Optimal. Leaf size=129 \[ \frac {2 a^6 \sqrt {d x}}{d}+\frac {12 a^5 b (d x)^{5/2}}{5 d^3}+\frac {10 a^4 b^2 (d x)^{9/2}}{3 d^5}+\frac {40 a^3 b^3 (d x)^{13/2}}{13 d^7}+\frac {30 a^2 b^4 (d x)^{17/2}}{17 d^9}+\frac {4 a b^5 (d x)^{21/2}}{7 d^{11}}+\frac {2 b^6 (d x)^{25/2}}{25 d^{13}} \]
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Rubi [A] time = 0.06, antiderivative size = 129, 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} \[ \frac {30 a^2 b^4 (d x)^{17/2}}{17 d^9}+\frac {40 a^3 b^3 (d x)^{13/2}}{13 d^7}+\frac {10 a^4 b^2 (d x)^{9/2}}{3 d^5}+\frac {12 a^5 b (d x)^{5/2}}{5 d^3}+\frac {2 a^6 \sqrt {d x}}{d}+\frac {4 a b^5 (d x)^{21/2}}{7 d^{11}}+\frac {2 b^6 (d x)^{25/2}}{25 d^{13}} \]
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 )^3}{\sqrt {d x}} \, dx &=\frac {\int \frac {\left (a b+b^2 x^2\right )^6}{\sqrt {d x}} \, dx}{b^6}\\ &=\frac {\int \left (\frac {a^6 b^6}{\sqrt {d x}}+\frac {6 a^5 b^7 (d x)^{3/2}}{d^2}+\frac {15 a^4 b^8 (d x)^{7/2}}{d^4}+\frac {20 a^3 b^9 (d x)^{11/2}}{d^6}+\frac {15 a^2 b^{10} (d x)^{15/2}}{d^8}+\frac {6 a b^{11} (d x)^{19/2}}{d^{10}}+\frac {b^{12} (d x)^{23/2}}{d^{12}}\right ) \, dx}{b^6}\\ &=\frac {2 a^6 \sqrt {d x}}{d}+\frac {12 a^5 b (d x)^{5/2}}{5 d^3}+\frac {10 a^4 b^2 (d x)^{9/2}}{3 d^5}+\frac {40 a^3 b^3 (d x)^{13/2}}{13 d^7}+\frac {30 a^2 b^4 (d x)^{17/2}}{17 d^9}+\frac {4 a b^5 (d x)^{21/2}}{7 d^{11}}+\frac {2 b^6 (d x)^{25/2}}{25 d^{13}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 77, normalized size = 0.60 \[ \frac {2 \left (116025 a^6 x+139230 a^5 b x^3+193375 a^4 b^2 x^5+178500 a^3 b^3 x^7+102375 a^2 b^4 x^9+33150 a b^5 x^{11}+4641 b^6 x^{13}\right )}{116025 \sqrt {d x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 75, normalized size = 0.58 \[ \frac {2 \, {\left (4641 \, b^{6} x^{12} + 33150 \, a b^{5} x^{10} + 102375 \, a^{2} b^{4} x^{8} + 178500 \, a^{3} b^{3} x^{6} + 193375 \, a^{4} b^{2} x^{4} + 139230 \, a^{5} b x^{2} + 116025 \, a^{6}\right )} \sqrt {d x}}{116025 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 105, normalized size = 0.81 \[ \frac {2 \, {\left (4641 \, \sqrt {d x} b^{6} x^{12} + 33150 \, \sqrt {d x} a b^{5} x^{10} + 102375 \, \sqrt {d x} a^{2} b^{4} x^{8} + 178500 \, \sqrt {d x} a^{3} b^{3} x^{6} + 193375 \, \sqrt {d x} a^{4} b^{2} x^{4} + 139230 \, \sqrt {d x} a^{5} b x^{2} + 116025 \, \sqrt {d x} a^{6}\right )}}{116025 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 74, normalized size = 0.57 \[ \frac {2 \left (4641 b^{6} x^{12}+33150 a \,b^{5} x^{10}+102375 a^{2} b^{4} x^{8}+178500 a^{3} b^{3} x^{6}+193375 a^{4} b^{2} x^{4}+139230 a^{5} b \,x^{2}+116025 a^{6}\right ) x}{116025 \sqrt {d x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 155, normalized size = 1.20 \[ \frac {2 \, {\left (116025 \, \sqrt {d x} a^{6} + \frac {4641 \, \left (d x\right )^{\frac {25}{2}} b^{6}}{d^{12}} + \frac {33150 \, \left (d x\right )^{\frac {21}{2}} a b^{5}}{d^{10}} + \frac {81900 \, \left (d x\right )^{\frac {17}{2}} a^{2} b^{4}}{d^{8}} + \frac {71400 \, \left (d x\right )^{\frac {13}{2}} a^{3} b^{3}}{d^{6}} + 7735 \, {\left (\frac {5 \, \left (d x\right )^{\frac {9}{2}} b^{2}}{d^{4}} + \frac {18 \, \left (d x\right )^{\frac {5}{2}} a b}{d^{2}}\right )} a^{4} + 175 \, {\left (\frac {117 \, \left (d x\right )^{\frac {17}{2}} b^{4}}{d^{8}} + \frac {612 \, \left (d x\right )^{\frac {13}{2}} a b^{3}}{d^{6}} + \frac {884 \, \left (d x\right )^{\frac {9}{2}} a^{2} b^{2}}{d^{4}}\right )} a^{2}\right )}}{116025 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 103, normalized size = 0.80 \[ \frac {2\,a^6\,\sqrt {d\,x}}{d}+\frac {2\,b^6\,{\left (d\,x\right )}^{25/2}}{25\,d^{13}}+\frac {10\,a^4\,b^2\,{\left (d\,x\right )}^{9/2}}{3\,d^5}+\frac {40\,a^3\,b^3\,{\left (d\,x\right )}^{13/2}}{13\,d^7}+\frac {30\,a^2\,b^4\,{\left (d\,x\right )}^{17/2}}{17\,d^9}+\frac {12\,a^5\,b\,{\left (d\,x\right )}^{5/2}}{5\,d^3}+\frac {4\,a\,b^5\,{\left (d\,x\right )}^{21/2}}{7\,d^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.94, size = 129, normalized size = 1.00 \[ \frac {2 a^{6} \sqrt {x}}{\sqrt {d}} + \frac {12 a^{5} b x^{\frac {5}{2}}}{5 \sqrt {d}} + \frac {10 a^{4} b^{2} x^{\frac {9}{2}}}{3 \sqrt {d}} + \frac {40 a^{3} b^{3} x^{\frac {13}{2}}}{13 \sqrt {d}} + \frac {30 a^{2} b^{4} x^{\frac {17}{2}}}{17 \sqrt {d}} + \frac {4 a b^{5} x^{\frac {21}{2}}}{7 \sqrt {d}} + \frac {2 b^{6} x^{\frac {25}{2}}}{25 \sqrt {d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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