Integrand size = 35, antiderivative size = 308 \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 (b d-a e)^3 (B d-A e) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}-\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x)}+\frac {6 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^5 (a+b x)}-\frac {2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^5 (a+b x)}+\frac {2 b^3 B (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 e^5 (a+b x)} \]
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Time = 0.10 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {784, 78} \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=-\frac {2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (-3 a B e-A b e+4 b B d)}{13 e^5 (a+b x)}+\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e) (-a B e-A b e+2 b B d)}{11 e^5 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{9 e^5 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3 (B d-A e)}{7 e^5 (a+b x)}+\frac {2 b^3 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{15/2}}{15 e^5 (a+b x)} \]
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Rule 78
Rule 784
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^3 (A+B x) (d+e x)^{5/2} \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^3 (b d-a e)^3 (-B d+A e) (d+e x)^{5/2}}{e^4}+\frac {b^3 (b d-a e)^2 (-4 b B d+3 A b e+a B e) (d+e x)^{7/2}}{e^4}-\frac {3 b^4 (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^{9/2}}{e^4}+\frac {b^5 (-4 b B d+A b e+3 a B e) (d+e x)^{11/2}}{e^4}+\frac {b^6 B (d+e x)^{13/2}}{e^4}\right ) \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = \frac {2 (b d-a e)^3 (B d-A e) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}-\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x)}+\frac {6 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^5 (a+b x)}-\frac {2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^5 (a+b x)}+\frac {2 b^3 B (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 e^5 (a+b x)} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.80 \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 \sqrt {(a+b x)^2} (d+e x)^{7/2} \left (715 a^3 e^3 (-2 B d+9 A e+7 B e x)+195 a^2 b e^2 \left (11 A e (-2 d+7 e x)+B \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )-15 a b^2 e \left (-13 A e \left (8 d^2-28 d e x+63 e^2 x^2\right )+3 B \left (16 d^3-56 d^2 e x+126 d e^2 x^2-231 e^3 x^3\right )\right )+b^3 \left (15 A e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+B \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )\right )}{45045 e^5 (a+b x)} \]
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Time = 0.25 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.03
