Optimal. Leaf size=183 \[ -\frac {2 A \sqrt {a+b x}}{11 a x^{11/2}}+\frac {2 (10 A b-11 a B) \sqrt {a+b x}}{99 a^2 x^{9/2}}-\frac {16 b (10 A b-11 a B) \sqrt {a+b x}}{693 a^3 x^{7/2}}+\frac {32 b^2 (10 A b-11 a B) \sqrt {a+b x}}{1155 a^4 x^{5/2}}-\frac {128 b^3 (10 A b-11 a B) \sqrt {a+b x}}{3465 a^5 x^{3/2}}+\frac {256 b^4 (10 A b-11 a B) \sqrt {a+b x}}{3465 a^6 \sqrt {x}} \]
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Rubi [A]
time = 0.05, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37}
\begin {gather*} \frac {256 b^4 \sqrt {a+b x} (10 A b-11 a B)}{3465 a^6 \sqrt {x}}-\frac {128 b^3 \sqrt {a+b x} (10 A b-11 a B)}{3465 a^5 x^{3/2}}+\frac {32 b^2 \sqrt {a+b x} (10 A b-11 a B)}{1155 a^4 x^{5/2}}-\frac {16 b \sqrt {a+b x} (10 A b-11 a B)}{693 a^3 x^{7/2}}+\frac {2 \sqrt {a+b x} (10 A b-11 a B)}{99 a^2 x^{9/2}}-\frac {2 A \sqrt {a+b x}}{11 a x^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 79
Rubi steps
\begin {align*} \int \frac {A+B x}{x^{13/2} \sqrt {a+b x}} \, dx &=-\frac {2 A \sqrt {a+b x}}{11 a x^{11/2}}+\frac {\left (2 \left (-5 A b+\frac {11 a B}{2}\right )\right ) \int \frac {1}{x^{11/2} \sqrt {a+b x}} \, dx}{11 a}\\ &=-\frac {2 A \sqrt {a+b x}}{11 a x^{11/2}}+\frac {2 (10 A b-11 a B) \sqrt {a+b x}}{99 a^2 x^{9/2}}+\frac {(8 b (10 A b-11 a B)) \int \frac {1}{x^{9/2} \sqrt {a+b x}} \, dx}{99 a^2}\\ &=-\frac {2 A \sqrt {a+b x}}{11 a x^{11/2}}+\frac {2 (10 A b-11 a B) \sqrt {a+b x}}{99 a^2 x^{9/2}}-\frac {16 b (10 A b-11 a B) \sqrt {a+b x}}{693 a^3 x^{7/2}}-\frac {\left (16 b^2 (10 A b-11 a B)\right ) \int \frac {1}{x^{7/2} \sqrt {a+b x}} \, dx}{231 a^3}\\ &=-\frac {2 A \sqrt {a+b x}}{11 a x^{11/2}}+\frac {2 (10 A b-11 a B) \sqrt {a+b x}}{99 a^2 x^{9/2}}-\frac {16 b (10 A b-11 a B) \sqrt {a+b x}}{693 a^3 x^{7/2}}+\frac {32 b^2 (10 A b-11 a B) \sqrt {a+b x}}{1155 a^4 x^{5/2}}+\frac {\left (64 b^3 (10 A b-11 a B)\right ) \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx}{1155 a^4}\\ &=-\frac {2 A \sqrt {a+b x}}{11 a x^{11/2}}+\frac {2 (10 A b-11 a B) \sqrt {a+b x}}{99 a^2 x^{9/2}}-\frac {16 b (10 A b-11 a B) \sqrt {a+b x}}{693 a^3 x^{7/2}}+\frac {32 b^2 (10 A b-11 a B) \sqrt {a+b x}}{1155 a^4 x^{5/2}}-\frac {128 b^3 (10 A b-11 a B) \sqrt {a+b x}}{3465 a^5 x^{3/2}}-\frac {\left (128 b^4 (10 A b-11 a B)\right ) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{3465 a^5}\\ &=-\frac {2 A \sqrt {a+b x}}{11 a x^{11/2}}+\frac {2 (10 A b-11 a B) \sqrt {a+b x}}{99 a^2 x^{9/2}}-\frac {16 b (10 A b-11 a B) \sqrt {a+b x}}{693 a^3 x^{7/2}}+\frac {32 b^2 (10 A b-11 a B) \sqrt {a+b x}}{1155 a^4 x^{5/2}}-\frac {128 b^3 (10 A b-11 a B) \sqrt {a+b x}}{3465 a^5 x^{3/2}}+\frac {256 b^4 (10 A b-11 a B) \sqrt {a+b x}}{3465 a^6 \sqrt {x}}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 114, normalized size = 0.62 \begin {gather*} -\frac {2 \sqrt {a+b x} \left (-1280 A b^5 x^5+128 a b^4 x^4 (5 A+11 B x)+35 a^5 (9 A+11 B x)-32 a^2 b^3 x^3 (15 A+22 B x)+16 a^3 b^2 x^2 (25 A+33 B x)-10 a^4 b x (35 A+44 B x)\right )}{3465 a^6 x^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 125, normalized size = 0.68
