Integrand size = 22, antiderivative size = 198 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a+b x} \, dx=\frac {2 (A b-a B) (b d-a e)^3 \sqrt {d+e x}}{b^5}+\frac {2 (A b-a B) (b d-a e)^2 (d+e x)^{3/2}}{3 b^4}+\frac {2 (A b-a B) (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {2 (A b-a B) (d+e x)^{7/2}}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e}-\frac {2 (A b-a B) (b d-a e)^{7/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{11/2}} \]
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Time = 0.17 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {81, 52, 65, 214} \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a+b x} \, dx=-\frac {2 (A b-a B) (b d-a e)^{7/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{11/2}}+\frac {2 \sqrt {d+e x} (A b-a B) (b d-a e)^3}{b^5}+\frac {2 (d+e x)^{3/2} (A b-a B) (b d-a e)^2}{3 b^4}+\frac {2 (d+e x)^{5/2} (A b-a B) (b d-a e)}{5 b^3}+\frac {2 (d+e x)^{7/2} (A b-a B)}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e} \]
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Rule 52
Rule 65
Rule 81
Rule 214
Rubi steps \begin{align*} \text {integral}& = \frac {2 B (d+e x)^{9/2}}{9 b e}+\frac {\left (2 \left (\frac {9 A b e}{2}-\frac {9 a B e}{2}\right )\right ) \int \frac {(d+e x)^{7/2}}{a+b x} \, dx}{9 b e} \\ & = \frac {2 (A b-a B) (d+e x)^{7/2}}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e}+\frac {((A b-a B) (b d-a e)) \int \frac {(d+e x)^{5/2}}{a+b x} \, dx}{b^2} \\ & = \frac {2 (A b-a B) (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {2 (A b-a B) (d+e x)^{7/2}}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e}+\frac {\left ((A b-a B) (b d-a e)^2\right ) \int \frac {(d+e x)^{3/2}}{a+b x} \, dx}{b^3} \\ & = \frac {2 (A b-a B) (b d-a e)^2 (d+e x)^{3/2}}{3 b^4}+\frac {2 (A b-a B) (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {2 (A b-a B) (d+e x)^{7/2}}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e}+\frac {\left ((A b-a B) (b d-a e)^3\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{b^4} \\ & = \frac {2 (A b-a B) (b d-a e)^3 \sqrt {d+e x}}{b^5}+\frac {2 (A b-a B) (b d-a e)^2 (d+e x)^{3/2}}{3 b^4}+\frac {2 (A b-a B) (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {2 (A b-a B) (d+e x)^{7/2}}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e}+\frac {\left ((A b-a B) (b d-a e)^4\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{b^5} \\ & = \frac {2 (A b-a B) (b d-a e)^3 \sqrt {d+e x}}{b^5}+\frac {2 (A b-a B) (b d-a e)^2 (d+e x)^{3/2}}{3 b^4}+\frac {2 (A b-a B) (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {2 (A b-a B) (d+e x)^{7/2}}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e}+\frac {\left (2 (A b-a B) (b d-a e)^4\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{b^5 e} \\ & = \frac {2 (A b-a B) (b d-a e)^3 \sqrt {d+e x}}{b^5}+\frac {2 (A b-a B) (b d-a e)^2 (d+e x)^{3/2}}{3 b^4}+\frac {2 (A b-a B) (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {2 (A b-a B) (d+e x)^{7/2}}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e}-\frac {2 (A b-a B) (b d-a