Optimal. Leaf size=198 \[ -\frac{c^2 d \left (2 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{7/2}}-\frac{c e \sqrt{a+c x^2} \left (11 c d^2-4 a e^2\right )}{6 (d+e x) \left (a e^2+c d^2\right )^3}-\frac{5 c d e \sqrt{a+c x^2}}{6 (d+e x)^2 \left (a e^2+c d^2\right )^2}-\frac{e \sqrt{a+c x^2}}{3 (d+e x)^3 \left (a e^2+c d^2\right )} \]
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Rubi [A] time = 0.16382, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {745, 835, 807, 725, 206} \[ -\frac{c^2 d \left (2 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{7/2}}-\frac{c e \sqrt{a+c x^2} \left (11 c d^2-4 a e^2\right )}{6 (d+e x) \left (a e^2+c d^2\right )^3}-\frac{5 c d e \sqrt{a+c x^2}}{6 (d+e x)^2 \left (a e^2+c d^2\right )^2}-\frac{e \sqrt{a+c x^2}}{3 (d+e x)^3 \left (a e^2+c d^2\right )} \]
Antiderivative was successfully verified.
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Rule 745
Rule 835
Rule 807
Rule 725
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^4 \sqrt{a+c x^2}} \, dx &=-\frac{e \sqrt{a+c x^2}}{3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac{c \int \frac{-3 d+2 e x}{(d+e x)^3 \sqrt{a+c x^2}} \, dx}{3 \left (c d^2+a e^2\right )}\\ &=-\frac{e \sqrt{a+c x^2}}{3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac{5 c d e \sqrt{a+c x^2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^2}+\frac{c \int \frac{2 \left (3 c d^2-2 a e^2\right )-5 c d e x}{(d+e x)^2 \sqrt{a+c x^2}} \, dx}{6 \left (c d^2+a e^2\right )^2}\\ &=-\frac{e \sqrt{a+c x^2}}{3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac{5 c d e \sqrt{a+c x^2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac{c e \left (11 c d^2-4 a e^2\right ) \sqrt{a+c x^2}}{6 \left (c d^2+a e^2\right )^3 (d+e x)}+\frac{\left (c^2 d \left (2 c d^2-3 a e^2\right )\right ) \int \frac{1}{(d+e x) \sqrt{a+c x^2}} \, dx}{2 \left (c d^2+a e^2\right )^3}\\ &=-\frac{e \sqrt{a+c x^2}}{3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac{5 c d e \sqrt{a+c x^2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac{c e \left (11 c d^2-4 a e^2\right ) \sqrt{a+c x^2}}{6 \left (c d^2+a e^2\right )^3 (d+e x)}-\frac{\left (c^2 d \left (2 c d^2-3 a e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c d^2+a e^2-x^2} \, dx,x,\frac{a e-c d x}{\sqrt{a+c x^2}}\right )}{2 \left (c d^2+a e^2\right )^3}\\ &=-\frac{e \sqrt{a+c x^2}}{3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac{5 c d e \sqrt{a+c x^2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac{c e \left (11 c d^2-4 a e^2\right ) \sqrt{a+c x^2}}{6 \left (c d^2+a e^2\right )^3 (d+e x)}-\frac{c^2 d \left (2 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{c d^2+a e^2} \sqrt{a+c x^2}}\right )}{2 \left (c d^2+a e^2\right )^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.178611, size = 209, normalized size = 1.06 \[ \frac{-3 c^2 d (d+e x)^3 \left (2 c d^2-3 a e^2\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )+3 c^2 d (d+e x)^3 \left (2 c d^2-3 a e^2\right ) \log (d+e x)-e \sqrt{a+c x^2} \sqrt{a e^2+c d^2} \left (5 c d (d+e x) \left (a e^2+c d^2\right )+c (d+e x)^2 \left (11 c d^2-4 a e^2\right )+2 \left (a e^2+c d^2\right )^2\right )}{6 (d+e x)^3 \left (a e^2+c d^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.196, size = 573, normalized size = 2.9 \begin{align*} -{\frac{1}{3\,{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) }\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \left ({\frac{d}{e}}+x \right ) ^{-3}}-{\frac{5\,cd}{6\,e \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \left ({\frac{d}{e}}+x \right ) ^{-2}}-{\frac{5\,{c}^{2}{d}^{2}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \left ({\frac{d}{e}}+x \right ) ^{-1}}-{\frac{5\,{c}^{3}{d}^{3}}{2\,e \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}\ln \left ({ \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{3\,{c}^{2}d}{2\,e \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\ln \left ({ \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{2\,c}{3\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \left ({\frac{d}{e}}+x \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 8.50196, size = 2290, normalized size = 11.57 \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{1}{\sqrt{a + c x^{2}} \left (d + e x\right )^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.37986, size = 780, normalized size = 3.94 \begin{align*} \frac{1}{3} \, c^{\frac{3}{2}}{\left (\frac{3 \,{\left (2 \, c^{\frac{3}{2}} d^{3} - 3 \, a \sqrt{c} d e^{2}\right )} \arctan \left (-\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} - a e^{2}}}\right )}{{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} \sqrt{-c d^{2} - a e^{2}}} - \frac{30 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} c^{2} d^{4} e + 44 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} c^{\frac{5}{2}} d^{5} + 6 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} c^{\frac{3}{2}} d^{3} e^{2} - 102 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} a c^{2} d^{4} e - 82 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} a c^{\frac{3}{2}} d^{3} e^{2} - 45 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} a c d^{2} e^{3} - 9 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} a \sqrt{c} d e^{4} + 60 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} a^{2} c^{\frac{3}{2}} d^{3} e^{2} + 36 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} a^{2} c d^{2} e^{3} + 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} a^{2} \sqrt{c} d e^{4} - 11 \, a^{3} c d^{2} e^{3} - 15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} a^{3} \sqrt{c} d e^{4} - 12 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} a^{3} e^{5} + 4 \, a^{4} e^{5}}{{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} e + 2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} \sqrt{c} d - a e\right )}^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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