method | result | size |
gosper | \(-\frac {2 \sqrt {b x +a}\, \left (-1280 A \,b^{5} x^{5}+1408 B a \,b^{4} x^{5}+640 a A \,b^{4} x^{4}-704 B \,a^{2} b^{3} x^{4}-480 a^{2} A \,b^{3} x^{3}+528 B \,a^{3} b^{2} x^{3}+400 a^{3} A \,b^{2} x^{2}-440 B \,a^{4} b \,x^{2}-350 a^{4} A b x +385 a^{5} B x +315 a^{5} A \right )}{3465 x^{\frac {11}{2}} a^{6}}\) | \(125\) |
default | \(-\frac {2 \sqrt {b x +a}\, \left (-1280 A \,b^{5} x^{5}+1408 B a \,b^{4} x^{5}+640 a A \,b^{4} x^{4}-704 B \,a^{2} b^{3} x^{4}-480 a^{2} A \,b^{3} x^{3}+528 B \,a^{3} b^{2} x^{3}+400 a^{3} A \,b^{2} x^{2}-440 B \,a^{4} b \,x^{2}-350 a^{4} A b x +385 a^{5} B x +315 a^{5} A \right )}{3465 x^{\frac {11}{2}} a^{6}}\) | \(125\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (-1280 A \,b^{5} x^{5}+1408 B a \,b^{4} x^{5}+640 a A \,b^{4} x^{4}-704 B \,a^{2} b^{3} x^{4}-480 a^{2} A \,b^{3} x^{3}+528 B \,a^{3} b^{2} x^{3}+400 a^{3} A \,b^{2} x^{2}-440 B \,a^{4} b \,x^{2}-350 a^{4} A b x +385 a^{5} B x +315 a^{5} A \right )}{3465 x^{\frac {11}{2}} a^{6}}\) | \(125\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 244, normalized size = 1.33 \begin {gather*} -\frac {256 \, \sqrt {b x^{2} + a x} B b^{4}}{315 \, a^{5} x} + \frac {512 \, \sqrt {b x^{2} + a x} A b^{5}}{693 \, a^{6} x} + \frac {128 \, \sqrt {b x^{2} + a x} B b^{3}}{315 \, a^{4} x^{2}} - \frac {256 \, \sqrt {b x^{2} + a x} A b^{4}}{693 \, a^{5} x^{2}} - \frac {32 \, \sqrt {b x^{2} + a x} B b^{2}}{105 \, a^{3} x^{3}} + \frac {64 \, \sqrt {b x^{2} + a x} A b^{3}}{231 \, a^{4} x^{3}} + \frac {16 \, \sqrt {b x^{2} + a x} B b}{63 \, a^{2} x^{4}} - \frac {160 \, \sqrt {b x^{2} + a x} A b^{2}}{693 \, a^{3} x^{4}} - \frac {2 \, \sqrt {b x^{2} + a x} B}{9 \, a x^{5}} + \frac {20 \, \sqrt {b x^{2} + a x} A b}{99 \, a^{2} x^{5}} - \frac {2 \, \sqrt {b x^{2} + a x} A}{11 \, a x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.47, size = 126, normalized size = 0.69 \begin {gather*} -\frac {2 \, {\left (315 \, A a^{5} + 128 \, {\left (11 \, B a b^{4} - 10 \, A b^{5}\right )} x^{5} - 64 \, {\left (11 \, B a^{2} b^{3} - 10 \, A a b^{4}\right )} x^{4} + 48 \, {\left (11 \, B a^{3} b^{2} - 10 \, A a^{2} b^{3}\right )} x^{3} - 40 \, {\left (11 \, B a^{4} b - 10 \, A a^{3} b^{2}\right )} x^{2} + 35 \, {\left (11 \, B a^{5} - 10 \, A a^{4} b\right )} x\right )} \sqrt {b x + a}}{3465 \, a^{6} x^{\frac {11}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1809 vs.
