Optimal. Leaf size=162 \[ \frac {9 e (b d-a e)^3 \sqrt {d+e x}}{b^5}+\frac {3 e (b d-a e)^2 (d+e x)^{3/2}}{b^4}+\frac {9 e (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {9 e (d+e x)^{7/2}}{7 b^2}-\frac {(d+e x)^{9/2}}{b (a+b x)}-\frac {9 e (b d-a e)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{11/2}} \]
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Rubi [A]
time = 0.14, antiderivative size = 162, 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, 43, 52, 65,
214} \begin {gather*} -\frac {9 e (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}}+\frac {9 e \sqrt {d+e x} (b d-a e)^3}{b^5}+\frac {3 e (d+e x)^{3/2} (b d-a e)^2}{b^4}+\frac {9 e (d+e x)^{5/2} (b d-a e)}{5 b^3}-\frac {(d+e x)^{9/2}}{b (a+b x)}+\frac {9 e (d+e x)^{7/2}}{7 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rule 52
Rule 65
Rule 214
Rubi steps
\begin {align*} \int \frac {(d+e x)^{9/2}}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac {(d+e x)^{9/2}}{(a+b x)^2} \, dx\\ &=-\frac {(d+e x)^{9/2}}{b (a+b x)}+\frac {(9 e) \int \frac {(d+e x)^{7/2}}{a+b x} \, dx}{2 b}\\ &=\frac {9 e (d+e x)^{7/2}}{7 b^2}-\frac {(d+e x)^{9/2}}{b (a+b x)}+\frac {(9 e (b d-a e)) \int \frac {(d+e x)^{5/2}}{a+b x} \, dx}{2 b^2}\\ &=\frac {9 e (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {9 e (d+e x)^{7/2}}{7 b^2}-\frac {(d+e x)^{9/2}}{b (a+b x)}+\frac {\left (9 e (b d-a e)^2\right ) \int \frac {(d+e x)^{3/2}}{a+b x} \, dx}{2 b^3}\\ &=\frac {3 e (b d-a e)^2 (d+e x)^{3/2}}{b^4}+\frac {9 e (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {9 e (d+e x)^{7/2}}{7 b^2}-\frac {(d+e x)^{9/2}}{b (a+b x)}+\frac {\left (9 e (b d-a e)^3\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{2 b^4}\\ &=\frac {9 e (b d-a e)^3 \sqrt {d+e x}}{b^5}+\frac {3 e (b d-a e)^2 (d+e x)^{3/2}}{b^4}+\frac {9 e (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {9 e (d+e x)^{7/2}}{7 b^2}-\frac {(d+e x)^{9/2}}{b (a+b x)}+\frac {\left (9 e (b d-a e)^4\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{2 b^5}\\ &=\frac {9 e (b d-a e)^3 \sqrt {d+e x}}{b^5}+\frac {3 e (b d-a e)^2 (d+e x)^{3/2}}{b^4}+\frac {9 e (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {9 e (d+e x)^{7/2}}{7 b^2}-\frac {(d+e x)^{9/2}}{b (a+b x)}+\frac {\left (9 (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}\\ &=\frac {9 e (b d-a e)^3 \sqrt {d+e x}}{b^5}+\frac {3 e (b d-a e)^2 (d+e x)^{3/2}}{b^4}+\frac {9 e (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {9 e (d+e x)^{7/2}}{7 b^2}-\frac {(d+e x)^{9/2}}{b (a+b x)}-\frac {9 e (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*}
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Mathematica [A]
time = 0.42, size = 208, normalized size = 1.28 \begin {gather*} -\frac {\sqrt {d+e x} \left (315 a^4 e^4+210 a^3 b e^3 (-5 d+e x)-42 a^2 b^2 e^2 \left (-29 d^2+17 d e x+e^2 x^2\right )+6 a b^3 e \left (-88 d^3+142 d^2 e x+23 d e^2 x^2+3 e^3 x^3\right )+b^4 \left (35 d^4-388 d^3 e x-156 d^2 e^2 x^2-58 d e^3 x^3-10 e^4 x^4\right )\right )}{35 b^5 (a+b x)}+\frac {9 e (-b d+a e)^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{b^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(325\) vs.
