Optimal. Leaf size=61 \[ -\frac {2 \left (a e^2+c d^2\right )}{5 e^3 (d+e x)^{5/2}}-\frac {2 c}{e^3 \sqrt {d+e x}}+\frac {4 c d}{3 e^3 (d+e x)^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 61, 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} \[ -\frac {2 \left (a e^2+c d^2\right )}{5 e^3 (d+e x)^{5/2}}-\frac {2 c}{e^3 \sqrt {d+e x}}+\frac {4 c d}{3 e^3 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin {align*} \int \frac {a+c x^2}{(d+e x)^{7/2}} \, dx &=\int \left (\frac {c d^2+a e^2}{e^2 (d+e x)^{7/2}}-\frac {2 c d}{e^2 (d+e x)^{5/2}}+\frac {c}{e^2 (d+e x)^{3/2}}\right ) \, dx\\ &=-\frac {2 \left (c d^2+a e^2\right )}{5 e^3 (d+e x)^{5/2}}+\frac {4 c d}{3 e^3 (d+e x)^{3/2}}-\frac {2 c}{e^3 \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 44, normalized size = 0.72 \[ -\frac {2 \left (3 a e^2+c \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )}{15 e^3 (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 72, normalized size = 1.18 \[ -\frac {2 \, {\left (15 \, c e^{2} x^{2} + 20 \, c d e x + 8 \, c d^{2} + 3 \, a e^{2}\right )} \sqrt {e x + d}}{15 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 45, normalized size = 0.74 \[ -\frac {2 \, {\left (15 \, {\left (x e + d\right )}^{2} c - 10 \, {\left (x e + d\right )} c d + 3 \, c d^{2} + 3 \, a e^{2}\right )} e^{\left (-3\right )}}{15 \, {\left (x e + d\right )}^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 41, normalized size = 0.67 \[ -\frac {2 \left (15 c \,e^{2} x^{2}+20 c d e x +3 a \,e^{2}+8 c \,d^{2}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 44, normalized size = 0.72 \[ -\frac {2 \, {\left (15 \, {\left (e x + d\right )}^{2} c - 10 \, {\left (e x + d\right )} c d + 3 \, c d^{2} + 3 \, a e^{2}\right )}}{15 \, {\left (e x + d\right )}^{\frac {5}{2}} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 44, normalized size = 0.72 \[ -\frac {30\,c\,{\left (d+e\,x\right )}^2+6\,a\,e^2+6\,c\,d^2-20\,c\,d\,\left (d+e\,x\right )}{15\,e^3\,{\left (d+e\,x\right )}^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.80, size = 252, normalized size = 4.13 \[ \begin {cases} - \frac {6 a e^{2}}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} - \frac {16 c d^{2}}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} - \frac {40 c d e x}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} - \frac {30 c e^{2} x^{2}}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {a x + \frac {c x^{3}}{3}}{d^{\frac {7}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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