left ( cx+{\frac{cd}{e}} \right ) }+{\frac{{c}^{2}{d}^{2}+{e}^{2}}{{e}^{2}}}} \left ( cx+{\frac{cd}{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{cd}{e} \left ( cx+{\frac{cd}{e}} \right ) }+2\,\sqrt{{\frac{{c}^{2}{d}^{2}+{e}^{2}}{{e}^{2}}}}\sqrt{ \left ( cx+{\frac{cd}{e}} \right ) ^{2}-2\,{\frac{cd}{e} \left ( cx+{\frac{cd}{e}} \right ) }+{\frac{{c}^{2}{d}^{2}+{e}^{2}}{{e}^{2}}}} \right ) \left ( cx+{\frac{cd}{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{cd}{e} \left ( cx+{\frac{cd}{e}} \right ) }+2\,\sqrt{{\frac{{c}^{2}{d}^{2}+{e}^{2}}{{e}^{2}}}}\sqrt{ \left ( cx+{\frac{cd}{e}} \right ) ^{2}-2\,{\frac{cd}{e} \left ( cx+{\frac{cd}{e}} \right ) }+{\frac{{c}^{2}{d}^{2}+{e}^{2}}{{e}^{2}}}} \right ) \left ( cx+{\frac{cd}{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*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 7.69333, size = 1960, normalized size = 10.71 \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{asinh}{\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 \operatorname{arsinh}\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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