Optimal. Leaf size=172 \[ -\frac {105 e^3 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{11/2}}+\frac {105 e^3 \sqrt {d+e x} (b d-a e)}{8 b^5}-\frac {21 e^2 (d+e x)^{5/2}}{8 b^3 (a+b x)}-\frac {3 e (d+e x)^{7/2}}{4 b^2 (a+b x)^2}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}+\frac {35 e^3 (d+e x)^{3/2}}{8 b^4} \]
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Rubi [A] time = 0.10, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ -\frac {21 e^2 (d+e x)^{5/2}}{8 b^3 (a+b x)}+\frac {105 e^3 \sqrt {d+e x} (b d-a e)}{8 b^5}-\frac {105 e^3 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{11/2}}-\frac {3 e (d+e x)^{7/2}}{4 b^2 (a+b x)^2}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}+\frac {35 e^3 (d+e x)^{3/2}}{8 b^4} \]
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)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {(d+e x)^{9/2}}{(a+b x)^4} \, dx\\ &=-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}+\frac {(3 e) \int \frac {(d+e x)^{7/2}}{(a+b x)^3} \, dx}{2 b}\\ &=-\frac {3 e (d+e x)^{7/2}}{4 b^2 (a+b x)^2}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}+\frac {\left (21 e^2\right ) \int \frac {(d+e x)^{5/2}}{(a+b x)^2} \, dx}{8 b^2}\\ &=-\frac {21 e^2 (d+e x)^{5/2}}{8 b^3 (a+b x)}-\frac {3 e (d+e x)^{7/2}}{4 b^2 (a+b x)^2}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}+\frac {\left (105 e^3\right ) \int \frac {(d+e x)^{3/2}}{a+b x} \, dx}{16 b^3}\\ &=\frac {35 e^3 (d+e x)^{3/2}}{8 b^4}-\frac {21 e^2 (d+e x)^{5/2}}{8 b^3 (a+b x)}-\frac {3 e (d+e x)^{7/2}}{4 b^2 (a+b x)^2}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}+\frac {\left (105 e^3 (b d-a e)\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{16 b^4}\\ &=\frac {105 e^3 (b d-a e) \sqrt {d+e x}}{8 b^5}+\frac {35 e^3 (d+e x)^{3/2}}{8 b^4}-\frac {21 e^2 (d+e x)^{5/2}}{8 b^3 (a+b x)}-\frac {3 e (d+e x)^{7/2}}{4 b^2 (a+b x)^2}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}+\frac {\left (105 e^3 (b d-a e)^2\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{16 b^5}\\ &=\frac {105 e^3 (b d-a e) \sqrt {d+e x}}{8 b^5}+\frac {35 e^3 (d+e x)^{3/2}}{8 b^4}-\frac {21 e^2 (d+e x)^{5/2}}{8 b^3 (a+b x)}-\frac {3 e (d+e x)^{7/2}}{4 b^2 (a+b x)^2}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}+\frac {\left (105 e^2 (b d-a e)^2\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^5}\\ &=\frac {105 e^3 (b d-a e) \sqrt {d+e x}}{8 b^5}+\frac {35 e^3 (d+e x)^{3/2}}{8 b^4}-\frac {21 e^2 (d+e x)^{5/2}}{8 b^3 (a+b x)}-\frac {3 e (d+e x)^{7/2}}{4 b^2 (a+b x)^2}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}-\frac {105 e^3 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{11/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 52, normalized size = 0.30 \[ \frac {2 e^3 (d+e x)^{11/2} \, _2F_1\left (4,\frac {11}{2};\frac {13}{2};-\frac {b (d+e x)}{a e-b d}\right )}{11 (a e-b