Optimal. Leaf size=191 \[ \frac{2 b^3 (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d^7 \left (a+b x^2\right )}+\frac{2 a b^2 (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^5 \left (a+b x^2\right )}-\frac{6 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^3 \sqrt{d x} \left (a+b x^2\right )}-\frac{2 a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d (d x)^{5/2} \left (a+b x^2\right )} \]
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Rubi [A] time = 0.0552309, antiderivative size = 191, normalized size of antiderivative = 1., 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 1112
Rule 270
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*}
Mathematica [A] time = 0.0260261, size = 66, normalized size = 0.35 \[ \frac{2 x \sqrt{\left (a+b x^2\right )^2} \left (-105 a^2 b x^2-7 a^3+35 a b^2 x^4+5 b^3 x^6\right )}{35 (d x)^{7/2} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.17, size = 61, normalized size = 0.3 \begin{align*} -{\frac{2\, \left ( -5\,{b}^{3}{x}^{6}-35\,a{x}^{4}{b}^{2}+105\,{a}^{2}b{x}^{2}+7\,{a}^{3} \right ) x}{35\, \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}} \left ( dx \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00227, size = 116, normalized size = 0.61 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4689, size = 104, normalized size = 0.54 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}}{\left (d x\right )^{\frac{7}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2064, size = 144, normalized size = 0.75 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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