Optimal. Leaf size=200 \[ \frac{6 c^2 (d+e x)^{5/2} \left (a e^2+5 c d^2\right )}{5 e^7}-\frac{8 c^2 d (d+e x)^{3/2} \left (3 a e^2+5 c d^2\right )}{3 e^7}+\frac{6 c \sqrt{d+e x} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7}+\frac{12 c d \left (a e^2+c d^2\right )^2}{e^7 \sqrt{d+e x}}-\frac{2 \left (a e^2+c d^2\right )^3}{3 e^7 (d+e x)^{3/2}}+\frac{2 c^3 (d+e x)^{9/2}}{9 e^7}-\frac{12 c^3 d (d+e x)^{7/2}}{7 e^7} \]
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Rubi [A] time = 0.082786, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {697} \[ \frac{6 c^2 (d+e x)^{5/2} \left (a e^2+5 c d^2\right )}{5 e^7}-\frac{8 c^2 d (d+e x)^{3/2} \left (3 a e^2+5 c d^2\right )}{3 e^7}+\frac{6 c \sqrt{d+e x} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7}+\frac{12 c d \left (a e^2+c d^2\right )^2}{e^7 \sqrt{d+e x}}-\frac{2 \left (a e^2+c d^2\right )^3}{3 e^7 (d+e x)^{3/2}}+\frac{2 c^3 (d+e x)^{9/2}}{9 e^7}-\frac{12 c^3 d (d+e x)^{7/2}}{7 e^7} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin{align*} \int \frac{\left (a+c x^2\right )^3}{(d+e x)^{5/2}} \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^3}{e^6 (d+e x)^{5/2}}-\frac{6 c d \left (c d^2+a e^2\right )^2}{e^6 (d+e x)^{3/2}}+\frac{3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{e^6 \sqrt{d+e x}}-\frac{4 c^2 d \left (5 c d^2+3 a e^2\right ) \sqrt{d+e x}}{e^6}+\frac{3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{3/2}}{e^6}-\frac{6 c^3 d (d+e x)^{5/2}}{e^6}+\frac{c^3 (d+e x)^{7/2}}{e^6}\right ) \, dx\\ &=-\frac{2 \left (c d^2+a e^2\right )^3}{3 e^7 (d+e x)^{3/2}}+\frac{12 c d \left (c d^2+a e^2\right )^2}{e^7 \sqrt{d+e x}}+\frac{6 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) \sqrt{d+e x}}{e^7}-\frac{8 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{3/2}}{3 e^7}+\frac{6 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{5/2}}{5 e^7}-\frac{12 c^3 d (d+e x)^{7/2}}{7 e^7}+\frac{2 c^3 (d+e x)^{9/2}}{9 e^7}\\ \end{align*}
Mathematica [A] time = 0.10891, size = 171, normalized size = 0.86 \[ \frac{2 \left (315 a^2 c e^4 \left (8 d^2+12 d e x+3 e^2 x^2\right )-105 a^3 e^6+63 a c^2 e^2 \left (48 d^2 e^2 x^2+192 d^3 e x+128 d^4-8 d e^3 x^3+3 e^4 x^4\right )+5 c^3 \left (384 d^4 e^2 x^2-64 d^3 e^3 x^3+24 d^2 e^4 x^4+1536 d^5 e x+1024 d^6-12 d e^5 x^5+7 e^6 x^6\right )\right )}{315 e^7 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 205, normalized size = 1. \begin{align*} -{\frac{-70\,{c}^{3}{x}^{6}{e}^{6}+120\,{c}^{3}d{x}^{5}{e}^{5}-378\,a{c}^{2}{e}^{6}{x}^{4}-240\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}+1008\,a{c}^{2}d{e}^{5}{x}^{3}+640\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}-1890\,{a}^{2}c{e}^{6}{x}^{2}-6048\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}-3840\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}-7560\,{a}^{2}cd{e}^{5}x-24192\,a{c}^{2}{d}^{3}{e}^{3}x-15360\,{c}^{3}{d}^{5}ex+210\,{a}^{3}{e}^{6}-5040\,{a}^{2}c{d}^{2}{e}^{4}-16128\,{d}^{4}{e}^{2}a{c}^{2}-10240\,{c}^{3}{d}^{6}}{315\,{e}^{7}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.60327, size = 290, normalized