Optimal. Leaf size=81 \[ -\frac {4 c d (d+e x)^{3/2} \left (c d^2-a e^2\right )}{3 e^3}+\frac {2 \sqrt {d+e x} \left (c d^2-a e^2\right )^2}{e^3}+\frac {2 c^2 d^2 (d+e x)^{5/2}}{5 e^3} \]
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Rubi [A] time = 0.04, antiderivative size = 81, 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} \begin {gather*} -\frac {4 c d (d+e x)^{3/2} \left (c d^2-a e^2\right )}{3 e^3}+\frac {2 \sqrt {d+e x} \left (c d^2-a e^2\right )^2}{e^3}+\frac {2 c^2 d^2 (d+e x)^{5/2}}{5 e^3} \end {gather*}
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)^{5/2}} \, dx &=\int \frac {(a e+c d x)^2}{\sqrt {d+e x}} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^2}{e^2 \sqrt {d+e x}}-\frac {2 c d \left (c d^2-a e^2\right ) \sqrt {d+e x}}{e^2}+\frac {c^2 d^2 (d+e x)^{3/2}}{e^2}\right ) \, dx\\ &=\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}{e^3}-\frac {4 c d \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{3 e^3}+\frac {2 c^2 d^2 (d+e x)^{5/2}}{5 e^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 66, normalized size = 0.81 \begin {gather*} \frac {2 \sqrt {d+e x} \left (15 a^2 e^4+10 a c d e^2 (e x-2 d)+c^2 d^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )}{15 e^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 84, normalized size = 1.04 \begin {gather*} \frac {2 \sqrt {d+e x} \left (15 a^2 e^4-30 a c d^2 e^2+10 a c d e^2 (d+e x)+15 c^2 d^4-10 c^2 d^3 (d+e x)+3 c^2 d^2 (d+e x)^2\right )}{15 e^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 74, normalized size = 0.91 \begin {gather*} \frac {2 \, {\left (3 \, c^{2} d^{2} e^{2} x^{2} + 8 \, c^{2} d^{4} - 20 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4} - 2 \, {\left (2 \, c^{2} d^{3} e - 5 \, a c d e^{3}\right )} x\right )} \sqrt {e x + d}}{15 \, e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.68, size = 106, normalized size = 1.31 \begin {gather*} \frac {2}{15} \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{2} d^{2} e^{12} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{2} d^{3} e^{12} + 15 \, \sqrt {x e + d} c^{2} d^{4} e^{12} + 10 \, {\left (x e + d\right )}^{\frac {3}{2}} a c d e^{14} - 30 \, \sqrt {x e + d} a c d^{2} e^{14} + 15 \, \sqrt {x e + d} a^{2} e^{16}\right )} e^{\left (-15\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 73, normalized size = 0.90 \begin {gather*} \frac {2 \sqrt {e x +d}\, \left (3 c^{2} d^{2} e^{2} x^{2}+10 a c d \,e^{3} x -4 c^{2} d^{3} e x +15 a^{2} e^{4}-20 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right )}{15 e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.13, size = 80, normalized size = 0.99 \begin {gather*} \frac {2 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{2} d^{2} - 10 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 15 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {e x + d}\right )}}{15 \, e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 80, normalized size = 0.99 \begin {gather*} \frac {2\,\sqrt {d+e\,x}\,\left (15\,a^2\,e^4+15\,c^2\,d^4+3\,c^2\,d^2\,{\left (d+e\,x\right )}^2-10\,c^2\,d^3\,\left (d+e\,x\right )-30\,a\,c\,d^2\,e^2+10\,a\,c\,d\,e^2\,\left (d+e\,x\right )\right )}{15\,e^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 44.16, size = 236, normalized size = 2.91 \begin {gather*} \begin {cases} \frac {- \frac {2 a^{2} d e^{2}}{\sqrt {d + e x}} - 2 a^{2} e^{2} \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - 4 a c d^{2} \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - 4 a c d \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 ) - \frac {2 c^{2} d^{3} \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 {2 c^{2} d^{2} \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}}}{e} & \text {for}\: e \neq 0 \\\frac {c^{2} d^{\frac {3}{2}} x^{3}}{3} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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