Optimal. Leaf size=48 \[ \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c d (d+e x)^{7/2}} \]
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Rubi [A]
time = 0.01, 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 = {662}
\begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 662
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 = 0.03, size = 37, normalized size = 0.77 \begin {gather*} \frac {2 ((a e+c d x) (d+e x))^{7/2}}{7 c d (d+e x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 42, normalized size = 0.88
method | result | size |
default | \(\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (c d x +a e \right )^{3}}{7 \sqrt {e x +d}\, c d}\) | \(42\) |
gosper | \(\frac {2 \left (c d x +a e \right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{7 c d \left (e x +d \right )^{\frac {5}{2}}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 60, normalized size = 1.25 \begin {gather*} \frac {2 \, {\left (c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} x^{2} e + 3 \, a^{2} c d x e^{2} + a^{3} e^{3}\right )} \sqrt {c d x + a e}}{7 \, c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 92 vs.
\(2 (44) = 88\).
time = 1.64, size = 92, normalized size = 1.92 \begin {gather*} \frac {2 \, {\left (c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} x^{2} e + 3 \, a^{2} c d x e^{2} + a^{3} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{7 \, {\left (c d x e + c d^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 469 vs.
\(2 (44) = 88\).
time = 6.04, size = 469, normalized size = 9.77 \begin {gather*} \frac {2}{105} \, {\left (c^{2} d^{2} {\left (\frac {{\left (15 \, \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{6} - 3 \, \sqrt {-c d^{2} e + a e^{3}} a c^{2} d^{4} e^{2} - 4 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c d^{2} e^{4} - 8 \, \sqrt {-c d^{2} e + a e^{3}} a^{3} e^{6}\right )} e^{\left (-2\right )}}{c^{3} d^{3}} + \frac {{\left (35 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} e^{6} - 42 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a e^{3} + 15 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}}\right )} e^{\left (-5\right )}}{c^{3} d^{3}}\right )} e^{\left (-1\right )} - 14 \, a c d {\left (\frac {{\left (5 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a e^{3} - 3 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}}\right )} e^{\left (-2\right )}}{c^{2} d^{2}} + \frac {3 \, \sqrt {-c d^{2} e + a e^{3}} c^{2} d^{4} - \sqrt {-c d^{2} e + a e^{3}} a c d^{2} e^{2} - 2 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} e^{4}}{c^{2} d^{2}}\right )} e^{\left (-1\right )} + 35 \, a^{2} {\left (\frac {{\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} e^{\left (-1\right )}}{c d} + \frac {\sqrt {-c d^{2} e + a e^{3}} c d^{2} - \sqrt {-c d^{2} e + a e^{3}} a e^{2}}{c d}\right )} e\right )} e^{\left (-1\right )} \end {gather*}
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 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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