Optimal. Leaf size=91 \[ \frac{2 x (4 a d+b c)}{15 c^3 d \sqrt{c+d x^2}}+\frac{x (4 a d+b c)}{15 c^2 d \left (c+d x^2\right )^{3/2}}-\frac{x (b c-a d)}{5 c d \left (c+d x^2\right )^{5/2}} \]
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Rubi [A] time = 0.0288756, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {385, 192, 191} \[ \frac{2 x (4 a d+b c)}{15 c^3 d \sqrt{c+d x^2}}+\frac{x (4 a d+b c)}{15 c^2 d \left (c+d x^2\right )^{3/2}}-\frac{x (b c-a d)}{5 c d \left (c+d x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 385
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{a+b x^2}{\left (c+d x^2\right )^{7/2}} \, dx &=-\frac{(b c-a d) x}{5 c d \left (c+d x^2\right )^{5/2}}+\frac{(b c+4 a d) \int \frac{1}{\left (c+d x^2\right )^{5/2}} \, dx}{5 c d}\\ &=-\frac{(b c-a d) x}{5 c d \left (c+d x^2\right )^{5/2}}+\frac{(b c+4 a d) x}{15 c^2 d \left (c+d x^2\right )^{3/2}}+\frac{(2 (b c+4 a d)) \int \frac{1}{\left (c+d x^2\right )^{3/2}} \, dx}{15 c^2 d}\\ &=-\frac{(b c-a d) x}{5 c d \left (c+d x^2\right )^{5/2}}+\frac{(b c+4 a d) x}{15 c^2 d \left (c+d x^2\right )^{3/2}}+\frac{2 (b c+4 a d) x}{15 c^3 d \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [A] time = 0.0224532, size = 59, normalized size = 0.65 \[ \frac{a \left (15 c^2 x+20 c d x^3+8 d^2 x^5\right )+b c x^3 \left (5 c+2 d x^2\right )}{15 c^3 \left (c+d x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 57, normalized size = 0.6 \begin{align*}{\frac{x \left ( 8\,a{d}^{2}{x}^{4}+2\,bcd{x}^{4}+20\,acd{x}^{2}+5\,b{c}^{2}{x}^{2}+15\,a{c}^{2} \right ) }{15\,{c}^{3}} \left ( d{x}^{2}+c \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.970178, size = 139, normalized size = 1.53 \begin{align*} \frac{8 \, a x}{15 \, \sqrt{d x^{2} + c} c^{3}} + \frac{4 \, a x}{15 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} c^{2}} + \frac{a x}{5 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} c} - \frac{b x}{5 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} d} + \frac{2 \, b x}{15 \, \sqrt{d x^{2} + c} c^{2} d} + \frac{b x}{15 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55856, size = 185, normalized size = 2.03 \begin{align*} \frac{{\left (2 \,{\left (b c d + 4 \, a d^{2}\right )} x^{5} + 15 \, a c^{2} x + 5 \,{\left (b c^{2} + 4 \, a c d\right )} x^{3}\right )} \sqrt{d x^{2} + c}}{15 \,{\left (c^{3} d^{3} x^{6} + 3 \, c^{4} d^{2} x^{4} + 3 \, c^{5} d x^{2} + c^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 36.6034, size = 566, normalized size = 6.22 \begin{align*} a \left (\frac{15 c^{5} x}{15 c^{\frac{17}{2}} \sqrt{1 + \frac{d x^{2}}{c}} + 45 c^{\frac{15}{2}} d x^{2} \sqrt{1 + \frac{d x^{2}}{c}} + 45 c^{\frac{13}{2}} d^{2} x^{4} \sqrt{1 + \frac{d x^{2}}{c}} + 15 c^{\frac{11}{2}} d^{3} x^{6} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{35 c^{4} d x^{3}}{15 c^{\frac{17}{2}} \sqrt{1 + \frac{d x^{2}}{c}} + 45 c^{\frac{15}{2}} d x^{2} \sqrt{1 + \frac{d x^{2}}{c}} + 45 c^{\frac{13}{2}} d^{2} x^{4} \sqrt{1 + \frac{d x^{2}}{c}} + 15 c^{\frac{11}{2}} d^{3} x^{6} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{28 c^{3} d^{2} x^{5}}{15 c^{\frac{17}{2}} \sqrt{1 + \frac{d x^{2}}{c}} + 45 c^{\frac{15}{2}} d x^{2} \sqrt{1 + \frac{d x^{2}}{c}} + 45 c^{\frac{13}{2}} d^{2} x^{4} \sqrt{1 + \frac{d x^{2}}{c}} + 15 c^{\frac{11}{2}} d^{3} x^{6} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{8 c^{2} d^{3} x^{7}}{15 c^{\frac{17}{2}} \sqrt{1 + \frac{d x^{2}}{c}} + 45 c^{\frac{15}{2}} d x^{2} \sqrt{1 + \frac{d x^{2}}{c}} + 45 c^{\frac{13}{2}} d^{2} x^{4} \sqrt{1 + \frac{d x^{2}}{c}} + 15 c^{\frac{11}{2}} d^{3} x^{6} \sqrt{1 + \frac{d x^{2}}{c}}}\right ) + b \left (\frac{5 c x^{3}}{15 c^{\frac{9}{2}} \sqrt{1 + \frac{d x^{2}}{c}} + 30 c^{\frac{7}{2}} d x^{2} \sqrt{1 + \frac{d x^{2}}{c}} + 15 c^{\frac{5}{2}} d^{2} x^{4} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{2 d x^{5}}{15 c^{\frac{9}{2}} \sqrt{1 + \frac{d x^{2}}{c}} + 30 c^{\frac{7}{2}} d x^{2} \sqrt{1 + \frac{d x^{2}}{c}} + 15 c^{\frac{5}{2}} d^{2} x^{4} \sqrt{1 + \frac{d x^{2}}{c}}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1769, size = 97, normalized size = 1.07 \begin{align*} \frac{{\left (x^{2}{\left (\frac{2 \,{\left (b c d^{3} + 4 \, a d^{4}\right )} x^{2}}{c^{3} d^{2}} + \frac{5 \,{\left (b c^{2} d^{2} + 4 \, a c d^{3}\right )}}{c^{3} d^{2}}\right )} + \frac{15 \, a}{c}\right )} x}{15 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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