Optimal. Leaf size=83 \[ -\frac {4 c d (d+e x)^{5/2} \left (c d^2-a e^2\right )}{5 e^3}+\frac {2 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}{3 e^3}+\frac {2 c^2 d^2 (d+e x)^{7/2}}{7 e^3} \]
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Rubi [A] time = 0.04, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {626, 43} \[ -\frac {4 c d (d+e x)^{5/2} \left (c d^2-a e^2\right )}{5 e^3}+\frac {2 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}{3 e^3}+\frac {2 c^2 d^2 (d+e x)^{7/2}}{7 e^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 626
Rubi steps
\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{3/2}} \, dx &=\int (a e+c d x)^2 \sqrt {d+e x} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^2 \sqrt {d+e x}}{e^2}-\frac {2 c d \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{e^2}+\frac {c^2 d^2 (d+e x)^{5/2}}{e^2}\right ) \, dx\\ &=\frac {2 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{3 e^3}-\frac {4 c d \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 e^3}+\frac {2 c^2 d^2 (d+e x)^{7/2}}{7 e^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 67, normalized size = 0.81 \[ \frac {2 (d+e x)^{3/2} \left (35 a^2 e^4+14 a c d e^2 (3 e x-2 d)+c^2 d^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )}{105 e^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 109, normalized size = 1.31 \[ \frac {2 \, {\left (15 \, c^{2} d^{2} e^{3} x^{3} + 8 \, c^{2} d^{5} - 28 \, a c d^{3} e^{2} + 35 \, a^{2} d e^{4} + 3 \, {\left (c^{2} d^{3} e^{2} + 14 \, a c d e^{4}\right )} x^{2} - {\left (4 \, c^{2} d^{4} e - 14 \, a c d^{2} e^{3} - 35 \, a^{2} e^{5}\right )} x\right )} \sqrt {e x + d}}{105 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 106, normalized size = 1.28 \[ \frac {2}{105} \, {\left (15 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{2} d^{2} e^{18} - 42 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{2} d^{3} e^{18} + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{2} d^{4} e^{18} + 42 \, {\left (x e + d\right )}^{\frac {5}{2}} a c d e^{20} - 70 \, {\left (x e + d\right )}^{\frac {3}{2}} a c d^{2} e^{20} + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} e^{22}\right )} e^{\left (-21\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 73, normalized size = 0.88 \[ \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (15 c^{2} d^{2} e^{2} x^{2}+42 a c d \,e^{3} x -12 c^{2} d^{3} e x +35 a^{2} e^{4}-28 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right )}{105 e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.14, size = 80, normalized size = 0.96 \[ \frac {2 \, {\left (15 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{2} d^{2} - 42 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 35 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{105 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.63, size = 80, normalized size = 0.96 \[ \frac {2\,{\left (d+e\,x\right )}^{3/2}\,\left (35\,a^2\,e^4+35\,c^2\,d^4+15\,c^2\,d^2\,{\left (d+e\,x\right )}^2-42\,c^2\,d^3\,\left (d+e\,x\right )-70\,a\,c\,d^2\,e^2+42\,a\,c\,d\,e^2\,\left (d+e\,x\right )\right )}{105\,e^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 43.92, size = 411, normalized size = 4.95 \[ \begin {cases} \frac {- \frac {2 a^{2} d^{2} e^{2}}{\sqrt {d + e x}} - 4 a^{2} d e^{2} \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - 2 a^{2} e^{2} \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right ) - 4 a c d^{3} \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - 8 a c d^{2} \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right ) - 4 a c d \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right ) - \frac {2 c^{2} d^{4} \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e^{2}} - \frac {4 c^{2} d^{3} \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} - \frac {2 c^{2} d^{2} \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{2}}}{e} & \text {for}\: e \neq 0 \\\frac {c^{2} d^{\frac {5}{2}} x^{3}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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