Optimal. Leaf size=127 \[ \frac{4 c (d+e x)^{11/2} \left (a e^2+3 c d^2\right )}{11 e^5}-\frac{8 c d (d+e x)^{9/2} \left (a e^2+c d^2\right )}{9 e^5}+\frac{2 (d+e x)^{7/2} \left (a e^2+c d^2\right )^2}{7 e^5}+\frac{2 c^2 (d+e x)^{15/2}}{15 e^5}-\frac{8 c^2 d (d+e x)^{13/2}}{13 e^5} \]
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Rubi [A] time = 0.0568408, antiderivative size = 127, 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} \[ \frac{4 c (d+e x)^{11/2} \left (a e^2+3 c d^2\right )}{11 e^5}-\frac{8 c d (d+e x)^{9/2} \left (a e^2+c d^2\right )}{9 e^5}+\frac{2 (d+e x)^{7/2} \left (a e^2+c d^2\right )^2}{7 e^5}+\frac{2 c^2 (d+e x)^{15/2}}{15 e^5}-\frac{8 c^2 d (d+e x)^{13/2}}{13 e^5} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin{align*} \int (d+e x)^{5/2} \left (a+c x^2\right )^2 \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^2 (d+e x)^{5/2}}{e^4}-\frac{4 c d \left (c d^2+a e^2\right ) (d+e x)^{7/2}}{e^4}+\frac{2 c \left (3 c d^2+a e^2\right ) (d+e x)^{9/2}}{e^4}-\frac{4 c^2 d (d+e x)^{11/2}}{e^4}+\frac{c^2 (d+e x)^{13/2}}{e^4}\right ) \, dx\\ &=\frac{2 \left (c d^2+a e^2\right )^2 (d+e x)^{7/2}}{7 e^5}-\frac{8 c d \left (c d^2+a e^2\right ) (d+e x)^{9/2}}{9 e^5}+\frac{4 c \left (3 c d^2+a e^2\right ) (d+e x)^{11/2}}{11 e^5}-\frac{8 c^2 d (d+e x)^{13/2}}{13 e^5}+\frac{2 c^2 (d+e x)^{15/2}}{15 e^5}\\ \end{align*}
Mathematica [A] time = 0.111509, size = 96, normalized size = 0.76 \[ \frac{2 (d+e x)^{7/2} \left (6435 a^2 e^4+130 a c e^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )+c^2 \left (1008 d^2 e^2 x^2-448 d^3 e x+128 d^4-1848 d e^3 x^3+3003 e^4 x^4\right )\right )}{45045 e^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 106, normalized size = 0.8 \begin{align*}{\frac{6006\,{c}^{2}{x}^{4}{e}^{4}-3696\,{c}^{2}d{x}^{3}{e}^{3}+16380\,ac{e}^{4}{x}^{2}+2016\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}-7280\,acd{e}^{3}x-896\,{c}^{2}{d}^{3}ex+12870\,{a}^{2}{e}^{4}+2080\,ac{d}^{2}{e}^{2}+256\,{c}^{2}{d}^{4}}{45045\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12371, size = 153, normalized size = 1.2 \begin{align*} \frac{2 \,{\left (3003 \,{\left (e x + d\right )}^{\frac{15}{2}} c^{2} - 13860 \,{\left (e x + d\right )}^{\frac{13}{2}} c^{2} d + 8190 \,{\left (3 \, c^{2} d^{2} + a c e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 20020 \,{\left (c^{2} d^{3} + a c d e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 6435 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{45045 \, e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.75913, size = 504, normalized size = 3.97 \begin{align*} \frac{2 \,{\left (3003 \, c^{2} e^{7} x^{7} + 7161 \, c^{2} d e^{6} x^{6} + 128 \, c^{2} d^{7} + 1040 \, a c d^{5} e^{2} + 6435 \, a^{2} d^{3} e^{4} + 63 \,{\left (71 \, c^{2} d^{2} e^{5} + 130 \, a c e^{7}\right )} x^{5} + 35 \,{\left (c^{2} d^{3} e^{4} + 598 \, a c d e^{6}\right )} x^{4} - 5 \,{\left (8 \, c^{2} d^{4} e^{3} - 2938 \, a c d^{2} e^{5} - 1287 \, a^{2} e^{7}\right )} x^{3} + 3 \,{\left (16 \, c^{2} d^{5} e^{2} + 130 \, a c d^{3} e^{4} + 6435 \, a^{2} d e^{6}\right )} x^{2} -{\left (64 \, c^{2} d^{6} e + 520 \, a c d^{4} e^{3} - 19305 \, a^{2} d^{2} e^{5}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 17.5292, size = 566, normalized size = 4.46 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.71189, size = 676, normalized size = 5.32 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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