Optimal. Leaf size=59 \[ -\frac {2 \left (a e^2+c d^2\right )}{3 e^3 (d+e x)^{3/2}}+\frac {2 c \sqrt {d+e x}}{e^3}+\frac {4 c d}{e^3 \sqrt {d+e x}} \]
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Rubi [A] time = 0.02, antiderivative size = 59, 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} \begin {gather*} -\frac {2 \left (a e^2+c d^2\right )}{3 e^3 (d+e x)^{3/2}}+\frac {2 c \sqrt {d+e x}}{e^3}+\frac {4 c d}{e^3 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin {align*} \int \frac {a+c x^2}{(d+e x)^{5/2}} \, dx &=\int \left (\frac {c d^2+a e^2}{e^2 (d+e x)^{5/2}}-\frac {2 c d}{e^2 (d+e x)^{3/2}}+\frac {c}{e^2 \sqrt {d+e x}}\right ) \, dx\\ &=-\frac {2 \left (c d^2+a e^2\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 c d}{e^3 \sqrt {d+e x}}+\frac {2 c \sqrt {d+e x}}{e^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 44, normalized size = 0.75 \begin {gather*} \frac {2 \left (c \left (8 d^2+12 d e x+3 e^2 x^2\right )-a e^2\right )}{3 e^3 (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 48, normalized size = 0.81 \begin {gather*} \frac {2 \left (-a e^2-c d^2+6 c d (d+e x)+3 c (d+e x)^2\right )}{3 e^3 (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 61, normalized size = 1.03 \begin {gather*} \frac {2 \, {\left (3 \, c e^{2} x^{2} + 12 \, c d e x + 8 \, c d^{2} - a e^{2}\right )} \sqrt {e x + d}}{3 \, {\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 48, normalized size = 0.81 \begin {gather*} 2 \, \sqrt {x e + d} c e^{\left (-3\right )} + \frac {2 \, {\left (6 \, {\left (x e + d\right )} c d - c d^{2} - a e^{2}\right )} e^{\left (-3\right )}}{3 \, {\left (x e + d\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 40, normalized size = 0.68 \begin {gather*} -\frac {2 \left (-3 c \,e^{2} x^{2}-12 c d e x +a \,e^{2}-8 c \,d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 52, normalized size = 0.88 \begin {gather*} \frac {2 \, {\left (\frac {3 \, \sqrt {e x + d} c}{e^{2}} + \frac {6 \, {\left (e x + d\right )} c d - c d^{2} - a e^{2}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{2}}\right )}}{3 \, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 44, normalized size = 0.75 \begin {gather*} \frac {6\,c\,{\left (d+e\,x\right )}^2-2\,a\,e^2-2\,c\,d^2+12\,c\,d\,\left (d+e\,x\right )}{3\,e^3\,{\left (d+e\,x\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.20, size = 168, normalized size = 2.85 \begin {gather*} \begin {cases} - \frac {2 a e^{2}}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} + \frac {16 c d^{2}}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} + \frac {24 c d e x}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} + \frac {6 c e^{2} x^{2}}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {a x + \frac {c x^{3}}{3}}{d^{\frac {5}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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