\(2(138)=276\).
time = 0.72, size = 326, normalized size = 2.01
method | result | size |
derivativedivides | \(2 e \left (-\frac {-\frac {\left (e x +d \right )^{\frac {7}{2}} b^{3}}{7}+\frac {2 a \,b^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 b^{3} d \left (e x +d \right )^{\frac {5}{2}}}{5}-a^{2} b \,e^{2} \left (e x +d \right )^{\frac {3}{2}}+2 a \,b^{2} d e \left (e x +d \right )^{\frac {3}{2}}-b^{3} d^{2} \left (e x +d \right )^{\frac {3}{2}}+4 a^{3} e^{3} \sqrt {e x +d}-12 a^{2} d \,e^{2} b \sqrt {e x +d}+12 a \,d^{2} e \,b^{2} \sqrt {e x +d}-4 d^{3} b^{3} \sqrt {e x +d}}{b^{5}}+\frac {\frac {\left (-\frac {1}{2} e^{4} a^{4}+2 a^{3} b d \,e^{3}-3 a^{2} b^{2} d^{2} e^{2}+2 a \,b^{3} d^{3} e -\frac {1}{2} b^{4} d^{4}\right ) \sqrt {e x +d}}{\left (e x +d \right ) b +a e -b d}+\frac {9 \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{2 \sqrt {b \left (a e -b d \right )}}}{b^{5}}\right )\) | \(326\) |
default | \(2 e \left (-\frac {-\frac {\left (e x +d \right )^{\frac {7}{2}} b^{3}}{7}+\frac {2 a \,b^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 b^{3} d \left (e x +d \right )^{\frac {5}{2}}}{5}-a^{2} b \,e^{2} \left (e x +d \right )^{\frac {3}{2}}+2 a \,b^{2} d e \left (e x +d \right )^{\frac {3}{2}}-b^{3} d^{2} \left (e x +d \right )^{\frac {3}{2}}+4 a^{3} e^{3} \sqrt {e x +d}-12 a^{2} d \,e^{2} b \sqrt {e x +d}+12 a \,d^{2} e \,b^{2} \sqrt {e x +d}-4 d^{3} b^{3} \sqrt {e x +d}}{b^{5}}+\frac {\frac {\left (-\frac {1}{2} e^{4} a^{4}+2 a^{3} b d \,e^{3}-3 a^{2} b^{2} d^{2} e^{2}+2 a \,b^{3} d^{3} e -\frac {1}{2} b^{4} d^{4}\right ) \sqrt {e x +d}}{\left (e x +d \right ) b +a e -b d}+\frac {9 \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{2 \sqrt {b \left (a e -b d \right )}}}{b^{5}}\right )\) | \(326\) |
risch | \(-\frac {2 e \left (-5 b^{3} e^{3} x^{3}+14 a \,b^{2} e^{3} x^{2}-29 b^{3} d \,e^{2} x^{2}-35 a^{2} b \,e^{3} x +98 a \,b^{2} d \,e^{2} x -78 b^{3} d^{2} e x +140 e^{3} a^{3}-455 a^{2} b d \,e^{2}+504 a \,b^{2} d^{2} e -194 b^{3} d^{3}\right ) \sqrt {e x +d}}{35 b^{5}}-\frac {e^{5} \sqrt {e x +d}\, a^{4}}{b^{5} \left (b e x +a e \right )}+\frac {4 e^{4} \sqrt {e x +d}\, a^{3} d}{b^{4} \left (b e x +a e \right )}-\frac {6 e^{3} \sqrt {e x +d}\, a^{2} d^{2}}{b^{3} \left (b e x +a e \right )}+\frac {4 e^{2} \sqrt {e x +d}\, a \,d^{3}}{b^{2} \left (b e x +a e \right )}-\frac {e \sqrt {e x +d}\, d^{4}}{b \left (b e x +a e \right )}+\frac {9 e^{5} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{4}}{b^{5} \sqrt {b \left (a e -b d \right )}}-\frac {36 e^{4} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{3} d}{b^{4} \sqrt {b \left (a e -b d \right )}}+\frac {54 e^{3} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{2} d^{2}}{b^{3} \sqrt {b \left (a e -b d \right )}}-\frac {36 e^{2} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a \,d^{3}}{b^{2} \sqrt {b \left (a e -b d \right )}}+\frac {9 e \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) d^{4}}{b \sqrt {b \left (a e -b d \right )}}\) | \(489\) |
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 317 vs.
\(2 (154) = 308\).
