Integrand size = 20, antiderivative size = 200 \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^{3/2}} \, dx=\frac {2 (A b-a B) x^{9/2}}{a b \sqrt {a+b x}}+\frac {35 a^2 (8 A b-9 a B) \sqrt {x} \sqrt {a+b x}}{64 b^5}-\frac {35 a (8 A b-9 a B) x^{3/2} \sqrt {a+b x}}{96 b^4}+\frac {7 (8 A b-9 a B) x^{5/2} \sqrt {a+b x}}{24 b^3}-\frac {(8 A b-9 a B) x^{7/2} \sqrt {a+b x}}{4 a b^2}-\frac {35 a^3 (8 A b-9 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^{11/2}} \]
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Time = 0.06 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {79, 52, 65, 223, 212} \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^{3/2}} \, dx=-\frac {35 a^3 (8 A b-9 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^{11/2}}+\frac {35 a^2 \sqrt {x} \sqrt {a+b x} (8 A b-9 a B)}{64 b^5}-\frac {35 a x^{3/2} \sqrt {a+b x} (8 A b-9 a B)}{96 b^4}+\frac {7 x^{5/2} \sqrt {a+b x} (8 A b-9 a B)}{24 b^3}-\frac {x^{7/2} \sqrt {a+b x} (8 A b-9 a B)}{4 a b^2}+\frac {2 x^{9/2} (A b-a B)}{a b \sqrt {a+b x}} \]
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Rule 52
Rule 65
Rule 79
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {2 (A b-a B) x^{9/2}}{a b \sqrt {a+b x}}-\frac {\left (2 \left (4 A b-\frac {9 a B}{2}\right )\right ) \int \frac {x^{7/2}}{\sqrt {a+b x}} \, dx}{a b} \\ & = \frac {2 (A b-a B) x^{9/2}}{a b \sqrt {a+b x}}-\frac {(8 A b-9 a B) x^{7/2} \sqrt {a+b x}}{4 a b^2}+\frac {(7 (8 A b-9 a B)) \int \frac {x^{5/2}}{\sqrt {a+b x}} \, dx}{8 b^2} \\ & = \frac {2 (A b-a B) x^{9/2}}{a b \sqrt {a+b x}}+\frac {7 (8 A b-9 a B) x^{5/2} \sqrt {a+b x}}{24 b^3}-\frac {(8 A b-9 a B) x^{7/2} \sqrt {a+b x}}{4 a b^2}-\frac {(35 a (8 A b-9 a B)) \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx}{48 b^3} \\ & = \frac {2 (A b-a B) x^{9/2}}{a b \sqrt {a+b x}}-\frac {35 a (8 A b-9 a B) x^{3/2} \sqrt {a+b x}}{96 b^4}+\frac {7 (8 A b-9 a B) x^{5/2} \sqrt {a+b x}}{24 b^3}-\frac {(8 A b-9 a B) x^{7/2} \sqrt {a+b x}}{4 a b^2}+\frac {\left (35 a^2 (8 A b-9 a B)\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{64 b^4} \\ & = \frac {2 (A b-a B) x^{9/2}}{a b \sqrt {a+b x}}+\frac {35 a^2 (8 A b-9 a B) \sqrt {x} \sqrt {a+b x}}{64 b^5}-\frac {35 a (8 A b-9 a B) x^{3/2} \sqrt {a+b x}}{96 b^4}+\frac {7 (8 A b-9 a B) x^{5/2} \sqrt {a+b x}}{24 b^3}-\frac {(8 A b-9 a B) x^{7/2} \sqrt {a+b x}}{4 a b^2}-\frac {\left (35 a^3 (8 A b-9 a B)\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{128 b^5} \\ & = \frac {2 (A b-a B) x^{9/2}}{a b \sqrt {a+b x}}+\frac {35 a^2 (8 A b-9 a B) \sqrt {x} \sqrt {a+b x}}{64 b^5}-\frac {35 a (8 A b-9 a B) x^{3/2} \sqrt {a+b x}}{96 b^4}+\frac {7 (8 A b-9 a B) x^{5/2} \sqrt {a+b x}}{24 b^3}-\frac {(8 A b-9 a B) x^{7/2} \sqrt {a+b x}}{4 a b^2}-\frac {\left (35 a^3 (8 A b-9 a B)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{64 b^5} \\ & = \frac {2 (A b-a B) x^{9/2}}{a b \sqrt {a+b x}}+\frac {35 a^2 (8 A b-9 a B) \sqrt {x} \sqrt {a+b x}}{64 b^5}-\frac {35 a (8 A b-9 a B) x^{3/2} \sqrt {a+b x}}{96 b^4}+\frac {7 (8 A b-9 a B) x^{5/2} \sqrt {a+b x}}{24 b^3}-\frac {(8 A b-9 a B) x^{7/2} \sqrt {a+b x}}{4 a b^2}-\frac {\left (35 a^3 (8 A b-9 a B)\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^5} \\ & = \frac {2 (A b-a B) x^{9/2}}{a b \sqrt {a+b x}}+\frac {35 a^2 (8 A b-9 a B) \sqrt {x} \sqrt {a+b x}}{64 b^5}-\frac {35 a (8 A b-9 a B) x^{3/2} \sqrt {a+b x}}{96 b^4}+\frac {7 (8 A b-9 a B) x^{5/2} \sqrt {a+b x}}{24 b^3}-\frac {(8 A b-9 a B) x^{7/2} \sqrt {a+b x}}{4 a b^2}-\frac {35 a^3 (8 A b-9 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^{11/2}} \\ \end{align*}
