Optimal. Leaf size=89 \[ -\frac {2 a^4}{d \sqrt {d x}}+\frac {8 a^3 b (d x)^{3/2}}{3 d^3}+\frac {12 a^2 b^2 (d x)^{7/2}}{7 d^5}+\frac {8 a b^3 (d x)^{11/2}}{11 d^7}+\frac {2 b^4 (d x)^{15/2}}{15 d^9} \]
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Rubi [A] time = 0.04, antiderivative size = 89, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.02, size = 55, normalized size = 0.62 \[ \frac {2 x \left (-1155 a^4+1540 a^3 b x^2+990 a^2 b^2 x^4+420 a b^3 x^6+77 b^4 x^8\right )}{1155 (d x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 56, normalized size = 0.63 \[ \frac {2 \, {\left (77 \, b^{4} x^{8} + 420 \, a b^{3} x^{6} + 990 \, a^{2} b^{2} x^{4} + 1540 \, a^{3} b x^{2} - 1155 \, a^{4}\right )} \sqrt {d x}}{1155 \, d^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 89, normalized size = 1.00 \[ -\frac {2 \, {\left (\frac {1155 \, a^{4}}{\sqrt {d x}} - \frac {77 \, \sqrt {d x} b^{4} d^{119} x^{7} + 420 \, \sqrt {d x} a b^{3} d^{119} x^{5} + 990 \, \sqrt {d x} a^{2} b^{2} d^{119} x^{3} + 1540 \, \sqrt {d x} a^{3} b d^{119} x}{d^{120}}\right )}}{1155 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 52, normalized size = 0.58 \[ -\frac {2 \left (-77 b^{4} x^{8}-420 a \,b^{3} x^{6}-990 a^{2} b^{2} x^{4}-1540 a^{3} b \,x^{2}+1155 a^{4}\right ) x}{1155 \left (d x \right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 76, normalized size = 0.85 \[ -\frac {2 \, {\left (\frac {1155 \, a^{4}}{\sqrt {d x}} - \frac {77 \, \left (d x\right )^{\frac {15}{2}} b^{4} + 420 \, \left (d x\right )^{\frac {11}{2}} a b^{3} d^{2} + 990 \, \left (d x\right )^{\frac {7}{2}} a^{2} b^{2} d^{4} + 1540 \, \left (d x\right )^{\frac {3}{2}} a^{3} b d^{6}}{d^{8}}\right )}}{1155 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 71, normalized size = 0.80 \[ \frac {2\,b^4\,{\left (d\,x\right )}^{15/2}}{15\,d^9}-\frac {2\,a^4}{d\,\sqrt {d\,x}}+\frac {12\,a^2\,b^2\,{\left (d\,x\right )}^{7/2}}{7\,d^5}+\frac {8\,a^3\,b\,{\left (d\,x\right )}^{3/2}}{3\,d^3}+\frac {8\,a\,b^3\,{\left (d\,x\right )}^{11/2}}{11\,d^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.38, size = 88, normalized size = 0.99 \[ - \frac {2 a^{4}}{d^{\frac {3}{2}} \sqrt {x}} + \frac {8 a^{3} b x^{\frac {3}{2}}}{3 d^{\frac {3}{2}}} + \frac {12 a^{2} b^{2} x^{\frac {7}{2}}}{7 d^{\frac {3}{2}}} + \frac {8 a b^{3} x^{\frac {11}{2}}}{11 d^{\frac {3}{2}}} + \frac {2 b^{4} x^{\frac {15}{2}}}{15 d^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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