e)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{11/2}} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.33 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a+b x} \, dx=\frac {2 \sqrt {d+e x} \left (315 a^4 B e^4-105 a^3 b e^3 (10 B d+3 A e+B e x)+21 a^2 b^2 e^2 \left (5 A e (10 d+e x)+B \left (58 d^2+16 d e x+3 e^2 x^2\right )\right )-3 a b^3 e \left (7 A e \left (58 d^2+16 d e x+3 e^2 x^2\right )+B \left (176 d^3+122 d^2 e x+66 d e^2 x^2+15 e^3 x^3\right )\right )+b^4 \left (35 B (d+e x)^4+3 A e \left (176 d^3+122 d^2 e x+66 d e^2 x^2+15 e^3 x^3\right )\right )\right )}{315 b^5 e}+\frac {2 (A b-a B) (-b d+a e)^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{b^{11/2}} \]
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Time = 1.26 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.36
method | result | size |
pseudoelliptic | \(\frac {-2 \left (\frac {\left (-\frac {\left (e x +d \right )^{4} B}{3}-\frac {176 \left (\frac {15}{176} e^{3} x^{3}+\frac {3}{8} d \,e^{2} x^{2}+\frac {61}{88} d^{2} e x +d^{3}\right ) e A}{35}\right ) b^{4}}{3}+\frac {58 e a \left (\frac {\left (\frac {15}{2} e^{3} x^{3}+33 d \,e^{2} x^{2}+61 d^{2} e x +88 d^{3}\right ) B}{203}+A e \left (\frac {3}{58} x^{2} e^{2}+\frac {8}{29} d e x +d^{2}\right )\right ) b^{3}}{15}-\frac {10 \left (\frac {\left (\frac {3}{2} x^{2} e^{2}+8 d e x +29 d^{2}\right ) B}{25}+A e \left (\frac {e x}{10}+d \right )\right ) e^{2} a^{2} b^{2}}{3}+e^{3} a^{3} \left (\frac {\left (e x +10 d \right ) B}{3}+A e \right ) b -B \,a^{4} e^{4}\right ) \sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}+2 e \left (a e -b d \right )^{4} \left (A b -B a \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{e \,b^{5} \sqrt {\left (a e -b d \right ) b}}\) | \(269\) |
risch | \(-\frac {2 \left (-35 B \,e^{4} b^{4} x^{4}-45 A \,b^{4} e^{4} x^{3}+45 B a \,b^{3} e^{4} x^{3}-140 B \,b^{4} d \,e^{3} x^{3}+63 A a \,b^{3} e^{4} x^{2}-198 A \,b^{4} d \,e^{3} x^{2}-63 B \,a^{2} b^{2} e^{4} x^{2}+198 B a \,b^{3} d \,e^{3} x^{2}-210 B \,b^{4} d^{2} e^{2} x^{2}-105 A \,a^{2} b^{2} e^{4} x +336 A a \,b^{3} d \,e^{3} x -366 A \,b^{4} d^{2} e^{2} x +105 B \,a^{3} b \,e^{4} x -336 B \,a^{2} b^{2} d \,e^{3} x +366 B a \,b^{3} d^{2} e^{2} x -140 B \,b^{4} d^{3} e x +315 A \,a^{3} b \,e^{4}-1050 A \,a^{2} b^{2} d \,e^{3}+1218 A a \,b^{3} d^{2} e^{2}-528 A \,b^{4} d^{3} e -315 B \,a^{4} e^{4}+1050 B \,a^{3} b d \,e^{3}-1218 B \,a^{2} b^{2} d^{2} e^{2}+528 B a \,b^{3} d^{3} e -35 B \,b^{4} d^{4}\right ) \sqrt {e x +d}}{315 e \,b^{5}}+\frac {2 \left (A \,a^{4} b \,e^{4}-4 A \,a^{3} b^{2} d \,e^{3}+6 A \,a^{2} b^{3} d^{2} e^{2}-4 A a \,b^{4} d^{3} e +A \,d^{4} b^{5}-a^{5} B \,e^{4}+4 B \,a^{4} b d \,e^{3}-6 B \,a^{3} b^{2} d^{2} e^{2}+4 B \,a^{2} b^{3} d^{3} e -B a \,b^{4} d^{4}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{b^{5} \sqrt {\left (a e -b d \right ) b}}\) | \(481\) |
