Optimal. Leaf size=293 \[ \frac {2 b^5 (d x)^{17/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{17 d^{11} \left (a+b x^2\right )}+\frac {10 a b^4 (d x)^{13/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{13 d^9 \left (a+b x^2\right )}+\frac {20 a^2 b^3 (d x)^{9/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 d^7 \left (a+b x^2\right )}-\frac {2 a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d (d x)^{3/2} \left (a+b x^2\right )}+\frac {10 a^4 b \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^3 \left (a+b x^2\right )}+\frac {4 a^3 b^2 (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^5 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.08, antiderivative size = 293, 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} \[ \frac {2 b^5 (d x)^{17/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{17 d^{11} \left (a+b x^2\right )}+\frac {10 a b^4 (d x)^{13/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{13 d^9 \left (a+b x^2\right )}+\frac {20 a^2 b^3 (d x)^{9/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 d^7 \left (a+b x^2\right )}+\frac {4 a^3 b^2 (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^5 \left (a+b x^2\right )}+\frac {10 a^4 b \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^3 \left (a+b x^2\right )}-\frac {2 a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d (d x)^{3/2} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 270
Rule 1112
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{(d x)^{5/2}} \, dx &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {\left (a b+b^2 x^2\right )^5}{(d x)^{5/2}} \, 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 (\frac {a^5 b^5}{(d x)^{5/2}}+\frac {5 a^4 b^6}{d^2 \sqrt {d x}}+\frac {10 a^3 b^7 (d x)^{3/2}}{d^4}+\frac {10 a^2 b^8 (d x)^{7/2}}{d^6}+\frac {5 a b^9 (d x)^{11/2}}{d^8}+\frac {b^{10} (d x)^{15/2}}{d^{10}}\right ) \, dx}{b^4 \left (a b+b^2 x^2\right )}\\ &=-\frac {2 a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d (d x)^{3/2} \left (a+b x^2\right )}+\frac {10 a^4 b \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^3 \left (a+b x^2\right )}+\frac {4 a^3 b^2 (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^5 \left (a+b x^2\right )}+\frac {20 a^2 b^3 (d x)^{9/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 d^7 \left (a+b x^2\right )}+\frac {10 a b^4 (d x)^{13/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{13 d^9 \left (a+b x^2\right )}+\frac {2 b^5 (d x)^{17/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{17 d^{11} \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 88, normalized size = 0.30 \[ \frac {2 x \sqrt {\left (a+b x^2\right )^2} \left (-663 a^5+9945 a^4 b x^2+3978 a^3 b^2 x^4+2210 a^2 b^3 x^6+765 a b^4 x^8+117 b^5 x^{10}\right )}{1989 (d x)^{5/2} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 67, normalized size = 0.23 \[ \frac {2 \, {\left (117 \, b^{5} x^{10} + 765 \, a b^{4} x^{8} + 2210 \, a^{2} b^{3} x^{6} + 3978 \, a^{3} b^{2} x^{4} + 9945 \, a^{4} b x^{2} - 663 \, a^{5}\right )} \sqrt {d x}}{1989 \, d^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 159, normalized size = 0.54 \[ -\frac {2 \, {\left (\frac {663 \, a^{5} d \mathrm {sgn}\left (b x^{2} + a\right )}{\sqrt {d x} x} - \frac {117 \, \sqrt {d x} b^{5} d^{136} x^{8} \mathrm {sgn}\left (b x^{2} + a\right ) + 765 \, \sqrt {d x} a b^{4} d^{136} x^{6} \mathrm {sgn}\left (b x^{2} + a\right ) + 2210 \, \sqrt {d x} a^{2} b^{3} d^{136} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 3978 \, \sqrt {d x} a^{3} b^{2} d^{136} x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 9945 \, \sqrt {d x} a^{4} b d^{136} \mathrm {sgn}\left (b x^{2} + a\right )}{d^{136}}\right )}}{1989 \, d^{3}} \]
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 \[ -\frac {2 \left (-117 b^{5} x^{10}-765 a \,b^{4} x^{8}-2210 a^{2} b^{3} x^{6}-3978 a^{3} b^{2} x^{4}-9945 a^{4} b \,x^{2}+663 a^{5}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}} x}{1989 \left (b \,x^{2}+a \right )^{5} \left (d x \right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.54, size = 151, normalized size = 0.52 \[ \frac {2 \, {\left (45 \, {\left (13 \, b^{5} \sqrt {d} x^{3} + 17 \, a b^{4} \sqrt {d} x\right )} x^{\frac {11}{2}} + 340 \, {\left (9 \, a b^{4} \sqrt {d} x^{3} + 13 \, a^{2} b^{3} \sqrt {d} x\right )} x^{\frac {7}{2}} + 1326 \, {\left (5 \, a^{2} b^{3} \sqrt {d} x^{3} + 9 \, a^{3} b^{2} \sqrt {d} x\right )} x^{\frac {3}{2}} + \frac {7956 \, {\left (a^{3} b^{2} \sqrt {d} x^{3} + 5 \, a^{4} b \sqrt {d} x\right )}}{\sqrt {x}} + \frac {3315 \, {\left (3 \, a^{4} b \sqrt {d} x^{3} - a^{5} \sqrt {d} x\right )}}{x^{\frac {5}{2}}}\right )}}{9945 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.56, size = 116, normalized size = 0.40 \[ \frac {\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}\,\left (\frac {10\,a^4\,x^2}{d^2}-\frac {2\,a^5}{3\,b\,d^2}+\frac {2\,b^4\,x^{10}}{17\,d^2}+\frac {4\,a^3\,b\,x^4}{d^2}+\frac {10\,a\,b^3\,x^8}{13\,d^2}+\frac {20\,a^2\,b^2\,x^6}{9\,d^2}\right )}{x^3\,\sqrt {d\,x}+\frac {a\,x\,\sqrt {d\,x}}{b}} \]
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 \[ \int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}{\left (d x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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