Integrand size = 30, antiderivative size = 400 \[ \int \frac {(d+e x)^{13/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {3003 e^4 (b d-a e)^2 (a+b x) \sqrt {d+e x}}{64 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1001 e^4 (b d-a e) (a+b x) (d+e x)^{3/2}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x) (d+e x)^{5/2}}{320 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {429 e^3 (d+e x)^{7/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2 (d+e x)^{9/2}}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3003 e^4 (b d-a e)^{5/2} (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
[Out]
Time = 0.17 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {660, 43, 52, 65, 214} \[ \int \frac {(d+e x)^{13/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {3003 e^4 (a+b x) (b d-a e)^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x) \sqrt {d+e x} (b d-a e)^2}{64 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1001 e^4 (a+b x) (d+e x)^{3/2} (b d-a e)}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x) (d+e x)^{5/2}}{320 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {429 e^3 (d+e x)^{7/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2 (d+e x)^{9/2}}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]
[In]
[Out]
Rule 43
Rule 52
Rule 65
Rule 214
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{13/2}}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (13 b^2 e \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{11/2}}{\left (a b+b^2 x\right )^4} \, dx}{8 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (143 e^2 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{9/2}}{\left (a b+b^2 x\right )^3} \, dx}{48 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {143 e^2 (d+e x)^{9/2}}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (429 e^3 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{7/2}}{\left (a b+b^2 x\right )^2} \, dx}{64 b^2 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {429 e^3 (d+e x)^{7/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2 (d+e x)^{9/2}}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3003 e^4 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{5/2}}{a b+b^2 x} \, dx}{128 b^4 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {3003 e^4 (a+b x) (d+e x)^{5/2}}{320 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {429 e^3 (d+e x)^{7/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2 (d+e x)^{9/2}}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3003 e^4 \left (b^2 d-a b e\right ) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{3/2}}{a b+b^2 x} \, dx}{128 b^6 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {1001 e^4 (b d-a e) (a+b x) (d+e x)^{3/2}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x) (d+e x)^{5/2}}{320 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {429 e^3 (d+e x)^{7/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2 (d+e x)^{9/2}}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3003 e^4 \left (b^2 d-a b e\right )^2 \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{a b+b^2 x} \, dx}{128 b^8 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {3003 e^4 (b d-a e)^2 (a+b x) \sqrt {d+e x}}{64 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1001 e^4 (b d-a e) (a+b x) (d+e x)^{3/2}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x) (d+e x)^{5/2}}{320 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {429 e^3 (d+e x)^{7/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2 (d+e x)^{9/2}}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3003 e^4 \left (b^2 d-a b e\right )^3 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{128 b^{10} \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {3003 e^4 (b d-a e)^2 (a+b x) \sqrt {d+e x}}{64 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1001 e^4 (b d-a e) (a+b x) (d+e x)^{3/2}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x) (d+e x)^{5/2}}{320 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {429 e^3 (d+e x)^{7/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2 (d+e x)^{9/2}}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3003 e^3 \left (b^2 d-a b e\right )^3 \left (a b+b^2 x\right )\right ) \text {Subst}\left (\int \frac {1}{a b-\frac {b^2 d}{e}+\frac {b^2 x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{64 b^{10} \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {3003 e^4 (b d-a e)^2 (a+b x) \sqrt {d+e x}}{64 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1001 e^4 (b d-a e) (a+b x) (d+e x)^{3/2}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x) (d+e x)^{5/2}}{320 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {429 e^3 (d+e x)^{7/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2 (d+e x)^{9/2}}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3003 e^4 (b d-a e)^{5/2} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 1.41 (sec) , antiderivative size = 374, normalized size of antiderivative = 0.94 \[ \int \frac {(d+e