Optimal. Leaf size=109 \[ \frac{(d x-c)^{5/2} (c+d x)^{5/2} \left (a d^2+2 b c^2\right )}{5 d^6}+\frac{c^2 (d x-c)^{3/2} (c+d x)^{3/2} \left (a d^2+b c^2\right )}{3 d^6}+\frac{b (d x-c)^{7/2} (c+d x)^{7/2}}{7 d^6} \]
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Rubi [A] time = 0.0867472, antiderivative size = 118, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {460, 100, 12, 74} \[ \frac{x^2 (d x-c)^{3/2} (c+d x)^{3/2} \left (7 a d^2+4 b c^2\right )}{35 d^4}+\frac{2 c^2 (d x-c)^{3/2} (c+d x)^{3/2} \left (7 a d^2+4 b c^2\right )}{105 d^6}+\frac{b x^4 (d x-c)^{3/2} (c+d x)^{3/2}}{7 d^2} \]
Antiderivative was successfully verified.
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Rule 460
Rule 100
Rule 12
Rule 74
Rubi steps
\begin{align*} \int x^3 \sqrt{-c+d x} \sqrt{c+d x} \left (a+b x^2\right ) \, dx &=\frac{b x^4 (-c+d x)^{3/2} (c+d x)^{3/2}}{7 d^2}-\frac{1}{7} \left (-7 a-\frac{4 b c^2}{d^2}\right ) \int x^3 \sqrt{-c+d x} \sqrt{c+d x} \, dx\\ &=\frac{\left (4 b c^2+7 a d^2\right ) x^2 (-c+d x)^{3/2} (c+d x)^{3/2}}{35 d^4}+\frac{b x^4 (-c+d x)^{3/2} (c+d x)^{3/2}}{7 d^2}+\frac{\left (4 b c^2+7 a d^2\right ) \int 2 c^2 x \sqrt{-c+d x} \sqrt{c+d x} \, dx}{35 d^4}\\ &=\frac{\left (4 b c^2+7 a d^2\right ) x^2 (-c+d x)^{3/2} (c+d x)^{3/2}}{35 d^4}+\frac{b x^4 (-c+d x)^{3/2} (c+d x)^{3/2}}{7 d^2}+\frac{\left (2 c^2 \left (4 b c^2+7 a d^2\right )\right ) \int x \sqrt{-c+d x} \sqrt{c+d x} \, dx}{35 d^4}\\ &=\frac{2 c^2 \left (4 b c^2+7 a d^2\right ) (-c+d x)^{3/2} (c+d x)^{3/2}}{105 d^6}+\frac{\left (4 b c^2+7 a d^2\right ) x^2 (-c+d x)^{3/2} (c+d x)^{3/2}}{35 d^4}+\frac{b x^4 (-c+d x)^{3/2} (c+d x)^{3/2}}{7 d^2}\\ \end{align*}
Mathematica [A] time = 0.0528235, size = 88, normalized size = 0.81 \[ \frac{\sqrt{d x-c} \sqrt{c+d x} \left (d^2 x^2-c^2\right ) \left (7 a d^2 \left (2 c^2+3 d^2 x^2\right )+b \left (12 c^2 d^2 x^2+8 c^4+15 d^4 x^4\right )\right )}{105 d^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 68, normalized size = 0.6 \begin{align*}{\frac{15\,b{d}^{4}{x}^{4}+21\,a{d}^{4}{x}^{2}+12\,b{c}^{2}{d}^{2}{x}^{2}+14\,a{c}^{2}{d}^{2}+8\,b{c}^{4}}{105\,{d}^{6}} \left ( dx+c \right ) ^{{\frac{3}{2}}} \left ( dx-c \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.984139, size = 167, normalized size = 1.53 \begin{align*} \frac{{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} b x^{4}}{7 \, d^{2}} + \frac{4 \,{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} b c^{2} x^{2}}{35 \, d^{4}} + \frac{{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} a x^{2}}{5 \, d^{2}} + \frac{8 \,{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} b c^{4}}{105 \, d^{6}} + \frac{2 \,{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} a c^{2}}{15 \, d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67887, size = 193, normalized size = 1.77 \begin{align*} \frac{{\left (15 \, b d^{6} x^{6} - 8 \, b c^{6} - 14 \, a c^{4} d^{2} - 3 \,{\left (b c^{2} d^{4} - 7 \, a d^{6}\right )} x^{4} -{\left (4 \, b c^{4} d^{2} + 7 \, a c^{2} d^{4}\right )} x^{2}\right )} \sqrt{d x + c} \sqrt{d x - c}}{105 \, d^{6}} \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 x^{3} \left (a + b x^{2}\right ) \sqrt{- c + d x} \sqrt{c + d x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19113, size = 227, normalized size = 2.08 \begin{align*} \frac{7 \,{\left ({\left (d x + c\right )}{\left (3 \,{\left (d x + c\right )}{\left (\frac{d x + c}{d^{3}} - \frac{4 \, c}{d^{3}}\right )} + \frac{17 \, c^{2}}{d^{3}}\right )} - \frac{10 \, c^{3}}{d^{3}}\right )}{\left (d x + c\right )}^{\frac{3}{2}} \sqrt{d x - c} a +{\left ({\left (3 \,{\left ({\left (d x + c\right )}{\left (5 \,{\left (d x + c\right )}{\left (\frac{d x + c}{d^{5}} - \frac{6 \, c}{d^{5}}\right )} + \frac{74 \, c^{2}}{d^{5}}\right )} - \frac{96 \, c^{3}}{d^{5}}\right )}{\left (d x + c\right )} + \frac{203 \, c^{4}}{d^{5}}\right )}{\left (d x + c\right )} - \frac{70 \, c^{5}}{d^{5}}\right )}{\left (d x + c\right )}^{\frac{3}{2}} \sqrt{d x - c} b}{105 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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