Optimal. Leaf size=127 \[ -\frac {2 a^6}{5 d (d x)^{5/2}}-\frac {12 a^5 b}{d^3 \sqrt {d x}}+\frac {10 a^4 b^2 (d x)^{3/2}}{d^5}+\frac {40 a^3 b^3 (d x)^{7/2}}{7 d^7}+\frac {30 a^2 b^4 (d x)^{11/2}}{11 d^9}+\frac {4 a b^5 (d x)^{15/2}}{5 d^{11}}+\frac {2 b^6 (d x)^{19/2}}{19 d^{13}} \]
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Rubi [A] time = 0.06, antiderivative size = 127, 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} \[ \frac {30 a^2 b^4 (d x)^{11/2}}{11 d^9}+\frac {40 a^3 b^3 (d x)^{7/2}}{7 d^7}+\frac {10 a^4 b^2 (d x)^{3/2}}{d^5}-\frac {12 a^5 b}{d^3 \sqrt {d x}}-\frac {2 a^6}{5 d (d x)^{5/2}}+\frac {4 a b^5 (d x)^{15/2}}{5 d^{11}}+\frac {2 b^6 (d x)^{19/2}}{19 d^{13}} \]
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}{(d x)^{7/2}} \, dx &=\frac {\int \frac {\left (a b+b^2 x^2\right )^6}{(d x)^{7/2}} \, dx}{b^6}\\ &=\frac {\int \left (\frac {a^6 b^6}{(d x)^{7/2}}+\frac {6 a^5 b^7}{d^2 (d x)^{3/2}}+\frac {15 a^4 b^8 \sqrt {d x}}{d^4}+\frac {20 a^3 b^9 (d x)^{5/2}}{d^6}+\frac {15 a^2 b^{10} (d x)^{9/2}}{d^8}+\frac {6 a b^{11} (d x)^{13/2}}{d^{10}}+\frac {b^{12} (d x)^{17/2}}{d^{12}}\right ) \, dx}{b^6}\\ &=-\frac {2 a^6}{5 d (d x)^{5/2}}-\frac {12 a^5 b}{d^3 \sqrt {d x}}+\frac {10 a^4 b^2 (d x)^{3/2}}{d^5}+\frac {40 a^3 b^3 (d x)^{7/2}}{7 d^7}+\frac {30 a^2 b^4 (d x)^{11/2}}{11 d^9}+\frac {4 a b^5 (d x)^{15/2}}{5 d^{11}}+\frac {2 b^6 (d x)^{19/2}}{19 d^{13}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 82, normalized size = 0.65 \[ \frac {2 \sqrt {d x} \left (-1463 a^6-43890 a^5 b x^2+36575 a^4 b^2 x^4+20900 a^3 b^3 x^6+9975 a^2 b^4 x^8+2926 a b^5 x^{10}+385 b^6 x^{12}\right )}{7315 d^4 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 78, normalized size = 0.61 \[ \frac {2 \, {\left (385 \, b^{6} x^{12} + 2926 \, a b^{5} x^{10} + 9975 \, a^{2} b^{4} x^{8} + 20900 \, a^{3} b^{3} x^{6} + 36575 \, a^{4} b^{2} x^{4} - 43890 \, a^{5} b x^{2} - 1463 \, a^{6}\right )} \sqrt {d x}}{7315 \, d^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 133, normalized size = 1.05 \[ -\frac {2 \, {\left (\frac {1463 \, {\left (30 \, a^{5} b d^{3} x^{2} + a^{6} d^{3}\right )}}{\sqrt {d x} d^{2} x^{2}} - \frac {385 \, \sqrt {d x} b^{6} d^{171} x^{9} + 2926 \, \sqrt {d x} a b^{5} d^{171} x^{7} + 9975 \, \sqrt {d x} a^{2} b^{4} d^{171} x^{5} + 20900 \, \sqrt {d x} a^{3} b^{3} d^{171} x^{3} + 36575 \, \sqrt {d x} a^{4} b^{2} d^{171} x}{d^{171}}\right )}}{7315 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 74, normalized size = 0.58 \[ -\frac {2 \left (-385 b^{6} x^{12}-2926 a \,b^{5} x^{10}-9975 a^{2} b^{4} x^{8}-20900 a^{3} b^{3} x^{6}-36575 a^{4} b^{2} x^{4}+43890 a^{5} b \,x^{2}+1463 a^{6}\right ) x}{7315 \left (d x \right )^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 114, normalized size = 0.90 \[ -\frac {2 \, {\left (\frac {1463 \, {\left (30 \, a^{5} b d^{2} x^{2} + a^{6} d^{2}\right )}}{\left (d x\right )^{\frac {5}{2}} d^{2}} - \frac {385 \, \left (d x\right )^{\frac {19}{2}} b^{6} + 2926 \, \left (d x\right )^{\frac {15}{2}} a b^{5} d^{2} + 9975 \, \left (d x\right )^{\frac {11}{2}} a^{2} b^{4} d^{4} + 20900 \, \left (d x\right )^{\frac {7}{2}} a^{3} b^{3} d^{6} + 36575 \, \left (d x\right )^{\frac {3}{2}} a^{4} b^{2} d^{8}}{d^{12}}\right )}}{7315 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 107, normalized size = 0.84 \[ \frac {2\,b^6\,{\left (d\,x\right )}^{19/2}}{19\,d^{13}}-\frac {\frac {2\,a^6\,d^2}{5}+12\,b\,a^5\,d^2\,x^2}{d^3\,{\left (d\,x\right )}^{5/2}}+\frac {10\,a^4\,b^2\,{\left (d\,x\right )}^{3/2}}{d^5}+\frac {40\,a^3\,b^3\,{\left (d\,x\right )}^{7/2}}{7\,d^7}+\frac {30\,a^2\,b^4\,{\left (d\,x\right )}^{11/2}}{11\,d^9}+\frac {4\,a\,b^5\,{\left (d\,x\right )}^{15/2}}{5\,d^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.56, size = 128, normalized size = 1.01 \[ - \frac {2 a^{6}}{5 d^{\frac {7}{2}} x^{\frac {5}{2}}} - \frac {12 a^{5} b}{d^{\frac {7}{2}} \sqrt {x}} + \frac {10 a^{4} b^{2} x^{\frac {3}{2}}}{d^{\frac {7}{2}}} + \frac {40 a^{3} b^{3} x^{\frac {7}{2}}}{7 d^{\frac {7}{2}}} + \frac {30 a^{2} b^{4} x^{\frac {11}{2}}}{11 d^{\frac {7}{2}}} + \frac {4 a b^{5} x^{\frac {15}{2}}}{5 d^{\frac {7}{2}}} + \frac {2 b^{6} x^{\frac {19}{2}}}{19 d^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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