Optimal. Leaf size=293 \[ \frac{2 b^5 (d x)^{21/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{21 d^{11} \left (a+b x^2\right )}+\frac{10 a b^4 (d x)^{17/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 d^9 \left (a+b x^2\right )}+\frac{20 a^2 b^3 (d x)^{13/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 d^7 \left (a+b x^2\right )}+\frac{20 a^3 b^2 (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 d^5 \left (a+b x^2\right )}+\frac{2 a^4 b (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^3 \left (a+b x^2\right )}+\frac{2 a^5 \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d \left (a+b x^2\right )} \]
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Rubi [A] time = 0.078523, antiderivative size = 293, normalized size of antiderivative = 1., 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} \[ \frac{2 b^5 (d x)^{21/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{21 d^{11} \left (a+b x^2\right )}+\frac{10 a b^4 (d x)^{17/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 d^9 \left (a+b x^2\right )}+\frac{20 a^2 b^3 (d x)^{13/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 d^7 \left (a+b x^2\right )}+\frac{20 a^3 b^2 (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 d^5 \left (a+b x^2\right )}+\frac{2 a^4 b (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^3 \left (a+b x^2\right )}+\frac{2 a^5 \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 1112
Rule 270
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{\sqrt{d x}} \, 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}{\sqrt{d x}} \, 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}{\sqrt{d x}}+\frac{5 a^4 b^6 (d x)^{3/2}}{d^2}+\frac{10 a^3 b^7 (d x)^{7/2}}{d^4}+\frac{10 a^2 b^8 (d x)^{11/2}}{d^6}+\frac{5 a b^9 (d x)^{15/2}}{d^8}+\frac{b^{10} (d x)^{19/2}}{d^{10}}\right ) \, dx}{b^4 \left (a b+b^2 x^2\right )}\\ &=\frac{2 a^5 \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d \left (a+b x^2\right )}+\frac{2 a^4 b (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^3 \left (a+b x^2\right )}+\frac{20 a^3 b^2 (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 d^5 \left (a+b x^2\right )}+\frac{20 a^2 b^3 (d x)^{13/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 d^7 \left (a+b x^2\right )}+\frac{10 a b^4 (d x)^{17/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 d^9 \left (a+b x^2\right )}+\frac{2 b^5 (d x)^{21/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{21 d^{11} \left (a+b x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.0308095, size = 88, normalized size = 0.3 \[ \frac{2 \sqrt{\left (a+b x^2\right )^2} \left (10710 a^2 b^3 x^7+15470 a^3 b^2 x^5+13923 a^4 b x^3+13923 a^5 x+4095 a b^4 x^9+663 b^5 x^{11}\right )}{13923 \sqrt{d x} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.166, size = 83, normalized size = 0.3 \begin{align*}{\frac{2\, \left ( 663\,{b}^{5}{x}^{10}+4095\,a{b}^{4}{x}^{8}+10710\,{a}^{2}{b}^{3}{x}^{6}+15470\,{b}^{2}{a}^{3}{x}^{4}+13923\,{a}^{4}b{x}^{2}+13923\,{a}^{5} \right ) x}{13923\, \left ( b{x}^{2}+a \right ) ^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}{\frac{1}{\sqrt{dx}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00069, size = 204, normalized size = 0.7 \begin{align*} \frac{2 \,{\left (195 \,{\left (17 \, b^{5} \sqrt{d} x^{3} + 21 \, a b^{4} \sqrt{d} x\right )} x^{\frac{15}{2}} + 1260 \,{\left (13 \, a b^{4} \sqrt{d} x^{3} + 17 \, a^{2} b^{3} \sqrt{d} x\right )} x^{\frac{11}{2}} + 3570 \,{\left (9 \, a^{2} b^{3} \sqrt{d} x^{3} + 13 \, a^{3} b^{2} \sqrt{d} x\right )} x^{\frac{7}{2}} + 6188 \,{\left (5 \, a^{3} b^{2} \sqrt{d} x^{3} + 9 \, a^{4} b \sqrt{d} x\right )} x^{\frac{3}{2}} + \frac{13923 \,{\left (a^{4} b \sqrt{d} x^{3} + 5 \, a^{5} \sqrt{d} x\right )}}{\sqrt{x}}\right )}}{69615 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49227, size = 166, normalized size = 0.57 \begin{align*} \frac{2 \,{\left (663 \, b^{5} x^{10} + 4095 \, a b^{4} x^{8} + 10710 \, a^{2} b^{3} x^{6} + 15470 \, a^{3} b^{2} x^{4} + 13923 \, a^{4} b x^{2} + 13923 \, a^{5}\right )} \sqrt{d x}}{13923 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}}{\sqrt{d x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27013, size = 185, normalized size = 0.63 \begin{align*} \frac{2 \,{\left (663 \, \sqrt{d x} b^{5} x^{10} \mathrm{sgn}\left (b x^{2} + a\right ) + 4095 \, \sqrt{d x} a b^{4} x^{8} \mathrm{sgn}\left (b x^{2} + a\right ) + 10710 \, \sqrt{d x} a^{2} b^{3} x^{6} \mathrm{sgn}\left (b x^{2} + a\right ) + 15470 \, \sqrt{d x} a^{3} b^{2} x^{4} \mathrm{sgn}\left (b x^{2} + a\right ) + 13923 \, \sqrt{d x} a^{4} b x^{2} \mathrm{sgn}\left (b x^{2} + a\right ) + 13923 \, \sqrt{d x} a^{5} \mathrm{sgn}\left (b x^{2} + a\right )\right )}}{13923 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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