time = 2.15, size = 645, normalized size = 3.98 \begin {gather*} \left [\frac {315 \, {\left ({\left (a^{3} b x + a^{4}\right )} e^{4} - 3 \, {\left (a^{2} b^{2} d x + a^{3} b d\right )} e^{3} + 3 \, {\left (a b^{3} d^{2} x + a^{2} b^{2} d^{2}\right )} e^{2} - {\left (b^{4} d^{3} x + a b^{3} d^{3}\right )} e\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {2 \, b d + 2 \, \sqrt {x e + d} b \sqrt {\frac {b d - a e}{b}} + {\left (b x - a\right )} e}{b x + a}\right ) - 2 \, {\left (35 \, b^{4} d^{4} - {\left (10 \, b^{4} x^{4} - 18 \, a b^{3} x^{3} + 42 \, a^{2} b^{2} x^{2} - 210 \, a^{3} b x - 315 \, a^{4}\right )} e^{4} - 2 \, {\left (29 \, b^{4} d x^{3} - 69 \, a b^{3} d x^{2} + 357 \, a^{2} b^{2} d x + 525 \, a^{3} b d\right )} e^{3} - 6 \, {\left (26 \, b^{4} d^{2} x^{2} - 142 \, a b^{3} d^{2} x - 203 \, a^{2} b^{2} d^{2}\right )} e^{2} - 4 \, {\left (97 \, b^{4} d^{3} x + 132 \, a b^{3} d^{3}\right )} e\right )} \sqrt {x e + d}}{70 \, {\left (b^{6} x + a b^{5}\right )}}, \frac {315 \, {\left ({\left (a^{3} b x + a^{4}\right )} e^{4} - 3 \, {\left (a^{2} b^{2} d x + a^{3} b d\right )} e^{3} + 3 \, {\left (a b^{3} d^{2} x + a^{2} b^{2} d^{2}\right )} e^{2} - {\left (b^{4} d^{3} x + a b^{3} d^{3}\right )} e\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {x e + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (35 \, b^{4} d^{4} - {\left (10 \, b^{4} x^{4} - 18 \, a b^{3} x^{3} + 42 \, a^{2} b^{2} x^{2} - 210 \, a^{3} b x - 315 \, a^{4}\right )} e^{4} - 2 \, {\left (29 \, b^{4} d x^{3} - 69 \, a b^{3} d x^{2} + 357 \, a^{2} b^{2} d x + 525 \, a^{3} b d\right )} e^{3} - 6 \, {\left (26 \, b^{4} d^{2} x^{2} - 142 \, a b^{3} d^{2} x - 203 \, a^{2} b^{2} d^{2}\right )} e^{2} - 4 \, {\left (97 \, b^{4} d^{3} x + 132 \, a b^{3} d^{3}\right )} e\right )} \sqrt {x e + d}}{35 \, {\left (b^{6} x + a b^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 387 vs.
\(2 (154) = 308\).
time = 0.92, size = 387, normalized size = 2.39 \begin {gather*} \frac {9 \, {\left (b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{5}} - \frac {\sqrt {x e + d} b^{4} d^{4} e - 4 \, \sqrt {x e + d} a b^{3} d^{3} e^{2} + 6 \, \sqrt {x e + d} a^{2} b^{2} d^{2} e^{3} - 4 \, \sqrt {x e + d} a^{3} b d e^{4} + \sqrt {x e + d} a^{4} e^{5}}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{5}} + \frac {2 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{12} e + 14 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{12} d e + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{12} d^{2} e + 140 \, \sqrt {x e + d} b^{12} d^{3} e - 14 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{11} e^{2} - 70 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{11} d e^{2} - 420 \, \sqrt {x e + d} a b^{11} d^{2} e^{2} + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{10} e^{3} + 420 \, \sqrt {x e + d} a^{2} b^{10} d e^{3} - 140 \, \sqrt {x e + d} a^{3} b^{9} e^{4}\right )}}{35 \, b^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.61, size = 352, normalized size = 2.17 \begin {gather*} \left (\frac {\left (\frac {2\,e\,{\left (2\,b^2\,d-2\,a\,b\,e\right )}^2}{b^6}-\frac {2\,e\,{\left (a\,e-b\,d\right )}^2}{b^4}\right )\,\left (2\,b^2\,d-2\,a\,b\,e\right )}{b^2}-\frac {2\,e\,\left (2\,b^2\,d-2\,a\,b\,e\right )\,{\left (a\,e-b\,d\right )}^2}{b^6}\right )\,\sqrt {d+e\,x}+\left (\frac {2\,e\,{\left (2\,b^2\,d-2\,a\,b\,e\right )}^2}{3\,b^6}-\frac {2\,e\,{\left (a\,e-b\,d\right )}^2}{3\,b^4}\right )\,{\left (d+e\,x\right )}^{3/2}-\frac {\sqrt {d+e\,x}\,\left (a^4\,e^5-4\,a^3\,b\,d\,e^4+6\,a^2\,b^2\,d^2\,e^3-4\,a\,b^3\,d^3\,e^2+b^4\,d^4\,e\right )}{b^6\,\left (d+e\,x\right )-b^6\,d+a\,b^5\,e}+\frac {2\,e\,{\left (d+e\,x\right )}^{7/2}}{7\,b^2}+\frac {2\,e\,\left (2\,b^2\,d-2\,a\,b\,e\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,b^4}+\frac {9\,e\,\mathrm {atan}\left (\frac {\sqrt {b}\,e\,{\left (a\,e-b\,d\right )}^{7/2}\,\sqrt {d+e\,x}}{a^4\,e^5-4\,a^3\,b\,d\,e^4+6\,a^2\,b^2\,d^2\,e^3-4\,a\,b^3\,d^3\,e^2+b^4\,d^4\,e}\right )\,{\left (a\,e-b\,d\right )}^{7/2}}{b^{11/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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