Integrand size = 17, antiderivative size = 107 \[ \int (d+e x) \left (a+c x^2\right )^{5/2} \, dx=\frac {5}{16} a^2 d x \sqrt {a+c x^2}+\frac {5}{24} a d x \left (a+c x^2\right )^{3/2}+\frac {1}{6} d x \left (a+c x^2\right )^{5/2}+\frac {e \left (a+c x^2\right )^{7/2}}{7 c}+\frac {5 a^3 d \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{16 \sqrt {c}} \]
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Time = 0.02 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {655, 201, 223, 212} \[ \int (d+e x) \left (a+c x^2\right )^{5/2} \, dx=\frac {5 a^3 d \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{16 \sqrt {c}}+\frac {5}{16} a^2 d x \sqrt {a+c x^2}+\frac {1}{6} d x \left (a+c x^2\right )^{5/2}+\frac {5}{24} a d x \left (a+c x^2\right )^{3/2}+\frac {e \left (a+c x^2\right )^{7/2}}{7 c} \]
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Rule 201
Rule 212
Rule 223
Rule 655
Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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*}
Time = 0.47 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00 \[ \int (d+e x) \left (a+c x^2\right )^{5/2} \, dx=\frac {\sqrt {a+c x^2} \left (48 a^3 e+8 c^3 x^5 (7 d+6 e x)+3 a^2 c x (77 d+48 e x)+2 a c^2 x^3 (91 d+72 e x)\right )-105 a^3 \sqrt {c} d \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{336 c} \]
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Time = 1.89 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.80
method | result | size |
default | \(d \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{6}\right )+\frac {e \left (c \,x^{2}+a \right )^{\frac {7}{2}}}{7 c}\) | \(86\) |
risch | \(\frac {\left (48 e \,c^{3} x^{6}+56 c^{3} d \,x^{5}+144 c^{2} a e \,x^{4}+182 a \,c^{2} d \,x^{3}+144 x^{2} a^{2} c e +231 a^{2} c d x +48 e \,a^{3}\right ) \sqrt {c \,x^{2}+a}}{336 c}+\frac {5 a^{3} d \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{16 \sqrt {c}}\) | \(104\) |
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Time = 0.29 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.09 \[ \int (d+e x) \left (a+c x^2\right )^{5/2} \, dx=\left [\frac {105 \, a^{3} \sqrt {c} d \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (48 \, c^{3} e x^{6} + 56 \, c^{3} d x^{5} + 144 \, a c^{2} e x^{4} + 182 \, a c^{2} d x^{3} + 144 \, a^{2} c e x^{2} + 231 \, a^{2} c d x + 48 \, a^{3} e\right )} \sqrt {c x^{2} + a}}{672 \, c}, -\frac {105 \, a^{3} \sqrt {-c} d \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (48 \, c^{3} e x^{6} + 56 \, c^{3} d x^{5} + 144 \, a c^{2} e x^{4} + 182 \, a c^{2} d x^{3} + 144 \, a^{2} c e x^{2} + 231 \, a^{2} c d x + 48 \, a^{3} e\right )} \sqrt {c x^{2} + a}}{336 \, c}\right ] \]
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Time = 0.48 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.40 \[ \int (d+e x) \left (a+c x^2\right )^{5/2} \, dx=\begin {cases} \frac {5 a^{3} d \left (\begin {cases} \frac {\log {\left (2 \sqrt {c} \sqrt {a + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {c x^{2}}} & \text {otherwise} \end {cases}\right )}{16} + \sqrt {a + c x^{2}} \left (\frac {a^{3} e}{7 c} + \frac {11 a^{2} d x}{16} + \frac {3 a^{2} e x^{2}}{7} + \frac {13 a c d x^{3}}{24} + \frac {3 a c e x^{4}}{7} + \frac {c^{2} d x^{5}}{6} + \frac {c^{2} e x^{6}}{7}\right ) & \text {for}\: c \neq 0 \\a^{\frac {5}{2}} \left (d x + \frac {e x^{2}}{2}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.72 \[ \int (d+e x) \left (a+c x^2\right )^{5/2} \, dx=\frac {1}{6} \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} d x + \frac {5}{24} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a d x + \frac {5}{16} \, \sqrt {c x^{2} + a} a^{2} d x + \frac {5 \, a^{3} d \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {c}} + \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} e}{7 \, c} \]
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Time = 0.28 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.94 \[ \int (d+e x) \left (a+c x^2\right )^{5/2} \, dx=-\frac {5 \, a^{3} d \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{16 \, \sqrt {c}} + \frac {1}{336} \, {\left (\frac {48 \, a^{3} e}{c} + {\left (231 \, a^{2} d + 2 \, {\left (72 \, a^{2} e + {\left (91 \, a c d + 4 \, {\left (18 \, a c e + {\left (6 \, c^{2} e x + 7 \, c^{2} d\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {c x^{2} + a} \]
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Time = 9.47 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.50 \[ \int (d+e x) \left (a+c x^2\right )^{5/2} \, dx=\frac {e\,{\left (c\,x^2+a\right )}^{7/2}}{7\,c}+\frac {d\,x\,{\left (c\,x^2+a\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{2},\frac {1}{2};\ \frac {3}{2};\ -\frac {c\,x^2}{a}\right )}{{\left (\frac {c\,x^2}{a}+1\right )}^{5/2}} \]
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