Optimal. Leaf size=304 \[ -\frac {a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{13 x^{13} (a+b x)}-\frac {a^4 (5 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{12 x^{12} (a+b x)}-\frac {5 a^3 b (2 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac {a^2 b^2 (A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{x^{10} (a+b x)}-\frac {5 a b^3 (A b+2 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac {b^4 (A b+5 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{8 x^8 (a+b x)}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)} \]
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Rubi [A]
time = 0.08, antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {784, 77}
\begin {gather*} -\frac {a^2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{x^{10} (a+b x)}-\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{8 x^8 (a+b x)}-\frac {5 a b^3 \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{9 x^9 (a+b x)}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac {a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{13 x^{13} (a+b x)}-\frac {a^4 \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{12 x^{12} (a+b x)}-\frac {5 a^3 b \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{11 x^{11} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 784
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{14}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5 (A+B x)}{x^{14}} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a^5 A b^5}{x^{14}}+\frac {a^4 b^5 (5 A b+a B)}{x^{13}}+\frac {5 a^3 b^6 (2 A b+a B)}{x^{12}}+\frac {10 a^2 b^7 (A b+a B)}{x^{11}}+\frac {5 a b^8 (A b+2 a B)}{x^{10}}+\frac {b^9 (A b+5 a B)}{x^9}+\frac {b^{10} B}{x^8}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac {a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{13 x^{13} (a+b x)}-\frac {a^4 (5 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{12 x^{12} (a+b x)}-\frac {5 a^3 b (2 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac {a^2 b^2 (A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{x^{10} (a+b x)}-\frac {5 a b^3 (A b+2 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac {b^4 (A b+5 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{8 x^8 (a+b x)}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 125, normalized size = 0.41 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (1287 b^5 x^5 (7 A+8 B x)+5005 a b^4 x^4 (8 A+9 B x)+8008 a^2 b^3 x^3 (9 A+10 B x)+6552 a^3 b^2 x^2 (10 A+11 B x)+2730 a^4 b x (11 A+12 B x)+462 a^5 (12 A+13 B x)\right )}{72072 x^{13} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.65, size = 140, normalized size = 0.46
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{5} B \,x^{6}}{7}+\left (-\frac {1}{8} b^{5} A -\frac {5}{8} a \,b^{4} B \right ) x^{5}+\left (-\frac {5}{9} a \,b^{4} A -\frac {10}{9} a^{2} b^{3} B \right ) x^{4}+\left (-a^{2} b^{3} A -a^{3} b^{2} B \right ) x^{3}+\left (-\frac {10}{11} a^{3} b^{2} A -\frac {5}{11} a^{4} b B \right ) x^{2}+\left (-\frac {5}{12} a^{4} b A -\frac {1}{12} a^{5} B \right ) x -\frac {a^{5} A}{13}\right )}{\left (b x +a \right ) x^{13}}\) | \(136\) |
gosper | \(-\frac {\left (10296 b^{5} B \,x^{6}+9009 A \,b^{5} x^{5}+45045 B a \,b^{4} x^{5}+40040 A a \,b^{4} x^{4}+80080 B \,a^{2} b^{3} x^{4}+72072 A \,a^{2} b^{3} x^{3}+72072 B \,a^{3} b^{2} x^{3}+65520 A \,a^{3} b^{2} x^{2}+32760 B \,a^{4} b \,x^{2}+30030 a^{4} b A x +6006 a^{5} B x +5544 a^{5} A \right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{72072 x^{13} \left (b x +a \right )^{5}}\) | \(140\) |
default | \(-\frac {\left (10296 b^{5} B \,x^{6}+9009 A \,b^{5} x^{5}+45045 B a \,b^{4} x^{5}+40040 A a \,b^{4} x^{4}+80080 B \,a^{2} b^{3} x^{4}+72072 A \,a^{2} b^{3} x^{3}+72072 B \,a^{3} b^{2} x^{3}+65520 A \,a^{3} b^{2} x^{2}+32760 B \,a^{4} b \,x^{2}+30030 a^{4} b A x +6006 a^{5} B x +5544 a^{5} A \right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{72072 x^{13} \left (b x +a \right )^{5}}\) | \(140\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 795 vs.
