Optimal. Leaf size=119 \[ \frac {a \left (4 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{3/2}}+\frac {x \sqrt {a+c x^2} \left (4 c d^2-a e^2\right )}{8 c}+\frac {5 d e \left (a+c x^2\right )^{3/2}}{12 c}+\frac {e \left (a+c x^2\right )^{3/2} (d+e x)}{4 c} \]
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Rubi [A] time = 0.05, antiderivative size = 119, normalized size of antiderivative = 1.00, 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 = {743, 641, 195, 217, 206} \[ \frac {a \left (4 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{3/2}}+\frac {x \sqrt {a+c x^2} \left (4 c d^2-a e^2\right )}{8 c}+\frac {5 d e \left (a+c x^2\right )^{3/2}}{12 c}+\frac {e \left (a+c x^2\right )^{3/2} (d+e x)}{4 c} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 641
Rule 743
Rubi steps
\begin {align*} \int (d+e x)^2 \sqrt {a+c x^2} \, dx &=\frac {e (d+e x) \left (a+c x^2\right )^{3/2}}{4 c}+\frac {\int \left (4 c d^2-a e^2+5 c d e x\right ) \sqrt {a+c x^2} \, dx}{4 c}\\ &=\frac {5 d e \left (a+c x^2\right )^{3/2}}{12 c}+\frac {e (d+e x) \left (a+c x^2\right )^{3/2}}{4 c}+\frac {\left (4 c d^2-a e^2\right ) \int \sqrt {a+c x^2} \, dx}{4 c}\\ &=\frac {\left (4 c d^2-a e^2\right ) x \sqrt {a+c x^2}}{8 c}+\frac {5 d e \left (a+c x^2\right )^{3/2}}{12 c}+\frac {e (d+e x) \left (a+c x^2\right )^{3/2}}{4 c}+\frac {\left (a \left (4 c d^2-a e^2\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{8 c}\\ &=\frac {\left (4 c d^2-a e^2\right ) x \sqrt {a+c x^2}}{8 c}+\frac {5 d e \left (a+c x^2\right )^{3/2}}{12 c}+\frac {e (d+e x) \left (a+c x^2\right )^{3/2}}{4 c}+\frac {\left (a \left (4 c d^2-a e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{8 c}\\ &=\frac {\left (4 c d^2-a e^2\right ) x \sqrt {a+c x^2}}{8 c}+\frac {5 d e \left (a+c x^2\right )^{3/2}}{12 c}+\frac {e (d+e x) \left (a+c x^2\right )^{3/2}}{4 c}+\frac {a \left (4 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 99, normalized size = 0.83 \[ \frac {\sqrt {c} \sqrt {a+c x^2} \left (a e (16 d+3 e x)+2 c x \left (6 d^2+8 d e x+3 e^2 x^2\right )\right )-3 a \left (a e^2-4 c d^2\right ) \log \left (\sqrt {c} \sqrt {a+c x^2}+c x\right )}{24 c^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 214, normalized size = 1.80 \[ \left [-\frac {3 \, {\left (4 \, a c d^{2} - a^{2} e^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - 2 \, {\left (6 \, c^{2} e^{2} x^{3} + 16 \, c^{2} d e x^{2} + 16 \, a c d e + 3 \, {\left (4 \, c^{2} d^{2} + a c e^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{48 \, c^{2}}, -\frac {3 \, {\left (4 \, a c d^{2} - a^{2} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (6 \, c^{2} e^{2} x^{3} + 16 \, c^{2} d e x^{2} + 16 \, a c d e + 3 \, {\left (4 \, c^{2} d^{2} + a c e^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{24 \, c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 96, normalized size = 0.81 \[ \frac {1}{24} \, \sqrt {c x^{2} + a} {\left ({\left (2 \, {\left (3 \, x e^{2} + 8 \, d e\right )} x + \frac {3 \, {\left (4 \, c^{2} d^{2} + a c e^{2}\right )}}{c^{2}}\right )} x + \frac {16 \, a d e}{c}\right )} - \frac {{\left (4 \, a c d^{2} - a^{2} e^{2}\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{8 \, c^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 122, normalized size = 1.03 \[ -\frac {a^{2} e^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{8 c^{\frac {3}{2}}}+\frac {a \,d^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}-\frac {\sqrt {c \,x^{2}+a}\, a \,e^{2} x}{8 c}+\frac {\sqrt {c \,x^{2}+a}\, d^{2} x}{2}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} e^{2} x}{4 c}+\frac {2 \left (c \,x^{2}+a \right )^{\frac {3}{2}} d e}{3 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 107, normalized size = 0.90 \[ \frac {1}{2} \, \sqrt {c x^{2} + a} d^{2} x + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} e^{2} x}{4 \, c} - \frac {\sqrt {c x^{2} + a} a e^{2} x}{8 \, c} + \frac {a d^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {c}} - \frac {a^{2} e^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{8 \, c^{\frac {3}{2}}} + \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} d e}{3 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.59, size = 184, normalized size = 1.55 \[ \frac {a^{\frac {3}{2}} e^{2} x}{8 c \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {\sqrt {a} d^{2} x \sqrt {1 + \frac {c x^{2}}{a}}}{2} + \frac {3 \sqrt {a} e^{2} x^{3}}{8 \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {a^{2} e^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{8 c^{\frac {3}{2}}} + \frac {a d^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{2 \sqrt {c}} + 2 d e \left (\begin {cases} \frac {\sqrt {a} x^{2}}{2} & \text {for}\: c = 0 \\\frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases}\right ) + \frac {c e^{2} x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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