Optimal. Leaf size=91 \[ -\frac {2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^3 \sqrt {d x} \left (a+b x^2\right )}-\frac {2 a \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 d (d x)^{5/2} \left (a+b x^2\right )} \]
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Rubi [A] time = 0.03, antiderivative size = 91, 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, 14} \begin {gather*} -\frac {2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^3 \sqrt {d x} \left (a+b x^2\right )}-\frac {2 a \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 d (d x)^{5/2} \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 1112
Rubi steps
\begin {align*} \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{(d x)^{7/2}} \, dx &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {a b+b^2 x^2}{(d x)^{7/2}} \, dx}{a b+b^2 x^2}\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (\frac {a b}{(d x)^{7/2}}+\frac {b^2}{d^2 (d x)^{3/2}}\right ) \, dx}{a b+b^2 x^2}\\ &=-\frac {2 a \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 d (d x)^{5/2} \left (a+b x^2\right )}-\frac {2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^3 \sqrt {d x} \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 42, normalized size = 0.46 \begin {gather*} -\frac {2 x \sqrt {\left (a+b x^2\right )^2} \left (a+5 b x^2\right )}{5 (d x)^{7/2} \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 29.29, size = 67, normalized size = 0.74 \begin {gather*} -\frac {2 \left (a d^2+b d^2 x^2\right ) \left (a d^2+5 b d^2 x^2\right )}{5 d^5 (d x)^{5/2} \sqrt {\frac {\left (a d^2+b d^2 x^2\right )^2}{d^4}}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.19, size = 21, normalized size = 0.23 \begin {gather*} -\frac {2 \, {\left (5 \, b x^{2} + a\right )} \sqrt {d x}}{5 \, d^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 44, normalized size = 0.48 \begin {gather*} -\frac {2 \, {\left (5 \, b d^{3} x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + a d^{3} \mathrm {sgn}\left (b x^{2} + a\right )\right )}}{5 \, \sqrt {d x} d^{6} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 37, normalized size = 0.41 \begin {gather*} -\frac {2 \left (5 b \,x^{2}+a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}\, x}{5 \left (b \,x^{2}+a \right ) \left (d x \right )^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 25, normalized size = 0.27 \begin {gather*} -\frac {2 \, {\left (5 \, b d^{2} x^{2} + a d^{2}\right )}}{5 \, \left (d x\right )^{\frac {5}{2}} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.32, size = 56, normalized size = 0.62 \begin {gather*} -\frac {\left (\frac {2\,x^2}{d^3}+\frac {2\,a}{5\,b\,d^3}\right )\,\sqrt {{\left (b\,x^2+a\right )}^2}}{x^4\,\sqrt {d\,x}+\frac {a\,x^2\,\sqrt {d\,x}}{b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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