\(2 (215) = 430\).
time = 0.31, size = 795, normalized size = 2.62 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{12}}{6 \, a^{12}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{13}}{6 \, a^{13}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{11}}{6 \, a^{11} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{12}}{6 \, a^{12} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{10}}{6 \, a^{12} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{11}}{6 \, a^{13} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{9}}{6 \, a^{11} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{10}}{6 \, a^{12} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{8}}{6 \, a^{10} x^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{9}}{6 \, a^{11} x^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{7}}{6 \, a^{9} x^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{8}}{6 \, a^{10} x^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{6}}{6 \, a^{8} x^{6}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{7}}{6 \, a^{9} x^{6}} + \frac {923 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{5}}{5544 \, a^{7} x^{7}} - \frac {1715 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{6}}{10296 \, a^{8} x^{7}} - \frac {131 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{4}}{792 \, a^{6} x^{8}} + \frac {1709 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{5}}{10296 \, a^{7} x^{8}} + \frac {16 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{3}}{99 \, a^{5} x^{9}} - \frac {211 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{4}}{1287 \, a^{6} x^{9}} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{2}}{33 \, a^{4} x^{10}} + \frac {68 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{3}}{429 \, a^{5} x^{10}} + \frac {17 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b}{132 \, a^{3} x^{11}} - \frac {251 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{2}}{1716 \, a^{4} x^{11}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B}{12 \, a^{2} x^{12}} + \frac {19 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b}{156 \, a^{3} x^{12}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A}{13 \, a^{2} x^{13}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.54, size = 119, normalized size = 0.39 \begin {gather*} -\frac {10296 \, B b^{5} x^{6} + 5544 \, A a^{5} + 9009 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 40040 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 72072 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 32760 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 6006 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{72072 \, x^{13}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{x^{14}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.71, size = 221, normalized size = 0.73 \begin {gather*} -\frac {{\left (13 \, B a b^{12} - 7 \, A b^{13}\right )} \mathrm {sgn}\left (b x + a\right )}{72072 \, a^{8}} - \frac {10296 \, B b^{5} x^{6} \mathrm {sgn}\left (b x + a\right ) + 45045 \, B a b^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + 9009 \, A b^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 80080 \, B a^{2} b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 40040 \, A a b^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 72072 \, B a^{3} b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 72072 \, A a^{2} b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 32760 \, B a^{4} b x^{2} \mathrm {sgn}\left (b x + a\right ) + 65520 \, A a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 6006 \, B a^{5} x \mathrm {sgn}\left (b x + a\right ) + 30030 \, A a^{4} b x \mathrm {sgn}\left (b x + a\right ) + 5544 \, A a^{5} \mathrm {sgn}\left (b x + a\right )}{72072 \, x^{13}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.23, size = 284, normalized size = 0.93 \begin {gather*} -\frac {\left (\frac {B\,a^5}{12}+\frac {5\,A\,b\,a^4}{12}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^{12}\,\left (a+b\,x\right )}-\frac {\left (\frac {A\,b^5}{8}+\frac {5\,B\,a\,b^4}{8}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^8\,\left (a+b\,x\right )}-\frac {A\,a^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{13\,x^{13}\,\left (a+b\,x\right )}-\frac {B\,b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{7\,x^7\,\left (a+b\,x\right )}-\frac {5\,a\,b^3\,\left (A\,b+2\,B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{9\,x^9\,\left (a+b\,x\right )}-\frac {5\,a^3\,b\,\left (2\,A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{11\,x^{11}\,\left (a+b\,x\right )}-\frac {a^2\,b^2\,\left (A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^{10}\,\left (a+b\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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