Optimal. Leaf size=196 \[ \frac{2 c^2 (d+e x)^{3/2} \left (a e^2+5 c d^2\right )}{e^7}-\frac{8 c^2 d \sqrt{d+e x} \left (3 a e^2+5 c d^2\right )}{e^7}-\frac{6 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7 \sqrt{d+e x}}+\frac{4 c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^{3/2}}-\frac{2 \left (a e^2+c d^2\right )^3}{5 e^7 (d+e x)^{5/2}}+\frac{2 c^3 (d+e x)^{7/2}}{7 e^7}-\frac{12 c^3 d (d+e x)^{5/2}}{5 e^7} \]
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Rubi [A] time = 0.0807837, antiderivative size = 196, 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} \[ \frac{2 c^2 (d+e x)^{3/2} \left (a e^2+5 c d^2\right )}{e^7}-\frac{8 c^2 d \sqrt{d+e x} \left (3 a e^2+5 c d^2\right )}{e^7}-\frac{6 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7 \sqrt{d+e x}}+\frac{4 c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^{3/2}}-\frac{2 \left (a e^2+c d^2\right )^3}{5 e^7 (d+e x)^{5/2}}+\frac{2 c^3 (d+e x)^{7/2}}{7 e^7}-\frac{12 c^3 d (d+e x)^{5/2}}{5 e^7} \]
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)^{7/2}} \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^3}{e^6 (d+e x)^{7/2}}-\frac{6 c d \left (c d^2+a e^2\right )^2}{e^6 (d+e x)^{5/2}}+\frac{3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{e^6 (d+e x)^{3/2}}-\frac{4 c^2 d \left (5 c d^2+3 a e^2\right )}{e^6 \sqrt{d+e x}}+\frac{3 c^2 \left (5 c d^2+a e^2\right ) \sqrt{d+e x}}{e^6}-\frac{6 c^3 d (d+e x)^{3/2}}{e^6}+\frac{c^3 (d+e x)^{5/2}}{e^6}\right ) \, dx\\ &=-\frac{2 \left (c d^2+a e^2\right )^3}{5 e^7 (d+e x)^{5/2}}+\frac{4 c d \left (c d^2+a e^2\right )^2}{e^7 (d+e x)^{3/2}}-\frac{6 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{e^7 \sqrt{d+e x}}-\frac{8 c^2 d \left (5 c d^2+3 a e^2\right ) \sqrt{d+e x}}{e^7}+\frac{2 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{3/2}}{e^7}-\frac{12 c^3 d (d+e x)^{5/2}}{5 e^7}+\frac{2 c^3 (d+e x)^{7/2}}{7 e^7}\\ \end{align*}
Mathematica [A] time = 0.115347, size = 170, normalized size = 0.87 \[ -\frac{2 \left (7 a^2 c e^4 \left (8 d^2+20 d e x+15 e^2 x^2\right )+7 a^3 e^6+7 a c^2 e^2 \left (240 d^2 e^2 x^2+320 d^3 e x+128 d^4+40 d e^3 x^3-5 e^4 x^4\right )+c^3 \left (1920 d^4 e^2 x^2+320 d^3 e^3 x^3-40 d^2 e^4 x^4+2560 d^5 e x+1024 d^6+12 d e^5 x^5-5 e^6 x^6\right )\right )}{35 e^7 (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 205, normalized size = 1.1 \begin{align*} -{\frac{-10\,{c}^{3}{x}^{6}{e}^{6}+24\,{c}^{3}d{x}^{5}{e}^{5}-70\,a{c}^{2}{e}^{6}{x}^{4}-80\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}+560\,a{c}^{2}d{e}^{5}{x}^{3}+640\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+210\,{a}^{2}c{e}^{6}{x}^{2}+3360\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}+3840\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}+280\,{a}^{2}cd{e}^{5}x+4480\,a{c}^{2}{d}^{3}{e}^{3}x+5120\,{c}^{3}{d}^{5}ex+14\,{a}^{3}{e}^{6}+112\,{a}^{2}c{d}^{2}{e}^{4}+1792\,{d}^{4}{e}^{2}a{c}^{2}+2048\,{c}^{3}{d}^{6}}{35\,{e}^{7}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.61279, size = 290, normalized size = 1.48 \begin{align*} \frac{2 \,{\left (\frac{5 \,{\left (e x + d\right )}^{\frac{7}{2}} c^{3} - 42 \,{\left (e x + d\right )}^{\frac{5}{2}} c^{3} d + 35 \,{\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 140 \,{\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} \sqrt{e x + d}}{e^{6}} - \frac{7 \,{\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} + 15 \,{\left (5 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )}{\left (e x + d\right )}^{2} - 10 \,{\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{5}{2}} e^{6}}\right )}}{35 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76616, size = 498, normalized size = 2.54 \begin{align*} \frac{2 \,{\left (5 \, c^{3} e^{6} x^{6} - 12 \, c^{3} d e^{5} x^{5} - 1024 \, c^{3} d^{6} - 896 \, a c^{2} d^{4} e^{2} - 56 \, a^{2} c d^{2} e^{4} - 7 \, a^{3} e^{6} + 5 \,{\left (8 \, c^{3} d^{2} e^{4} + 7 \, a c^{2} e^{6}\right )} x^{4} - 40 \,{\left (8 \, c^{3} d^{3} e^{3} + 7 \, a c^{2} d e^{5}\right )} x^{3} - 15 \,{\left (128 \, c^{3} d^{4} e^{2} + 112 \, a c^{2} d^{2} e^{4} + 7 \, a^{2} c e^{6}\right )} x^{2} - 20 \,{\left (128 \, c^{3} d^{5} e + 112 \, a c^{2} d^{3} e^{3} + 7 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{e x + d}}{35 \,{\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 48.5068, size = 197, normalized size = 1.01 \begin{align*} - \frac{12 c^{3} d \left (d + e x\right )^{\frac{5}{2}}}{5 e^{7}} + \frac{2 c^{3} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{7}} + \frac{4 c d \left (a e^{2} + c d^{2}\right )^{2}}{e^{7} \left (d + e x\right )^{\frac{3}{2}}} - \frac{6 c \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right )}{e^{7} \sqrt{d + e x}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (6 a c^{2} e^{2} + 30 c^{3} d^{2}\right )}{3 e^{7}} + \frac{\sqrt{d + e x} \left (- 24 a c^{2} d e^{2} - 40 c^{3} d^{3}\right )}{e^{7}} - \frac{2 \left (a e^{2} + c d^{2}\right )^{3}}{5 e^{7} \left (d + e x\right )^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.34764, size = 339, normalized size = 1.73 \begin{align*} \frac{2}{35} \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{3} e^{42} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{3} d e^{42} + 175 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{2} e^{42} - 700 \, \sqrt{x e + d} c^{3} d^{3} e^{42} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} a c^{2} e^{44} - 420 \, \sqrt{x e + d} a c^{2} d e^{44}\right )} e^{\left (-49\right )} - \frac{2 \,{\left (75 \,{\left (x e + d\right )}^{2} c^{3} d^{4} - 10 \,{\left (x e + d\right )} c^{3} d^{5} + c^{3} d^{6} + 90 \,{\left (x e + d\right )}^{2} a c^{2} d^{2} e^{2} - 20 \,{\left (x e + d\right )} a c^{2} d^{3} e^{2} + 3 \, a c^{2} d^{4} e^{2} + 15 \,{\left (x e + d\right )}^{2} a^{2} c e^{4} - 10 \,{\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 )}}{5 \,{\left (x e + d\right )}^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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