Time = 0.59 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.72 \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^{3/2}} \, dx=\frac {\sqrt {x} \left (-945 a^4 B+105 a^3 b (8 A-3 B x)+16 b^4 x^3 (4 A+3 B x)-8 a b^3 x^2 (14 A+9 B x)+14 a^2 b^2 x (20 A+9 B x)\right )}{192 b^5 \sqrt {a+b x}}+\frac {35 a^3 (-8 A b+9 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )}{32 b^{11/2}} \]
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Time = 1.46 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.05
method | result | size |
risch | \(\frac {\left (48 b^{3} B \,x^{3}+64 A \,b^{3} x^{2}-120 B a \,b^{2} x^{2}-176 a \,b^{2} A x +246 a^{2} b B x +456 a^{2} b A -561 a^{3} B \right ) \sqrt {x}\, \sqrt {b x +a}}{192 b^{5}}-\frac {a^{3} \left (280 A \sqrt {b}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )-\frac {315 B a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{\sqrt {b}}-\frac {256 \left (A b -B a \right ) \sqrt {b \left (x +\frac {a}{b}\right )^{2}-\left (x +\frac {a}{b}\right ) a}}{b \left (x +\frac {a}{b}\right )}\right ) \sqrt {x \left (b x +a \right )}}{128 b^{5} \sqrt {x}\, \sqrt {b x +a}}\) | \(210\) |
default | \(-\frac {\left (-96 B \,b^{\frac {9}{2}} x^{4} \sqrt {x \left (b x +a \right )}-128 A \,b^{\frac {9}{2}} x^{3} \sqrt {x \left (b x +a \right )}+144 B a \,b^{\frac {7}{2}} x^{3} \sqrt {x \left (b x +a \right )}+224 A a \,b^{\frac {7}{2}} x^{2} \sqrt {x \left (b x +a \right )}-252 B \,a^{2} b^{\frac {5}{2}} x^{2} \sqrt {x \left (b x +a \right )}+840 A \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{3} b^{2} x -560 A \,b^{\frac {5}{2}} \sqrt {x \left (b x +a \right )}\, a^{2} x -945 B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{4} b x +630 B \,b^{\frac {3}{2}} \sqrt {x \left (b x +a \right )}\, a^{3} x +840 A \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{4} b -1680 A \,b^{\frac {3}{2}} \sqrt {x \left (b x +a \right )}\, a^{3}-945 B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{5}+1890 B \sqrt {b}\, \sqrt {x \left (b x +a \right )}\, a^{4}\right ) \sqrt {x}}{384 b^{\frac {11}{2}} \sqrt {x \left (b x +a \right )}\, \sqrt {b x +a}}\) | \(330\) |
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Time = 0.24 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.78 \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^{3/2}} \, dx=\left [-\frac {105 \, {\left (9 \, B a^{5} - 8 \, A a^{4} b + {\left (9 \, B a^{4} b - 8 \, A a^{3} b^{2}\right )} x\right )} \sqrt {b} \log \left (2 \, b x - 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (48 \, B b^{5} x^{4} - 945 \, B a^{4} b + 840 \, A a^{3} b^{2} - 8 \, {\left (9 \, B a b^{4} - 8 \, A b^{5}\right )} x^{3} + 14 \, {\left (9 \, B