Optimal. Leaf size=218 \[ -\frac {7 e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{3/2} (b d-a e)^{9/2}}+\frac {7 e^4 \sqrt {d+e x}}{128 b (a+b x) (b d-a e)^4}-\frac {7 e^3 \sqrt {d+e x}}{192 b (a+b x)^2 (b d-a e)^3}+\frac {7 e^2 \sqrt {d+e x}}{240 b (a+b x)^3 (b d-a e)^2}-\frac {e \sqrt {d+e x}}{40 b (a+b x)^4 (b d-a e)}-\frac {\sqrt {d+e x}}{5 b (a+b x)^5} \]
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Rubi [A] time = 0.15, antiderivative size = 218, 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, 51, 63, 208} \begin {gather*} -\frac {7 e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{3/2} (b d-a e)^{9/2}}+\frac {7 e^4 \sqrt {d+e x}}{128 b (a+b x) (b d-a e)^4}-\frac {7 e^3 \sqrt {d+e x}}{192 b (a+b x)^2 (b d-a e)^3}+\frac {7 e^2 \sqrt {d+e x}}{240 b (a+b x)^3 (b d-a e)^2}-\frac {e \sqrt {d+e x}}{40 b (a+b x)^4 (b d-a e)}-\frac {\sqrt {d+e x}}{5 b (a+b x)^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {\sqrt {d+e x}}{(a+b x)^6} \, dx\\ &=-\frac {\sqrt {d+e x}}{5 b (a+b x)^5}+\frac {e \int \frac {1}{(a+b x)^5 \sqrt {d+e x}} \, dx}{10 b}\\ &=-\frac {\sqrt {d+e x}}{5 b (a+b x)^5}-\frac {e \sqrt {d+e x}}{40 b (b d-a e) (a+b x)^4}-\frac {\left (7 e^2\right ) \int \frac {1}{(a+b x)^4 \sqrt {d+e x}} \, dx}{80 b (b d-a e)}\\ &=-\frac {\sqrt {d+e x}}{5 b (a+b x)^5}-\frac {e \sqrt {d+e x}}{40 b (b d-a e) (a+b x)^4}+\frac {7 e^2 \sqrt {d+e x}}{240 b (b d-a e)^2 (a+b x)^3}+\frac {\left (7 e^3\right ) \int \frac {1}{(a+b x)^3 \sqrt {d+e x}} \, dx}{96 b (b d-a e)^2}\\ &=-\frac {\sqrt {d+e x}}{5 b (a+b x)^5}-\frac {e \sqrt {d+e x}}{40 b (b d-a e) (a+b x)^4}+\frac {7 e^2 \sqrt {d+e x}}{240 b (b d-a e)^2 (a+b x)^3}-\frac {7 e^3 \sqrt {d+e x}}{192 b (b d-a e)^3 (a+b x)^2}-\frac {\left (7 e^4\right ) \int \frac {1}{(a+b x)^2 \sqrt {d+e x}} \, dx}{128 b (b d-a e)^3}\\ &=-\frac {\sqrt {d+e x}}{5 b (a+b x)^5}-\frac {e \sqrt {d+e x}}{40 b (b d-a e) (a+b x)^4}+\frac {7 e^2 \sqrt {d+e x}}{240 b (b d-a e)^2 (a+b x)^3}-\frac {7 e^3 \sqrt {d+e x}}{192 b (b d-a e)^3 (a+b x)^2}+\frac {7 e^4 \sqrt {d+e x}}{128 b (b d-a e)^4 (a+b x)}+\frac {\left (7 e^5\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 b (b d-a e)^4}\\ &=-\frac {\sqrt {d+e x}}{5 b (a+b x)^5}-\frac {e \sqrt {d+e x}}{40 b (b d-a e) (a+b x)^4}+\frac {7 e^2 \sqrt {d+e x}}{240 b (b d-a e)^2 (a+b x)^3}-\frac {7 e^3 \sqrt {d+e x}}{192 b (b d-a e)^3 (a+b x)^2}+\frac {7 e^4 \sqrt {d+e x}}{128 b (b d-a e)^4 (a+b x)}+\frac {\left (7 e^4\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{128 b (b d-a e)^4}\\ &=-\frac {\sqrt {d+e x}}{5 b (a+b x)^5}-\frac {e \sqrt {d+e x}}{40 b (b d-a e) (a+b x)^4}+\frac {7 e^2 \sqrt {d+e x}}{240 b (b d-a e)^2 (a+b x)^3}-\frac {7 e^3 \sqrt {d+e x}}{192 b (b d-a e)^3 (a+b x)^2}+\frac {7 e^4 \sqrt {d+e x}}{128 b (b d-a e)^4 (a+b x)}-\frac {7 e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{3/2} (b d-a e)^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 52, normalized size = 0.24 \begin {gather*} \frac {2 e^5 (d+e x)^{3/2} \, _2F_1\left (\frac {3}{2},6;\frac {5}{2};-\frac {b (d+e x)}{a e-b d}\right )}{3 (a e-b d)^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.70, size = 317, normalized size = 1.45 \begin {gather*} \frac {e^5 \sqrt {d+e x} \left (105 a^4 e^4-790 a^3 b e^3 (d+e x)-420 a^3 b d e^3+630 a^2 b^2 d^2 e^2-896 a^2 b^2 e^2 (d+e x)^2+2370 a^2 b^2 d e^2 (d+e x)-420 a b^3 d^3 e-2370 a b^3 d^2 e (d+e x)-490 a b^3 e (d+e x)^3+1792 a b^3 d e (d+e x)^2+105 b^4 d^4+790 b^4 d^3 (d+e x)-896 b^4 d^2 (d+e x)^2-105 b^4 (d+e x)^4+490 b^4 d (d+e x)^3\right )}{1920 b (b d-a e)^4 (-a e-b (d+e x)+b d)^5}-\frac {7 e^5 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{128 b^{3/2} (b d-a e)^4 \sqrt {a e-b d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 1673, normalized size = 7.67
