Optimal. Leaf size=63 \[ \frac {2 (d+e x)^{7/2} \left (a e^2+c d^2\right )}{7 e^3}+\frac {2 c (d+e x)^{11/2}}{11 e^3}-\frac {4 c d (d+e x)^{9/2}}{9 e^3} \]
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Rubi [A] time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {697} \[ \frac {2 (d+e x)^{7/2} \left (a e^2+c d^2\right )}{7 e^3}+\frac {2 c (d+e x)^{11/2}}{11 e^3}-\frac {4 c d (d+e x)^{9/2}}{9 e^3} \]
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 ) \, dx &=\int \left (\frac {\left (c d^2+a e^2\right ) (d+e x)^{5/2}}{e^2}-\frac {2 c d (d+e x)^{7/2}}{e^2}+\frac {c (d+e x)^{9/2}}{e^2}\right ) \, dx\\ &=\frac {2 \left (c d^2+a e^2\right ) (d+e x)^{7/2}}{7 e^3}-\frac {4 c d (d+e x)^{9/2}}{9 e^3}+\frac {2 c (d+e x)^{11/2}}{11 e^3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 44, normalized size = 0.70 \[ \frac {2 (d+e x)^{7/2} \left (99 a e^2+c \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )}{693 e^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.05, size = 108, normalized size = 1.71 \[ \frac {2 \, {\left (63 \, c e^{5} x^{5} + 161 \, c d e^{4} x^{4} + 8 \, c d^{5} + 99 \, a d^{3} e^{2} + {\left (113 \, c d^{2} e^{3} + 99 \, a e^{5}\right )} x^{3} + 3 \, {\left (c d^{3} e^{2} + 99 \, a d e^{4}\right )} x^{2} - {\left (4 \, c d^{4} e - 297 \, a d^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{693 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 380, normalized size = 6.03 \[ \frac {2}{3465} \, {\left (231 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} c d^{3} e^{\left (-2\right )} + 297 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} c d^{2} e^{\left (-2\right )} + 3465 \, \sqrt {x e + d} a d^{3} + 3465 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a d^{2} + 33 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} c d e^{\left (-2\right )} + 693 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a d + 5 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} c e^{\left (-2\right )} + 99 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 41, normalized size = 0.65 \[ \frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (63 c \,e^{2} x^{2}-28 c d e x +99 a \,e^{2}+8 c \,d^{2}\right )}{693 e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 47, normalized size = 0.75 \[ \frac {2 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} c - 154 \, {\left (e x + d\right )}^{\frac {9}{2}} c d + 99 \, {\left (c d^{2} + a e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{693 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 44, normalized size = 0.70 \[ \frac {2\,{\left (d+e\,x\right )}^{7/2}\,\left (63\,c\,{\left (d+e\,x\right )}^2+99\,a\,e^2+99\,c\,d^2-154\,c\,d\,\left (d+e\,x\right )\right )}{693\,e^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.34, size = 218, normalized size = 3.46 \[ \begin {cases} \frac {2 a d^{3} \sqrt {d + e x}}{7 e} + \frac {6 a d^{2} x \sqrt {d + e x}}{7} + \frac {6 a d e x^{2} \sqrt {d + e x}}{7} + \frac {2 a e^{2} x^{3} \sqrt {d + e x}}{7} + \frac {16 c d^{5} \sqrt {d + e x}}{693 e^{3}} - \frac {8 c d^{4} x \sqrt {d + e x}}{693 e^{2}} + \frac {2 c d^{3} x^{2} \sqrt {d + e x}}{231 e} + \frac {226 c d^{2} x^{3} \sqrt {d + e x}}{693} + \frac {46 c d e x^{4} \sqrt {d + e x}}{99} + \frac {2 c e^{2} x^{5} \sqrt {d + e x}}{11} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (a x + \frac {c x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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