Integrand size = 31, antiderivative size = 306 \[ \int \frac {x^{7/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {35 (A b-9 a B) x^{3/2}}{192 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) x^{9/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-9 a B) x^{7/2}}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (A b-9 a B) x^{5/2}}{96 a b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (A b-9 a B) \sqrt {x} (a+b x)}{64 a b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 (A b-9 a B) (a+b x) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 \sqrt {a} b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
[Out]
Time = 0.10 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {784, 79, 43, 52, 65, 211} \[ \int \frac {x^{7/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {35 (a+b x) (A b-9 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 \sqrt {a} b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {x^{9/2} (A b-a B)}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {x^{7/2} (A b-9 a B)}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 \sqrt {x} (a+b x) (A b-9 a B)}{64 a b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 x^{3/2} (A b-9 a B)}{192 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 x^{5/2} (A b-9 a B)}{96 a b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]
[In]
[Out]
Rule 43
Rule 52
Rule 65
Rule 79
Rule 211
Rule 784
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {x^{7/2} (A+B x)}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {(A b-a B) x^{9/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (b^2 (A b-9 a B) \left (a b+b^2 x\right )\right ) \int \frac {x^{7/2}}{\left (a b+b^2 x\right )^4} \, dx}{8 a \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {(A b-a B) x^{9/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-9 a B) x^{7/2}}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (7 (A b-9 a B) \left (a b+b^2 x\right )\right ) \int \frac {x^{5/2}}{\left (a b+b^2 x\right )^3} \, dx}{48 a \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {(A b-a B) x^{9/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-9 a B) x^{7/2}}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (A b-9 a B) x^{5/2}}{96 a b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (35 (A b-9 a B) \left (a b+b^2 x\right )\right ) \int \frac {x^{3/2}}{\left (a b+b^2 x\right )^2} \, dx}{192 a b^2 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {35 (A b-9 a B) x^{3/2}}{192 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) x^{9/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-9 a B) x^{7/2}}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (A b-9 a B) x^{5/2}}{96 a b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (35 (A b-9 a B) \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {x}}{a b+b^2 x} \, dx}{128 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {35 (A b-9 a B) x^{3/2}}{192 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) x^{9/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-9 a B) x^{7/2}}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (A b-9 a B) x^{5/2}}{96 a b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (A b-9 a B) \sqrt {x} (a+b x)}{64 a b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 (A b-9 a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{\sqrt {x} \left (a b+b^2 x\right )} \, dx}{128 b^5 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {35 (A b-9 a B) x^{3/2}}{192 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) x^{9/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-9 a B) x^{7/2}}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (A b-9 a B) x^{5/2}}{96 a b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (A b-9 a B) \sqrt {x} (a+b x)}{64 a b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 (A b-9 a B) \left (a b+b^2 x\right )\right ) \text {Subst}\left (\int \frac {1}{a b+b^2 x^2} \, dx,x,\sqrt {x}\right )}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {35 (A b-9 a B) x^{3/2}}{192 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) x^{9/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-9 a B) x^{7/2}}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (A b-9 a B) x^{5/2}}{96 a b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (A b-9 a B) \sqrt {x} (a+b x)}{64 a b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 (A b-9 a B) (a+b x) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 \sqrt {a} b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.48 \[ \int \frac {x^{7/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {\sqrt {b} \sqrt {x} \left (945 a^4 B-105 a^3 b (A-33 B x)+3 b^4 x^3 (-93 A+128 B x)+7 a^2 b^2 x (-55 A+657 B x)+a b^3 x^2 (-511 A+2511 B x)\right )+\frac {105 (A b-9 a B) (a+b x)^4 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a}}}{192 b^{11/2} (a+b x)^3 \sqrt {(a+b x)^2}} \]
[In]
[Out]
Time = 0.18 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.52
method | result | size |
risch | \(\frac {2 B \sqrt {x}\, \sqrt {\left (b x +a \right )^{2}}}{b^{5} \left (b x +a \right )}+\frac {\left (\frac {2 \left (-\frac {93}{128} A \,b^{4}+\frac {325}{128} B \,b^{3} a \right ) x^{\frac {7}{2}}+2 \left (\frac {765}{128} B \,a^{2} b^{2}-\frac {511}{384} A \,b^{3} a \right ) x^{\frac {5}{2}}-\frac {a^{2} b \left (385 A b -1929 B a \right ) x^{\frac {3}{2}}}{192}+2 \left (\frac {187}{128} B \,a^{4}-\frac {35}{128} A \,a^{3} b \right ) \sqrt {x}}{\left (b x +a \right )^{4}}+\frac {35 \left (A b -9 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{64 \sqrt {b a}}\right ) \sqrt {\left (b x +a \right )^{2}}}{b^{5} \left (b x +a \right )}\) | \(159\) |
