Integrand size = 19, antiderivative size = 154 \[ \int (d+e x)^2 \left (a+c x^2\right )^{3/2} \, dx=\frac {a \left (6 c d^2-a e^2\right ) x \sqrt {a+c x^2}}{16 c}+\frac {\left (6 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac {7 d e \left (a+c x^2\right )^{5/2}}{30 c}+\frac {e (d+e x) \left (a+c x^2\right )^{5/2}}{6 c}+\frac {a^2 \left (6 c d^2-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{16 c^{3/2}} \]
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Time = 0.05 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {757, 655, 201, 223, 212} \[ \int (d+e x)^2 \left (a+c x^2\right )^{3/2} \, dx=\frac {a^2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (6 c d^2-a e^2\right )}{16 c^{3/2}}+\frac {x \left (a+c x^2\right )^{3/2} \left (6 c d^2-a e^2\right )}{24 c}+\frac {a x \sqrt {a+c x^2} \left (6 c d^2-a e^2\right )}{16 c}+\frac {7 d e \left (a+c x^2\right )^{5/2}}{30 c}+\frac {e \left (a+c x^2\right )^{5/2} (d+e x)}{6 c} \]
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Rule 201
Rule 212
Rule 223
Rule 655
Rule 757
Rubi steps \begin{align*} \text {integral}& = \frac {e (d+e x) \left (a+c x^2\right )^{5/2}}{6 c}+\frac {\int \left (6 c d^2-a e^2+7 c d e x\right ) \left (a+c x^2\right )^{3/2} \, dx}{6 c} \\ & = \frac {7 d e \left (a+c x^2\right )^{5/2}}{30 c}+\frac {e (d+e x) \left (a+c x^2\right )^{5/2}}{6 c}+\frac {\left (6 c d^2-a e^2\right ) \int \left (a+c x^2\right )^{3/2} \, dx}{6 c} \\ & = \frac {\left (6 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac {7 d e \left (a+c x^2\right )^{5/2}}{30 c}+\frac {e (d+e x) \left (a+c x^2\right )^{5/2}}{6 c}+\frac {\left (a \left (6 c d^2-a e^2\right )\right ) \int \sqrt {a+c x^2} \, dx}{8 c} \\ & = \frac {a \left (6 c d^2-a e^2\right ) x \sqrt {a+c x^2}}{16 c}+\frac {\left (6 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac {7 d e \left (a+c x^2\right )^{5/2}}{30 c}+\frac {e (d+e x) \left (a+c x^2\right )^{5/2}}{6 c}+\frac {\left (a^2 \left (6 c d^2-a e^2\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{16 c} \\ & = \frac {a \left (6 c d^2-a e^2\right ) x \sqrt {a+c x^2}}{16 c}+\frac {\left (6 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac {7 d e \left (a+c x^2\right )^{5/2}}{30 c}+\frac {e (d+e x) \left (a+c x^2\right )^{5/2}}{6 c}+\frac {\left (a^2 \left (6 c d^2-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 )}{16 c} \\ & = \frac {a \left (6 c d^2-a e^2\right ) x \sqrt {a+c x^2}}{16 c}+\frac {\left (6 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac {7 d e \left (a+c x^2\right )^{5/2}}{30 c}+\frac {e (d+e x) \left (a+c x^2\right )^{5/2}}{6 c}+\frac {a^2 \left (6 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{16 c^{3/2}} \\ \end{align*}
Time = 0.52 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.85 \[ \int (d+e x)^2 \left (a+c x^2\right )^{3/2} \, dx=\frac {\sqrt {c} \sqrt {a+c x^2} \left (3 a^2 e (32 d+5 e x)+4 c^2 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )+2 a c x \left (75 d^2+96 d e x+35 e^2 x^2\right )\right )+15 a^2 \left (-6 c d^2+a e^2\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{240 c^{3/2}} \]
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Time = 2.30 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.83
method | result | size |
risch | \(\frac {\left (40 c^{2} e^{2} x^{5}+96 d e \,c^{2} x^{4}+70 a c \,e^{2} x^{3}+60 c^{2} d^{2} x^{3}+192 a c d e \,x^{2}+15 a^{2} e^{2} x +150 a c \,d^{2} x +96 a^{2} d e \right ) \sqrt {c \,x^{2}+a}}{240 c}-\frac {a^{2} \left (e^{2} a -6 c \,d^{2}\right ) \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{16 c^{\frac {3}{2}}}\) | \(128\) |
