Optimal. Leaf size=67 \[ \frac {2 x}{3 a^4 c^2 \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {x}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {40, 39} \[ \frac {2 x}{3 a^4 c^2 \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {x}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 39
Rule 40
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx &=\frac {x}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {2 \int \frac {1}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx}{3 a^2 c}\\ &=\frac {x}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {2 x}{3 a^4 c^2 \sqrt {a+b x} \sqrt {a c-b c x}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 46, normalized size = 0.69 \[ \frac {3 a^2 x-2 b^2 x^3}{3 a^4 c (a+b x)^{3/2} (c (a-b x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 72, normalized size = 1.07 \[ -\frac {{\left (2 \, b^{2} x^{3} - 3 \, a^{2} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{3 \, {\left (a^{4} b^{4} c^{3} x^{4} - 2 \, a^{6} b^{2} c^{3} x^{2} + a^{8} c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.38, size = 251, normalized size = 3.75 \[ -\frac {\sqrt {-b c x + a c} {\left (\frac {9 \, {\left | c \right |}}{a^{3} b c} + \frac {4 \, {\left (b c x - a c\right )} {\left | c \right |}}{a^{4} b c^{2}}\right )}}{12 \, {\left (2 \, a c^{2} + {\left (b c x - a c\right )} c\right )}^{\frac {3}{2}}} + \frac {16 \, a^{2} \sqrt {-c} c^{4} - 18 \, a {\left (\sqrt {-b c x + a c} \sqrt {-c} - \sqrt {2 \, a c^{2} + {\left (b c x - a c\right )} c}\right )}^{2} \sqrt {-c} c^{2} + 3 \, {\left (\sqrt {-b c x + a c} \sqrt {-c} - \sqrt {2 \, a c^{2} + {\left (b c x - a c\right )} c}\right )}^{4} \sqrt {-c}}{3 \, {\left (2 \, a c^{2} - {\left (\sqrt {-b c x + a c} \sqrt {-c} - \sqrt {2 \, a c^{2} + {\left (b c x - a c\right )} c}\right )}^{2}\right )}^{3} a^{3} b {\left | c \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 45, normalized size = 0.67 \[ \frac {\left (-b x +a \right ) \left (-2 b^{2} x^{2}+3 a^{2}\right ) x}{3 \left (b x +a \right )^{\frac {3}{2}} \left (-b c x +a c \right )^{\frac {5}{2}} a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 53, normalized size = 0.79 \[ \frac {x}{3 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} a^{2} c} + \frac {2 \, x}{3 \, \sqrt {-b^{2} c x^{2} + a^{2} c} a^{4} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.58, size = 80, normalized size = 1.19 \[ -\frac {3\,a^2\,x\,\sqrt {a\,c-b\,c\,x}-2\,b^2\,x^3\,\sqrt {a\,c-b\,c\,x}}{{\left (a\,c-b\,c\,x\right )}^2\,\left (3\,a^4\,\left (a\,c-b\,c\,x\right )-6\,a^5\,c\right )\,\sqrt {a+b\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 15.85, size = 94, normalized size = 1.40 \[ \frac {i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {5}{4}, \frac {7}{4}, 1 & \frac {1}{2}, \frac {5}{2}, 3 \\\frac {5}{4}, \frac {7}{4}, 2, \frac {5}{2}, 3 & 0 \end {matrix} \middle | {\frac {a^{2}}{b^{2} x^{2}}} \right )}}{3 \pi ^{\frac {3}{2}} a^{4} b c^{\frac {5}{2}}} + \frac {{G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, 0, \frac {1}{2}, \frac {3}{4}, \frac {5}{4}, 1 & \\\frac {3}{4}, \frac {5}{4} & - \frac {1}{2}, 0, 2, 0 \end {matrix} \middle | {\frac {a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{3 \pi ^{\frac {3}{2}} a^{4} b c^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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