Optimal. Leaf size=78 \[ -\frac {8 d \left (c d^2-c e^2 x^2\right )^{3/2}}{15 c e (d+e x)^{3/2}}-\frac {2 \left (c d^2-c e^2 x^2\right )^{3/2}}{5 c e \sqrt {d+e x}} \]
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Rubi [A]
time = 0.03, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {671, 663}
\begin {gather*} -\frac {2 \left (c d^2-c e^2 x^2\right )^{3/2}}{5 c e \sqrt {d+e x}}-\frac {8 d \left (c d^2-c e^2 x^2\right )^{3/2}}{15 c e (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 663
Rule 671
Rubi steps
\begin {align*} \int \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2} \, dx &=-\frac {2 \left (c d^2-c e^2 x^2\right )^{3/2}}{5 c e \sqrt {d+e x}}+\frac {1}{5} (4 d) \int \frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {d+e x}} \, dx\\ &=-\frac {8 d \left (c d^2-c e^2 x^2\right )^{3/2}}{15 c e (d+e x)^{3/2}}-\frac {2 \left (c d^2-c e^2 x^2\right )^{3/2}}{5 c e \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 53, normalized size = 0.68 \begin {gather*} -\frac {2 \left (7 d^2-4 d e x-3 e^2 x^2\right ) \sqrt {c \left (d^2-e^2 x^2\right )}}{15 e \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.47, size = 43, normalized size = 0.55
method | result | size |
default | \(-\frac {2 \sqrt {c \left (-e^{2} x^{2}+d^{2}\right )}\, \left (-e x +d \right ) \left (3 e x +7 d \right )}{15 \sqrt {e x +d}\, e}\) | \(43\) |
gosper | \(-\frac {2 \left (-e x +d \right ) \left (3 e x +7 d \right ) \sqrt {-x^{2} c \,e^{2}+c \,d^{2}}}{15 e \sqrt {e x +d}}\) | \(44\) |
risch | \(-\frac {2 \sqrt {-\frac {c \left (e^{2} x^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}\, c \left (-3 e^{2} x^{2}-4 d x e +7 d^{2}\right ) \left (-e x +d \right )}{15 \sqrt {-c \left (e^{2} x^{2}-d^{2}\right )}\, e \sqrt {-c \left (e x -d \right )}}\) | \(94\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 56, normalized size = 0.72 \begin {gather*} \frac {2 \, {\left (3 \, \sqrt {c} x^{2} e^{2} + 4 \, \sqrt {c} d x e - 7 \, \sqrt {c} d^{2}\right )} {\left (x e + d\right )} \sqrt {-x e + d}}{15 \, {\left (x e^{2} + d e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.58, size = 56, normalized size = 0.72 \begin {gather*} \frac {2 \, \sqrt {-c x^{2} e^{2} + c d^{2}} {\left (3 \, x^{2} e^{2} + 4 \, d x e - 7 \, d^{2}\right )} \sqrt {x e + d}}{15 \, {\left (x e^{2} + d e\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 {- c \left (- d + e x\right ) \left (d + e x\right )} \sqrt {d + e x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.25, size = 112, normalized size = 1.44 \begin {gather*} -\frac {2}{15} \, {\left (2 \, \sqrt {2} \sqrt {c d} d^{2} - 5 \, {\left (2 \, \sqrt {2} \sqrt {c d} d - \frac {{\left (-{\left (x e + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}}}{c}\right )} d + \frac {5 \, {\left (-{\left (x e + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c d - 3 \, {\left ({\left (x e + d\right )} c - 2 \, c d\right )}^{2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d}}{c^{2}}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.52, size = 69, normalized size = 0.88 \begin {gather*} \frac {\sqrt {c\,d^2-c\,e^2\,x^2}\,\left (\frac {2\,x^2\,\sqrt {d+e\,x}}{5}-\frac {14\,d^2\,\sqrt {d+e\,x}}{15\,e^2}+\frac {8\,d\,x\,\sqrt {d+e\,x}}{15\,e}\right )}{x+\frac {d}{e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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