Optimal. Leaf size=61 \[ -\frac{2 a^2 c}{5 x^{5/2}}+\frac{2}{3} b x^{3/2} (2 a d+b c)-\frac{2 a (a d+2 b c)}{\sqrt{x}}+\frac{2}{7} b^2 d x^{7/2} \]
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Rubi [A] time = 0.0294659, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {448} \[ -\frac{2 a^2 c}{5 x^{5/2}}+\frac{2}{3} b x^{3/2} (2 a d+b c)-\frac{2 a (a d+2 b c)}{\sqrt{x}}+\frac{2}{7} b^2 d x^{7/2} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x^{7/2}} \, dx &=\int \left (\frac{a^2 c}{x^{7/2}}+\frac{a (2 b c+a d)}{x^{3/2}}+b (b c+2 a d) \sqrt{x}+b^2 d x^{5/2}\right ) \, dx\\ &=-\frac{2 a^2 c}{5 x^{5/2}}-\frac{2 a (2 b c+a d)}{\sqrt{x}}+\frac{2}{3} b (b c+2 a d) x^{3/2}+\frac{2}{7} b^2 d x^{7/2}\\ \end{align*}
Mathematica [A] time = 0.0163813, size = 57, normalized size = 0.93 \[ \frac{-42 a^2 \left (c+5 d x^2\right )+140 a b x^2 \left (d x^2-3 c\right )+10 b^2 x^4 \left (7 c+3 d x^2\right )}{105 x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 56, normalized size = 0.9 \begin{align*} -{\frac{-30\,{b}^{2}d{x}^{6}-140\,{x}^{4}abd-70\,{b}^{2}c{x}^{4}+210\,{x}^{2}{a}^{2}d+420\,abc{x}^{2}+42\,{a}^{2}c}{105}{x}^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11314, size = 72, normalized size = 1.18 \begin{align*} \frac{2}{7} \, b^{2} d x^{\frac{7}{2}} + \frac{2}{3} \,{\left (b^{2} c + 2 \, a b d\right )} x^{\frac{3}{2}} - \frac{2 \,{\left (a^{2} c + 5 \,{\left (2 \, a b c + a^{2} d\right )} x^{2}\right )}}{5 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.861375, size = 130, normalized size = 2.13 \begin{align*} \frac{2 \,{\left (15 \, b^{2} d x^{6} + 35 \,{\left (b^{2} c + 2 \, a b d\right )} x^{4} - 21 \, a^{2} c - 105 \,{\left (2 \, a b c + a^{2} d\right )} x^{2}\right )}}{105 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.1883, size = 76, normalized size = 1.25 \begin{align*} - \frac{2 a^{2} c}{5 x^{\frac{5}{2}}} - \frac{2 a^{2} d}{\sqrt{x}} - \frac{4 a b c}{\sqrt{x}} + \frac{4 a b d x^{\frac{3}{2}}}{3} + \frac{2 b^{2} c x^{\frac{3}{2}}}{3} + \frac{2 b^{2} d x^{\frac{7}{2}}}{7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13856, size = 74, normalized size = 1.21 \begin{align*} \frac{2}{7} \, b^{2} d x^{\frac{7}{2}} + \frac{2}{3} \, b^{2} c x^{\frac{3}{2}} + \frac{4}{3} \, a b d x^{\frac{3}{2}} - \frac{2 \,{\left (10 \, a b c x^{2} + 5 \, a^{2} d x^{2} + a^{2} c\right )}}{5 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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