Integrand size = 30, antiderivative size = 193 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{(d x)^{5/2}} \, dx=-\frac {2 a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d (d x)^{3/2} \left (a+b x^2\right )}+\frac {6 a^2 b \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^3 \left (a+b x^2\right )}+\frac {6 a b^2 (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 d^5 \left (a+b x^2\right )}+\frac {2 b^3 (d x)^{9/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 d^7 \left (a+b x^2\right )} \]
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Time = 0.03 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1126, 276} \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{(d x)^{5/2}} \, dx=\frac {6 a b^2 (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 d^5 \left (a+b x^2\right )}+\frac {6 a^2 b \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^3 \left (a+b x^2\right )}+\frac {2 b^3 (d x)^{9/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 d^7 \left (a+b x^2\right )}-\frac {2 a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d (d x)^{3/2} \left (a+b x^2\right )} \]
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Rule 276
Rule 1126
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {\left (a b+b^2 x^2\right )^3}{(d x)^{5/2}} \, dx}{b^2 \left (a b+b^2 x^2\right )} \\ & = \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (\frac {a^3 b^3}{(d x)^{5/2}}+\frac {3 a^2 b^4}{d^2 \sqrt {d x}}+\frac {3 a b^5 (d x)^{3/2}}{d^4}+\frac {b^6 (d x)^{7/2}}{d^6}\right ) \, dx}{b^2 \left (a b+b^2 x^2\right )} \\ & = -\frac {2 a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d (d x)^{3/2} \left (a+b x^2\right )}+\frac {6 a^2 b \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^3 \left (a+b x^2\right )}+\frac {6 a b^2 (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 d^5 \left (a+b x^2\right )}+\frac {2 b^3 (d x)^{9/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 d^7 \left (a+b x^2\right )} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.34 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{(d x)^{5/2}} \, dx=-\frac {2 x \left (\left (a+b x^2\right )^2\right )^{3/2} \left (15 a^3-135 a^2 b x^2-27 a b^2 x^4-5 b^3 x^6\right )}{45 (d x)^{5/2} \left (a+b x^2\right )^3} \]
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Time = 0.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.32
method | result | size |
gosper | \(-\frac {2 x \left (-5 b^{3} x^{6}-27 b^{2} x^{4} a -135 a^{2} b \,x^{2}+15 a^{3}\right ) {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {3}{2}}}{45 \left (b \,x^{2}+a \right )^{3} \left (d x \right )^{\frac {5}{2}}}\) | \(61\) |
default | \(-\frac {2 {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {3}{2}} \left (-5 b^{3} x^{6}-27 b^{2} x^{4} a -135 a^{2} b \,x^{2}+15 a^{3}\right )}{45 d \left (b \,x^{2}+a \right )^{3} \left (d x \right )^{\frac {3}{2}}}\) | \(63\) |
risch | \(-\frac {2 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (-5 b^{3} x^{6}-27 b^{2} x^{4} a -135 a^{2} b \,x^{2}+15 a^{3}\right )}{45 d^{2} \left (b \,x^{2}+a \right ) x \sqrt {d x}}\) | \(66\) |
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Time = 0.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.23 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{(d x)^{5/2}} \, dx=\frac {2 \, {\left (5 \, b^{3} x^{6} + 27 \, a b^{2} x^{4} + 135 \, a^{2} b x^{2} - 15 \, a^{3}\right )} \sqrt {d x}}{45 \, d^{3} x^{2}} \]
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\[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{(d x)^{5/2}} \, dx=\int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}{\left (d x\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.45 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{(d x)^{5/2}} \, dx=\frac {2 \, {\left ({\left (5 \, b^{3} \sqrt {d} x^{3} + 9 \, a b^{2} \sqrt {d} x\right )} x^{\frac {3}{2}} + \frac {18 \, {\left (a b^{2} \sqrt {d} x^{3} + 5 \, a^{2} b \sqrt {d} x\right )}}{\sqrt {x}} + \frac {15 \, {\left (3 \, a^{2} b \sqrt {d} x^{3} - a^{3} \sqrt {d} x\right )}}{x^{\frac {5}{2}}}\right )}}{45 \, d^{3}} \]
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Time = 0.30 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.54 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{(d x)^{5/2}} \, dx=-\frac {2 \, {\left (\frac {15 \, a^{3} d \mathrm {sgn}\left (b x^{2} + a\right )}{\sqrt {d x} x} - \frac {5 \, \sqrt {d x} b^{3} d^{36} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 27 \, \sqrt {d x} a b^{2} d^{36} x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 135 \, \sqrt {d x} a^{2} b d^{36} \mathrm {sgn}\left (b x^{2} + a\right )}{d^{36}}\right )}}{45 \, d^{3}} \]
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Time = 13.73 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.46 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{(d x)^{5/2}} \, dx=\frac {\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}\,\left (\frac {6\,a^2\,x^2}{d^2}-\frac {2\,a^3}{3\,b\,d^2}+\frac {2\,b^2\,x^6}{9\,d^2}+\frac {6\,a\,b\,x^4}{5\,d^2}\right )}{x^3\,\sqrt {d\,x}+\frac {a\,x\,\sqrt {d\,x}}{b}} \]
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