default | \(d^{2} \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )+e^{2} \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{6 c}-\frac {a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{6 c}\right )+\frac {2 d e \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{5 c}\) | \(150\) |
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Time = 0.30 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.92 \[ \int (d+e x)^2 \left (a+c x^2\right )^{3/2} \, dx=\left [-\frac {15 \, {\left (6 \, a^{2} c d^{2} - a^{3} e^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - 2 \, {\left (40 \, c^{3} e^{2} x^{5} + 96 \, c^{3} d e x^{4} + 192 \, a c^{2} d e x^{2} + 96 \, a^{2} c d e + 10 \, {\left (6 \, c^{3} d^{2} + 7 \, a c^{2} e^{2}\right )} x^{3} + 15 \, {\left (10 \, a c^{2} d^{2} + a^{2} c e^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{480 \, c^{2}}, -\frac {15 \, {\left (6 \, a^{2} c d^{2} - a^{3} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (40 \, c^{3} e^{2} x^{5} + 96 \, c^{3} d e x^{4} + 192 \, a c^{2} d e x^{2} + 96 \, a^{2} c d e + 10 \, {\left (6 \, c^{3} d^{2} + 7 \, a c^{2} e^{2}\right )} x^{3} + 15 \, {\left (10 \, a c^{2} d^{2} + a^{2} c e^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{240 \, c^{2}}\right ] \]
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Time = 0.54 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.53 \[ \int (d+e x)^2 \left (a+c x^2\right )^{3/2} \, dx=\begin {cases} \sqrt {a + c x^{2}} \cdot \left (\frac {2 a^{2} d e}{5 c} + \frac {4 a d e x^{2}}{5} + \frac {2 c d e x^{4}}{5} + \frac {c e^{2} x^{5}}{6} + \frac {x^{3} \cdot \left (\frac {7 a c e^{2}}{6} + c^{2} d^{2}\right )}{4 c} + \frac {x \left (a^{2} e^{2} + 2 a c d^{2} - \frac {3 a \left (\frac {7 a c e^{2}}{6} + c^{2} d^{2}\right )}{4 c}\right )}{2 c}\right ) + \left (a^{2} d^{2} - \frac {a \left (a^{2} e^{2} + 2 a c d^{2} - \frac {3 a \left (\frac {7 a c e^{2}}{6} + c^{2} d^{2}\right )}{4 c}\right )}{2 c}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {c} \sqrt {a + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {c x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: c \neq 0 \\a^{\frac {3}{2}} \left (\begin {cases} d^{2} x & \text {for}\: e = 0 \\\frac {\left (d + e x\right )^{3}}{3 e} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.95 \[ \int (d+e x)^2 \left (a+c x^2\right )^{3/2} \, dx=\frac {1}{4} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} d^{2} x + \frac {3}{8} \, \sqrt {c x^{2} + a} a d^{2} x + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} e^{2} x}{6 \, c} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} a e^{2} x}{24 \, c} - \frac {\sqrt {c x^{2} + a} a^{2} e^{2} x}{16 \, c} + \frac {3 \, a^{2} d^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {c}} - \frac {a^{3} e^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{16 \, c^{\frac {3}{2}}} + \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} d e}{5 \, c} \]
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Time = 0.28 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.93 \[ \int (d+e x)^2 \left (a+c x^2\right )^{3/2} \, dx=\frac {1}{240} \, {\left (\frac {96 \, a^{2} d e}{c} + {\left (2 \, {\left (96 \, a d e + {\left (4 \, {\left (5 \, c e^{2} x + 12 \, c d e\right )} x + \frac {5 \, {\left (6 \, c^{5} d^{2} + 7 \, a c^{4} e^{2}\right )}}{c^{4}}\right )} x\right )} x + \frac {15 \, {\left (10 \, a c^{4} d^{2} + a^{2} c^{3} e^{2}\right )}}{c^{4}}\right )} x\right )} \sqrt {c x^{2} + a} - \frac {{\left (6 \, a^{2} c d^{2} - a^{3} e^{2}\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{16 \, c^{\frac {3}{2}}} \]
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Timed out. \[ \int (d+e x)^2 \left (a+c x^2\right )^{3/2} \, dx=\int {\left (c\,x^2+a\right )}^{3/2}\,{\left (d+e\,x\right )}^2 \,d x \]
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