x)^{13/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {e^4 (a+b x)^5 \left (-\frac {\sqrt {b} \sqrt {d+e x} \left (-45045 a^6 e^6+15015 a^5 b e^5 (7 d-11 e x)-3003 a^4 b^2 e^4 \left (23 d^2-129 d e x+73 e^2 x^2\right )+429 a^3 b^3 e^3 \left (15 d^3-599 d^2 e x+1207 d e^2 x^2-279 e^3 x^3\right )+143 a^2 b^4 e^2 \left (10 d^4+175 d^3 e x-2433 d^2 e^2 x^2+1999 d e^3 x^3-128 e^4 x^4\right )+13 a b^5 e \left (40 d^5+420 d^4 e x+2765 d^3 e^2 x^2-15077 d^2 e^3 x^3+3456 d e^4 x^4+128 e^5 x^5\right )+b^6 \left (240 d^6+1960 d^5 e x+7630 d^4 e^2 x^2+22155 d^3 e^3 x^3-32384 d^2 e^4 x^4-3968 d e^5 x^5-384 e^6 x^6\right )\right )}{e^4 (a+b x)^4}-45045 (-b d+a e)^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )\right )}{960 b^{15/2} \left ((a+b x)^2\right )^{5/2}} \]
[In]
[Out]
Time = 2.32 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.78
method | result | size |
risch | \(\frac {2 e^{4} \left (3 x^{2} b^{2} e^{2}-25 x a b \,e^{2}+31 b^{2} d e x +225 a^{2} e^{2}-475 a b d e +253 b^{2} d^{2}\right ) \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{15 b^{7} \left (b x +a \right )}-\frac {\left (2 a^{3} e^{3}-6 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e -2 b^{3} d^{3}\right ) e^{4} \left (\frac {-\frac {1477 \left (e x +d \right )^{\frac {7}{2}} b^{3}}{128}-\frac {11767 \left (a e -b d \right ) b^{2} \left (e x +d \right )^{\frac {5}{2}}}{384}+\left (-\frac {10633}{384} e^{2} a^{2} b +\frac {10633}{192} a d e \,b^{2}-\frac {10633}{384} b^{3} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {1083}{128} a^{3} e^{3}+\frac {3249}{128} a^{2} b d \,e^{2}-\frac {3249}{128} a \,b^{2} d^{2} e +\frac {1083}{128} b^{3} d^{3}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{4}}+\frac {3003 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \sqrt {\left (a e -b d \right ) b}}\right ) \sqrt {\left (b x +a \right )^{2}}}{b^{7} \left (b x +a \right )}\) | \(311\) |
default | \(\text {Expression too large to display}\) | \(2192\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 638 vs. \(2 (272) = 544\).
Time = 0.57 (sec) , antiderivative size = 1286, normalized size of antiderivative = 3.22 \[ \int \frac {(d+e x)^{13/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
[In]
[Out]
Timed out. \[ \int \frac {(d+e x)^{13/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {(d+e x)^{13/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {13}{2}}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 651 vs. \(2 (272) = 544\).
Time = 0.31 (sec) , antiderivative size = 651, normalized size of antiderivative = 1.63 \[ \int \frac {(d+e x)^{13/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {3003 \, {\left (b^{3} d^{3} e^{4} - 3 \, a b^{2} d^{2} e^{5} + 3 \, a^{2} b d e^{6} - a^{3} e^{7}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, \sqrt {-b^{2} d + a b e} b^{7} \mathrm {sgn}\left (b x + a\right )} - \frac {4431 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{6} d^{3} e^{4} - 11767 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{6} d^{4} e^{4} + 10633 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{6} d^{5} e^{4} - 3249 \, \sqrt {e x + d} b^{6} d^{6} e^{4} - 13293 \, {\left (e x + d\right )}^{\frac {7}{2}} a b^{5} d^{2} e^{5} + 47068 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{5} d^{3} e^{5} - 53165 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{5} d^{4} e^{5} + 19494 \, \sqrt {e x + d} a b^{5} d^{5} e^{5} + 13293 \, {\left (e x + d\right )}^{\frac {7}{2}} a^{2} b^{4} d e^{6} - 70602 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} b^{4} d^{2} e^{6} + 106330 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{4} d^{3} e^{6} - 48735 \, \sqrt {e x + d} a^{2} b^{4} d^{4} e^{6} - 4431 \, {\left (e x + d\right )}^{\frac {7}{2}} a^{3} b^{3} e^{7} + 47068 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{3} b^{3} d e^{7} - 106330 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b^{3} d^{2} e^{7} + 64980 \, \sqrt {e x + d} a^{3} b^{3} d^{3} e^{7} - 11767 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{4} b^{2} e^{8} + 53165 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{4} b^{2} d e^{8} - 48735 \, \sqrt {e x + d} a^{4} b^{2} d^{2} e^{8} - 10633 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{5} b e^{9} + 19494 \, \sqrt {e x + d} a^{5} b d e^{9} - 3249 \, \sqrt {e x + d} a^{6} e^{10}}{192 \, {\left ({\left (e x + d\right )} b - b d + a e\right )}^{4} b^{7} \mathrm {sgn}\left (b x + a\right )} + \frac {2 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{20} e^{4} + 25 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{20} d e^{4} + 225 \, \sqrt {e x + d} b^{20} d^{2} e^{4} - 25 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{19} e^{5} - 450 \, \sqrt {e x + d} a b^{19} d e^{5} + 225 \, \sqrt {e x + d} a^{2} b^{18} e^{6}\right )}}{15 \, b^{25} \mathrm {sgn}\left (b x + a\right )} \]
[In]
[Out]
Timed out. \[ \int \frac {(d+e x)^{13/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{13/2}}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]
[In]
[Out]