default | \(-\frac {\left (279 A \,x^{\frac {7}{2}} \sqrt {b a}\, b^{4}-2511 B \,x^{\frac {7}{2}} \sqrt {b a}\, a \,b^{3}+511 A \,x^{\frac {5}{2}} \sqrt {b a}\, a \,b^{3}-105 A \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) b^{5} x^{4}-4599 B \,x^{\frac {5}{2}} \sqrt {b a}\, a^{2} b^{2}-384 B \,x^{\frac {9}{2}} \sqrt {b a}\, b^{4}+945 B \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a \,b^{4} x^{4}-420 A \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a \,b^{4} x^{3}+3780 B \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{2} b^{3} x^{3}+385 A \,x^{\frac {3}{2}} \sqrt {b a}\, a^{2} b^{2}-630 A \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{2} b^{3} x^{2}-3465 B \,x^{\frac {3}{2}} \sqrt {b a}\, a^{3} b +5670 B \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{3} b^{2} x^{2}-420 A \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{3} b^{2} x +3780 B \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{4} b x +105 A \sqrt {x}\, \sqrt {b a}\, a^{3} b -105 A \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{4} b -945 B \sqrt {x}\, \sqrt {b a}\, a^{4}+945 B \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{5}\right ) \left (b x +a \right )}{192 \sqrt {b a}\, b^{5} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(368\) |
[In]
[Out]
none
Time = 0.35 (sec) , antiderivative size = 555, normalized size of antiderivative = 1.81 \[ \int \frac {x^{7/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\left [\frac {105 \, {\left (9 \, B a^{5} - A a^{4} b + {\left (9 \, B a b^{4} - A b^{5}\right )} x^{4} + 4 \, {\left (9 \, B a^{2} b^{3} - A a b^{4}\right )} x^{3} + 6 \, {\left (9 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{2} + 4 \, {\left (9 \, B a^{4} b - A a^{3} b^{2}\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a - 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) + 2 \, {\left (384 \, B a b^{5} x^{4} + 945 \, B a^{5} b - 105 \, A a^{4} b^{2} + 279 \, {\left (9 \, B a^{2} b^{4} - A a b^{5}\right )} x^{3} + 511 \, {\left (9 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} x^{2} + 385 \, {\left (9 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} x\right )} \sqrt {x}}{384 \, {\left (a b^{10} x^{4} + 4 \, a^{2} b^{9} x^{3} + 6 \, a^{3} b^{8} x^{2} + 4 \, a^{4} b^{7} x + a^{5} b^{6}\right )}}, \frac {105 \, {\left (9 \, B a^{5} - A a^{4} b + {\left (9 \, B a b^{4} - A b^{5}\right )} x^{4} + 4 \, {\left (9 \, B a^{2} b^{3} - A a b^{4}\right )} x^{3} + 6 \, {\left (9 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{2} + 4 \, {\left (9 \, B a^{4} b - A a^{3} b^{2}\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (384 \, B a b^{5} x^{4} + 945 \, B a^{5} b - 105 \, A a^{4} b^{2} + 279 \, {\left (9 \, B a^{2} b^{4} - A a b^{5}\right )} x^{3} + 511 \, {\left (9 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} x^{2} + 385 \, {\left (9 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} x\right )} \sqrt {x}}{192 \, {\left (a b^{10} x^{4} + 4 \, a^{2} b^{9} x^{3} + 6 \, a^{3} b^{8} x^{2} + 4 \, a^{4} b^{7} x + a^{5} b^{6}\right )}}\right ] \]
[In]
[Out]
Timed out. \[ \int \frac {x^{7/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.40 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.22 \[ \int \frac {x^{7/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {105 \, {\left (3 \, {\left (11 \, B a b^{5} - A b^{6}\right )} x^{2} + {\left (9 \, B a^{2} b^{4} + A a b^{5}\right )} x\right )} x^{\frac {9}{2}} + 30 \, {\left ({\left (359 \, B a^{2} b^{4} - 21 \, A a b^{5}\right )} x^{2} + {\left (61 \, B a^{3} b^{3} + 21 \, A a^{2} b^{4}\right )} x\right )} x^{\frac {7}{2}} + 20 \, {\left (66 \, {\left (11 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} x^{2} + 13 \, {\left (9 \, B a^{4} b^{2} + A a^{3} b^{3}\right )} x\right )} x^{\frac {5}{2}} + 2 \, {\left (405 \, {\left (11 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} x^{2} + 77 \, {\left (9 \, B a^{5} b + A a^{4} b^{2}\right )} x\right )} x^{\frac {3}{2}} + 7 \, {\left (27 \, {\left (11 \, B a^{5} b - A a^{4} b^{2}\right )} x^{2} + 5 \, {\left (9 \, B a^{6} + A a^{5} b\right )} x\right )} \sqrt {x}}{1920 \, {\left (a^{2} b^{9} x^{5} + 5 \, a^{3} b^{8} x^{4} + 10 \, a^{4} b^{7} x^{3} + 10 \, a^{5} b^{6} x^{2} + 5 \, a^{6} b^{5} x + a^{7} b^{4}\right )}} - \frac {35 \, {\left (9 \, B a - A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} b^{5}} - \frac {7 \, {\left (3 \, {\left (11 \, B a b - A b^{2}\right )} x^{\frac {3}{2}} - 10 \, {\left (9 \, B a^{2} - A a b\right )} \sqrt {x}\right )}}{128 \, a^{2} b^{5}} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.52 \[ \int \frac {x^{7/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {2 \, B \sqrt {x}}{b^{5} \mathrm {sgn}\left (b x + a\right )} - \frac {35 \, {\left (9 \, B a - A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} b^{5} \mathrm {sgn}\left (b x + a\right )} + \frac {975 \, B a b^{3} x^{\frac {7}{2}} - 279 \, A b^{4} x^{\frac {7}{2}} + 2295 \, B a^{2} b^{2} x^{\frac {5}{2}} - 511 \, A a b^{3} x^{\frac {5}{2}} + 1929 \, B a^{3} b x^{\frac {3}{2}} - 385 \, A a^{2} b^{2} x^{\frac {3}{2}} + 561 \, B a^{4} \sqrt {x} - 105 \, A a^{3} b \sqrt {x}}{192 \, {\left (b x + a\right )}^{4} b^{5} \mathrm {sgn}\left (b x + a\right )} \]
[In]
[Out]
Timed out. \[ \int \frac {x^{7/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {x^{7/2}\,\left (A+B\,x\right )}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]
[In]
[Out]