Optimal. Leaf size=297 \[ \frac {2 b^5 (d x)^{23/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{23 d^{11} \left (a+b x^2\right )}+\frac {10 a b^4 (d x)^{19/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{19 d^9 \left (a+b x^2\right )}+\frac {4 a^2 b^3 (d x)^{15/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d^7 \left (a+b x^2\right )}+\frac {2 a^5 (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d \left (a+b x^2\right )}+\frac {10 a^4 b (d x)^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 d^3 \left (a+b x^2\right )}+\frac {20 a^3 b^2 (d x)^{11/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 d^5 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.08, antiderivative size = 297, normalized size of antiderivative = 1.00, 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} \[ \frac {2 b^5 (d x)^{23/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{23 d^{11} \left (a+b x^2\right )}+\frac {10 a b^4 (d x)^{19/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{19 d^9 \left (a+b x^2\right )}+\frac {4 a^2 b^3 (d x)^{15/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d^7 \left (a+b x^2\right )}+\frac {20 a^3 b^2 (d x)^{11/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 d^5 \left (a+b x^2\right )}+\frac {10 a^4 b (d x)^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 d^3 \left (a+b x^2\right )}+\frac {2 a^5 (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 270
Rule 1112
Rubi steps
\begin {align*} \int \sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \sqrt {d x} \left (a b+b^2 x^2\right )^5 \, dx}{b^4 \left (a b+b^2 x^2\right )}\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (a^5 b^5 \sqrt {d x}+\frac {5 a^4 b^6 (d x)^{5/2}}{d^2}+\frac {10 a^3 b^7 (d x)^{9/2}}{d^4}+\frac {10 a^2 b^8 (d x)^{13/2}}{d^6}+\frac {5 a b^9 (d x)^{17/2}}{d^8}+\frac {b^{10} (d x)^{21/2}}{d^{10}}\right ) \, dx}{b^4 \left (a b+b^2 x^2\right )}\\ &=\frac {2 a^5 (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d \left (a+b x^2\right )}+\frac {10 a^4 b (d x)^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 d^3 \left (a+b x^2\right )}+\frac {20 a^3 b^2 (d x)^{11/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 d^5 \left (a+b x^2\right )}+\frac {4 a^2 b^3 (d x)^{15/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d^7 \left (a+b x^2\right )}+\frac {10 a b^4 (d x)^{19/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{19 d^9 \left (a+b x^2\right )}+\frac {2 b^5 (d x)^{23/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{23 d^{11} \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 88, normalized size = 0.30 \[ \frac {2 \sqrt {d x} \sqrt {\left (a+b x^2\right )^2} \left (33649 a^5 x+72105 a^4 b x^3+91770 a^3 b^2 x^5+67298 a^2 b^3 x^7+26565 a b^4 x^9+4389 b^5 x^{11}\right )}{100947 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 62, normalized size = 0.21 \[ \frac {2}{100947} \, {\left (4389 \, b^{5} x^{11} + 26565 \, a b^{4} x^{9} + 67298 \, a^{2} b^{3} x^{7} + 91770 \, a^{3} b^{2} x^{5} + 72105 \, a^{4} b x^{3} + 33649 \, a^{5} x\right )} \sqrt {d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 133, normalized size = 0.45 \[ \frac {2}{23} \, \sqrt {d x} b^{5} x^{11} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {10}{19} \, \sqrt {d x} a b^{4} x^{9} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {4}{3} \, \sqrt {d x} a^{2} b^{3} x^{7} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {20}{11} \, \sqrt {d x} a^{3} b^{2} x^{5} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {10}{7} \, \sqrt {d x} a^{4} b x^{3} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {2}{3} \, \sqrt {d x} a^{5} x \mathrm {sgn}\left (b x^{2} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 83, normalized size = 0.28 \[ \frac {2 \left (4389 b^{5} x^{10}+26565 a \,b^{4} x^{8}+67298 a^{2} b^{3} x^{6}+91770 a^{3} b^{2} x^{4}+72105 a^{4} b \,x^{2}+33649 a^{5}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}} \sqrt {d x}\, x}{100947 \left (b \,x^{2}+a \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.53, size = 147, normalized size = 0.49 \[ \frac {2}{437} \, {\left (19 \, b^{5} \sqrt {d} x^{3} + 23 \, a b^{4} \sqrt {d} x\right )} x^{\frac {17}{2}} + \frac {8}{285} \, {\left (15 \, a b^{4} \sqrt {d} x^{3} + 19 \, a^{2} b^{3} \sqrt {d} x\right )} x^{\frac {13}{2}} + \frac {4}{55} \, {\left (11 \, a^{2} b^{3} \sqrt {d} x^{3} + 15 \, a^{3} b^{2} \sqrt {d} x\right )} x^{\frac {9}{2}} + \frac {8}{77} \, {\left (7 \, a^{3} b^{2} \sqrt {d} x^{3} + 11 \, a^{4} b \sqrt {d} x\right )} x^{\frac {5}{2}} + \frac {2}{21} \, {\left (3 \, a^{4} b \sqrt {d} x^{3} + 7 \, a^{5} \sqrt {d} x\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {d\,x}\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {d x} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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