Optimal. Leaf size=133 \[ \frac {16 x}{35 a^8 c^4 \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {8 x}{35 a^6 c^3 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {6 x}{35 a^4 c^2 (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac {x}{7 a^2 c (a+b x)^{7/2} (a c-b c x)^{7/2}} \]
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Rubi [A] time = 0.03, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {40, 39} \[ \frac {16 x}{35 a^8 c^4 \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {8 x}{35 a^6 c^3 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {6 x}{35 a^4 c^2 (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac {x}{7 a^2 c (a+b x)^{7/2} (a c-b c x)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 39
Rule 40
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{9/2} (a c-b c x)^{9/2}} \, dx &=\frac {x}{7 a^2 c (a+b x)^{7/2} (a c-b c x)^{7/2}}+\frac {6 \int \frac {1}{(a+b x)^{7/2} (a c-b c x)^{7/2}} \, dx}{7 a^2 c}\\ &=\frac {x}{7 a^2 c (a+b x)^{7/2} (a c-b c x)^{7/2}}+\frac {6 x}{35 a^4 c^2 (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac {24 \int \frac {1}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx}{35 a^4 c^2}\\ &=\frac {x}{7 a^2 c (a+b x)^{7/2} (a c-b c x)^{7/2}}+\frac {6 x}{35 a^4 c^2 (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac {8 x}{35 a^6 c^3 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {16 \int \frac {1}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx}{35 a^6 c^3}\\ &=\frac {x}{7 a^2 c (a+b x)^{7/2} (a c-b c x)^{7/2}}+\frac {6 x}{35 a^4 c^2 (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac {8 x}{35 a^6 c^3 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {16 x}{35 a^8 c^4 \sqrt {a+b x} \sqrt {a c-b c x}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 76, normalized size = 0.57 \[ \frac {x \left (35 a^6-70 a^4 b^2 x^2+56 a^2 b^4 x^4-16 b^6 x^6\right ) \sqrt {c (a-b x)}}{35 a^8 c^5 (a-b x)^4 (a+b x)^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 122, normalized size = 0.92 \[ -\frac {{\left (16 \, b^{6} x^{7} - 56 \, a^{2} b^{4} x^{5} + 70 \, a^{4} b^{2} x^{3} - 35 \, a^{6} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{35 \, {\left (a^{8} b^{8} c^{5} x^{8} - 4 \, a^{10} b^{6} c^{5} x^{6} + 6 \, a^{12} b^{4} c^{5} x^{4} - 4 \, a^{14} b^{2} c^{5} x^{2} + a^{16} c^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.30, size = 487, normalized size = 3.66 \[ -\frac {\sqrt {-b c x + a c} {\left ({\left (b c x - a c\right )} {\left ({\left (b c x - a c\right )} {\left (\frac {1617 \, {\left | c \right |}}{a^{7} b c} + \frac {256 \, {\left (b c x - a c\right )} {\left | c \right |}}{a^{8} b c^{2}}\right )} + \frac {3430 \, {\left | c \right |}}{a^{6} b}\right )} + \frac {2450 \, c {\left | c \right |}}{a^{5} b}\right )}}{1120 \, {\left (2 \, a c^{2} + {\left (b c x - a c\right )} c\right )}^{\frac {7}{2}}} - \frac {16384 \, a^{6} c^{12} - 51744 \, a^{5} {\left (\sqrt {-b c x + a c} \sqrt {-c} - \sqrt {2 \, a c^{2} + {\left (b c x - a c\right )} c}\right )}^{2} c^{10} + 66416 \, a^{4} {\left (\sqrt {-b c x + a c} \sqrt {-c} - \sqrt {2 \, a c^{2} + {\left (b c x - a c\right )} c}\right )}^{4} c^{8} - 43120 \, a^{3} {\left (\sqrt {-b c x + a c} \sqrt {-c} - \sqrt {2 \, a c^{2} + {\left (b c x - a c\right )} c}\right )}^{6} c^{6} + 14280 \, a^{2} {\left (\sqrt {-b c x + a c} \sqrt {-c} - \sqrt {2 \, a c^{2} + {\left (b c x - a c\right )} c}\right )}^{8} c^{4} - 2450 \, a {\left (\sqrt {-b c x + a c} \sqrt {-c} - \sqrt {2 \, a c^{2} + {\left (b c x - a c\right )} c}\right )}^{10} c^{2} + 175 \, {\left (\sqrt {-b c x + a c} \sqrt {-c} - \sqrt {2 \, a c^{2} + {\left (b c x - a c\right )} c}\right )}^{12}}{280 \, {\left (2 \, a c^{2} - {\left (\sqrt {-b c x + a c} \sqrt {-c} - \sqrt {2 \, a c^{2} + {\left (b c x - a c\right )} c}\right )}^{2}\right )}^{7} a^{7} b \sqrt {-c} c {\left | c \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 67, normalized size = 0.50 \[ \frac {\left (-b x +a \right ) \left (-16 b^{6} x^{6}+56 b^{4} x^{4} a^{2}-70 b^{2} x^{2} a^{4}+35 a^{6}\right ) x}{35 \left (b x +a \right )^{\frac {7}{2}} \left (-b c x +a c \right )^{\frac {9}{2}} a^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 105, normalized size = 0.79 \[ \frac {x}{7 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {7}{2}} a^{2} c} + \frac {6 \, x}{35 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {5}{2}} a^{4} c^{2}} + \frac {8 \, x}{35 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} a^{6} c^{3}} + \frac {16 \, x}{35 \, \sqrt {-b^{2} c x^{2} + a^{2} c} a^{8} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.71, size = 170, normalized size = 1.28 \[ -\frac {35\,a^6\,x\,\sqrt {a\,c-b\,c\,x}-16\,b^6\,x^7\,\sqrt {a\,c-b\,c\,x}-70\,a^4\,b^2\,x^3\,\sqrt {a\,c-b\,c\,x}+56\,a^2\,b^4\,x^5\,\sqrt {a\,c-b\,c\,x}}{\left (\left (70\,a^9\,{\left (a\,c-b\,c\,x\right )}^5+35\,a^8\,{\left (a\,c-b\,c\,x\right )}^5\,\left (a+b\,x\right )\right )\,\left (a+b\,x\right )+{\left (a\,c-b\,c\,x\right )}^4\,\left (140\,a^{10}\,\left (a\,c-b\,c\,x\right )-280\,a^{11}\,c\right )\right )\,\sqrt {a+b\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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