Optimal. Leaf size=89 \[ \frac{12 a^2 b^2 (d x)^{7/2}}{7 d^5}+\frac{8 a^3 b (d x)^{3/2}}{3 d^3}-\frac{2 a^4}{d \sqrt{d x}}+\frac{8 a b^3 (d x)^{11/2}}{11 d^7}+\frac{2 b^4 (d x)^{15/2}}{15 d^9} \]
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Rubi [A] time = 0.0420059, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {28, 270} \[ \frac{12 a^2 b^2 (d x)^{7/2}}{7 d^5}+\frac{8 a^3 b (d x)^{3/2}}{3 d^3}-\frac{2 a^4}{d \sqrt{d x}}+\frac{8 a b^3 (d x)^{11/2}}{11 d^7}+\frac{2 b^4 (d x)^{15/2}}{15 d^9} \]
Antiderivative was successfully verified.
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Rule 28
Rule 270
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^2}{(d x)^{3/2}} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^4}{(d x)^{3/2}} \, dx}{b^4}\\ &=\frac{\int \left (\frac{a^4 b^4}{(d x)^{3/2}}+\frac{4 a^3 b^5 \sqrt{d x}}{d^2}+\frac{6 a^2 b^6 (d x)^{5/2}}{d^4}+\frac{4 a b^7 (d x)^{9/2}}{d^6}+\frac{b^8 (d x)^{13/2}}{d^8}\right ) \, dx}{b^4}\\ &=-\frac{2 a^4}{d \sqrt{d x}}+\frac{8 a^3 b (d x)^{3/2}}{3 d^3}+\frac{12 a^2 b^2 (d x)^{7/2}}{7 d^5}+\frac{8 a b^3 (d x)^{11/2}}{11 d^7}+\frac{2 b^4 (d x)^{15/2}}{15 d^9}\\ \end{align*}
Mathematica [A] time = 0.0160557, size = 55, normalized size = 0.62 \[ \frac{2 x \left (990 a^2 b^2 x^4+1540 a^3 b x^2-1155 a^4+420 a b^3 x^6+77 b^4 x^8\right )}{1155 (d x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 52, normalized size = 0.6 \begin{align*} -{\frac{ \left ( -154\,{b}^{4}{x}^{8}-840\,a{b}^{3}{x}^{6}-1980\,{a}^{2}{b}^{2}{x}^{4}-3080\,{a}^{3}b{x}^{2}+2310\,{a}^{4} \right ) x}{1155} \left ( dx \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.983885, size = 103, normalized size = 1.16 \begin{align*} -\frac{2 \,{\left (\frac{1155 \, a^{4}}{\sqrt{d x}} - \frac{77 \, \left (d x\right )^{\frac{15}{2}} b^{4} + 420 \, \left (d x\right )^{\frac{11}{2}} a b^{3} d^{2} + 990 \, \left (d x\right )^{\frac{7}{2}} a^{2} b^{2} d^{4} + 1540 \, \left (d x\right )^{\frac{3}{2}} a^{3} b d^{6}}{d^{8}}\right )}}{1155 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.19973, size = 136, normalized size = 1.53 \begin{align*} \frac{2 \,{\left (77 \, b^{4} x^{8} + 420 \, a b^{3} x^{6} + 990 \, a^{2} b^{2} x^{4} + 1540 \, a^{3} b x^{2} - 1155 \, a^{4}\right )} \sqrt{d x}}{1155 \, d^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.3449, size = 88, normalized size = 0.99 \begin{align*} - \frac{2 a^{4}}{d^{\frac{3}{2}} \sqrt{x}} + \frac{8 a^{3} b x^{\frac{3}{2}}}{3 d^{\frac{3}{2}}} + \frac{12 a^{2} b^{2} x^{\frac{7}{2}}}{7 d^{\frac{3}{2}}} + \frac{8 a b^{3} x^{\frac{11}{2}}}{11 d^{\frac{3}{2}}} + \frac{2 b^{4} x^{\frac{15}{2}}}{15 d^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12669, size = 120, normalized size = 1.35 \begin{align*} -\frac{2 \,{\left (\frac{1155 \, a^{4}}{\sqrt{d x}} - \frac{77 \, \sqrt{d x} b^{4} d^{119} x^{7} + 420 \, \sqrt{d x} a b^{3} d^{119} x^{5} + 990 \, \sqrt{d x} a^{2} b^{2} d^{119} x^{3} + 1540 \, \sqrt{d x} a^{3} b d^{119} x}{d^{120}}\right )}}{1155 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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