Optimal. Leaf size=193 \[ \frac{2 b^3 (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 d^7 \left (a+b x^2\right )}+\frac{6 a b^2 (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d^5 \left (a+b x^2\right )}+\frac{6 a^2 b \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^3 \left (a+b x^2\right )}-\frac{2 a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d (d x)^{3/2} \left (a+b x^2\right )} \]
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Rubi [A] time = 0.0547985, antiderivative size = 193, 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} \[ \frac{2 b^3 (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 d^7 \left (a+b x^2\right )}+\frac{6 a b^2 (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d^5 \left (a+b x^2\right )}+\frac{6 a^2 b \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^3 \left (a+b x^2\right )}-\frac{2 a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d (d x)^{3/2} \left (a+b x^2\right )} \]
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)^{5/2}} \, 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)^{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*}
Mathematica [A] time = 0.0269476, size = 66, normalized size = 0.34 \[ \frac{2 x \sqrt{\left (a+b x^2\right )^2} \left (135 a^2 b x^2-15 a^3+27 a b^2 x^4+5 b^3 x^6\right )}{45 (d x)^{5/2} \left (a+b x^2\right )} \]
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}-27\,a{x}^{4}{b}^{2}-135\,{a}^{2}b{x}^{2}+15\,{a}^{3} \right ) x}{45\, \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}} \left ( dx \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02564, size = 116, normalized size = 0.6 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49424, size = 105, normalized size = 0.54 \begin{align*} \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}} \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{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28162, size = 142, normalized size = 0.74 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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