Optimal. Leaf size=201 \[ -\frac {231 e^3 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{13/2}}+\frac {231 e^3 \sqrt {d+e x} (b d-a e)^2}{8 b^6}+\frac {77 e^3 (d+e x)^{3/2} (b d-a e)}{8 b^5}-\frac {33 e^2 (d+e x)^{7/2}}{8 b^3 (a+b x)}-\frac {11 e (d+e x)^{9/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}+\frac {231 e^3 (d+e x)^{5/2}}{40 b^4} \]
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Rubi [A] time = 0.13, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 47, 50, 63, 208} \[ -\frac {33 e^2 (d+e x)^{7/2}}{8 b^3 (a+b x)}+\frac {77 e^3 (d+e x)^{3/2} (b d-a e)}{8 b^5}+\frac {231 e^3 \sqrt {d+e x} (b d-a e)^2}{8 b^6}-\frac {231 e^3 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{13/2}}-\frac {11 e (d+e x)^{9/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}+\frac {231 e^3 (d+e x)^{5/2}}{40 b^4} \]
Antiderivative was successfully verified.
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Rule 27
Rule 47
Rule 50
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {(d+e x)^{11/2}}{(a+b x)^4} \, dx\\ &=-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}+\frac {(11 e) \int \frac {(d+e x)^{9/2}}{(a+b x)^3} \, dx}{6 b}\\ &=-\frac {11 e (d+e x)^{9/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}+\frac {\left (33 e^2\right ) \int \frac {(d+e x)^{7/2}}{(a+b x)^2} \, dx}{8 b^2}\\ &=-\frac {33 e^2 (d+e x)^{7/2}}{8 b^3 (a+b x)}-\frac {11 e (d+e x)^{9/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}+\frac {\left (231 e^3\right ) \int \frac {(d+e x)^{5/2}}{a+b x} \, dx}{16 b^3}\\ &=\frac {231 e^3 (d+e x)^{5/2}}{40 b^4}-\frac {33 e^2 (d+e x)^{7/2}}{8 b^3 (a+b x)}-\frac {11 e (d+e x)^{9/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}+\frac {\left (231 e^3 (b d-a e)\right ) \int \frac {(d+e x)^{3/2}}{a+b x} \, dx}{16 b^4}\\ &=\frac {77 e^3 (b d-a e) (d+e x)^{3/2}}{8 b^5}+\frac {231 e^3 (d+e x)^{5/2}}{40 b^4}-\frac {33 e^2 (d+e x)^{7/2}}{8 b^3 (a+b x)}-\frac {11 e (d+e x)^{9/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}+\frac {\left (231 e^3 (b d-a e)^2\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{16 b^5}\\ &=\frac {231 e^3 (b d-a e)^2 \sqrt {d+e x}}{8 b^6}+\frac {77 e^3 (b d-a e) (d+e x)^{3/2}}{8 b^5}+\frac {231 e^3 (d+e x)^{5/2}}{40 b^4}-\frac {33 e^2 (d+e x)^{7/2}}{8 b^3 (a+b x)}-\frac {11 e (d+e x)^{9/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}+\frac {\left (231 e^3 (b d-a e)^3\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{16 b^6}\\ &=\frac {231 e^3 (b d-a e)^2 \sqrt {d+e x}}{8 b^6}+\frac {77 e^3 (b d-a e) (d+e x)^{3/2}}{8 b^5}+\frac {231 e^3 (d+e x)^{5/2}}{40 b^4}-\frac {33 e^2 (d+e x)^{7/2}}{8 b^3 (a+b x)}-\frac {11 e (d+e x)^{9/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}+\frac {\left (231 e^2 (b d-a e)^3\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{8 