Integrand size = 21, antiderivative size = 169 \[ \int \frac {\left (c+d x^2\right )^3}{\sqrt {a+b x^2}} \, dx=\frac {d \left (44 b^2 c^2-44 a b c d+15 a^2 d^2\right ) x \sqrt {a+b x^2}}{48 b^3}+\frac {5 d (2 b c-a d) x \sqrt {a+b x^2} \left (c+d x^2\right )}{24 b^2}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )^2}{6 b}+\frac {(2 b c-a d) \left (8 b^2 c^2-8 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{7/2}} \]
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Time = 0.10 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {427, 542, 396, 223, 212} \[ \int \frac {\left (c+d x^2\right )^3}{\sqrt {a+b x^2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b c-a d) \left (5 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{16 b^{7/2}}+\frac {d x \sqrt {a+b x^2} \left (15 a^2 d^2-44 a b c d+44 b^2 c^2\right )}{48 b^3}+\frac {5 d x \sqrt {a+b x^2} \left (c+d x^2\right ) (2 b c-a d)}{24 b^2}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )^2}{6 b} \]
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Rule 212
Rule 223
Rule 396
Rule 427
Rule 542
Rubi steps \begin{align*} \text {integral}& = \frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )^2}{6 b}+\frac {\int \frac {\left (c+d x^2\right ) \left (c (6 b c-a d)+5 d (2 b c-a d) x^2\right )}{\sqrt {a+b x^2}} \, dx}{6 b} \\ & = \frac {5 d (2 b c-a d) x \sqrt {a+b x^2} \left (c+d x^2\right )}{24 b^2}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )^2}{6 b}+\frac {\int \frac {c \left (24 b^2 c^2-14 a b c d+5 a^2 d^2\right )+d \left (44 b^2 c^2-44 a b c d+15 a^2 d^2\right ) x^2}{\sqrt {a+b x^2}} \, dx}{24 b^2} \\ & = \frac {d \left (44 b^2 c^2-44 a b c d+15 a^2 d^2\right ) x \sqrt {a+b x^2}}{48 b^3}+\frac {5 d (2 b c-a d) x \sqrt {a+b x^2} \left (c+d x^2\right )}{24 b^2}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )^2}{6 b}+\frac {\left ((2 b c-a d) \left (8 b^2 c^2-8 a b c d+5 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{16 b^3} \\ & = \frac {d \left (44 b^2 c^2-44 a b c d+15 a^2 d^2\right ) x \sqrt {a+b x^2}}{48 b^3}+\frac {5 d (2 b c-a d) x \sqrt {a+b x^2} \left (c+d x^2\right )}{24 b^2}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )^2}{6 b}+\frac {\left ((2 b c-a d) \left (8 b^2 c^2-8 a b c d+5 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{16 b^3} \\ & = \frac {d \left (44 b^2 c^2-44 a b c d+15 a^2 d^2\right ) x \sqrt {a+b x^2}}{48 b^3}+\frac {5 d (2 b c-a d) x \sqrt {a+b x^2} \left (c+d x^2\right )}{24 b^2}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )^2}{6 b}+\frac {(2 b c-a d) \left (8 b^2 c^2-8 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{7/2}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.82 \[ \int \frac {\left (c+d x^2\right )^3}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {b} d x \sqrt {a+b x^2} \left (15 a^2 d^2-2 a b d \left (27 c+5 d x^2\right )+4 b^2 \left (18 c^2+9 c d x^2+2 d^2 x^4\right )\right )+\left (-48 b^3 c^3+72 a b^2 c^2 d-54 a^2 b c d^2+15 a^3 d^3\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{48 b^{7/2}} \]
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Time = 2.42 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.70
method | result | size |
pseudoelliptic | \(-\frac {5 \left (\left (a d -2 b c \right ) \left (a^{2} d^{2}-\frac {8}{5} a b c d +\frac {8}{5} b^{2} c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )-\left (\left (\frac {8}{15} d^{2} x^{4}+\frac {12}{5} c d \,x^{2}+\frac {24}{5} c^{2}\right ) b^{\frac {5}{2}}+\left (\left (-\frac {2 d \,x^{2}}{3}-\frac {18 c}{5}\right ) b^{\frac {3}{2}}+a d \sqrt {b}\right ) d a \right ) x \sqrt {b \,x^{2}+a}\, d \right )}{16 b^{\frac {7}{2}}}\) | \(118\) |