derivativedivides | \(\frac {-\frac {2 \left (-\frac {B \left (e x +d \right )^{\frac {9}{2}} b^{4}}{9}-\frac {A \,b^{4} e \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {B a \,b^{3} e \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {A a \,b^{3} e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {A \,b^{4} d e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {B \,a^{2} b^{2} e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {B a \,b^{3} d e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {A \,a^{2} b^{2} e^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 A a \,b^{3} d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {A \,b^{4} d^{2} e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {B \,a^{3} b \,e^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {2 B \,a^{2} b^{2} d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {B a \,b^{3} d^{2} e \left (e x +d \right )^{\frac {3}{2}}}{3}+A \,a^{3} b \,e^{4} \sqrt {e x +d}-3 A \,a^{2} b^{2} d \,e^{3} \sqrt {e x +d}+3 A a \,b^{3} d^{2} e^{2} \sqrt {e x +d}-A \,b^{4} d^{3} e \sqrt {e x +d}-B \,a^{4} e^{4} \sqrt {e x +d}+3 B \,a^{3} b d \,e^{3} \sqrt {e x +d}-3 B \,a^{2} b^{2} d^{2} e^{2} \sqrt {e x +d}+B a \,b^{3} d^{3} e \sqrt {e x +d}\right )}{b^{5}}+\frac {2 e \left (A \,a^{4} b \,e^{4}-4 A \,a^{3} b^{2} d \,e^{3}+6 A \,a^{2} b^{3} d^{2} e^{2}-4 A a \,b^{4} d^{3} e +A \,d^{4} b^{5}-a^{5} B \,e^{4}+4 B \,a^{4} b d \,e^{3}-6 B \,a^{3} b^{2} d^{2} e^{2}+4 B \,a^{2} b^{3} d^{3} e -B a \,b^{4} d^{4}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{b^{5} \sqrt {\left (a e -b d \right ) b}}}{e}\) | \(531\) |
default | \(\frac {-\frac {2 \left (-\frac {B \left (e x +d \right )^{\frac {9}{2}} b^{4}}{9}-\frac {A \,b^{4} e \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {B a \,b^{3} e \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {A a \,b^{3} e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {A \,b^{4} d e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {B \,a^{2} b^{2} e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {B a \,b^{3} d e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {A \,a^{2} b^{2} e^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 A a \,b^{3} d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {A \,b^{4} d^{2} e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {B \,a^{3} b \,e^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {2 B \,a^{2} b^{2} d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {B a \,b^{3} d^{2} e \left (e x +d \right )^{\frac {3}{2}}}{3}+A \,a^{3} b \,e^{4} \sqrt {e x +d}-3 A \,a^{2} b^{2} d \,e^{3} \sqrt {e x +d}+3 A a \,b^{3} d^{2} e^{2} \sqrt {e x +d}-A \,b^{4} d^{3} e \sqrt {e x +d}-B \,a^{4} e^{4} \sqrt {e x +d}+3 B \,a^{3} b d \,e^{3} \sqrt {e x +d}-3 B \,a^{2} b^{2} d^{2} e^{2} \sqrt {e x +d}+B a \,b^{3} d^{3} e \sqrt {e x +d}\right )}{b^{5}}+\frac {2 e \left (A \,a^{4} b \,e^{4}-4 A \,a^{3} b^{2} d \,e^{3}+6 A \,a^{2} b^{3} d^{2} e^{2}-4 A a \,b^{4} d^{3} e +A \,d^{4} b^{5}-a^{5} B \,e^{4}+4 B \,a^{4} b d \,e^{3}-6 B \,a^{3} b^{2} d^{2} e^{2}+4 B \,a^{2} b^{3} d^{3} e -B a \,b^{4} d^{4}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{b^{5} \sqrt {\left (a e -b d \right ) b}}}{e}\) | \(531\) |
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Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (170) = 340\).
