Optimal. Leaf size=213 \[ -\frac {\sqrt {d+e x}}{5 (b d-a e) (a+b x)^5}+\frac {9 e \sqrt {d+e x}}{40 (b d-a e)^2 (a+b x)^4}-\frac {21 e^2 \sqrt {d+e x}}{80 (b d-a e)^3 (a+b x)^3}+\frac {21 e^3 \sqrt {d+e x}}{64 (b d-a e)^4 (a+b x)^2}-\frac {63 e^4 \sqrt {d+e x}}{128 (b d-a e)^5 (a+b x)}+\frac {63 e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 \sqrt {b} (b d-a e)^{11/2}} \]
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Rubi [A]
time = 0.08, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {27, 44, 65, 214}
\begin {gather*} \frac {63 e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 \sqrt {b} (b d-a e)^{11/2}}-\frac {63 e^4 \sqrt {d+e x}}{128 (a+b x) (b d-a e)^5}+\frac {21 e^3 \sqrt {d+e x}}{64 (a+b x)^2 (b d-a e)^4}-\frac {21 e^2 \sqrt {d+e x}}{80 (a+b x)^3 (b d-a e)^3}+\frac {9 e \sqrt {d+e x}}{40 (a+b x)^4 (b d-a e)^2}-\frac {\sqrt {d+e x}}{5 (a+b x)^5 (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 44
Rule 65
Rule 214
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {1}{(a+b x)^6 \sqrt {d+e x}} \, dx\\ &=-\frac {\sqrt {d+e x}}{5 (b d-a e) (a+b x)^5}-\frac {(9 e) \int \frac {1}{(a+b x)^5 \sqrt {d+e x}} \, dx}{10 (b d-a e)}\\ &=-\frac {\sqrt {d+e x}}{5 (b d-a e) (a+b x)^5}+\frac {9 e \sqrt {d+e x}}{40 (b d-a e)^2 (a+b x)^4}+\frac {\left (63 e^2\right ) \int \frac {1}{(a+b x)^4 \sqrt {d+e x}} \, dx}{80 (b d-a e)^2}\\ &=-\frac {\sqrt {d+e x}}{5 (b d-a e) (a+b x)^5}+\frac {9 e \sqrt {d+e x}}{40 (b d-a e)^2 (a+b x)^4}-\frac {21 e^2 \sqrt {d+e x}}{80 (b d-a e)^3 (a+b x)^3}-\frac {\left (21 e^3\right ) \int \frac {1}{(a+b x)^3 \sqrt {d+e x}} \, dx}{32 (b d-a e)^3}\\ &=-\frac {\sqrt {d+e x}}{5 (b d-a e) (a+b x)^5}+\frac {9 e \sqrt {d+e x}}{40 (b d-a e)^2 (a+b x)^4}-\frac {21 e^2 \sqrt {d+e x}}{80 (b d-a e)^3 (a+b x)^3}+\frac {21 e^3 \sqrt {d+e x}}{64 (b d-a e)^4 (a+b x)^2}+\frac {\left (63 e^4\right ) \int \frac {1}{(a+b x)^2 \sqrt {d+e x}} \, dx}{128 (b d-a e)^4}\\ &=-\frac {\sqrt {d+e x}}{5 (b d-a e) (a+b x)^5}+\frac {9 e \sqrt {d+e x}}{40 (b d-a e)^2 (a+b x)^4}-\frac {21 e^2 \sqrt {d+e x}}{80 (b d-a e)^3 (a+b x)^3}+\frac {21 e^3 \sqrt {d+e x}}{64 (b d-a e)^4 (a+b x)^2}-\frac {63 e^4 \sqrt {d+e x}}{128 (b d-a e)^5 (a+b x)}-\frac {\left (63 e^5\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 (b d-a e)^5}\\ &=-\frac {\sqrt {d+e x}}{5 (b d-a e) (a+b x)^5}+\frac {9 e \sqrt {d+e x}}{40 (b d-a e)^2 (a+b x)^4}-\frac {21 e^2 \sqrt {d+e x}}{80 (b d-a e)^3 (a+b x)^3}+\frac {21 e^3 \sqrt {d+e x}}{64 (b d-a e)^4 (a+b x)^2}-\frac {63 e^4 \sqrt {d+e x}}{128 (b d-a e)^5 (a+b x)}-\frac {\left (63 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 )}{128 (b d-a e)^5}\\ &=-\frac {\sqrt {d+e x}}{5 (b d-a e) (a+b x)^5}+\frac {9 e \sqrt {d+e x}}{40 (b d-a e)^2 (a+b x)^4}-\frac {21 e^2 \sqrt {d+e x}}{80 (b d-a e)^3 (a+b x)^3}+\frac {21 e^3 \sqrt {d+e x}}{64 (b d-a e)^4 (a+b x)^2}-\frac {63 e^4 \sqrt {d+e x}}{128 (b d-a e)^5 (a+b x)}+\frac {63 e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 \sqrt {b} (b d-a e)^{11/2}}\\ \end {align*}
