Integrand size = 29, antiderivative size = 154 \[ \int \frac {x^{7/2} (A+B x)}{a^2+2 a b x+b^2 x^2} \, dx=\frac {a^2 (7 A b-9 a B) \sqrt {x}}{b^5}-\frac {a (7 A b-9 a B) x^{3/2}}{3 b^4}+\frac {(7 A b-9 a B) x^{5/2}}{5 b^3}-\frac {(7 A b-9 a B) x^{7/2}}{7 a b^2}+\frac {(A b-a B) x^{9/2}}{a b (a+b x)}-\frac {a^{5/2} (7 A b-9 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{11/2}} \]
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Time = 0.06 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {27, 79, 52, 65, 211} \[ \int \frac {x^{7/2} (A+B x)}{a^2+2 a b x+b^2 x^2} \, dx=-\frac {a^{5/2} (7 A b-9 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{11/2}}+\frac {a^2 \sqrt {x} (7 A b-9 a B)}{b^5}-\frac {a x^{3/2} (7 A b-9 a B)}{3 b^4}+\frac {x^{5/2} (7 A b-9 a B)}{5 b^3}-\frac {x^{7/2} (7 A b-9 a B)}{7 a b^2}+\frac {x^{9/2} (A b-a B)}{a b (a+b x)} \]
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Rule 27
Rule 52
Rule 65
Rule 79
Rule 211
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^{7/2} (A+B x)}{(a+b x)^2} \, dx \\ & = \frac {(A b-a B) x^{9/2}}{a b (a+b x)}-\frac {\left (\frac {7 A b}{2}-\frac {9 a B}{2}\right ) \int \frac {x^{7/2}}{a+b x} \, dx}{a b} \\ & = -\frac {(7 A b-9 a B) x^{7/2}}{7 a b^2}+\frac {(A b-a B) x^{9/2}}{a b (a+b x)}+\frac {(7 A b-9 a B) \int \frac {x^{5/2}}{a+b x} \, dx}{2 b^2} \\ & = \frac {(7 A b-9 a B) x^{5/2}}{5 b^3}-\frac {(7 A b-9 a B) x^{7/2}}{7 a b^2}+\frac {(A b-a B) x^{9/2}}{a b (a+b x)}-\frac {(a (7 A b-9 a B)) \int \frac {x^{3/2}}{a+b x} \, dx}{2 b^3} \\ & = -\frac {a (7 A b-9 a B) x^{3/2}}{3 b^4}+\frac {(7 A b-9 a B) x^{5/2}}{5 b^3}-\frac {(7 A b-9 a B) x^{7/2}}{7 a b^2}+\frac {(A b-a B) x^{9/2}}{a b (a+b x)}+\frac {\left (a^2 (7 A b-9 a B)\right ) \int \frac {\sqrt {x}}{a+b x} \, dx}{2 b^4} \\ & = \frac {a^2 (7 A b-9 a B) \sqrt {x}}{b^5}-\frac {a (7 A b-9 a B) x^{3/2}}{3 b^4}+\frac {(7 A b-9 a B) x^{5/2}}{5 b^3}-\frac {(7 A b-9 a B) x^{7/2}}{7 a b^2}+\frac {(A b-a B) x^{9/2}}{a b (a+b x)}-\frac {\left (a^3 (7 A b-9 a B)\right ) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{2 b^5} \\ & = \frac {a^2 (7 A b-9 a B) \sqrt {x}}{b^5}-\frac {a (7 A b-9 a B) x^{3/2}}{3 b^4}+\frac {(7 A b-9 a B) x^{5/2}}{5 b^3}-\frac {(7 A b-9 a B) x^{7/2}}{7 a b^2}+\frac {(A b-a B) x^{9/2}}{a b (a+b x)}-\frac {\left (a^3 (7 A b-9 a B)\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{b^5} \\ & = \frac {a^2 (7 A b-9 a B) \sqrt {x}}{b^5}-\frac {a (7 A b-9 a B) x^{3/2}}{3 b^4}+\frac {(7 A b-9 a B) x^{5/2}}{5 b^3}-\frac {(7 A b-9 a B) x^{7/2}}{7 a b^2}+\frac {(A b-a B) x^{9/2}}{a b (a+b x)}-\frac {a^{5/2} (7 A b-9 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{11/2}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.83 \[ \int \frac {x^{7/2} (A+B x)}{a^2+2 a b x+b^2 x^2} \, dx=\frac {\sqrt {x} \left (-945 a^4 B+105 a^3 b (7 A-6 B x)+6 b^4 x^3 (7 A+5 B x)+14 a^2 b^2 x (35 A+9 B x)-2 a b^3 x^2 (49 A+27 B x)\right )}{105 b^5 (a+b x)}+\frac {a^{5/2} (-7 A b+9 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{11/2}} \]
