Optimal. Leaf size=61 \[ \frac {x}{3 a c (a+a x)^{3/2} (c-c x)^{3/2}}+\frac {2 x}{3 a^2 c^2 \sqrt {a+a x} \sqrt {c-c x}} \]
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Rubi [A]
time = 0.01, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {40, 39}
\begin {gather*} \frac {2 x}{3 a^2 c^2 \sqrt {a x+a} \sqrt {c-c x}}+\frac {x}{3 a c (a x+a)^{3/2} (c-c x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 39
Rule 40
Rubi steps
\begin {align*} \int \frac {1}{(a+a x)^{5/2} (c-c x)^{5/2}} \, dx &=\frac {x}{3 a c (a+a x)^{3/2} (c-c x)^{3/2}}+\frac {2 \int \frac {1}{(a+a x)^{3/2} (c-c x)^{3/2}} \, dx}{3 a c}\\ &=\frac {x}{3 a c (a+a x)^{3/2} (c-c x)^{3/2}}+\frac {2 x}{3 a^2 c^2 \sqrt {a+a x} \sqrt {c-c x}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 42, normalized size = 0.69 \begin {gather*} \frac {x (1+x) \left (-3+2 x^2\right )}{3 c^2 (-1+x) (a (1+x))^{5/2} \sqrt {c-c x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(104\) vs.
\(2(49)=98\).
time = 0.14, size = 105, normalized size = 1.72
method | result | size |
gosper | \(\frac {\left (1+x \right ) \left (-1+x \right ) x \left (2 x^{2}-3\right )}{3 \left (a x +a \right )^{\frac {5}{2}} \left (-c x +c \right )^{\frac {5}{2}}}\) | \(32\) |
default | \(-\frac {1}{3 a c \left (a x +a \right )^{\frac {3}{2}} \left (-c x +c \right )^{\frac {3}{2}}}+\frac {-\frac {1}{a c \sqrt {a x +a}\, \left (-c x +c \right )^{\frac {3}{2}}}+\frac {\frac {2 \sqrt {a x +a}}{3 a c \left (-c x +c \right )^{\frac {3}{2}}}+\frac {2 \sqrt {a x +a}}{3 a \,c^{2} \sqrt {-c x +c}}}{a}}{a}\) | \(105\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 45, normalized size = 0.74 \begin {gather*} \frac {x}{3 \, {\left (-a c x^{2} + a c\right )}^{\frac {3}{2}} a c} + \frac {2 \, x}{3 \, \sqrt {-a c x^{2} + a c} a^{2} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.97, size = 57, normalized size = 0.93 \begin {gather*} -\frac {{\left (2 \, x^{3} - 3 \, x\right )} \sqrt {a x + a} \sqrt {-c x + c}}{3 \, {\left (a^{3} c^{3} x^{4} - 2 \, a^{3} c^{3} x^{2} + a^{3} c^{3}\right )}} \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 = 7.26, size = 82, normalized size = 1.34 \begin {gather*} \frac {i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {5}{4}, \frac {7}{4}, 1 & \frac {1}{2}, \frac {5}{2}, 3 \\\frac {5}{4}, \frac {7}{4}, 2, \frac {5}{2}, 3 & 0 \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{3 \pi ^{\frac {3}{2}} a^{\frac {5}{2}} c^{\frac {5}{2}}} + \frac {{G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, 0, \frac {1}{2}, \frac {3}{4}, \frac {5}{4}, 1 & \\\frac {3}{4}, \frac {5}{4} & - \frac {1}{2}, 0, 2, 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{x^{2}}} \right )}}{3 \pi ^{\frac {3}{2}} a^{\frac {5}{2}} c^{\frac {5}{2}}} \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. 237 vs.
\(2 (49) = 98\).
time = 1.12, size = 237, normalized size = 3.89 \begin {gather*} -\frac {\sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \sqrt {a x + a} {\left (\frac {4 \, {\left (a x + a\right )} {\left | a \right |}}{a^{2} c} - \frac {9 \, {\left | a \right |}}{a c}\right )}}{12 \, {\left ({\left (a x + a\right )} a c - 2 \, a^{2} c\right )}^{2}} - \frac {16 \, \sqrt {-a c} a^{4} c^{2} - 18 \, \sqrt {-a c} {\left (\sqrt {-a c} \sqrt {a x + a} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{2} a^{2} c + 3 \, \sqrt {-a c} {\left (\sqrt {-a c} \sqrt {a x + a} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{4}}{3 \, {\left (2 \, a^{2} c - {\left (\sqrt {-a c} \sqrt {a x + a} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{2}\right )}^{3} c^{2} {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.41, size = 62, normalized size = 1.02 \begin {gather*} -\frac {3\,x\,\sqrt {c-c\,x}-2\,x^3\,\sqrt {c-c\,x}}{\sqrt {a+a\,x}\,{\left (c-c\,x\right )}^2\,\left (3\,a^2\,\left (c-c\,x\right )-6\,a^2\,c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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