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 432, normalized size = 1.98 \begin {gather*} \frac {7 \, \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{5}}{128 \, {\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )} \sqrt {-b^{2} d + a b e}} + \frac {105 \, {\left (x e + d\right )}^{\frac {9}{2}} b^{4} e^{5} - 490 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{4} d e^{5} + 896 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{4} d^{2} e^{5} - 790 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{4} d^{3} e^{5} - 105 \, \sqrt {x e + d} b^{4} d^{4} e^{5} + 490 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{3} e^{6} - 1792 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{3} d e^{6} + 2370 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{3} d^{2} e^{6} + 420 \, \sqrt {x e + d} a b^{3} d^{3} e^{6} + 896 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{2} e^{7} - 2370 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{2} d e^{7} - 630 \, \sqrt {x e + d} a^{2} b^{2} d^{2} e^{7} + 790 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b e^{8} + 420 \, \sqrt {x e + d} a^{3} b d e^{8} - 105 \, \sqrt {x e + d} a^{4} e^{9}}{1920 \, {\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\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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maple [A] time = 0.06, size = 337, normalized size = 1.55 \begin {gather*} \frac {7 \left (e x +d \right )^{\frac {9}{2}} b^{3} e^{5}}{128 \left (b e x +a e \right )^{5} \left (a^{4} e^{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 )}+\frac {49 \left (e x +d \right )^{\frac {7}{2}} b^{2} e^{5}}{192 \left (b e x +a e \right )^{5} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}+\frac {7 \left (e x +d \right )^{\frac {5}{2}} b \,e^{5}}{15 \left (b e x +a e \right )^{5} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}+\frac {7 e^{5} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \left (a^{4} e^{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 ) \sqrt {\left (a e -b d \right ) b}\, b}+\frac {79 \left (e x +d \right )^{\frac {3}{2}} e^{5}}{192 \left (b e x +a e \right )^{5} \left (a e -b d \right )}-\frac {7 \sqrt {e x +d}\, e^{5}}{128 \left (b e x +a e \right )^{5} b} \end {gather*}
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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mupad [B] time = 0.70, size = 401, normalized size = 1.84 \begin {gather*} \frac {\frac {79\,e^5\,{\left (d+e\,x\right )}^{3/2}}{192\,\left (a\,e-b\,d\right )}-\frac {7\,e^5\,\sqrt {d+e\,x}}{128\,b}+\frac {49\,b^2\,e^5\,{\left (d+e\,x\right )}^{7/2}}{192\,{\left (a\,e-b\,d\right )}^3}+\frac {7\,b^3\,e^5\,{\left (d+e\,x\right )}^{9/2}}{128\,{\left (a\,e-b\,d\right )}^4}+\frac {7\,b\,e^5\,{\left (d+e\,x\right )}^{5/2}}{15\,{\left (a\,e-b\,d\right )}^2}}{\left (d+e\,x\right )\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^2\,e^3+30\,a^2\,b^3\,d\,e^2-30\,a\,b^4\,d^2\,e+10\,b^5\,d^3\right )+b^5\,{\left (d+e\,x\right )}^5-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^4+a^5\,e^5-b^5\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )-10\,a^2\,b^3\,d^3\,e^2+10\,a^3\,b^2\,d^2\,e^3+5\,a\,b^4\,d^4\,e-5\,a^4\,b\,d\,e^4}+\frac {7\,e^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{128\,b^{3/2}\,{\left (a\,e-b\,d\right )}^{9/2}} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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