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Time = 0.19 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.81
method | result | size |
risch | \(\frac {2 \left (15 x^{3} B \,b^{3}+21 A \,b^{3} x^{2}-42 B \,x^{2} a \,b^{2}-70 A x a \,b^{2}+105 B x \,a^{2} b +315 A \,a^{2} b -420 B \,a^{3}\right ) \sqrt {x}}{105 b^{5}}-\frac {a^{3} \left (\frac {2 \left (-\frac {A b}{2}+\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (7 A b -9 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{\sqrt {b a}}\right )}{b^{5}}\) | \(124\) |
derivativedivides | \(\frac {\frac {2 B \,b^{3} x^{\frac {7}{2}}}{7}+\frac {2 A \,b^{3} x^{\frac {5}{2}}}{5}-\frac {4 B a \,b^{2} x^{\frac {5}{2}}}{5}-\frac {4 A a \,b^{2} x^{\frac {3}{2}}}{3}+2 B \,a^{2} b \,x^{\frac {3}{2}}+6 A \,a^{2} b \sqrt {x}-8 a^{3} B \sqrt {x}}{b^{5}}-\frac {2 a^{3} \left (\frac {\left (-\frac {A b}{2}+\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (7 A b -9 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{2 \sqrt {b a}}\right )}{b^{5}}\) | \(130\) |
default | \(\frac {\frac {2 B \,b^{3} x^{\frac {7}{2}}}{7}+\frac {2 A \,b^{3} x^{\frac {5}{2}}}{5}-\frac {4 B a \,b^{2} x^{\frac {5}{2}}}{5}-\frac {4 A a \,b^{2} x^{\frac {3}{2}}}{3}+2 B \,a^{2} b \,x^{\frac {3}{2}}+6 A \,a^{2} b \sqrt {x}-8 a^{3} B \sqrt {x}}{b^{5}}-\frac {2 a^{3} \left (\frac {\left (-\frac {A b}{2}+\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (7 A b -9 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{2 \sqrt {b a}}\right )}{b^{5}}\) | \(130\) |
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Time = 0.27 (sec) , antiderivative size = 341, normalized size of antiderivative = 2.21 \[ \int \frac {x^{7/2} (A+B x)}{a^2+2 a b x+b^2 x^2} \, dx=\left [-\frac {105 \, {\left (9 \, B a^{4} - 7 \, A a^{3} b + {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x - 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) - 2 \, {\left (30 \, B b^{4} x^{4} - 945 \, B a^{4} + 735 \, A a^{3} b - 6 \, {\left (9 \, B a b^{3} - 7 \, A b^{4}\right )} x^{3} + 14 \, {\left (9 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{2} - 70 \, {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x\right )} \sqrt {x}}{210 \, {\left (b^{6} x + a b^{5}\right )}}, \frac {105 \, {\left (9 \, B a^{4} - 7 \, A a^{3} b + {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) + {\left (30 \, B b^{4} x^{4} - 945 \, B a^{4} + 735 \, A a^{3} b - 6 \, {\left (9 \, B a b^{3} - 7 \, A b^{4}\right )} x^{3} + 14 \, {\left (9 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{2} - 70 \, {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x\right )} \sqrt {x}}{105 \, {\left (b^{6} x + a b^{5}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 986 vs. \(2 (143) = 286\).
