Optimal. Leaf size=87 \[ \frac {3 a^2 d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 \sqrt {c}}+\frac {1}{4} d x \left (a+c x^2\right )^{3/2}+\frac {3}{8} a d x \sqrt {a+c x^2}+\frac {e \left (a+c x^2\right )^{5/2}}{5 c} \]
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Rubi [A] time = 0.03, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {641, 195, 217, 206} \[ \frac {3 a^2 d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 \sqrt {c}}+\frac {1}{4} d x \left (a+c x^2\right )^{3/2}+\frac {3}{8} a d x \sqrt {a+c x^2}+\frac {e \left (a+c x^2\right )^{5/2}}{5 c} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 641
Rubi steps
\begin {align*} \int (d+e x) \left (a+c x^2\right )^{3/2} \, dx &=\frac {e \left (a+c x^2\right )^{5/2}}{5 c}+d \int \left (a+c x^2\right )^{3/2} \, dx\\ &=\frac {1}{4} d x \left (a+c x^2\right )^{3/2}+\frac {e \left (a+c x^2\right )^{5/2}}{5 c}+\frac {1}{4} (3 a d) \int \sqrt {a+c x^2} \, dx\\ &=\frac {3}{8} a d x \sqrt {a+c x^2}+\frac {1}{4} d x \left (a+c x^2\right )^{3/2}+\frac {e \left (a+c x^2\right )^{5/2}}{5 c}+\frac {1}{8} \left (3 a^2 d\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx\\ &=\frac {3}{8} a d x \sqrt {a+c x^2}+\frac {1}{4} d x \left (a+c x^2\right )^{3/2}+\frac {e \left (a+c x^2\right )^{5/2}}{5 c}+\frac {1}{8} \left (3 a^2 d\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )\\ &=\frac {3}{8} a d x \sqrt {a+c x^2}+\frac {1}{4} d x \left (a+c x^2\right )^{3/2}+\frac {e \left (a+c x^2\right )^{5/2}}{5 c}+\frac {3 a^2 d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 88, normalized size = 1.01 \[ \frac {\sqrt {a+c x^2} \left (8 a^2 e+a c x (25 d+16 e x)+2 c^2 x^3 (5 d+4 e x)\right )+15 a^2 \sqrt {c} d \log \left (\sqrt {c} \sqrt {a+c x^2}+c x\right )}{40 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 176, normalized size = 2.02 \[ \left [\frac {15 \, a^{2} \sqrt {c} d \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (8 \, c^{2} e x^{4} + 10 \, c^{2} d x^{3} + 16 \, a c e x^{2} + 25 \, a c d x + 8 \, a^{2} e\right )} \sqrt {c x^{2} + a}}{80 \, c}, -\frac {15 \, a^{2} \sqrt {-c} d \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (8 \, c^{2} e x^{4} + 10 \, c^{2} d x^{3} + 16 \, a c e x^{2} + 25 \, a c d x + 8 \, a^{2} e\right )} \sqrt {c x^{2} + a}}{40 \, c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 79, normalized size = 0.91 \[ -\frac {3 \, a^{2} d \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{8 \, \sqrt {c}} + \frac {1}{40} \, \sqrt {c x^{2} + a} {\left ({\left (25 \, a d + 2 \, {\left ({\left (4 \, c x e + 5 \, c d\right )} x + 8 \, a e\right )} x\right )} x + \frac {8 \, a^{2} e}{c}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 69, normalized size = 0.79 \[ \frac {3 a^{2} d \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{8 \sqrt {c}}+\frac {3 \sqrt {c \,x^{2}+a}\, a d x}{8}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} d x}{4}+\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} e}{5 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.41, size = 61, normalized size = 0.70 \[ \frac {1}{4} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} d x + \frac {3}{8} \, \sqrt {c x^{2} + a} a d x + \frac {3 \, a^{2} d \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {c}} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} e}{5 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.54, size = 54, normalized size = 0.62 \[ \frac {e\,{\left (c\,x^2+a\right )}^{5/2}}{5\,c}+\frac {d\,x\,{\left (c\,x^2+a\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {1}{2};\ \frac {3}{2};\ -\frac {c\,x^2}{a}\right )}{{\left (\frac {c\,x^2}{a}+1\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.03, size = 219, normalized size = 2.52 \[ \frac {a^{\frac {3}{2}} d x \sqrt {1 + \frac {c x^{2}}{a}}}{2} + \frac {a^{\frac {3}{2}} d x}{8 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 \sqrt {a} c d x^{3}}{8 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 a^{2} d \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{8 \sqrt {c}} + a e \left (\begin {cases} \frac {\sqrt {a} x^{2}}{2} & \text {for}\: c = 0 \\\frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases}\right ) + c e \left (\begin {cases} - \frac {2 a^{2} \sqrt {a + c x^{2}}}{15 c^{2}} + \frac {a x^{2} \sqrt {a + c x^{2}}}{15 c} + \frac {x^{4} \sqrt {a + c x^{2}}}{5} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + \frac {c^{2} d x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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