a^{2} b^{3} - 8 \, A a b^{4}\right )} x^{2} - 35 \, {\left (9 \, B a^{3} b^{2} - 8 \, A a^{2} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{384 \, {\left (b^{7} x + a b^{6}\right )}}, -\frac {105 \, {\left (9 \, B a^{5} - 8 \, A a^{4} b + {\left (9 \, B a^{4} b - 8 \, A a^{3} b^{2}\right )} x\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (48 \, B b^{5} x^{4} - 945 \, B a^{4} b + 840 \, A a^{3} b^{2} - 8 \, {\left (9 \, B a b^{4} - 8 \, A b^{5}\right )} x^{3} + 14 \, {\left (9 \, B a^{2} b^{3} - 8 \, A a b^{4}\right )} x^{2} - 35 \, {\left (9 \, B a^{3} b^{2} - 8 \, A a^{2} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{192 \, {\left (b^{7} x + a b^{6}\right )}}\right ] \]
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Timed out. \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^{3/2}} \, dx=\text {Timed out} \]
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Time = 0.19 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.29 \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^{3/2}} \, dx=\frac {B x^{5}}{4 \, \sqrt {b x^{2} + a x} b} - \frac {3 \, B a x^{4}}{8 \, \sqrt {b x^{2} + a x} b^{2}} + \frac {A x^{4}}{3 \, \sqrt {b x^{2} + a x} b} + \frac {21 \, B a^{2} x^{3}}{32 \, \sqrt {b x^{2} + a x} b^{3}} - \frac {7 \, A a x^{3}}{12 \, \sqrt {b x^{2} + a x} b^{2}} - \frac {105 \, B a^{3} x^{2}}{64 \, \sqrt {b x^{2} + a x} b^{4}} + \frac {35 \, A a^{2} x^{2}}{24 \, \sqrt {b x^{2} + a x} b^{3}} - \frac {315 \, B a^{4} x}{64 \, \sqrt {b x^{2} + a x} b^{5}} + \frac {35 \, A a^{3} x}{8 \, \sqrt {b x^{2} + a x} b^{4}} + \frac {315 \, B a^{4} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{128 \, b^{\frac {11}{2}}} - \frac {35 \, A a^{3} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{16 \, b^{\frac {9}{2}}} \]
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Time = 16.50 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.21 \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^{3/2}} \, dx=\frac {1}{192} \, {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b x + a\right )} B {\left | b \right |}}{b^{7}} - \frac {33 \, B a b^{27} {\left | b \right |} - 8 \, A b^{28} {\left | b \right |}}{b^{34}}\right )} + \frac {315 \, B a^{2} b^{27} {\left | b \right |} - 152 \, A a b^{28} {\left | b \right |}}{b^{34}}\right )} - \frac {3 \, {\left (325 \, B a^{3} b^{27} {\left | b \right |} - 232 \, A a^{2} b^{28} {\left | b \right |}\right )}}{b^{34}}\right )} \sqrt {{\left (b x + a\right )} b - a b} \sqrt {b x + a} - \frac {35 \, {\left (9 \, B a^{4} {\left | b \right |} - 8 \, A a^{3} b {\left | b \right |}\right )} \log \left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{128 \, b^{\frac {13}{2}}} - \frac {4 \, {\left (B a^{5} {\left | b \right |} - A a^{4} b {\left | b \right |}\right )}}{{\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} b^{\frac {11}{2}}} \]
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Timed out. \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^{3/2}} \, dx=\int \frac {x^{7/2}\,\left (A+B\,x\right )}{{\left (a+b\,x\right )}^{3/2}} \,d x \]
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