risch | \(\frac {d x \left (8 b^{2} d^{2} x^{4}-10 x^{2} a b \,d^{2}+36 x^{2} b^{2} c d +15 a^{2} d^{2}-54 a b c d +72 b^{2} c^{2}\right ) \sqrt {b \,x^{2}+a}}{48 b^{3}}-\frac {\left (5 a^{3} d^{3}-18 a^{2} b c \,d^{2}+24 a \,b^{2} c^{2} d -16 b^{3} c^{3}\right ) \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{16 b^{\frac {7}{2}}}\) | \(130\) |
default | \(\frac {c^{3} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}+d^{3} \left (\frac {x^{5} \sqrt {b \,x^{2}+a}}{6 b}-\frac {5 a \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )}{6 b}\right )+3 c \,d^{2} \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )+3 c^{2} d \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )\) | \(227\) |
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Time = 0.29 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.78 \[ \int \frac {\left (c+d x^2\right )^3}{\sqrt {a+b x^2}} \, dx=\left [-\frac {3 \, {\left (16 \, b^{3} c^{3} - 24 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (8 \, b^{3} d^{3} x^{5} + 2 \, {\left (18 \, b^{3} c d^{2} - 5 \, a b^{2} d^{3}\right )} x^{3} + 3 \, {\left (24 \, b^{3} c^{2} d - 18 \, a b^{2} c d^{2} + 5 \, a^{2} b d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{96 \, b^{4}}, -\frac {3 \, {\left (16 \, b^{3} c^{3} - 24 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (8 \, b^{3} d^{3} x^{5} + 2 \, {\left (18 \, b^{3} c d^{2} - 5 \, a b^{2} d^{3}\right )} x^{3} + 3 \, {\left (24 \, b^{3} c^{2} d - 18 \, a b^{2} c d^{2} + 5 \, a^{2} b d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{48 \, b^{4}}\right ] \]
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Time = 0.36 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.18 \[ \int \frac {\left (c+d x^2\right )^3}{\sqrt {a+b x^2}} \, dx=\begin {cases} \sqrt {a + b x^{2}} \left (\frac {d^{3} x^{5}}{6 b} + \frac {x^{3} \left (- \frac {5 a d^{3}}{6 b} + 3 c d^{2}\right )}{4 b} + \frac {x \left (- \frac {3 a \left (- \frac {5 a d^{3}}{6 b} + 3 c d^{2}\right )}{4 b} + 3 c^{2} d\right )}{2 b}\right ) + \left (- \frac {a \left (- \frac {3 a \left (- \frac {5 a d^{3}}{6 b} + 3 c d^{2}\right )}{4 b} + 3 c^{2} d\right )}{2 b} + c^{3}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: b \neq 0 \\\frac {c^{3} x + c^{2} d x^{3} + \frac {3 c d^{2} x^{5}}{5} + \frac {d^{3} x^{7}}{7}}{\sqrt {a}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.18 \[ \int \frac {\left (c+d x^2\right )^3}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {b x^{2} + a} d^{3} x^{5}}{6 \, b} + \frac {3 \, \sqrt {b x^{2} + a} c d^{2} x^{3}}{4 \, b} - \frac {5 \, \sqrt {b x^{2} + a} a d^{3} x^{3}}{24 \, b^{2}} + \frac {3 \, \sqrt {b x^{2} + a} c^{2} d x}{2 \, b} - \frac {9 \, \sqrt {b x^{2} + a} a c d^{2} x}{8 \, b^{2}} + \frac {5 \, \sqrt {b x^{2} + a} a^{2} d^{3} x}{16 \, b^{3}} + \frac {c^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b}} - \frac {3 \, a c^{2} d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {3}{2}}} + \frac {9 \, a^{2} c d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {5}{2}}} - \frac {5 \, a^{3} d^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {7}{2}}} \]
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Time = 0.29 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.89 \[ \int \frac {\left (c+d x^2\right )^3}{\sqrt {a+b x^2}} \, dx=\frac {1}{48} \, {\left (2 \, {\left (\frac {4 \, d^{3} x^{2}}{b} + \frac {18 \, b^{4} c d^{2} - 5 \, a b^{3} d^{3}}{b^{5}}\right )} x^{2} + \frac {3 \, {\left (24 \, b^{4} c^{2} d - 18 \, a b^{3} c d^{2} + 5 \, a^{2} b^{2} d^{3}\right )}}{b^{5}}\right )} \sqrt {b x^{2} + a} x - \frac {{\left (16 \, b^{3} c^{3} - 24 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{16 \, b^{\frac {7}{2}}} \]
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Timed out. \[ \int \frac {\left (c+d x^2\right )^3}{\sqrt {a+b x^2}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^3}{\sqrt {b\,x^2+a}} \,d x \]
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