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 = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.42, size = 53, normalized size = 0.87 \[ \frac {2 \, {\left (15 \, b^{2} d x^{6} + 35 \, {\left (b^{2} c + 2 \, a b d\right )} x^{4} - 21 \, a^{2} c - 105 \, {\left (2 \, a b c + a^{2} d\right )} x^{2}\right )}}{105 \, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 55, normalized size = 0.90 \[ \frac {2}{7} \, b^{2} d x^{\frac {7}{2}} + \frac {2}{3} \, b^{2} c x^{\frac {3}{2}} + \frac {4}{3} \, a b d x^{\frac {3}{2}} - \frac {2 \, {\left (10 \, a b c x^{2} + 5 \, a^{2} d x^{2} + a^{2} c\right )}}{5 \, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 56, normalized size = 0.92 \[ -\frac {2 \left (-15 b^{2} d \,x^{6}-70 a b d \,x^{4}-35 b^{2} c \,x^{4}+105 a^{2} d \,x^{2}+210 a b c \,x^{2}+21 a^{2} c \right )}{105 x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 53, normalized size = 0.87 \[ \frac {2}{7} \, b^{2} d x^{\frac {7}{2}} + \frac {2}{3} \, {\left (b^{2} c + 2 \, a b d\right )} x^{\frac {3}{2}} - \frac {2 \, {\left (a^{2} c + 5 \, {\left (2 \, a b c + a^{2} d\right )} x^{2}\right )}}{5 \, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 55, normalized size = 0.90 \[ -\frac {210\,d\,a^2\,x^2+42\,c\,a^2-140\,d\,a\,b\,x^4+420\,c\,a\,b\,x^2-30\,d\,b^2\,x^6-70\,c\,b^2\,x^4}{105\,x^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.83, size = 76, normalized size = 1.25 \[ - \frac {2 a^{2} c}{5 x^{\frac {5}{2}}} - \frac {2 a^{2} d}{\sqrt {x}} - \frac {4 a b c}{\sqrt {x}} + \frac {4 a b d x^{\frac {3}{2}}}{3} + \frac {2 b^{2} c x^{\frac {3}{2}}}{3} + \frac {2 b^{2} d x^{\frac {7}{2}}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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