Optimal. Leaf size=107 \[ \frac{5}{16} a^2 d x \sqrt{a+c x^2}+\frac{5 a^3 d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 \sqrt{c}}+\frac{1}{6} d x \left (a+c x^2\right )^{5/2}+\frac{5}{24} a d x \left (a+c x^2\right )^{3/2}+\frac{e \left (a+c x^2\right )^{7/2}}{7 c} \]
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Rubi [A] time = 0.0337156, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {641, 195, 217, 206} \[ \frac{5}{16} a^2 d x \sqrt{a+c x^2}+\frac{5 a^3 d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 \sqrt{c}}+\frac{1}{6} d x \left (a+c x^2\right )^{5/2}+\frac{5}{24} a d x \left (a+c x^2\right )^{3/2}+\frac{e \left (a+c x^2\right )^{7/2}}{7 c} \]
Antiderivative was successfully verified.
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Rule 641
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int (d+e x) \left (a+c x^2\right )^{5/2} \, dx &=\frac{e \left (a+c x^2\right )^{7/2}}{7 c}+d \int \left (a+c x^2\right )^{5/2} \, dx\\ &=\frac{1}{6} d x \left (a+c x^2\right )^{5/2}+\frac{e \left (a+c x^2\right )^{7/2}}{7 c}+\frac{1}{6} (5 a d) \int \left (a+c x^2\right )^{3/2} \, dx\\ &=\frac{5}{24} a d x \left (a+c x^2\right )^{3/2}+\frac{1}{6} d x \left (a+c x^2\right )^{5/2}+\frac{e \left (a+c x^2\right )^{7/2}}{7 c}+\frac{1}{8} \left (5 a^2 d\right ) \int \sqrt{a+c x^2} \, dx\\ &=\frac{5}{16} a^2 d x \sqrt{a+c x^2}+\frac{5}{24} a d x \left (a+c x^2\right )^{3/2}+\frac{1}{6} d x \left (a+c x^2\right )^{5/2}+\frac{e \left (a+c x^2\right )^{7/2}}{7 c}+\frac{1}{16} \left (5 a^3 d\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx\\ &=\frac{5}{16} a^2 d x \sqrt{a+c x^2}+\frac{5}{24} a d x \left (a+c x^2\right )^{3/2}+\frac{1}{6} d x \left (a+c x^2\right )^{5/2}+\frac{e \left (a+c x^2\right )^{7/2}}{7 c}+\frac{1}{16} \left (5 a^3 d\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )\\ &=\frac{5}{16} a^2 d x \sqrt{a+c x^2}+\frac{5}{24} a d x \left (a+c x^2\right )^{3/2}+\frac{1}{6} d x \left (a+c x^2\right )^{5/2}+\frac{e \left (a+c x^2\right )^{7/2}}{7 c}+\frac{5 a^3 d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.0702851, size = 108, normalized size = 1.01 \[ \frac{\sqrt{a+c x^2} \left (3 a^2 c x (77 d+48 e x)+48 a^3 e+2 a c^2 x^3 (91 d+72 e x)+8 c^3 x^5 (7 d+6 e x)\right )+105 a^3 \sqrt{c} d \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{336 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 85, normalized size = 0.8 \begin{align*}{\frac{e}{7\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{dx}{6} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,adx}{24} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}dx}{16}\sqrt{c{x}^{2}+a}}+{\frac{5\,d{a}^{3}}{16}\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 = 1.67628, size = 549, normalized size = 5.13 \begin{align*} \left [\frac{105 \, a^{3} \sqrt{c} d \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 2 \,{\left (48 \, c^{3} e x^{6} + 56 \, c^{3} d x^{5} + 144 \, a c^{2} e x^{4} + 182 \, a c^{2} d x^{3} + 144 \, a^{2} c e x^{2} + 231 \, a^{2} c d x + 48 \, a^{3} e\right )} \sqrt{c x^{2} + a}}{672 \, c}, -\frac{105 \, a^{3} \sqrt{-c} d \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) -{\left (48 \, c^{3} e x^{6} + 56 \, c^{3} d x^{5} + 144 \, a c^{2} e x^{4} + 182 \, a c^{2} d x^{3} + 144 \, a^{2} c e x^{2} + 231 \, a^{2} c d x + 48 \, a^{3} e\right )} \sqrt{c x^{2} + a}}{336 \, c}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 17.0691, size = 348, normalized size = 3.25 \begin{align*} \frac{a^{\frac{5}{2}} d x \sqrt{1 + \frac{c x^{2}}{a}}}{2} + \frac{3 a^{\frac{5}{2}} d x}{16 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{35 a^{\frac{3}{2}} c d x^{3}}{48 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{17 \sqrt{a} c^{2} d x^{5}}{24 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{5 a^{3} d \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{16 \sqrt{c}} + a^{2} 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 ) + 2 a c e \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 ) + c^{2} e \left (\begin{cases} \frac{8 a^{3} \sqrt{a + c x^{2}}}{105 c^{3}} - \frac{4 a^{2} x^{2} \sqrt{a + c x^{2}}}{105 c^{2}} + \frac{a x^{4} \sqrt{a + c x^{2}}}{35 c} + \frac{x^{6} \sqrt{a + c x^{2}}}{7} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{6}}{6} & \text{otherwise} \end{cases}\right ) + \frac{c^{3} d 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 = 1.30598, size = 142, normalized size = 1.33 \begin{align*} -\frac{5 \, a^{3} d \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{16 \, \sqrt{c}} + \frac{1}{336} \, \sqrt{c x^{2} + a}{\left (\frac{48 \, a^{3} e}{c} +{\left (231 \, a^{2} d + 2 \,{\left (72 \, a^{2} e +{\left (91 \, a c d + 4 \,{\left (18 \, a c e +{\left (6 \, c^{2} x e + 7 \, c^{2} d\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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