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Mathematica [A]
time = 0.60, size = 220, normalized size = 1.03 \begin {gather*} \frac {1}{640} \left (\frac {\sqrt {d+e x} \left (965 a^4 e^4+10 a^3 b e^3 (-149 d+237 e x)+6 a^2 b^2 e^2 \left (228 d^2-289 d e x+448 e^2 x^2\right )+2 a b^3 e \left (-328 d^3+384 d^2 e x-483 d e^2 x^2+735 e^3 x^3\right )+b^4 \left (128 d^4-144 d^3 e x+168 d^2 e^2 x^2-210 d e^3 x^3+315 e^4 x^4\right )\right )}{(-b d+a e)^5 (a+b x)^5}+\frac {315 e^5 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{\sqrt {b} (-b d+a e)^{11/2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.66, size = 285, normalized size = 1.34
method | result | size |
derivativedivides | \(2 e^{5} \left (\frac {\sqrt {e x +d}}{10 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )^{5}}+\frac {\frac {9 \sqrt {e x +d}}{80 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )^{4}}+\frac {9 \left (\frac {7 \sqrt {e x +d}}{48 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )^{3}}+\frac {7 \left (\frac {5 \sqrt {e x +d}}{24 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )^{2}}+\frac {5 \left (\frac {3 \sqrt {e x +d}}{8 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )}+\frac {3 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{8 \left (a e -b d \right ) \sqrt {b \left (a e -b d \right )}}\right )}{6 \left (a e -b d \right )}\right )}{8 \left (a e -b d \right )}\right )}{10 \left (a e -b d \right )}}{a e -b d}\right )\) | \(285\) |
default | \(2 e^{5} \left (\frac {\sqrt {e x +d}}{10 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )^{5}}+\frac {\frac {9 \sqrt {e x +d}}{80 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )^{4}}+\frac {9 \left (\frac {7 \sqrt {e x +d}}{48 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )^{3}}+\frac {7 \left (\frac {5 \sqrt {e x +d}}{24 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )^{2}}+\frac {5 \left (\frac {3 \sqrt {e x +d}}{8 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )}+\frac {3 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{8 \left (a e -b d \right ) \sqrt {b \left (a e -b d \right )}}\right )}{6 \left (a e -b d \right )}\right )}{8 \left (a e -b d \right )}\right )}{10 \left (a e -b d \right )}}{a e -b d}\right )\) | \(285\) |
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. 871 vs.
\(2 (191) = 382\).
time = 2.85, size = 1757, normalized size = 8.25 \begin {gather*} \left [-\frac {315 \, {\left (b^{5} x^{5} + 5 \, a b^{4} x^{4} + 10 \, a^{2} b^{3} x^{3} + 10 \, a^{3} b^{2} x^{2} + 5 \, a^{4} b x + a^{5}\right )} \sqrt {b^{2} d - a b e} e^{5} \log \left (\frac {2 \, b d + {\left (b x - a\right )} e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {x e + d}}{b x + a}\right ) + 2 \, {\left (128 \, b^{6} d^{5} - {\left (315 \, a b^{5} x^{4} + 1470 \, a^{2} b^{4} x^{3} + 2688 \, a^{3} b^{3} x^{2} + 2370 \, a^{4} b^{2} x + 965 \, a^{5} b\right )} e^{5} + {\left (315 \, b^{6} d x^{4} + 1680 \, a b^{5} d x^{3} + 3654 \, a^{2} b^{4} d x^{2} + 4104 \, a^{3} b^{3} d x + 2455 \, a^{4} b^{2} d\right )} e^{4} - 2 \, {\left (105 \, b^{6} d^{2} x^{3} + 567 \, a b^{5} d^{2} x^{2} + 1251 \, a^{2} b^{4} d^{2} x + 1429 \, a^{3} b^{3} d^{2}\right )} e^{3} + 8 \, {\left (21 \, b^{6} d^{3} x^{2} + 114 \, a b^{5} d^{3} x + 253 \, a^{2} b^{4} d^{3}\right )} e^{2} - 16 \, {\left (9 \, b^{6} d^{4} x + 49 \, a b^{5} d^{4}\right )} e\right )} \sqrt {x e + d}}{1280 \, {\left (b^{12} d^{6} x^{5} + 5 \, a b^{11} d^{6} x^{4} + 10 \, a^{2} b^{10} d^{6} x^{3} + 10 \, a^{3} b^{9} d^{6} x^{2} + 5 \, a^{4} b^{8} d^{6} x + a^{5} b^{7} d^{6} + {\left (a^{6} b^{6} x^{5} + 5 \, a^{7} b^{5} x^{4} + 10 \, a^{8} b^{4} x^{3} + 10 \, a^{9} b^{3} x^{2} + 5 \, a^{10} b^{2} x + a^{11} b\right )} e^{6} - 6 \, {\left (a^{5} b^{7} d x^{5} + 5 \, a^{6} b^{6} d x^{4} + 10 \, a^{7} b^{5} d x^{3} + 10 \, a^{8} b^{4} d x^{2} + 5 \, a^{9} b^{3} d x + a^{10} b^{2} d\right )} e^{5} + 15 \, {\left (a^{4} b^{8} d^{2} x^{5} + 5 \, a^{5} b^{7} d^{2} x^{4} + 10 \, a^{6} b^{6} d^{2} x^{3} + 10 \, a^{7} b^{5} d^{2} x^{2} + 5 \, a^{8} b^{4} d^{2} x + a^{9} b^{3} d^{2}\right )} e^{4} - 20 \, {\left (a^{3} b^{9} d^{3} x^{5} + 5 \, a^{4} b^{8} d^{3} x^{4} + 10 \, a^{5} b^{7} d^{3} x^{3} + 10 \, a^{6} b^{6} d^{3} x^{2} + 5 \, a^{7} b^{5} d^{3} x + a^{8} b^{4} d^{3}\right )} e^{3} + 15 \, {\left (a^{2} b^{10} d^{4} x^{5} + 5 \, a^{3} b^{9} d^{4} x^{4} + 10 \, a^{4} b^{8} d^{4} x^{3} + 10 \, a^{5} b^{7} d^{4} x^{2} + 5 \, a^{6} b^{6} d^{4} x + a^{7} b^{5} d^{4}\right )} e^{2} - 6 \, {\left (a b^{11} d^{5} x^{5} + 5 \, a^{2} b^{10} d^{5} x^{4} + 10 \, a^{3} b^{9} d^{5} x^{3} + 10 \, a^{4} b^{8} d^{5} x^{2} + 5 \, a^{5} b^{7} d^{5} x + a^{6} b^{6} d^{5}\right )} e\right )}}, -\frac {315 \, {\left (b^{5} x^{5} + 5 \, a b^{4} x^{4} + 10 \, a^{2} b^{3} x^{3} + 10 \, a^{3} b^{2} x^{2} + 5 \, a^{4} b x + a^{5}\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {x e + d}}{b x e + b d}\right ) e^{5} + {\left (128 \, b^{6} d^{5} - {\left (315 \, a b^{5} x^{4} + 1470 \, a^{2} b^{4} x^{3} + 2688 \, a^{3} b^{3} x^{2} + 2370 \, a^{4} b^{2} x + 965 \, a^{5} b\right )} e^{5} + {\left (315 \, b^{6} d x^{4} + 1680 \, a b^{5} d x^{3} + 3654 \, a^{2} b^{4} d x^{2} + 4104 \, a^{3} b^{3} d x + 2455 \, a^{4} b^{2} d\right )} e^{4} - 2 \, {\left (105 \, b^{6} d^{2} x^{3} + 567 \, a b^{5} d^{2} x^{2} + 1251 \, a^{2} b^{4} d^{2} x + 1429 \, a^{3} b^{3} d^{2}\right )} e^{3} + 8 \, {\left (21 \, b^{6} d^{3} x^{2} + 114 \, a b^{5} d^{3} x + 253 \, a^{2} b^{4} d^{3}\right )} e^{2} - 16 \, {\left (9 \, b^{6} d^{4} x + 49 \, a b^{5} d^{4}\right )} e\right )} \sqrt {x e + d}}{640 \, {\left (b^{12} d^{6} x^{5} + 5 \, a b^{11} d^{6} x^{4} + 10 \, a^{2} b^{10} d^{6} x^{3} + 10 \, a^{3} b^{9} d^{6} x^{2} + 5 \, a^{4} b^{8} d^{6} x + a^{5} b^{7} d^{6} + {\left (a^{6} b^{6} x^{5} + 5 \, a^{7} b^{5} x^{4} + 10 \, a^{8} b^{4} x^{3} + 10 \, a^{9} b^{3} x^{2} + 5 \, a^{10} b^{2} x + a^{11} b\right )} e^{6} - 6 \, {\left (a^{5} b^{7} d x^{5} + 5 \, a^{6} b^{6} d x^{4} + 10 \, a^{7} b^{5} d x^{3} + 10 \, a^{8} b^{4} d x^{2} + 5 \, a^{9} b^{3} d x + a^{10} b^{2} d\right )} e^{5} + 15 \, {\left (a^{4} b^{8} d^{2} x^{5} + 5 \, a^{5} b^{7} d^{2} x^{4} + 10 \, a^{6} b^{6} d^{2} x^{3} + 10 \, a^{7} b^{5} d^{2} x^{2} + 5 \, a^{8} b^{4} d^{2} x + a^{9} b^{3} d^{2}\right )} e^{4} - 20 \, {\left (a^{3} b^{9} d^{3} x^{5} + 5 \, a^{4} b^{8} d^{3} x^{4} + 10 \, a^{5} b^{7} d^{3} x^{3} + 10 \, a^{6} b^{6} d^{3} x^{2} + 5 \, a^{7} b^{5} d^{3} x + a^{8} b^{4} d^{3}\right )} e^{3} + 15 \, {\left (a^{2} b^{10} d^{4} x^{5} + 5 \, a^{3} b^{9} d^{4} x^{4} + 10 \, a^{4} b^{8} d^{4} x^{3} + 10 \, a^{5} b^{7} d^{4} x^{2} + 5 \, a^{6} b^{6} d^{4} x + a^{7} b^{5} d^{4}\right )} e^{2} - 6 \, {\left (a b^{11} d^{5} x^{5} + 5 \, a^{2} b^{10} d^{5} x^{4} + 10 \, a^{3} b^{9} d^{5} x^{3} + 10 \, a^{4} b^{8} d^{5} x^{2} + 5 \, a^{5} b^{7} d^{5} x + a^{6} b^{6} d^{5}\right )} e\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. 454 vs.
\(2 (191) = 382\).
time = 0.92, size = 454, normalized size = 2.13 \begin {gather*} -\frac {63 \, \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{5}}{128 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \sqrt {-b^{2} d + a b e}} - \frac {315 \, {\left (x e + d\right )}^{\frac {9}{2}} b^{4} e^{5} - 1470 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{4} d e^{5} + 2688 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{4} d^{2} e^{5} - 2370 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{4} d^{3} e^{5} + 965 \, \sqrt {x e + d} b^{4} d^{4} e^{5} + 1470 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{3} e^{6} - 5376 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{3} d e^{6} + 7110 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{3} d^{2} e^{6} - 3860 \, \sqrt {x e + d} a b^{3} d^{3} e^{6} + 2688 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{2} e^{7} - 7110 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{2} d e^{7} + 5790 \, \sqrt {x e + d} a^{2} b^{2} d^{2} e^{7} + 2370 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b e^{8} - 3860 \, \sqrt {x e + d} a^{3} b d e^{8} + 965 \, \sqrt {x e + d} a^{4} e^{9}}{640 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.74, size = 252, normalized size = 1.18 \begin {gather*} \frac {\frac {965\,\sqrt {b}\,{\left (a\,e-b\,d\right )}^{9/2}\,\sqrt {d+e\,x}+2370\,b^{3/2}\,{\left (a\,e-b\,d\right )}^{7/2}\,{\left (d+e\,x\right )}^{3/2}+2688\,b^{5/2}\,{\left (a\,e-b\,d\right )}^{5/2}\,{\left (d+e\,x\right )}^{5/2}+1470\,b^{7/2}\,{\left (a\,e-b\,d\right )}^{3/2}\,{\left (d+e\,x\right )}^{7/2}+315\,b^{9/2}\,\sqrt {a\,e-b\,d}\,{\left (d+e\,x\right )}^{9/2}+315\,b^5\,e^5\,x^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{640\,\sqrt {b}\,{\left (a\,e-b\,d\right )}^{11/2}}-\frac {63\,b^{9/2}\,e^5\,x^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{128\,{\left (a\,e-b\,d\right )}^{11/2}}}{{\left (a+b\,x\right )}^5}+\frac {63\,e^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{128\,\sqrt {b}\,{\left (a\,e-b\,d\right )}^{11/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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