size = 1.45 \begin{align*} \frac{2 \,{\left (\frac{35 \,{\left (e x + d\right )}^{\frac{9}{2}} c^{3} - 270 \,{\left (e x + d\right )}^{\frac{7}{2}} c^{3} d + 189 \,{\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 420 \,{\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 945 \,{\left (5 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} \sqrt{e x + d}}{e^{6}} - \frac{105 \,{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} - 18 \,{\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )}{\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{6}}\right )}}{315 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84655, size = 495, normalized size = 2.48 \begin{align*} \frac{2 \,{\left (35 \, c^{3} e^{6} x^{6} - 60 \, c^{3} d e^{5} x^{5} + 5120 \, c^{3} d^{6} + 8064 \, a c^{2} d^{4} e^{2} + 2520 \, a^{2} c d^{2} e^{4} - 105 \, a^{3} e^{6} + 3 \,{\left (40 \, c^{3} d^{2} e^{4} + 63 \, a c^{2} e^{6}\right )} x^{4} - 8 \,{\left (40 \, c^{3} d^{3} e^{3} + 63 \, a c^{2} d e^{5}\right )} x^{3} + 3 \,{\left (640 \, c^{3} d^{4} e^{2} + 1008 \, a c^{2} d^{2} e^{4} + 315 \, a^{2} c e^{6}\right )} x^{2} + 12 \,{\left (640 \, c^{3} d^{5} e + 1008 \, a c^{2} d^{3} e^{3} + 315 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{e x + d}}{315 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 31.1395, size = 206, normalized size = 1.03 \begin{align*} - \frac{12 c^{3} d \left (d + e x\right )^{\frac{7}{2}}}{7 e^{7}} + \frac{2 c^{3} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{7}} + \frac{12 c d \left (a e^{2} + c d^{2}\right )^{2}}{e^{7} \sqrt{d + e x}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (6 a c^{2} e^{2} + 30 c^{3} d^{2}\right )}{5 e^{7}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (- 24 a c^{2} d e^{2} - 40 c^{3} d^{3}\right )}{3 e^{7}} + \frac{\sqrt{d + e x} \left (6 a^{2} c e^{4} + 36 a c^{2} d^{2} e^{2} + 30 c^{3} d^{4}\right )}{e^{7}} - \frac{2 \left (a e^{2} + c d^{2}\right )^{3}}{3 e^{7} \left (d + e x\right )^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.40015, size = 344, normalized size = 1.72 \begin{align*} \frac{2}{315} \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} c^{3} e^{56} - 270 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{3} d e^{56} + 945 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{3} d^{2} e^{56} - 2100 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{3} e^{56} + 4725 \, \sqrt{x e + d} c^{3} d^{4} e^{56} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} a c^{2} e^{58} - 1260 \,{\left (x e + d\right )}^{\frac{3}{2}} a c^{2} d e^{58} + 5670 \, \sqrt{x e + d} a c^{2} d^{2} e^{58} + 945 \, \sqrt{x e + d} a^{2} c e^{60}\right )} e^{\left (-63\right )} + \frac{2 \,{\left (18 \,{\left (x e + d\right )} c^{3} d^{5} - c^{3} d^{6} + 36 \,{\left (x e + d\right )} a c^{2} d^{3} e^{2} - 3 \, a c^{2} d^{4} e^{2} + 18 \,{\left (x e + d\right )} a^{2} c d e^{4} - 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} e^{\left (-7\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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