Optimal. Leaf size=109 \[ \frac {4 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{15 c^2 d^2 (d+e x)^{3/2}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 c d \sqrt {d+e x}} \]
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Rubi [A]
time = 0.04, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {670, 662}
\begin {gather*} \frac {4 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{15 c^2 d^2 (d+e x)^{3/2}}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 662
Rule 670
Rubi steps
\begin {align*} \int \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 c d \sqrt {d+e x}}+\frac {\left (2 \left (d^2-\frac {a e^2}{c}\right )\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx}{5 d}\\ &=\frac {4 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{15 c^2 d^2 (d+e x)^{3/2}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 c d \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 55, normalized size = 0.50 \begin {gather*} \frac {2 ((a e+c d x) (d+e x))^{3/2} \left (-2 a e^2+c d (5 d+3 e x)\right )}{15 c^2 d^2 (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.89, size = 59, normalized size = 0.54
method | result | size |
default | \(-\frac {2 \left (c d x +a e \right ) \left (-3 c d e x +2 e^{2} a -5 c \,d^{2}\right ) \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}}{15 c^{2} d^{2} \sqrt {e x +d}}\) | \(59\) |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-3 c d e x +2 e^{2} a -5 c \,d^{2}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{15 c^{2} d^{2} \sqrt {e x +d}}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 86, normalized size = 0.79 \begin {gather*} \frac {2 \, {\left (3 \, c^{2} d^{2} x^{2} e + 5 \, a c d^{2} e - 2 \, a^{2} e^{3} + {\left (5 \, c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt {c d x + a e} {\left (x e + d\right )}}{15 \, {\left (c^{2} d^{2} x e + c^{2} d^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.17, size = 103, normalized size = 0.94 \begin {gather*} \frac {2 \, {\left (5 \, c^{2} d^{3} x + a c d x e^{2} - 2 \, a^{2} e^{3} + {\left (3 \, c^{2} d^{2} x^{2} + 5 \, a c d^{2}\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{15 \, {\left (c^{2} d^{2} x e + c^{2} d^{3}\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 \sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \sqrt {d + e x}\, 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. 245 vs.
\(2 (100) = 200\).
time = 1.30, size = 245, normalized size = 2.25 \begin {gather*} \frac {2}{15} \, {\left (5 \, d {\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^{\left (-1\right )} - {\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 )}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.80, size = 121, normalized size = 1.11 \begin {gather*} \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,x^2\,\sqrt {d+e\,x}}{5}-\frac {\left (4\,a^2\,e^3-10\,a\,c\,d^2\,e\right )\,\sqrt {d+e\,x}}{15\,c^2\,d^2\,e}+\frac {x\,\left (10\,c^2\,d^3+2\,a\,c\,d\,e^2\right )\,\sqrt {d+e\,x}}{15\,c^2\,d^2\,e}\right )}{x+\frac {d}{e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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