method | result | size |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (3003 B \,b^{3} e^{4} x^{4}+3465 A \,b^{3} e^{4} x^{3}+10395 B a \,b^{2} e^{4} x^{3}-1848 B \,b^{3} d \,e^{3} x^{3}+12285 A a \,b^{2} e^{4} x^{2}-1890 A \,b^{3} d \,e^{3} x^{2}+12285 B \,a^{2} b \,e^{4} x^{2}-5670 B a \,b^{2} d \,e^{3} x^{2}+1008 B \,b^{3} d^{2} e^{2} x^{2}+15015 A \,a^{2} b \,e^{4} x -5460 A a \,b^{2} d \,e^{3} x +840 A \,b^{3} d^{2} e^{2} x +5005 B \,a^{3} e^{4} x -5460 B \,a^{2} b d \,e^{3} x +2520 B a \,b^{2} d^{2} e^{2} x -448 B \,b^{3} d^{3} e x +6435 A \,a^{3} e^{4}-4290 A \,a^{2} b d \,e^{3}+1560 A a \,b^{2} d^{2} e^{2}-240 A \,b^{3} d^{3} e -1430 B \,a^{3} d \,e^{3}+1560 B \,a^{2} b \,d^{2} e^{2}-720 B a \,b^{2} d^{3} e +128 B \,b^{3} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{45045 e^{5} \left (b x +a \right )^{3}}\) | \(317\) |
default | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (3003 B \,b^{3} e^{4} x^{4}+3465 A \,b^{3} e^{4} x^{3}+10395 B a \,b^{2} e^{4} x^{3}-1848 B \,b^{3} d \,e^{3} x^{3}+12285 A a \,b^{2} e^{4} x^{2}-1890 A \,b^{3} d \,e^{3} x^{2}+12285 B \,a^{2} b \,e^{4} x^{2}-5670 B a \,b^{2} d \,e^{3} x^{2}+1008 B \,b^{3} d^{2} e^{2} x^{2}+15015 A \,a^{2} b \,e^{4} x -5460 A a \,b^{2} d \,e^{3} x +840 A \,b^{3} d^{2} e^{2} x +5005 B \,a^{3} e^{4} x -5460 B \,a^{2} b d \,e^{3} x +2520 B a \,b^{2} d^{2} e^{2} x -448 B \,b^{3} d^{3} e x +6435 A \,a^{3} e^{4}-4290 A \,a^{2} b d \,e^{3}+1560 A a \,b^{2} d^{2} e^{2}-240 A \,b^{3} d^{3} e -1430 B \,a^{3} d \,e^{3}+1560 B \,a^{2} b \,d^{2} e^{2}-720 B a \,b^{2} d^{3} e +128 B \,b^{3} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{45045 e^{5} \left (b x +a \right )^{3}}\) | \(317\) |
risch | \(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (3003 b^{3} B \,e^{7} x^{7}+3465 A \,b^{3} e^{7} x^{6}+10395 B a \,b^{2} e^{7} x^{6}+7161 B \,b^{3} d \,e^{6} x^{6}+12285 A a \,b^{2} e^{7} x^{5}+8505 A \,b^{3} d \,e^{6} x^{5}+12285 B \,a^{2} b \,e^{7} x^{5}+25515 B a \,b^{2} d \,e^{6} x^{5}+4473 B \,b^{3} d^{2} e^{5} x^{5}+15015 A \,a^{2} b \,e^{7} x^{4}+31395 A a \,b^{2} d \,e^{6} x^{4}+5565 A \,b^{3} d^{2} e^{5} x^{4}+5005 B \,a^{3} e^{7} x^{4}+31395 B \,a^{2} b d \,e^{6} x^{4}+16695 B a \,b^{2} d^{2} e^{5} x^{4}+35 B \,b^{3} d^{3} e^{4} x^{4}+6435 A \,a^{3} e^{7} x^{3}+40755 A \,a^{2} b d \,e^{6} x^{3}+22035 A a \,b^{2} d^{2} e^{5} x^{3}+75 A \,b^{3} d^{3} e^{4} x^{3}+13585 B \,a^{3} d \,e^{6} x^{3}+22035 B \,a^{2} b \,d^{2} e^{5} x^{3}+225 B a \,b^{2} d^{3} e^{4} x^{3}-40 B \,b^{3} d^{4} e^{3} x^{3}+19305 A \,a^{3} d \,e^{6} x^{2}+32175 A \,a^{2} b \,d^{2} e^{5} x^{2}+585 A a \,b^{2} d^{3} e^{4} x^{2}-90 A \,b^{3} d^{4} e^{3} x^{2}+10725 B \,a^{3} d^{2} e^{5} x^{2}+585 B \,a^{2} b \,d^{3} e^{4} x^{2}-270 B a \,b^{2} d^{4} e^{3} x^{2}+48 B \,b^{3} d^{5} e^{2} x^{2}+19305 A \,a^{3} d^{2} e^{5} x +2145 A \,a^{2} b \,d^{3} e^{4} x -780 A a \,b^{2} d^{4} e^{3} x +120 A \,b^{3} d^{5} e^{2} x +715 B \,a^{3} d^{3} e^{4} x -780 B \,a^{2} b \,d^{4} e^{3} x +360 B a \,b^{2} d^{5} e^{2} x -64 B \,b^{3} d^{6} e x +6435 A \,a^{3} d^{3} e^{4}-4290 A \,a^{2} b \,d^{4} e^{3}+1560 A a \,b^{2} d^{5} e^{2}-240 A \,b^{3} d^{6} e -1430 B \,a^{3} d^{4} e^{3}+1560 B \,a^{2} b \,d^{5} e^{2}-720 B a \,b^{2} d^{6} e +128 B \,b^{3} d^{7}\right ) \sqrt {e x +d}}{45045 \left (b x +a \right ) e^{5}}\) | \(685\) |
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Leaf count of result is larger than twice the leaf count of optimal. 539 vs. \(2 (233) = 466\).