\(2 (184) = 368\).
time = 157.67, size = 1809, normalized size = 9.89 \begin {gather*} - \frac {126 A a^{10} b^{\frac {51}{2}} \sqrt {\frac {a}{b x} + 1}}{693 a^{11} b^{25} x^{5} + 3465 a^{10} b^{26} x^{6} + 6930 a^{9} b^{27} x^{7} + 6930 a^{8} b^{28} x^{8} + 3465 a^{7} b^{29} x^{9} + 693 a^{6} b^{30} x^{10}} - \frac {490 A a^{9} b^{\frac {53}{2}} x \sqrt {\frac {a}{b x} + 1}}{693 a^{11} b^{25} x^{5} + 3465 a^{10} b^{26} x^{6} + 6930 a^{9} b^{27} x^{7} + 6930 a^{8} b^{28} x^{8} + 3465 a^{7} b^{29} x^{9} + 693 a^{6} b^{30} x^{10}} - \frac {720 A a^{8} b^{\frac {55}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{693 a^{11} b^{25} x^{5} + 3465 a^{10} b^{26} x^{6} + 6930 a^{9} b^{27} x^{7} + 6930 a^{8} b^{28} x^{8} + 3465 a^{7} b^{29} x^{9} + 693 a^{6} b^{30} x^{10}} - \frac {468 A a^{7} b^{\frac {57}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{693 a^{11} b^{25} x^{5} + 3465 a^{10} b^{26} x^{6} + 6930 a^{9} b^{27} x^{7} + 6930 a^{8} b^{28} x^{8} + 3465 a^{7} b^{29} x^{9} + 693 a^{6} b^{30} x^{10}} - \frac {126 A a^{6} b^{\frac {59}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{693 a^{11} b^{25} x^{5} + 3465 a^{10} b^{26} x^{6} + 6930 a^{9} b^{27} x^{7} + 6930 a^{8} b^{28} x^{8} + 3465 a^{7} b^{29} x^{9} + 693 a^{6} b^{30} x^{10}} + \frac {126 A a^{5} b^{\frac {61}{2}} x^{5} \sqrt {\frac {a}{b x} + 1}}{693 a^{11} b^{25} x^{5} + 3465 a^{10} b^{26} x^{6} + 6930 a^{9} b^{27} x^{7} + 6930 a^{8} b^{28} x^{8} + 3465 a^{7} b^{29} x^{9} + 693 a^{6} b^{30} x^{10}} + \frac {1260 A a^{4} b^{\frac {63}{2}} x^{6} \sqrt {\frac {a}{b x} + 1}}{693 a^{11} b^{25} x^{5} + 3465 a^{10} b^{26} x^{6} + 6930 a^{9} b^{27} x^{7} + 6930 a^{8} b^{28} x^{8} + 3465 a^{7} b^{29} x^{9} + 693 a^{6} b^{30} x^{10}} + \frac {3360 A a^{3} b^{\frac {65}{2}} x^{7} \sqrt {\frac {a}{b x} + 1}}{693 a^{11} b^{25} x^{5} + 3465 a^{10} b^{26} x^{6} + 6930 a^{9} b^{27} x^{7} + 6930 a^{8} b^{28} x^{8} + 3465 a^{7} b^{29} x^{9} + 693 a^{6} b^{30} x^{10}} + \frac {4032 A a^{2} b^{\frac {67}{2}} x^{8} \sqrt {\frac {a}{b x} + 1}}{693 a^{11} b^{25} x^{5} + 3465 a^{10} b^{26} x^{6} + 6930 a^{9} b^{27} x^{7} + 6930 a^{8} b^{28} x^{8} + 3465 a^{7} b^{29} x^{9} + 693 a^{6} b^{30} x^{10}} + \frac {2304 A a b^{\frac {69}{2}} x^{9} \sqrt {\frac {a}{b x} + 1}}{693 a^{11} b^{25} x^{5} + 3465 a^{10} b^{26} x^{6} + 6930 a^{9} b^{27} x^{7} + 6930 a^{8} b^{28} x^{8} + 3465 a^{7} b^{29} x^{9} + 693 a^{6} b^{30} x^{10}} + \frac {512 A b^{\frac {71}{2}} x^{10} \sqrt {\frac {a}{b x} + 1}}{693 a^{11} b^{25} x^{5} + 3465 a^{10} b^{26} x^{6} + 6930 a^{9} b^{27} x^{7} + 6930 a^{8} b^{28} x^{8} + 3465 a^{7} b^{29} x^{9} + 693 a^{6} b^{30} x^{10}} - \frac {70 B a^{8} b^{\frac {33}{2}} \sqrt {\frac {a}{b x} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{5} + 1890 a^{7} b^{18} x^{6} + 1260 a^{6} b^{19} x^{7} + 315 a^{5} b^{20} x^{8}} - \frac {200 B a^{7} b^{\frac {35}{2}} x \sqrt {\frac {a}{b x} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{5} + 1890 a^{7} b^{18} x^{6} + 1260 a^{6} b^{19} x^{7} + 315 a^{5} b^{20} x^{8}} - \frac {196 B a^{6} b^{\frac {37}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{5} + 1890 a^{7} b^{18} x^{6} + 1260 a^{6} b^{19} x^{7} + 315 a^{5} b^{20} x^{8}} - \frac {56 B a^{5} b^{\frac {39}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{5} + 1890 a^{7} b^{18} x^{6} + 1260 a^{6} b^{19} x^{7} + 315 a^{5} b^{20} x^{8}} - \frac {70 B a^{4} b^{\frac {41}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{5} + 1890 a^{7} b^{18} x^{6} + 1260 a^{6} b^{19} x^{7} + 315 a^{5} b^{20} x^{8}} - \frac {560 B a^{3} b^{\frac {43}{2}} x^{5} \sqrt {\frac {a}{b x} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{5} + 1890 a^{7} b^{18} x^{6} + 1260 a^{6} b^{19} x^{7} + 315 a^{5} b^{20} x^{8}} - \frac {1120 B a^{2} b^{\frac {45}{2}} x^{6} \sqrt {\frac {a}{b x} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{5} + 1890 a^{7} b^{18} x^{6} + 1260 a^{6} b^{19} x^{7} + 315 a^{5} b^{20} x^{8}} - \frac {896 B a b^{\frac {47}{2}} x^{7} \sqrt {\frac {a}{b x} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{5} + 1890 a^{7} b^{18} x^{6} + 1260 a^{6} b^{19} x^{7} + 315 a^{5} b^{20} x^{8}} - \frac {256 B b^{\frac {49}{2}} x^{8} \sqrt {\frac {a}{b x} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{5} + 1890 a^{7} b^{18} x^{6} + 1260 a^{6} b^{19} x^{7} + 315 a^{5} b^{20} x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.08, size = 201, normalized size = 1.10 \begin {gather*} -\frac {2 \, {\left ({\left (8 \, {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (11 \, B a b^{10} - 10 \, A b^{11}\right )} {\left (b x + a\right )}}{a^{6}} - \frac {11 \, {\left (11 \, B a^{2} b^{10} - 10 \, A a b^{11}\right )}}{a^{6}}\right )} + \frac {99 \, {\left (11 \, B a^{3} b^{10} - 10 \, A a^{2} b^{11}\right )}}{a^{6}}\right )} - \frac {231 \, {\left (11 \, B a^{4} b^{10} - 10 \, A a^{3} b^{11}\right )}}{a^{6}}\right )} {\left (b x + a\right )} + \frac {1155 \, {\left (11 \, B a^{5} b^{10} - 10 \, A a^{4} b^{11}\right )}}{a^{6}}\right )} {\left (b x + a\right )} - \frac {3465 \, {\left (B a^{6} b^{10} - A a^{5} b^{11}\right )}}{a^{6}}\right )} \sqrt {b x + a} b}{3465 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {11}{2}} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.05, size = 117, normalized size = 0.64 \begin {gather*} -\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{11\,a}+\frac {x\,\left (770\,B\,a^5-700\,A\,a^4\,b\right )}{3465\,a^6}-\frac {32\,b^2\,x^3\,\left (10\,A\,b-11\,B\,a\right )}{1155\,a^4}+\frac {128\,b^3\,x^4\,\left (10\,A\,b-11\,B\,a\right )}{3465\,a^5}-\frac {256\,b^4\,x^5\,\left (10\,A\,b-11\,B\,a\right )}{3465\,a^6}+\frac {16\,b\,x^2\,\left (10\,A\,b-11\,B\,a\right )}{693\,a^3}\right )}{x^{11/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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