Optimal. Leaf size=48 \[ \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d (d+e x)^{7/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {648} \[ \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d (d+e x)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 648
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 )^{5/2}}{(d+e x)^{5/2}} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c d (d+e x)^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 37, normalized size = 0.77 \[ \frac {2 ((d+e x) (a e+c d x))^{7/2}}{7 c d (d+e x)^{7/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.87, size = 91, normalized size = 1.90 \[ \frac {2 \, {\left (c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} e x^{2} + 3 \, a^{2} c d e^{2} x + a^{3} e^{3}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{7 \, {\left (c d e x + c d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 50, normalized size = 1.04 \[ \frac {2 \left (c d x +a e \right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{7 \left (e x +d \right )^{\frac {5}{2}} c d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 60, normalized size = 1.25 \[ \frac {2 \, {\left (c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} e x^{2} + 3 \, a^{2} c d e^{2} x + a^{3} e^{3}\right )} \sqrt {c d x + a e}}{7 \, c d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.16, size = 79, normalized size = 1.65 \[ \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {6\,a^2\,e^2\,x}{7}+\frac {2\,c^2\,d^2\,x^3}{7}+\frac {2\,a^3\,e^3}{7\,c\,d}+\frac {6\,a\,c\,d\,e\,x^2}{7}\right )}{\sqrt {d+e\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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