Integrand size = 19, antiderivative size = 110 \[ \int \frac {(d+e x)^3}{\sqrt {a+c x^2}} \, dx=\frac {e (d+e x)^2 \sqrt {a+c x^2}}{3 c}+\frac {e \left (4 \left (4 c d^2-a e^2\right )+5 c d e x\right ) \sqrt {a+c x^2}}{6 c^2}+\frac {d \left (2 c d^2-3 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{3/2}} \]
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Time = 0.05 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {757, 794, 223, 212} \[ \int \frac {(d+e x)^3}{\sqrt {a+c x^2}} \, dx=\frac {d \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (2 c d^2-3 a e^2\right )}{2 c^{3/2}}+\frac {e \sqrt {a+c x^2} \left (4 \left (4 c d^2-a e^2\right )+5 c d e x\right )}{6 c^2}+\frac {e \sqrt {a+c x^2} (d+e x)^2}{3 c} \]
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Rule 212
Rule 223
Rule 757
Rule 794
Rubi steps \begin{align*} \text {integral}& = \frac {e (d+e x)^2 \sqrt {a+c x^2}}{3 c}+\frac {\int \frac {(d+e x) \left (3 c d^2-2 a e^2+5 c d e x\right )}{\sqrt {a+c x^2}} \, dx}{3 c} \\ & = \frac {e (d+e x)^2 \sqrt {a+c x^2}}{3 c}+\frac {e \left (4 \left (4 c d^2-a e^2\right )+5 c d e x\right ) \sqrt {a+c x^2}}{6 c^2}+\frac {\left (d \left (2 c d^2-3 a e^2\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{2 c} \\ & = \frac {e (d+e x)^2 \sqrt {a+c x^2}}{3 c}+\frac {e \left (4 \left (4 c d^2-a e^2\right )+5 c d e x\right ) \sqrt {a+c x^2}}{6 c^2}+\frac {\left (d \left (2 c d^2-3 a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{2 c} \\ & = \frac {e (d+e x)^2 \sqrt {a+c x^2}}{3 c}+\frac {e \left (4 \left (4 c d^2-a e^2\right )+5 c d e x\right ) \sqrt {a+c x^2}}{6 c^2}+\frac {d \left (2 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{3/2}} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.83 \[ \int \frac {(d+e x)^3}{\sqrt {a+c x^2}} \, dx=\frac {e \sqrt {a+c x^2} \left (-4 a e^2+c \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+3 \sqrt {c} d \left (-2 c d^2+3 a e^2\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{6 c^2} \]
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Time = 2.14 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.73
method | result | size |
risch | \(-\frac {e \left (-2 c \,x^{2} e^{2}-9 x c d e +4 e^{2} a -18 c \,d^{2}\right ) \sqrt {c \,x^{2}+a}}{6 c^{2}}-\frac {d \left (3 e^{2} a -2 c \,d^{2}\right ) \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 c^{\frac {3}{2}}}\) | \(80\) |
default | \(\frac {d^{3} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{\sqrt {c}}+e^{3} \left (\frac {x^{2} \sqrt {c \,x^{2}+a}}{3 c}-\frac {2 a \sqrt {c \,x^{2}+a}}{3 c^{2}}\right )+3 d \,e^{2} \left (\frac {x \sqrt {c \,x^{2}+a}}{2 c}-\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 c^{\frac {3}{2}}}\right )+\frac {3 d^{2} e \sqrt {c \,x^{2}+a}}{c}\) | \(124\) |
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Time = 0.29 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.64 \[ \int \frac {(d+e x)^3}{\sqrt {a+c x^2}} \, dx=\left [-\frac {3 \, {\left (2 \, c d^{3} - 3 \, a d e^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - 2 \, {\left (2 \, c e^{3} x^{2} + 9 \, c d e^{2} x + 18 \, c d^{2} e - 4 \, a e^{3}\right )} \sqrt {c x^{2} + a}}{12 \, c^{2}}, -\frac {3 \, {\left (2 \, c d^{3} - 3 \, a d e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (2 \, c e^{3} x^{2} + 9 \, c d e^{2} x + 18 \, c d^{2} e - 4 \, a e^{3}\right )} \sqrt {c x^{2} + a}}{6 \, c^{2}}\right ] \]
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Time = 0.47 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.16 \[ \int \frac {(d+e x)^3}{\sqrt {a+c x^2}} \, dx=\begin {cases} \sqrt {a + c x^{2}} \cdot \left (\frac {3 d e^{2} x}{2 c} + \frac {e^{3} x^{2}}{3 c} + \frac {- \frac {2 a e^{3}}{3 c} + 3 d^{2} e}{c}\right ) + \left (- \frac {3 a d e^{2}}{2 c} + d^{3}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {c} \sqrt {a + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {c x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: c \neq 0 \\\frac {\begin {cases} d^{3} x & \text {for}\: e = 0 \\\frac {\left (d + e x\right )^{4}}{4 e} & \text {otherwise} \end {cases}}{\sqrt {a}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.01 \[ \int \frac {(d+e x)^3}{\sqrt {a+c x^2}} \, dx=\frac {\sqrt {c x^{2} + a} e^{3} x^{2}}{3 \, c} + \frac {3 \, \sqrt {c x^{2} + a} d e^{2} x}{2 \, c} + \frac {d^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {c}} - \frac {3 \, a d e^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{2 \, c^{\frac {3}{2}}} + \frac {3 \, \sqrt {c x^{2} + a} d^{2} e}{c} - \frac {2 \, \sqrt {c x^{2} + a} a e^{3}}{3 \, c^{2}} \]
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Time = 0.27 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.85 \[ \int \frac {(d+e x)^3}{\sqrt {a+c x^2}} \, dx=\frac {1}{6} \, \sqrt {c x^{2} + a} {\left ({\left (\frac {2 \, e^{3} x}{c} + \frac {9 \, d e^{2}}{c}\right )} x + \frac {2 \, {\left (9 \, c^{2} d^{2} e - 2 \, a c e^{3}\right )}}{c^{3}}\right )} - \frac {{\left (2 \, c d^{3} - 3 \, a d e^{2}\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{2 \, c^{\frac {3}{2}}} \]
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Timed out. \[ \int \frac {(d+e x)^3}{\sqrt {a+c x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^3}{\sqrt {c\,x^2+a}} \,d x \]
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