Time = 0.24 (sec) , antiderivative size = 865, normalized size of antiderivative = 4.37 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a+b x} \, dx=\left [-\frac {315 \, {\left ({\left (B a b^{3} - A b^{4}\right )} d^{3} e - 3 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2} e^{2} + 3 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} d e^{3} - {\left (B a^{4} - A a^{3} b\right )} e^{4}\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) - 2 \, {\left (35 \, B b^{4} e^{4} x^{4} + 35 \, B b^{4} d^{4} - 528 \, {\left (B a b^{3} - A b^{4}\right )} d^{3} e + 1218 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2} e^{2} - 1050 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} d e^{3} + 315 \, {\left (B a^{4} - A a^{3} b\right )} e^{4} + 5 \, {\left (28 \, B b^{4} d e^{3} - 9 \, {\left (B a b^{3} - A b^{4}\right )} e^{4}\right )} x^{3} + 3 \, {\left (70 \, B b^{4} d^{2} e^{2} - 66 \, {\left (B a b^{3} - A b^{4}\right )} d e^{3} + 21 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} e^{4}\right )} x^{2} + {\left (140 \, B b^{4} d^{3} e - 366 \, {\left (B a b^{3} - A b^{4}\right )} d^{2} e^{2} + 336 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d e^{3} - 105 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} e^{4}\right )} x\right )} \sqrt {e x + d}}{315 \, b^{5} e}, \frac {2 \, {\left (315 \, {\left ({\left (B a b^{3} - A b^{4}\right )} d^{3} e - 3 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2} e^{2} + 3 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} d e^{3} - {\left (B a^{4} - A a^{3} b\right )} e^{4}\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) + {\left (35 \, B b^{4} e^{4} x^{4} + 35 \, B b^{4} d^{4} - 528 \, {\left (B a b^{3} - A b^{4}\right )} d^{3} e + 1218 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2} e^{2} - 1050 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} d e^{3} + 315 \, {\left (B a^{4} - A a^{3} b\right )} e^{4} + 5 \, {\left (28 \, B b^{4} d e^{3} - 9 \, {\left (B a b^{3} - A b^{4}\right )} e^{4}\right )} x^{3} + 3 \, {\left (70 \, B b^{4} d^{2} e^{2} - 66 \, {\left (B a b^{3} - A b^{4}\right )} d e^{3} + 21 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} e^{4}\right )} x^{2} + {\left (140 \, B b^{4} d^{3} e - 366 \, {\left (B a b^{3} - A b^{4}\right )} d^{2} e^{2} + 336 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d e^{3} - 105 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} e^{4}\right )} x\right )} \sqrt {e x + d}\right )}}{315 \, b^{5} e}\right ] \]
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Time = 4.15 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.90 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a+b x} \, dx=\begin {cases} \frac {2 \left (\frac {B \left (d + e x\right )^{\frac {9}{2}}}{9 b} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (A b e - B a e\right )}{7 b^{2}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (- A a b e^{2} + A b^{2} d e + B a^{2} e^{2} - B a b d e\right )}{5 b^{3}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (A a^{2} b e^{3} - 2 A a b^{2} d e^{2} + A b^{3} d^{2} e - B a^{3} e^{3} + 2 B a^{2} b d e^{2} - B a b^{2} d^{2} e\right )}{3 b^{4}} + \frac {\sqrt {d + e x} \left (- A a^{3} b e^{4} + 3 A a^{2} b^{2} d e^{3} - 3 A a b^{3} d^{2} e^{2} + A b^{4} d^{3} e + B a^{4} e^{4} - 3 B a^{3} b d e^{3} + 3 B a^{2} b^{2} d^{2} e^{2} - B a b^{3} d^{3} e\right )}{b^{5}} - \frac {e \left (- A b + B a\right ) \left (a e - b d\right )^{4} \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e - b d}{b}}} \right )}}{b^{6} \sqrt {\frac {a e - b d}{b}}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {7}{2}} \left (\frac {B x}{b} - \frac {\left (- A b + B a\right ) \left (\begin {cases} \frac {x}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b x \right )}}{b} & \text {otherwise} \end {cases}\right )}{b}\right ) & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a+b x} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 560 vs. \(2 (170) = 340\).