Time = 111.01 (sec) , antiderivative size = 986, normalized size of antiderivative = 6.40 \[ \int \frac {x^{7/2} (A+B x)}{a^2+2 a b x+b^2 x^2} \, dx=\begin {cases} \tilde {\infty } \left (\frac {2 A x^{\frac {5}{2}}}{5} + \frac {2 B x^{\frac {7}{2}}}{7}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {\frac {2 A x^{\frac {9}{2}}}{9} + \frac {2 B x^{\frac {11}{2}}}{11}}{a^{2}} & \text {for}\: b = 0 \\\frac {\frac {2 A x^{\frac {5}{2}}}{5} + \frac {2 B x^{\frac {7}{2}}}{7}}{b^{2}} & \text {for}\: a = 0 \\- \frac {735 A a^{4} b \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} + \frac {735 A a^{4} b \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} + \frac {1470 A a^{3} b^{2} \sqrt {x} \sqrt {- \frac {a}{b}}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} - \frac {735 A a^{3} b^{2} x \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} + \frac {735 A a^{3} b^{2} x \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} + \frac {980 A a^{2} b^{3} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} - \frac {196 A a b^{4} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} + \frac {84 A b^{5} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} + \frac {945 B a^{5} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} - \frac {945 B a^{5} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} - \frac {1890 B a^{4} b \sqrt {x} \sqrt {- \frac {a}{b}}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} + \frac {945 B a^{4} b x \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} - \frac {945 B a^{4} b x \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} - \frac {1260 B a^{3} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} + \frac {252 B a^{2} b^{3} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} - \frac {108 B a b^{4} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} + \frac {60 B b^{5} x^{\frac {9}{2}} \sqrt {- \frac {a}{b}}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.90 \[ \int \frac {x^{7/2} (A+B x)}{a^2+2 a b x+b^2 x^2} \, dx=-\frac {{\left (B a^{4} - A a^{3} b\right )} \sqrt {x}}{b^{6} x + a b^{5}} + \frac {{\left (9 \, B a^{4} - 7 \, A a^{3} b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{5}} + \frac {2 \, {\left (15 \, B b^{3} x^{\frac {7}{2}} - 21 \, {\left (2 \, B a b^{2} - A b^{3}\right )} x^{\frac {5}{2}} + 35 \, {\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{\frac {3}{2}} - 105 \, {\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} \sqrt {x}\right )}}{105 \, b^{5}} \]
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Time = 0.26 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.95 \[ \int \frac {x^{7/2} (A+B x)}{a^2+2 a b x+b^2 x^2} \, dx=\frac {{\left (9 \, B a^{4} - 7 \, A a^{3} b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{5}} - \frac {B a^{4} \sqrt {x} - A a^{3} b \sqrt {x}}{{\left (b x + a\right )} b^{5}} + \frac {2 \, {\left (15 \, B b^{12} x^{\frac {7}{2}} - 42 \, B a b^{11} x^{\frac {5}{2}} + 21 \, A b^{12} x^{\frac {5}{2}} + 105 \, B a^{2} b^{10} x^{\frac {3}{2}} - 70 \, A a b^{11} x^{\frac {3}{2}} - 420 \, B a^{3} b^{9} \sqrt {x} + 315 \, A a^{2} b^{10} \sqrt {x}\right )}}{105 \, b^{14}} \]
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Time = 0.07 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.36 \[ \int \frac {x^{7/2} (A+B x)}{a^2+2 a b x+b^2 x^2} \, dx=\sqrt {x}\,\left (\frac {2\,a\,\left (\frac {2\,a\,\left (\frac {2\,A}{b^2}-\frac {4\,B\,a}{b^3}\right )}{b}+\frac {2\,B\,a^2}{b^4}\right )}{b}-\frac {a^2\,\left (\frac {2\,A}{b^2}-\frac {4\,B\,a}{b^3}\right )}{b^2}\right )+x^{5/2}\,\left (\frac {2\,A}{5\,b^2}-\frac {4\,B\,a}{5\,b^3}\right )-x^{3/2}\,\left (\frac {2\,a\,\left (\frac {2\,A}{b^2}-\frac {4\,B\,a}{b^3}\right )}{3\,b}+\frac {2\,B\,a^2}{3\,b^4}\right )+\frac {2\,B\,x^{7/2}}{7\,b^2}-\frac {\sqrt {x}\,\left (B\,a^4-A\,a^3\,b\right )}{x\,b^6+a\,b^5}+\frac {a^{5/2}\,\mathrm {atan}\left (\frac {a^{5/2}\,\sqrt {b}\,\sqrt {x}\,\left (7\,A\,b-9\,B\,a\right )}{9\,B\,a^4-7\,A\,a^3\,b}\right )\,\left (7\,A\,b-9\,B\,a\right )}{b^{11/2}} \]
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