Time = 0.30 (sec) , antiderivative size = 539, normalized size of antiderivative = 1.75 \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 \, {\left (3003 \, B b^{3} e^{7} x^{7} + 128 \, B b^{3} d^{7} + 6435 \, A a^{3} d^{3} e^{4} - 240 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{6} e + 1560 \, {\left (B a^{2} b + A a b^{2}\right )} d^{5} e^{2} - 1430 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4} e^{3} + 231 \, {\left (31 \, B b^{3} d e^{6} + 15 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{7}\right )} x^{6} + 63 \, {\left (71 \, B b^{3} d^{2} e^{5} + 135 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{6} + 195 \, {\left (B a^{2} b + A a b^{2}\right )} e^{7}\right )} x^{5} + 35 \, {\left (B b^{3} d^{3} e^{4} + 159 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{5} + 897 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{6} + 143 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{7}\right )} x^{4} - 5 \, {\left (8 \, B b^{3} d^{4} e^{3} - 1287 \, A a^{3} e^{7} - 15 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{4} - 4407 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{5} - 2717 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{6}\right )} x^{3} + 3 \, {\left (16 \, B b^{3} d^{5} e^{2} + 6435 \, A a^{3} d e^{6} - 30 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e^{3} + 195 \, {\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{4} + 3575 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{5}\right )} x^{2} - {\left (64 \, B b^{3} d^{6} e - 19305 \, A a^{3} d^{2} e^{5} - 120 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} e^{2} + 780 \, {\left (B a^{2} b + A a b^{2}\right )} d^{4} e^{3} - 715 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e^{4}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{5}} \]
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Timed out. \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 592 vs. \(2 (233) = 466\).
Time = 0.22 (sec) , antiderivative size = 592, normalized size of antiderivative = 1.92 \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 \, {\left (231 \, b^{3} e^{6} x^{6} - 16 \, b^{3} d^{6} + 104 \, a b^{2} d^{5} e - 286 \, a^{2} b d^{4} e^{2} + 429 \, a^{3} d^{3} e^{3} + 63 \, {\left (9 \, b^{3} d e^{5} + 13 \, a b^{2} e^{6}\right )} x^{5} + 7 \, {\left (53 \, b^{3} d^{2} e^{4} + 299 \, a b^{2} d e^{5} + 143 \, a^{2} b e^{6}\right )} x^{4} + {\left (5 \, b^{3} d^{3} e^{3} + 1469 \, a b^{2} d^{2} e^{4} + 2717 \, a^{2} b d e^{5} + 429 \, a^{3} e^{6}\right )} x^{3} - 3 \, {\left (2 \, b^{3} d^{4} e^{2} - 13 \, a b^{2} d^{3} e^{3} - 715 \, a^{2} b d^{2} e^{4} - 429 \, a^{3} d e^{5}\right )} x^{2} + {\left (8 \, b^{3} d^{5} e - 52 \, a b^{2} d^{4} e^{2} + 143 \, a^{2} b d^{3} e^{3} + 1287 \, a^{3} d^{2} e^{4}\right )} x\right )} \sqrt {e x + d} A}{3003 \, e^{4}} + \frac {2 \, {\left (3003 \, b^{3} e^{7} x^{7} + 128 \, b^{3} d^{7} - 720 \, a b^{2} d^{6} e + 1560 \, a^{2} b d^{5} e^{2} - 1430 \, a^{3} d^{4} e^{3} + 231 \, {\left (31 \, b^{3} d e^{6} + 45 \, a b^{2} e^{7}\right )} x^{6} + 63 \, {\left (71 \, b^{3} d^{2} e^{5} + 405 \, a b^{2} d e^{6} + 195 \, a^{2} b e^{7}\right )} x^{5} + 35 \, {\left (b^{3} d^{3} e^{4} + 477 \, a b^{2} d^{2} e^{5} + 897 \, a^{2} b d e^{6} + 143 \, a^{3} e^{7}\right )} x^{4} - 5 \, {\left (8 \, b^{3} d^{4} e^{3} - 45 \, a b^{2} d^{3} e^{4} - 4407 \, a^{2} b d^{2} e^{5} - 2717 \, a^{3} d e^{6}\right )} x^{3} + 3 \, {\left (16 \, b^{3} d^{5} e^{2} - 90 \, a b^{2} d^{4} e^{3} + 195 \, a^{2} b d^{3} e^{4} + 3575 \, a^{3} d^{2} e^{5}\right )} x^{2} - {\left (64 \, b^{3} d^{6} e - 360 \, a b^{2} d^{5} e^{2} + 780 \, a^{2} b d^{4} e^{3} - 715 \, a^{3} d^{3} e^{4}\right )} x\right )} \sqrt {e x + d} B}{45045 \, e^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2139 vs. \(2 (233) = 466\).
Time = 0.35 (sec) , antiderivative size = 2139, normalized size of antiderivative = 6.94 \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\int \left (A+B\,x\right )\,{\left (d+e\,x\right )}^{5/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \]
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