Optimal. Leaf size=72 \[ \frac {\left (2 b c^2+3 a d^2\right ) \sqrt {-c+d x} \sqrt {c+d x}}{3 d^4}+\frac {b x^2 \sqrt {-c+d x} \sqrt {c+d x}}{3 d^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 72, 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 = {471, 75}
\begin {gather*} \frac {\sqrt {d x-c} \sqrt {c+d x} \left (3 a d^2+2 b c^2\right )}{3 d^4}+\frac {b x^2 \sqrt {d x-c} \sqrt {c+d x}}{3 d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 75
Rule 471
Rubi steps
\begin {align*} \int \frac {x \left (a+b x^2\right )}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx &=\frac {b x^2 \sqrt {-c+d x} \sqrt {c+d x}}{3 d^2}-\frac {1}{3} \left (-3 a-\frac {2 b c^2}{d^2}\right ) \int \frac {x}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx\\ &=\frac {\left (2 b c^2+3 a d^2\right ) \sqrt {-c+d x} \sqrt {c+d x}}{3 d^4}+\frac {b x^2 \sqrt {-c+d x} \sqrt {c+d x}}{3 d^2}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 48, normalized size = 0.67 \begin {gather*} \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (2 b c^2+3 a d^2+b d^2 x^2\right )}{3 d^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 43, normalized size = 0.60
method | result | size |
gosper | \(\frac {\sqrt {d x +c}\, \left (b \,d^{2} x^{2}+3 a \,d^{2}+2 b \,c^{2}\right ) \sqrt {d x -c}}{3 d^{4}}\) | \(43\) |
default | \(\frac {\sqrt {d x +c}\, \left (b \,d^{2} x^{2}+3 a \,d^{2}+2 b \,c^{2}\right ) \sqrt {d x -c}}{3 d^{4}}\) | \(43\) |
risch | \(-\frac {\sqrt {d x +c}\, \left (-d x +c \right ) \left (b \,d^{2} x^{2}+3 a \,d^{2}+2 b \,c^{2}\right )}{3 d^{4} \sqrt {d x -c}}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 69, normalized size = 0.96 \begin {gather*} \frac {\sqrt {d^{2} x^{2} - c^{2}} b x^{2}}{3 \, d^{2}} + \frac {2 \, \sqrt {d^{2} x^{2} - c^{2}} b c^{2}}{3 \, d^{4}} + \frac {\sqrt {d^{2} x^{2} - c^{2}} a}{d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.84, size = 42, normalized size = 0.58 \begin {gather*} \frac {{\left (b d^{2} x^{2} + 2 \, b c^{2} + 3 \, a d^{2}\right )} \sqrt {d x + c} \sqrt {d x - c}}{3 \, d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 19.78, size = 223, normalized size = 3.10 \begin {gather*} \frac {a c {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{4} & 0, 0, \frac {1}{2}, 1 \\- \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 0 & \end {matrix} \middle | {\frac {c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{2}} + \frac {i a c {G_{6, 6}^{2, 6}\left (\begin {matrix} -1, - \frac {3}{4}, - \frac {1}{2}, - \frac {1}{4}, 0, 1 & \\- \frac {3}{4}, - \frac {1}{4} & -1, - \frac {1}{2}, - \frac {1}{2}, 0 \end {matrix} \middle | {\frac {c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{2}} + \frac {b c^{3} {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {5}{4}, - \frac {3}{4} & -1, -1, - \frac {1}{2}, 1 \\- \frac {3}{2}, - \frac {5}{4}, -1, - \frac {3}{4}, - \frac {1}{2}, 0 & \end {matrix} \middle | {\frac {c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{4}} + \frac {i b c^{3} {G_{6, 6}^{2, 6}\left (\begin {matrix} -2, - \frac {7}{4}, - \frac {3}{2}, - \frac {5}{4}, -1, 1 & \\- \frac {7}{4}, - \frac {5}{4} & -2, - \frac {3}{2}, - \frac {3}{2}, 0 \end {matrix} \middle | {\frac {c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.61, size = 65, normalized size = 0.90 \begin {gather*} \frac {\sqrt {d x + c} \sqrt {d x - c} {\left ({\left (d x + c\right )} {\left (\frac {{\left (d x + c\right )} b}{d^{3}} - \frac {2 \, b c}{d^{3}}\right )} + \frac {3 \, {\left (b c^{2} d^{9} + a d^{11}\right )}}{d^{12}}\right )}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.66, size = 76, normalized size = 1.06 \begin {gather*} \frac {\sqrt {d\,x-c}\,\left (\frac {2\,b\,c^3+3\,a\,c\,d^2}{3\,d^4}+\frac {b\,x^3}{3\,d}+\frac {x\,\left (2\,b\,c^2\,d+3\,a\,d^3\right )}{3\,d^4}+\frac {b\,c\,x^2}{3\,d^2}\right )}{\sqrt {c+d\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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