Optimal. Leaf size=191 \[ \frac {2 a b^2 (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^5 \left (a+b x^2\right )}-\frac {6 a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^3 \sqrt {d x} \left (a+b x^2\right )}+\frac {2 b^3 (d x)^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 d^7 \left (a+b x^2\right )}-\frac {2 a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 d (d x)^{5/2} \left (a+b x^2\right )} \]
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Rubi [A] time = 0.06, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1112, 270} \[ \frac {2 b^3 (d x)^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 d^7 \left (a+b x^2\right )}+\frac {2 a b^2 (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^5 \left (a+b x^2\right )}-\frac {6 a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^3 \sqrt {d x} \left (a+b x^2\right )}-\frac {2 a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 d (d x)^{5/2} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 270
Rule 1112
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{(d x)^{7/2}} \, dx &=\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)^{7/2}} \, 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)^{7/2}}+\frac {3 a^2 b^4}{d^2 (d x)^{3/2}}+\frac {3 a b^5 \sqrt {d x}}{d^4}+\frac {b^6 (d x)^{5/2}}{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}}{5 d (d x)^{5/2} \left (a+b x^2\right )}-\frac {6 a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^3 \sqrt {d x} \left (a+b x^2\right )}+\frac {2 a b^2 (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^5 \left (a+b x^2\right )}+\frac {2 b^3 (d x)^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 d^7 \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 66, normalized size = 0.35 \[ \frac {2 x \sqrt {\left (a+b x^2\right )^2} \left (-7 a^3-105 a^2 b x^2+35 a b^2 x^4+5 b^3 x^6\right )}{35 (d x)^{7/2} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.07, size = 45, normalized size = 0.24 \[ \frac {2 \, {\left (5 \, b^{3} x^{6} + 35 \, a b^{2} x^{4} - 105 \, a^{2} b x^{2} - 7 \, a^{3}\right )} \sqrt {d x}}{35 \, d^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 107, normalized size = 0.56 \[ -\frac {2 \, {\left (\frac {7 \, {\left (15 \, a^{2} b d^{3} x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + a^{3} d^{3} \mathrm {sgn}\left (b x^{2} + a\right )\right )}}{\sqrt {d x} d^{2} x^{2}} - \frac {5 \, {\left (\sqrt {d x} b^{3} d^{21} x^{3} \mathrm {sgn}\left (b x^{2} + a\right ) + 7 \, \sqrt {d x} a b^{2} d^{21} x \mathrm {sgn}\left (b x^{2} + a\right )\right )}}{d^{21}}\right )}}{35 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 61, normalized size = 0.32 \[ -\frac {2 \left (-5 b^{3} x^{6}-35 a \,b^{2} x^{4}+105 a^{2} b \,x^{2}+7 a^{3}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}} x}{35 \left (b \,x^{2}+a \right )^{3} \left (d x \right )^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.45, size = 86, normalized size = 0.45 \[ \frac {2 \, {\left (5 \, {\left (3 \, b^{3} \sqrt {d} x^{3} + 7 \, a b^{2} \sqrt {d} x\right )} \sqrt {x} + \frac {70 \, {\left (a b^{2} \sqrt {d} x^{3} - 3 \, a^{2} b \sqrt {d} x\right )}}{x^{\frac {3}{2}}} - \frac {21 \, {\left (5 \, a^{2} b \sqrt {d} x^{3} + a^{3} \sqrt {d} x\right )}}{x^{\frac {7}{2}}}\right )}}{105 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.53, size = 91, normalized size = 0.48 \[ -\frac {\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}\,\left (\frac {2\,a^3}{5\,b\,d^3}+\frac {6\,a^2\,x^2}{d^3}-\frac {2\,b^2\,x^6}{7\,d^3}-\frac {2\,a\,b\,x^4}{d^3}\right )}{x^4\,\sqrt {d\,x}+\frac {a\,x^2\,\sqrt {d\,x}}{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}{\left (d x\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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