Time = 0.29 (sec) , antiderivative size = 560, normalized size of antiderivative = 2.83 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a+b x} \, dx=-\frac {2 \, {\left (B a b^{4} d^{4} - A b^{5} d^{4} - 4 \, B a^{2} b^{3} d^{3} e + 4 \, A a b^{4} d^{3} e + 6 \, B a^{3} b^{2} d^{2} e^{2} - 6 \, A a^{2} b^{3} d^{2} e^{2} - 4 \, B a^{4} b d e^{3} + 4 \, A a^{3} b^{2} d e^{3} + B a^{5} e^{4} - A a^{4} b e^{4}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{5}} + \frac {2 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} B b^{8} e^{8} - 45 \, {\left (e x + d\right )}^{\frac {7}{2}} B a b^{7} e^{9} + 45 \, {\left (e x + d\right )}^{\frac {7}{2}} A b^{8} e^{9} - 63 \, {\left (e x + d\right )}^{\frac {5}{2}} B a b^{7} d e^{9} + 63 \, {\left (e x + d\right )}^{\frac {5}{2}} A b^{8} d e^{9} - 105 \, {\left (e x + d\right )}^{\frac {3}{2}} B a b^{7} d^{2} e^{9} + 105 \, {\left (e x + d\right )}^{\frac {3}{2}} A b^{8} d^{2} e^{9} - 315 \, \sqrt {e x + d} B a b^{7} d^{3} e^{9} + 315 \, \sqrt {e x + d} A b^{8} d^{3} e^{9} + 63 \, {\left (e x + d\right )}^{\frac {5}{2}} B a^{2} b^{6} e^{10} - 63 \, {\left (e x + d\right )}^{\frac {5}{2}} A a b^{7} e^{10} + 210 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{2} b^{6} d e^{10} - 210 \, {\left (e x + d\right )}^{\frac {3}{2}} A a b^{7} d e^{10} + 945 \, \sqrt {e x + d} B a^{2} b^{6} d^{2} e^{10} - 945 \, \sqrt {e x + d} A a b^{7} d^{2} e^{10} - 105 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{3} b^{5} e^{11} + 105 \, {\left (e x + d\right )}^{\frac {3}{2}} A a^{2} b^{6} e^{11} - 945 \, \sqrt {e x + d} B a^{3} b^{5} d e^{11} + 945 \, \sqrt {e x + d} A a^{2} b^{6} d e^{11} + 315 \, \sqrt {e x + d} B a^{4} b^{4} e^{12} - 315 \, \sqrt {e x + d} A a^{3} b^{5} e^{12}\right )}}{315 \, b^{9} e^{9}} \]
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Time = 0.19 (sec) , antiderivative size = 426, normalized size of antiderivative = 2.15 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a+b x} \, dx=\left (\frac {2\,A\,e-2\,B\,d}{7\,b\,e}-\frac {2\,B\,\left (a\,e^2-b\,d\,e\right )}{7\,b^2\,e^2}\right )\,{\left (d+e\,x\right )}^{7/2}+\frac {2\,B\,{\left (d+e\,x\right )}^{9/2}}{9\,b\,e}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\left (A\,b-B\,a\right )\,{\left (a\,e-b\,d\right )}^{7/2}\,\sqrt {d+e\,x}}{-B\,a^5\,e^4+4\,B\,a^4\,b\,d\,e^3+A\,a^4\,b\,e^4-6\,B\,a^3\,b^2\,d^2\,e^2-4\,A\,a^3\,b^2\,d\,e^3+4\,B\,a^2\,b^3\,d^3\,e+6\,A\,a^2\,b^3\,d^2\,e^2-B\,a\,b^4\,d^4-4\,A\,a\,b^4\,d^3\,e+A\,b^5\,d^4}\right )\,\left (A\,b-B\,a\right )\,{\left (a\,e-b\,d\right )}^{7/2}}{b^{11/2}}+\frac {\left (\frac {2\,A\,e-2\,B\,d}{b\,e}-\frac {2\,B\,\left (a\,e^2-b\,d\,e\right )}{b^2\,e^2}\right )\,{\left (a\,e^2-b\,d\,e\right )}^2\,{\left (d+e\,x\right )}^{3/2}}{3\,b^2\,e^2}-\frac {\left (\frac {2\,A\,e-2\,B\,d}{b\,e}-\frac {2\,B\,\left (a\,e^2-b\,d\,e\right )}{b^2\,e^2}\right )\,{\left (a\,e^2-b\,d\,e\right )}^3\,\sqrt {d+e\,x}}{b^3\,e^3}-\frac {\left (\frac {2\,A\,e-2\,B\,d}{b\,e}-\frac {2\,B\,\left (a\,e^2-b\,d\,e\right )}{b^2\,e^2}\right )\,\left (a\,e^2-b\,d\,e\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,b\,e} \]
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