d)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.36, size = 730, normalized size = 4.24 \[ \left [-\frac {315 \, {\left (a^{3} b d e^{3} - a^{4} e^{4} + {\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 3 \, {\left (a b^{3} d e^{3} - a^{2} b^{2} e^{4}\right )} x^{2} + 3 \, {\left (a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e + 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) - 2 \, {\left (16 \, b^{4} e^{4} x^{4} - 8 \, b^{4} d^{4} - 18 \, a b^{3} d^{3} e - 63 \, a^{2} b^{2} d^{2} e^{2} + 420 \, a^{3} b d e^{3} - 315 \, a^{4} e^{4} + 16 \, {\left (13 \, b^{4} d e^{3} - 9 \, a b^{3} e^{4}\right )} x^{3} - 3 \, {\left (55 \, b^{4} d^{2} e^{2} - 318 \, a b^{3} d e^{3} + 231 \, a^{2} b^{2} e^{4}\right )} x^{2} - 2 \, {\left (25 \, b^{4} d^{3} e + 90 \, a b^{3} d^{2} e^{2} - 567 \, a^{2} b^{2} d e^{3} + 420 \, a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{48 \, {\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}}, -\frac {315 \, {\left (a^{3} b d e^{3} - a^{4} e^{4} + {\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 3 \, {\left (a b^{3} d e^{3} - a^{2} b^{2} e^{4}\right )} x^{2} + 3 \, {\left (a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (16 \, b^{4} e^{4} x^{4} - 8 \, b^{4} d^{4} - 18 \, a b^{3} d^{3} e - 63 \, a^{2} b^{2} d^{2} e^{2} + 420 \, a^{3} b d e^{3} - 315 \, a^{4} e^{4} + 16 \, {\left (13 \, b^{4} d e^{3} - 9 \, a b^{3} e^{4}\right )} x^{3} - 3 \, {\left (55 \, b^{4} d^{2} e^{2} - 318 \, a b^{3} d e^{3} + 231 \, a^{2} b^{2} e^{4}\right )} x^{2} - 2 \, {\left (25 \, b^{4} d^{3} e + 90 \, a b^{3} d^{2} e^{2} - 567 \, a^{2} b^{2} d e^{3} + 420 \, a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{24 \, {\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 360, normalized size = 2.09 \[ \frac {105 \, {\left (b^{2} d^{2} e^{3} - 2 \, a b d e^{4} + a^{2} e^{5}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{8 \, \sqrt {-b^{2} d + a b e} b^{5}} - \frac {165 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{4} d^{2} e^{3} - 280 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{4} d^{3} e^{3} + 123 \, \sqrt {x e + d} b^{4} d^{4} e^{3} - 330 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{3} d e^{4} + 840 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{3} d^{2} e^{4} - 492 \, \sqrt {x e + d} a b^{3} d^{3} e^{4} + 165 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{2} e^{5} - 840 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{2} d e^{5} + 738 \, \sqrt {x e + d} a^{2} b^{2} d^{2} e^{5} + 280 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b e^{6} - 492 \, \sqrt {x e + d} a^{3} b d e^{6} + 123 \, \sqrt {x e + d} a^{4} e^{7}}{24 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{3} b^{5}} + \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} b^{8} e^{3} + 12 \, \sqrt {x e + d} b^{8} d e^{3} - 12 \, \sqrt {x e + d} a b^{7} e^{4}\right )}}{3 \, b^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 525, normalized size = 3.05 \[ -\frac {41 \sqrt {e x +d}\, a^{4} e^{7}}{8 \left (b e x +a e \right )^{3} b^{5}}+\frac {41 \sqrt {e x +d}\, a^{3} d \,e^{6}}{2 \left (b e x +a e \right )^{3} b^{4}}-\frac {123 \sqrt {e x +d}\, a^{2} d^{2} e^{5}}{4 \left (b e x +a e \right )^{3} b^{3}}+\frac {41 \sqrt {e x +d}\, a \,d^{3} e^{4}}{2 \left (b e x +a e \right )^{3} b^{2}}-\frac {41 \sqrt {e x +d}\, d^{4} e^{3}}{8 \left (b e x +a e \right )^{3} b}-\frac {35 \left (e x +d \right )^{\frac {3}{2}} a^{3} e^{6}}{3 \left (b e x +a e \right )^{3} b^{4}}+\frac {35 \left (e x +d \right )^{\frac {3}{2}} a^{2} d \,e^{5}}{\left (b e x +a e \right )^{3} b^{3}}-\frac {35 \left (e x +d \right )^{\frac {3}{2}} a \,d^{2} e^{4}}{\left (b e x +a e \right )^{3} b^{2}}+\frac {35 \left (e x +d \right )^{\frac {3}{2}} d^{3} e^{3}}{3 \left (b e x +a e \right )^{3} b}-\frac {55 \left (e x +d \right )^{\frac {5}{2}} a^{2} e^{5}}{8 \left (b e x +a e \right )^{3} b^{3}}+\frac {55 \left (e x +d \right )^{\frac {5}{2}} a d \,e^{4}}{4 \left (b e x +a e \right )^{3} b^{2}}-\frac {55 \left (e x +d \right )^{\frac {5}{2}} d^{2} e^{3}}{8 \left (b e x +a e \right )^{3} b}+\frac {105 a^{2} e^{5} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}\, b^{5}}-\frac {105 a d \,e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{4 \sqrt {\left (a e -b d \right ) b}\, b^{4}}+\frac {105 d^{2} e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}\, b^{3}}-\frac {8 \sqrt {e x +d}\, a \,e^{4}}{b^{5}}+\frac {8 \sqrt {e x +d}\, d \,e^{3}}{b^{4}}+\frac {2 \left (e x +d \right )^{\frac {3}{2}} e^{3}}{3 b^{4}} \]
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 = 388, normalized size = 2.26 \[ \frac {2\,e^3\,{\left (d+e\,x\right )}^{3/2}}{3\,b^4}-\frac {{\left (d+e\,x\right )}^{5/2}\,\left (\frac {55\,a^2\,b^2\,e^5}{8}-\frac {55\,a\,b^3\,d\,e^4}{4}+\frac {55\,b^4\,d^2\,e^3}{8}\right )+{\left (d+e\,x\right )}^{3/2}\,\left (\frac {35\,a^3\,b\,e^6}{3}-35\,a^2\,b^2\,d\,e^5+35\,a\,b^3\,d^2\,e^4-\frac {35\,b^4\,d^3\,e^3}{3}\right )+\sqrt {d+e\,x}\,\left (\frac {41\,a^4\,e^7}{8}-\frac {41\,a^3\,b\,d\,e^6}{2}+\frac {123\,a^2\,b^2\,d^2\,e^5}{4}-\frac {41\,a\,b^3\,d^3\,e^4}{2}+\frac {41\,b^4\,d^4\,e^3}{8}\right )}{b^8\,{\left (d+e\,x\right )}^3-\left (3\,b^8\,d-3\,a\,b^7\,e\right )\,{\left (d+e\,x\right )}^2+\left (d+e\,x\right )\,\left (3\,a^2\,b^6\,e^2-6\,a\,b^7\,d\,e+3\,b^8\,d^2\right )-b^8\,d^3+a^3\,b^5\,e^3-3\,a^2\,b^6\,d\,e^2+3\,a\,b^7\,d^2\,e}+\frac {2\,e^3\,\left (4\,b^4\,d-4\,a\,b^3\,e\right )\,\sqrt {d+e\,x}}{b^8}+\frac {105\,e^3\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^3\,{\left (a\,e-b\,d\right )}^{3/2}\,\sqrt {d+e\,x}}{a^2\,e^5-2\,a\,b\,d\,e^4+b^2\,d^2\,e^3}\right )\,{\left (a\,e-b\,d\right )}^{3/2}}{8\,b^{11/2}} \]
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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