Optimal. Leaf size=210 \[ -\frac {512 b^3 \sqrt {a+b x} (4 A b-3 a B)}{63 a^7 \sqrt {x}}+\frac {256 b^2 \sqrt {a+b x} (4 A b-3 a B)}{63 a^6 x^{3/2}}-\frac {64 b \sqrt {a+b x} (4 A b-3 a B)}{21 a^5 x^{5/2}}+\frac {160 \sqrt {a+b x} (4 A b-3 a B)}{63 a^4 x^{7/2}}-\frac {20 (4 A b-3 a B)}{9 a^3 x^{7/2} \sqrt {a+b x}}-\frac {2 (4 A b-3 a B)}{9 a^2 x^{7/2} (a+b x)^{3/2}}-\frac {2 A}{9 a x^{9/2} (a+b x)^{3/2}} \]
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Rubi [A] time = 0.08, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {78, 45, 37} \begin {gather*} \frac {256 b^2 \sqrt {a+b x} (4 A b-3 a B)}{63 a^6 x^{3/2}}-\frac {512 b^3 \sqrt {a+b x} (4 A b-3 a B)}{63 a^7 \sqrt {x}}-\frac {64 b \sqrt {a+b x} (4 A b-3 a B)}{21 a^5 x^{5/2}}+\frac {160 \sqrt {a+b x} (4 A b-3 a B)}{63 a^4 x^{7/2}}-\frac {20 (4 A b-3 a B)}{9 a^3 x^{7/2} \sqrt {a+b x}}-\frac {2 (4 A b-3 a B)}{9 a^2 x^{7/2} (a+b x)^{3/2}}-\frac {2 A}{9 a x^{9/2} (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 78
Rubi steps
\begin {align*} \int \frac {A+B x}{x^{11/2} (a+b x)^{5/2}} \, dx &=-\frac {2 A}{9 a x^{9/2} (a+b x)^{3/2}}+\frac {\left (2 \left (-6 A b+\frac {9 a B}{2}\right )\right ) \int \frac {1}{x^{9/2} (a+b x)^{5/2}} \, dx}{9 a}\\ &=-\frac {2 A}{9 a x^{9/2} (a+b x)^{3/2}}-\frac {2 (4 A b-3 a B)}{9 a^2 x^{7/2} (a+b x)^{3/2}}-\frac {(10 (4 A b-3 a B)) \int \frac {1}{x^{9/2} (a+b x)^{3/2}} \, dx}{9 a^2}\\ &=-\frac {2 A}{9 a x^{9/2} (a+b x)^{3/2}}-\frac {2 (4 A b-3 a B)}{9 a^2 x^{7/2} (a+b x)^{3/2}}-\frac {20 (4 A b-3 a B)}{9 a^3 x^{7/2} \sqrt {a+b x}}-\frac {(80 (4 A b-3 a B)) \int \frac {1}{x^{9/2} \sqrt {a+b x}} \, dx}{9 a^3}\\ &=-\frac {2 A}{9 a x^{9/2} (a+b x)^{3/2}}-\frac {2 (4 A b-3 a B)}{9 a^2 x^{7/2} (a+b x)^{3/2}}-\frac {20 (4 A b-3 a B)}{9 a^3 x^{7/2} \sqrt {a+b x}}+\frac {160 (4 A b-3 a B) \sqrt {a+b x}}{63 a^4 x^{7/2}}+\frac {(160 b (4 A b-3 a B)) \int \frac {1}{x^{7/2} \sqrt {a+b x}} \, dx}{21 a^4}\\ &=-\frac {2 A}{9 a x^{9/2} (a+b x)^{3/2}}-\frac {2 (4 A b-3 a B)}{9 a^2 x^{7/2} (a+b x)^{3/2}}-\frac {20 (4 A b-3 a B)}{9 a^3 x^{7/2} \sqrt {a+b x}}+\frac {160 (4 A b-3 a B) \sqrt {a+b x}}{63 a^4 x^{7/2}}-\frac {64 b (4 A b-3 a B) \sqrt {a+b x}}{21 a^5 x^{5/2}}-\frac {\left (128 b^2 (4 A b-3 a B)\right ) \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx}{21 a^5}\\ &=-\frac {2 A}{9 a x^{9/2} (a+b x)^{3/2}}-\frac {2 (4 A b-3 a B)}{9 a^2 x^{7/2} (a+b x)^{3/2}}-\frac {20 (4 A b-3 a B)}{9 a^3 x^{7/2} \sqrt {a+b x}}+\frac {160 (4 A b-3 a B) \sqrt {a+b x}}{63 a^4 x^{7/2}}-\frac {64 b (4 A b-3 a B) \sqrt {a+b x}}{21 a^5 x^{5/2}}+\frac {256 b^2 (4 A b-3 a B) \sqrt {a+b x}}{63 a^6 x^{3/2}}+\frac {\left (256 b^3 (4 A b-3 a B)\right ) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{63 a^6}\\ &=-\frac {2 A}{9 a x^{9/2} (a+b x)^{3/2}}-\frac {2 (4 A b-3 a B)}{9 a^2 x^{7/2} (a+b x)^{3/2}}-\frac {20 (4 A b-3 a B)}{9 a^3 x^{7/2} \sqrt {a+b x}}+\frac {160 (4 A b-3 a B) \sqrt {a+b x}}{63 a^4 x^{7/2}}-\frac {64 b (4 A b-3 a B) \sqrt {a+b x}}{21 a^5 x^{5/2}}+\frac {256 b^2 (4 A b-3 a B) \sqrt {a+b x}}{63 a^6 x^{3/2}}-\frac {512 b^3 (4 A b-3 a B) \sqrt {a+b x}}{63 a^7 \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 127, normalized size = 0.60 \begin {gather*} -\frac {2 \left (a^6 (7 A+9 B x)-6 a^5 b x (2 A+3 B x)+24 a^4 b^2 x^2 (A+2 B x)-32 a^3 b^3 x^3 (2 A+9 B x)+384 a^2 b^4 x^4 (A-3 B x)-768 a b^5 x^5 (B x-2 A)+1024 A b^6 x^6\right )}{63 a^7 x^{9/2} (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.34, size = 154, normalized size = 0.73 \begin {gather*} \frac {2 \left (-7 a^6 A-9 a^6 B x+12 a^5 A b x+18 a^5 b B x^2-24 a^4 A b^2 x^2-48 a^4 b^2 B x^3+64 a^3 A b^3 x^3+288 a^3 b^3 B x^4-384 a^2 A b^4 x^4+1152 a^2 b^4 B x^5-1536 a A b^5 x^5+768 a b^5 B x^6-1024 A b^6 x^6\right )}{63 a^7 x^{9/2} (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 176, normalized size = 0.84 \begin {gather*} -\frac {2 \, {\left (7 \, A a^{6} - 256 \, {\left (3 \, B a b^{5} - 4 \, A b^{6}\right )} x^{6} - 384 \, {\left (3 \, B a^{2} b^{4} - 4 \, A a b^{5}\right )} x^{5} - 96 \, {\left (3 \, B a^{3} b^{3} - 4 \, A a^{2} b^{4}\right )} x^{4} + 16 \, {\left (3 \, B a^{4} b^{2} - 4 \, A a^{3} b^{3}\right )} x^{3} - 6 \, {\left (3 \, B a^{5} b - 4 \, A a^{4} b^{2}\right )} x^{2} + 3 \, {\left (3 \, B a^{6} - 4 \, A a^{5} b\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{63 \, {\left (a^{7} b^{2} x^{7} + 2 \, a^{8} b x^{6} + a^{9} x^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.80, size = 418, normalized size = 1.99 \begin {gather*} \frac {2 \, {\left ({\left ({\left (b x + a\right )} {\left ({\left (b x + a\right )} {\left (\frac {{\left (474 \, B a^{19} b^{13} - 667 \, A a^{18} b^{14}\right )} {\left (b x + a\right )}}{a^{25} b^{4} {\left | b \right |}} - \frac {9 \, {\left (223 \, B a^{20} b^{13} - 316 \, A a^{19} b^{14}\right )}}{a^{25} b^{4} {\left | b \right |}}\right )} + \frac {63 \, {\left (51 \, B a^{21} b^{13} - 73 \, A a^{20} b^{14}\right )}}{a^{25} b^{4} {\left | b \right |}}\right )} - \frac {210 \, {\left (11 \, B a^{22} b^{13} - 16 \, A a^{21} b^{14}\right )}}{a^{25} b^{4} {\left | b \right |}}\right )} {\left (b x + a\right )} + \frac {315 \, {\left (2 \, B a^{23} b^{13} - 3 \, A a^{22} b^{14}\right )}}{a^{25} b^{4} {\left | b \right |}}\right )} \sqrt {b x + a}}{63 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {9}{2}}} + \frac {4 \, {\left (12 \, B a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac {11}{2}} + 30 \, B a^{2} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {13}{2}} - 15 \, A {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac {13}{2}} + 14 \, B a^{3} b^{\frac {15}{2}} - 36 \, A a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {15}{2}} - 17 \, A a^{2} b^{\frac {17}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} a^{6} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 149, normalized size = 0.71 \begin {gather*} -\frac {2 \left (1024 A \,b^{6} x^{6}-768 B a \,b^{5} x^{6}+1536 A a \,b^{5} x^{5}-1152 B \,a^{2} b^{4} x^{5}+384 A \,a^{2} b^{4} x^{4}-288 B \,a^{3} b^{3} x^{4}-64 A \,a^{3} b^{3} x^{3}+48 B \,a^{4} b^{2} x^{3}+24 A \,a^{4} b^{2} x^{2}-18 B \,a^{5} b \,x^{2}-12 A \,a^{5} b x +9 B \,a^{6} x +7 A \,a^{6}\right )}{63 \left (b x +a \right )^{\frac {3}{2}} a^{7} x^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 270, normalized size = 1.29 \begin {gather*} -\frac {64 \, B b^{3} x}{21 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{4}} + \frac {512 \, B b^{4} x}{21 \, \sqrt {b x^{2} + a x} a^{6}} + \frac {256 \, A b^{4} x}{63 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{5}} - \frac {2048 \, A b^{5} x}{63 \, \sqrt {b x^{2} + a x} a^{7}} - \frac {32 \, B b^{2}}{21 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{3}} + \frac {256 \, B b^{3}}{21 \, \sqrt {b x^{2} + a x} a^{5}} + \frac {128 \, A b^{3}}{63 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{4}} - \frac {1024 \, A b^{4}}{63 \, \sqrt {b x^{2} + a x} a^{6}} + \frac {4 \, B b}{7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2} x} - \frac {16 \, A b^{2}}{21 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{3} x} - \frac {2 \, B}{7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a x^{2}} + \frac {8 \, A b}{21 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2} x^{2}} - \frac {2 \, A}{9 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.08, size = 162, normalized size = 0.77 \begin {gather*} -\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{9\,a\,b^2}-\frac {32\,x^3\,\left (4\,A\,b-3\,B\,a\right )}{63\,a^4}+\frac {4\,x^2\,\left (4\,A\,b-3\,B\,a\right )}{21\,a^3\,b}+\frac {256\,b^2\,x^5\,\left (4\,A\,b-3\,B\,a\right )}{21\,a^6}+\frac {512\,b^3\,x^6\,\left (4\,A\,b-3\,B\,a\right )}{63\,a^7}+\frac {64\,b\,x^4\,\left (4\,A\,b-3\,B\,a\right )}{21\,a^5}+\frac {x\,\left (18\,B\,a^6-24\,A\,a^5\,b\right )}{63\,a^7\,b^2}\right )}{x^{13/2}+\frac {2\,a\,x^{11/2}}{b}+\frac {a^2\,x^{9/2}}{b^2}} \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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