Optimal. Leaf size=207 \[ \frac{x \sqrt{a+c x^2} \left (a^2 e^4-12 a c d^2 e^2+8 c^2 d^4\right )}{16 c^2}+\frac{a \left (a^2 e^4-12 a c d^2 e^2+8 c^2 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 c^{5/2}}+\frac{e \left (a+c x^2\right )^{3/2} \left (3 e x \left (16 c d^2-5 a e^2\right )+8 d \left (13 c d^2-8 a e^2\right )\right )}{120 c^2}+\frac{e \left (a+c x^2\right )^{3/2} (d+e x)^3}{6 c}+\frac{3 d e \left (a+c x^2\right )^{3/2} (d+e x)^2}{10 c} \]
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Rubi [A] time = 0.208813, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {743, 833, 780, 195, 217, 206} \[ \frac{x \sqrt{a+c x^2} \left (a^2 e^4-12 a c d^2 e^2+8 c^2 d^4\right )}{16 c^2}+\frac{a \left (a^2 e^4-12 a c d^2 e^2+8 c^2 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 c^{5/2}}+\frac{e \left (a+c x^2\right )^{3/2} \left (3 e x \left (16 c d^2-5 a e^2\right )+8 d \left (13 c d^2-8 a e^2\right )\right )}{120 c^2}+\frac{e \left (a+c x^2\right )^{3/2} (d+e x)^3}{6 c}+\frac{3 d e \left (a+c x^2\right )^{3/2} (d+e x)^2}{10 c} \]
Antiderivative was successfully verified.
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Rule 743
Rule 833
Rule 780
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int (d+e x)^4 \sqrt{a+c x^2} \, dx &=\frac{e (d+e x)^3 \left (a+c x^2\right )^{3/2}}{6 c}+\frac{\int (d+e x)^2 \left (3 \left (2 c d^2-a e^2\right )+9 c d e x\right ) \sqrt{a+c x^2} \, dx}{6 c}\\ &=\frac{3 d e (d+e x)^2 \left (a+c x^2\right )^{3/2}}{10 c}+\frac{e (d+e x)^3 \left (a+c x^2\right )^{3/2}}{6 c}+\frac{\int (d+e x) \left (3 c d \left (10 c d^2-11 a e^2\right )+3 c e \left (16 c d^2-5 a e^2\right ) x\right ) \sqrt{a+c x^2} \, dx}{30 c^2}\\ &=\frac{3 d e (d+e x)^2 \left (a+c x^2\right )^{3/2}}{10 c}+\frac{e (d+e x)^3 \left (a+c x^2\right )^{3/2}}{6 c}+\frac{e \left (8 d \left (13 c d^2-8 a e^2\right )+3 e \left (16 c d^2-5 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{120 c^2}+\frac{\left (8 c^2 d^4-12 a c d^2 e^2+a^2 e^4\right ) \int \sqrt{a+c x^2} \, dx}{8 c^2}\\ &=\frac{\left (8 c^2 d^4-12 a c d^2 e^2+a^2 e^4\right ) x \sqrt{a+c x^2}}{16 c^2}+\frac{3 d e (d+e x)^2 \left (a+c x^2\right )^{3/2}}{10 c}+\frac{e (d+e x)^3 \left (a+c x^2\right )^{3/2}}{6 c}+\frac{e \left (8 d \left (13 c d^2-8 a e^2\right )+3 e \left (16 c d^2-5 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{120 c^2}+\frac{\left (a \left (8 c^2 d^4-12 a c d^2 e^2+a^2 e^4\right )\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{16 c^2}\\ &=\frac{\left (8 c^2 d^4-12 a c d^2 e^2+a^2 e^4\right ) x \sqrt{a+c x^2}}{16 c^2}+\frac{3 d e (d+e x)^2 \left (a+c x^2\right )^{3/2}}{10 c}+\frac{e (d+e x)^3 \left (a+c x^2\right )^{3/2}}{6 c}+\frac{e \left (8 d \left (13 c d^2-8 a e^2\right )+3 e \left (16 c d^2-5 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{120 c^2}+\frac{\left (a \left (8 c^2 d^4-12 a c d^2 e^2+a^2 e^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{16 c^2}\\ &=\frac{\left (8 c^2 d^4-12 a c d^2 e^2+a^2 e^4\right ) x \sqrt{a+c x^2}}{16 c^2}+\frac{3 d e (d+e x)^2 \left (a+c x^2\right )^{3/2}}{10 c}+\frac{e (d+e x)^3 \left (a+c x^2\right )^{3/2}}{6 c}+\frac{e \left (8 d \left (13 c d^2-8 a e^2\right )+3 e \left (16 c d^2-5 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{120 c^2}+\frac{a \left (8 c^2 d^4-12 a c d^2 e^2+a^2 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.133395, size = 177, normalized size = 0.86 \[ \frac{\sqrt{c} \sqrt{a+c x^2} \left (-a^2 e^3 (128 d+15 e x)+2 a c e \left (90 d^2 e x+160 d^3+32 d e^2 x^2+5 e^3 x^3\right )+8 c^2 x \left (45 d^2 e^2 x^2+40 d^3 e x+15 d^4+24 d e^3 x^3+5 e^4 x^4\right )\right )+15 a \left (a^2 e^4-12 a c d^2 e^2+8 c^2 d^4\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{240 c^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 260, normalized size = 1.3 \begin{align*}{\frac{{e}^{4}{x}^{3}}{6\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{{e}^{4}ax}{8\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{2}{e}^{4}x}{16\,{c}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{{e}^{4}{a}^{3}}{16}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{4\,d{e}^{3}{x}^{2}}{5\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{8\,d{e}^{3}a}{15\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{d}^{2}{e}^{2}x}{2\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{d}^{2}{e}^{2}ax}{4\,c}\sqrt{c{x}^{2}+a}}-{\frac{3\,{d}^{2}{e}^{2}{a}^{2}}{4}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{4\,{d}^{3}e}{3\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{{d}^{4}x}{2}\sqrt{c{x}^{2}+a}}+{\frac{{d}^{4}a}{2}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}} \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 [A] time = 2.71369, size = 887, normalized size = 4.29 \begin{align*} \left [\frac{15 \,{\left (8 \, a c^{2} d^{4} - 12 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} \sqrt{c} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 2 \,{\left (40 \, c^{3} e^{4} x^{5} + 192 \, c^{3} d e^{3} x^{4} + 320 \, a c^{2} d^{3} e - 128 \, a^{2} c d e^{3} + 10 \,{\left (36 \, c^{3} d^{2} e^{2} + a c^{2} e^{4}\right )} x^{3} + 64 \,{\left (5 \, c^{3} d^{3} e + a c^{2} d e^{3}\right )} x^{2} + 15 \,{\left (8 \, c^{3} d^{4} + 12 \, a c^{2} d^{2} e^{2} - a^{2} c e^{4}\right )} x\right )} \sqrt{c x^{2} + a}}{480 \, c^{3}}, -\frac{15 \,{\left (8 \, a c^{2} d^{4} - 12 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) -{\left (40 \, c^{3} e^{4} x^{5} + 192 \, c^{3} d e^{3} x^{4} + 320 \, a c^{2} d^{3} e - 128 \, a^{2} c d e^{3} + 10 \,{\left (36 \, c^{3} d^{2} e^{2} + a c^{2} e^{4}\right )} x^{3} + 64 \,{\left (5 \, c^{3} d^{3} e + a c^{2} d e^{3}\right )} x^{2} + 15 \,{\left (8 \, c^{3} d^{4} + 12 \, a c^{2} d^{2} e^{2} - a^{2} c e^{4}\right )} x\right )} \sqrt{c x^{2} + a}}{240 \, c^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 11.85, size = 411, normalized size = 1.99 \begin{align*} - \frac{a^{\frac{5}{2}} e^{4} x}{16 c^{2} \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 a^{\frac{3}{2}} d^{2} e^{2} x}{4 c \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{a^{\frac{3}{2}} e^{4} x^{3}}{48 c \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{\sqrt{a} d^{4} x \sqrt{1 + \frac{c x^{2}}{a}}}{2} + \frac{9 \sqrt{a} d^{2} e^{2} x^{3}}{4 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{5 \sqrt{a} e^{4} x^{5}}{24 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{a^{3} e^{4} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{16 c^{\frac{5}{2}}} - \frac{3 a^{2} d^{2} e^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{4 c^{\frac{3}{2}}} + \frac{a d^{4} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 \sqrt{c}} + 4 d^{3} e \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) + 4 d e^{3} \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + c x^{2}}}{15 c^{2}} + \frac{a x^{2} \sqrt{a + c x^{2}}}{15 c} + \frac{x^{4} \sqrt{a + c x^{2}}}{5} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) + \frac{3 c d^{2} e^{2} x^{5}}{2 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{c e^{4} x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36844, size = 266, normalized size = 1.29 \begin{align*} \frac{1}{240} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left ({\left (4 \,{\left (5 \, x e^{4} + 24 \, d e^{3}\right )} x + \frac{5 \,{\left (36 \, c^{4} d^{2} e^{2} + a c^{3} e^{4}\right )}}{c^{4}}\right )} x + \frac{32 \,{\left (5 \, c^{4} d^{3} e + a c^{3} d e^{3}\right )}}{c^{4}}\right )} x + \frac{15 \,{\left (8 \, c^{4} d^{4} + 12 \, a c^{3} d^{2} e^{2} - a^{2} c^{2} e^{4}\right )}}{c^{4}}\right )} x + \frac{64 \,{\left (5 \, a c^{3} d^{3} e - 2 \, a^{2} c^{2} d e^{3}\right )}}{c^{4}}\right )} - \frac{{\left (8 \, a c^{2} d^{4} - 12 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{16 \, c^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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