b^6}\\ &=\frac {231 e^3 (b d-a e)^2 \sqrt {d+e x}}{8 b^6}+\frac {77 e^3 (b d-a e) (d+e x)^{3/2}}{8 b^5}+\frac {231 e^3 (d+e x)^{5/2}}{40 b^4}-\frac {33 e^2 (d+e x)^{7/2}}{8 b^3 (a+b x)}-\frac {11 e (d+e x)^{9/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}-\frac {231 e^3 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{13/2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 52, normalized size = 0.26 \[ \frac {2 e^3 (d+e x)^{13/2} \, _2F_1\left (4,\frac {13}{2};\frac {15}{2};-\frac {b (d+e x)}{a e-b d}\right )}{13 (a e-b d)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.81, size = 994, normalized size = 4.95 \[ \left [\frac {3465 \, {\left (a^{3} b^{2} d^{2} e^{3} - 2 \, a^{4} b d e^{4} + a^{5} e^{5} + {\left (b^{5} d^{2} e^{3} - 2 \, a b^{4} d e^{4} + a^{2} b^{3} e^{5}\right )} x^{3} + 3 \, {\left (a b^{4} d^{2} e^{3} - 2 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} + 3 \, {\left (a^{2} b^{3} d^{2} e^{3} - 2 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x\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 (48 \, b^{5} e^{5} x^{5} - 40 \, b^{5} d^{5} - 110 \, a b^{4} d^{4} e - 495 \, a^{2} b^{3} d^{3} e^{2} + 5313 \, a^{3} b^{2} d^{2} e^{3} - 8085 \, a^{4} b d e^{4} + 3465 \, a^{5} e^{5} + 16 \, {\left (26 \, b^{5} d e^{4} - 11 \, a b^{4} e^{5}\right )} x^{4} + 16 \, {\left (173 \, b^{5} d^{2} e^{3} - 242 \, a b^{4} d e^{4} + 99 \, a^{2} b^{3} e^{5}\right )} x^{3} - 3 \, {\left (445 \, b^{5} d^{3} e^{2} - 4103 \, a b^{4} d^{2} e^{3} + 6039 \, a^{2} b^{3} d e^{4} - 2541 \, a^{3} b^{2} e^{5}\right )} x^{2} - 2 \, {\left (155 \, b^{5} d^{4} e + 715 \, a b^{4} d^{3} e^{2} - 7227 \, a^{2} b^{3} d^{2} e^{3} + 10857 \, a^{3} b^{2} d e^{4} - 4620 \, a^{4} b e^{5}\right )} x\right )} \sqrt {e x + d}}{240 \, {\left (b^{9} x^{3} + 3 \, a b^{8} x^{2} + 3 \, a^{2} b^{7} x + a^{3} b^{6}\right )}}, -\frac {3465 \, {\left (a^{3} b^{2} d^{2} e^{3} - 2 \, a^{4} b d e^{4} + a^{5} e^{5} + {\left (b^{5} d^{2} e^{3} - 2 \, a b^{4} d e^{4} + a^{2} b^{3} e^{5}\right )} x^{3} + 3 \, {\left (a b^{4} d^{2} e^{3} - 2 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} + 3 \, {\left (a^{2} b^{3} d^{2} e^{3} - 2 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x\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 (48 \, b^{5} e^{5} x^{5} - 40 \, b^{5} d^{5} - 110 \, a b^{4} d^{4} e - 495 \, a^{2} b^{3} d^{3} e^{2} + 5313 \, a^{3} b^{2} d^{2} e^{3} - 8085 \, a^{4} b d e^{4} + 3465 \, a^{5} e^{5} + 16 \, {\left (26 \, b^{5} d e^{4} - 11 \, a b^{4} e^{5}\right )} x^{4} + 16 \, {\left (173 \, b^{5} d^{2} e^{3} - 242 \, a b^{4} d e^{4} + 99 \, a^{2} b^{3} e^{5}\right )} x^{3} - 3 \, {\left (445 \, b^{5} d^{3} e^{2} - 4103 \, a b^{4} d^{2} e^{3} + 6039 \, a^{2} b^{3} d e^{4} - 2541 \, a^{3} b^{2} e^{5}\right )} x^{2} - 2 \, {\left (155 \, b^{5} d^{4} e + 715 \, a b^{4} d^{3} e^{2} - 7227 \, a^{2} b^{3} d^{2} e^{3} + 10857 \, a^{3} b^{2} d e^{4} - 4620 \, a^{4} b e^{5}\right )} x\right )} \sqrt {e x + d}}{120 \, {\left (b^{9} x^{3} + 3 \, a b^{8} x^{2} + 3 \, a^{2} b^{7} x + a^{3} b^{6}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 491, normalized size = 2.44 \[ \frac {231 \, {\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{8 \, \sqrt {-b^{2} d + a b e} b^{6}} - \frac {267 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{5} d^{3} e^{3} - 472 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{5} d^{4} e^{3} + 213 \, \sqrt {x e + d} b^{5} d^{5} e^{3} - 801 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{4} d^{2} e^{4} + 1888 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{4} d^{3} e^{4} - 1065 \, \sqrt {x e + d} a b^{4} d^{4} e^{4} + 801 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{3} d e^{5} - 2832 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{3} d^{2} e^{5} + 2130 \, \sqrt {x e + d} a^{2} b^{3} d^{3} e^{5} - 267 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{3} b^{2} e^{6} + 1888 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b^{2} d e^{6} - 2130 \, \sqrt {x e + d} a^{3} b^{2} d^{2} e^{6} - 472 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{4} b e^{7} + 1065 \, \sqrt {x e + d} a^{4} b d e^{7} - 213 \, \sqrt {x e + d} a^{5} e^{8}}{24 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{3} b^{6}} + \frac {2 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{16} e^{3} + 20 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{16} d e^{3} + 150 \, \sqrt {x e + d} b^{16} d^{2} e^{3} - 20 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{15} e^{4} - 300 \, \sqrt {x e + d} a b^{15} d e^{4} + 150 \, \sqrt {x e + d} a^{2} b^{14} e^{5}\right )}}{15 \, b^{20}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 719, normalized size = 3.58 \[ \frac {71 \sqrt {e x +d}\, a^{5} e^{8}}{8 \left (b e x +a e \right )^{3} b^{6}}-\frac {355 \sqrt {e x +d}\, a^{4} d \,e^{7}}{8 \left (b e x +a e \right )^{3} b^{5}}+\frac {355 \sqrt {e x +d}\, a^{3} d^{2} e^{6}}{4 \left (b e x +a e \right )^{3} b^{4}}-\frac {355 \sqrt {e x +d}\, a^{2} d^{3} e^{5}}{4 \left (b e x +a e \right )^{3} b^{3}}+\frac {355 \sqrt {e x +d}\, a \,d^{4} e^{4}}{8 \left (b e x +a e \right )^{3} b^{2}}-\frac {71 \sqrt {e x +d}\, d^{5} e^{3}}{8 \left (b e x +a e \right )^{3} b}+\frac {59 \left (e x +d \right )^{\frac {3}{2}} a^{4} e^{7}}{3 \left (b e x +a e \right )^{3} b^{5}}-\frac {236 \left (e x +d \right )^{\frac {3}{2}} a^{3} d \,e^{6}}{3 \left (b e x +a e \right )^{3} b^{4}}+\frac {118 \left (e x +d \right )^{\frac {3}{2}} a^{2} d^{2} e^{5}}{\left (b e x +a e \right )^{3} b^{3}}-\frac {236 \left (e x +d \right )^{\frac {3}{2}} a \,d^{3} e^{4}}{3 \left (b e x +a e \right )^{3} b^{2}}+\frac {59 \left (e x +d \right )^{\frac {3}{2}} d^{4} e^{3}}{3 \left (b e x +a e \right )^{3} b}+\frac {89 \left (e x +d \right )^{\frac {5}{2}} a^{3} e^{6}}{8 \left (b e x +a e \right )^{3} b^{4}}-\frac {267 \left (e x +d \right )^{\frac {5}{2}} a^{2} d \,e^{5}}{8 \left (b e x +a e \right )^{3} b^{3}}+\frac {267 \left (e x +d \right )^{\frac {5}{2}} a \,d^{2} e^{4}}{8 \left (b e x +a e \right )^{3} b^{2}}-\frac {89 \left (e x +d \right )^{\frac {5}{2}} d^{3} e^{3}}{8 \left (b e x +a e \right )^{3} b}-\frac {231 a^{3} e^{6} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}\, b^{6}}+\frac {693 a^{2} d \,e^{5} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}\, b^{5}}-\frac {693 a \,d^{2} e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}\, b^{4}}+\frac {231 d^{3} e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}\, b^{3}}+\frac {20 \sqrt {e x +d}\, a^{2} e^{5}}{b^{6}}-\frac {40 \sqrt {e x +d}\, a d \,e^{4}}{b^{5}}+\frac {20 \sqrt {e x +d}\, d^{2} e^{3}}{b^{4}}-\frac {8 \left (e x +d \right )^{\frac {3}{2}} a \,e^{4}}{3 b^{5}}+\frac {8 \left (e x +d \right )^{\frac {3}{2}} d \,e^{3}}{3 b^{4}}+\frac {2 \left (e x +d \right )^{\frac {5}{2}} e^{3}}{5 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.68, size = 495, normalized size = 2.46 \[ \left (\frac {2\,e^3\,{\left (4\,b^4\,d-4\,a\,b^3\,e\right )}^2}{b^{12}}-\frac {12\,e^3\,{\left (a\,e-b\,d\right )}^2}{b^6}\right )\,\sqrt {d+e\,x}+\frac {\sqrt {d+e\,x}\,\left (\frac {71\,a^5\,e^8}{8}-\frac {355\,a^4\,b\,d\,e^7}{8}+\frac {355\,a^3\,b^2\,d^2\,e^6}{4}-\frac {355\,a^2\,b^3\,d^3\,e^5}{4}+\frac {355\,a\,b^4\,d^4\,e^4}{8}-\frac {71\,b^5\,d^5\,e^3}{8}\right )+{\left (d+e\,x\right )}^{5/2}\,\left (\frac {89\,a^3\,b^2\,e^6}{8}-\frac {267\,a^2\,b^3\,d\,e^5}{8}+\frac {267\,a\,b^4\,d^2\,e^4}{8}-\frac {89\,b^5\,d^3\,e^3}{8}\right )+{\left (d+e\,x\right )}^{3/2}\,\left (\frac {59\,a^4\,b\,e^7}{3}-\frac {236\,a^3\,b^2\,d\,e^6}{3}+118\,a^2\,b^3\,d^2\,e^5-\frac {236\,a\,b^4\,d^3\,e^4}{3}+\frac {59\,b^5\,d^4\,e^3}{3}\right )}{b^9\,{\left (d+e\,x\right )}^3-\left (3\,b^9\,d-3\,a\,b^8\,e\right )\,{\left (d+e\,x\right )}^2+\left (d+e\,x\right )\,\left (3\,a^2\,b^7\,e^2-6\,a\,b^8\,d\,e+3\,b^9\,d^2\right )-b^9\,d^3+a^3\,b^6\,e^3-3\,a^2\,b^7\,d\,e^2+3\,a\,b^8\,d^2\,e}+\frac {2\,e^3\,{\left (d+e\,x\right )}^{5/2}}{5\,b^4}+\frac {2\,e^3\,\left (4\,b^4\,d-4\,a\,b^3\,e\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,b^8}-\frac {231\,e^3\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^3\,{\left (a\,e-b\,d\right )}^{5/2}\,\sqrt {d+e\,x}}{a^3\,e^6-3\,a^2\,b\,d\,e^5+3\,a\,b^2\,d^2\,e^4-b^3\,d^3\,e^3}\right )\,{\left (a\,e-b\,d\right )}^{5/2}}{8\,b^{13/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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