Optimal. Leaf size=133 \[ \frac {x}{7 a^2 c (a+b x)^{7/2} (a c-b c x)^{7/2}}+\frac {6 x}{35 a^4 c^2 (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac {8 x}{35 a^6 c^3 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {16 x}{35 a^8 c^4 \sqrt {a+b x} \sqrt {a c-b c x}} \]
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Rubi [A]
time = 0.02, 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}
\begin {gather*} \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}} \end {gather*}
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 = 0.09, size = 76, normalized size = 0.57 \begin {gather*} \frac {\sqrt {c (a-b x)} \left (35 a^6 x-70 a^4 b^2 x^3+56 a^2 b^4 x^5-16 b^6 x^7\right )}{35 a^8 c^5 (a-b x)^4 (a+b x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in
optimal.
time = 190.34, size = 79, normalized size = 0.59 \begin {gather*} \frac {4 \left (I \text {meijerg}\left [\left \{\left \{\frac {9}{4},\frac {11}{4},1\right \},\left \{\frac {1}{2},\frac {9}{2},5\right \}\right \},\left \{\left \{\frac {9}{4},\frac {11}{4},4,\frac {9}{2},5\right \},\left \{0\right \}\right \},\frac {a^2}{b^2 x^2}\right ]+\text {meijerg}\left [\left \{\left \{-\frac {1}{2},0,\frac {1}{2},\frac {7}{4},\frac {9}{4},1\right \},\left \{\right \}\right \},\left \{\left \{\frac {7}{4},\frac {9}{4}\right \},\left \{-\frac {1}{2},0,4,0\right \}\right \},\frac {a^2 \text {exp\_polar}\left [-2 I \text {Pi}\right ]}{b^2 x^2}\right ]\right )}{105 \text {Pi}^{\frac {3}{2}} a^8 b c^{\frac {9}{2}}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(274\) vs.
\(2(109)=218\).
time = 0.15, size = 275, normalized size = 2.07
method | result | size |
gosper | \(\frac {\left (-b x +a \right ) x \left (-16 x^{6} b^{6}+56 a^{2} x^{4} b^{4}-70 a^{4} x^{2} b^{2}+35 a^{6}\right )}{35 \left (b x +a \right )^{\frac {7}{2}} a^{8} \left (-b c x +a c \right )^{\frac {9}{2}}}\) | \(67\) |
default | \(-\frac {1}{7 a b c \left (b x +a \right )^{\frac {7}{2}} \left (-b c x +a c \right )^{\frac {7}{2}}}+\frac {-\frac {1}{5 a b c \left (b x +a \right )^{\frac {5}{2}} \left (-b c x +a c \right )^{\frac {7}{2}}}+\frac {-\frac {2}{5 a b c \left (b x +a \right )^{\frac {3}{2}} \left (-b c x +a c \right )^{\frac {7}{2}}}+\frac {6 \left (-\frac {5}{3 a b c \sqrt {b x +a}\, \left (-b c x +a c \right )^{\frac {7}{2}}}+\frac {5 \left (\frac {4 \sqrt {b x +a}}{7 a b c \left (-b c x +a c \right )^{\frac {7}{2}}}+\frac {4 \left (\frac {3 \sqrt {b x +a}}{35 a b c \left (-b c x +a c \right )^{\frac {5}{2}}}+\frac {3 \left (\frac {2 \sqrt {b x +a}}{15 a b c \left (-b c x +a c \right )^{\frac {3}{2}}}+\frac {2 \sqrt {b x +a}}{15 b \,a^{2} c^{2} \sqrt {-b c x +a c}}\right )}{7 a c}\right )}{a c}\right )}{3 a}\right )}{5 a}}{a}}{a}\) | \(275\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 105, normalized size = 0.79 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 122, normalized size = 0.92 \begin {gather*} -\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 )}} \end {gather*}
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 \begin {gather*} \text {Timed out} \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. 393 vs.
\(2 (109) = 218\).
time = 0.15, size = 521, normalized size = 3.92 \begin {gather*} \frac {2 \left (\frac {2 \left (\left (\left (-\frac {7046430720 c^{3} a^{18} \sqrt {a+b x} \sqrt {a+b x}}{123312537600 c^{4} a^{26}}+\frac {44508119040 c^{3} a^{19}}{123312537600 c^{4} a^{26}}\right ) \sqrt {a+b x} \sqrt {a+b x}-\frac {94411161600 c^{3} a^{20}}{123312537600 c^{4} a^{26}}\right ) \sqrt {a+b x} \sqrt {a+b x}+\frac {67436544000 c^{3} a^{21}}{123312537600 c^{4} a^{26}}\right ) \sqrt {a+b x} \sqrt {2 a c-c \left (a+b x\right )}}{\left (2 a c-c \left (a+b x\right )\right )^{4}}-\frac {2 \left (-175 \left (\sqrt {2 a c-c \left (a+b x\right )}-\sqrt {-c} \sqrt {a+b x}\right )^{12}+2450 c \left (\sqrt {2 a c-c \left (a+b x\right )}-\sqrt {-c} \sqrt {a+b x}\right )^{10} a-14280 c^{2} \left (\sqrt {2 a c-c \left (a+b x\right )}-\sqrt {-c} \sqrt {a+b x}\right )^{8} a^{2}+43120 c^{3} \left (\sqrt {2 a c-c \left (a+b x\right )}-\sqrt {-c} \sqrt {a+b x}\right )^{6} a^{3}-66416 c^{4} \left (\sqrt {2 a c-c \left (a+b x\right )}-\sqrt {-c} \sqrt {a+b x}\right )^{4} a^{4}+51744 c^{5} \left (\sqrt {2 a c-c \left (a+b x\right )}-\sqrt {-c} \sqrt {a+b x}\right )^{2} a^{5}-16384 c^{6} a^{6}\right )}{1120 c^{3} \sqrt {-c} a^{7} \left (-\left (\sqrt {2 a c-c \left (a+b x\right )}-\sqrt {-c} \sqrt {a+b x}\right )^{2}+2 c a\right )^{7}}\right )}{b} \end {gather*}
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 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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