Optimal. Leaf size=239 \[ -\frac {693 e^5}{128 \sqrt {d+e x} (b d-a e)^6}+\frac {693 \sqrt {b} e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{13/2}}-\frac {231 e^4}{128 (a+b x) \sqrt {d+e x} (b d-a e)^5}+\frac {231 e^3}{320 (a+b x)^2 \sqrt {d+e x} (b d-a e)^4}-\frac {33 e^2}{80 (a+b x)^3 \sqrt {d+e x} (b d-a e)^3}+\frac {11 e}{40 (a+b x)^4 \sqrt {d+e x} (b d-a e)^2}-\frac {1}{5 (a+b x)^5 \sqrt {d+e x} (b d-a e)} \]
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Rubi [A] time = 0.16, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {27, 51, 63, 208} \begin {gather*} -\frac {693 e^5}{128 \sqrt {d+e x} (b d-a e)^6}-\frac {231 e^4}{128 (a+b x) \sqrt {d+e x} (b d-a e)^5}+\frac {231 e^3}{320 (a+b x)^2 \sqrt {d+e x} (b d-a e)^4}-\frac {33 e^2}{80 (a+b x)^3 \sqrt {d+e x} (b d-a e)^3}+\frac {693 \sqrt {b} e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{13/2}}+\frac {11 e}{40 (a+b x)^4 \sqrt {d+e x} (b d-a e)^2}-\frac {1}{5 (a+b x)^5 \sqrt {d+e x} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 51
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {1}{(a+b x)^6 (d+e x)^{3/2}} \, dx\\ &=-\frac {1}{5 (b d-a e) (a+b x)^5 \sqrt {d+e x}}-\frac {(11 e) \int \frac {1}{(a+b x)^5 (d+e x)^{3/2}} \, dx}{10 (b d-a e)}\\ &=-\frac {1}{5 (b d-a e) (a+b x)^5 \sqrt {d+e x}}+\frac {11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}+\frac {\left (99 e^2\right ) \int \frac {1}{(a+b x)^4 (d+e x)^{3/2}} \, dx}{80 (b d-a e)^2}\\ &=-\frac {1}{5 (b d-a e) (a+b x)^5 \sqrt {d+e x}}+\frac {11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}-\frac {33 e^2}{80 (b d-a e)^3 (a+b x)^3 \sqrt {d+e x}}-\frac {\left (231 e^3\right ) \int \frac {1}{(a+b x)^3 (d+e x)^{3/2}} \, dx}{160 (b d-a e)^3}\\ &=-\frac {1}{5 (b d-a e) (a+b x)^5 \sqrt {d+e x}}+\frac {11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}-\frac {33 e^2}{80 (b d-a e)^3 (a+b x)^3 \sqrt {d+e x}}+\frac {231 e^3}{320 (b d-a e)^4 (a+b x)^2 \sqrt {d+e x}}+\frac {\left (231 e^4\right ) \int \frac {1}{(a+b x)^2 (d+e x)^{3/2}} \, dx}{128 (b d-a e)^4}\\ &=-\frac {1}{5 (b d-a e) (a+b x)^5 \sqrt {d+e x}}+\frac {11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}-\frac {33 e^2}{80 (b d-a e)^3 (a+b x)^3 \sqrt {d+e x}}+\frac {231 e^3}{320 (b d-a e)^4 (a+b x)^2 \sqrt {d+e x}}-\frac {231 e^4}{128 (b d-a e)^5 (a+b x) \sqrt {d+e x}}-\frac {\left (693 e^5\right ) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{256 (b d-a e)^5}\\ &=-\frac {693 e^5}{128 (b d-a e)^6 \sqrt {d+e x}}-\frac {1}{5 (b d-a e) (a+b x)^5 \sqrt {d+e x}}+\frac {11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}-\frac {33 e^2}{80 (b d-a e)^3 (a+b x)^3 \sqrt {d+e x}}+\frac {231 e^3}{320 (b d-a e)^4 (a+b x)^2 \sqrt {d+e x}}-\frac {231 e^4}{128 (b d-a e)^5 (a+b x) \sqrt {d+e x}}-\frac {\left (693 b e^5\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 (b d-a e)^6}\\ &=-\frac {693 e^5}{128 (b d-a e)^6 \sqrt {d+e x}}-\frac {1}{5 (b d-a e) (a+b x)^5 \sqrt {d+e x}}+\frac {11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}-\frac {33 e^2}{80 (b d-a e)^3 (a+b x)^3 \sqrt {d+e x}}+\frac {231 e^3}{320 (b d-a e)^4 (a+b x)^2 \sqrt {d+e x}}-\frac {231 e^4}{128 (b d-a e)^5 (a+b x) \sqrt {d+e x}}-\frac {\left (693 b 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 d-a e)^6}\\ &=-\frac {693 e^5}{128 (b d-a e)^6 \sqrt {d+e x}}-\frac {1}{5 (b d-a e) (a+b x)^5 \sqrt {d+e x}}+\frac {11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}-\frac {33 e^2}{80 (b d-a e)^3 (a+b x)^3 \sqrt {d+e x}}+\frac {231 e^3}{320 (b d-a e)^4 (a+b x)^2 \sqrt {d+e x}}-\frac {231 e^4}{128 (b d-a e)^5 (a+b x) \sqrt {d+e x}}+\frac {693 \sqrt {b} e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{13/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 50, normalized size = 0.21 \begin {gather*} -\frac {2 e^5 \, _2F_1\left (-\frac {1}{2},6;\frac {1}{2};-\frac {b (d+e x)}{a e-b d}\right )}{\sqrt {d+e x} (a e-b d)^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.91, size = 406, normalized size = 1.70 \begin {gather*} \frac {e^5 \left (1280 a^5 e^5+10615 a^4 b e^4 (d+e x)-6400 a^4 b d e^4+12800 a^3 b^2 d^2 e^3+26070 a^3 b^2 e^3 (d+e x)^2-42460 a^3 b^2 d e^3 (d+e x)-12800 a^2 b^3 d^3 e^2+63690 a^2 b^3 d^2 e^2 (d+e x)+29568 a^2 b^3 e^2 (d+e x)^3-78210 a^2 b^3 d e^2 (d+e x)^2+6400 a b^4 d^4 e-42460 a b^4 d^3 e (d+e x)+78210 a b^4 d^2 e (d+e x)^2+16170 a b^4 e (d+e x)^4-59136 a b^4 d e (d+e x)^3-1280 b^5 d^5+10615 b^5 d^4 (d+e x)-26070 b^5 d^3 (d+e x)^2+29568 b^5 d^2 (d+e x)^3+3465 b^5 (d+e x)^5-16170 b^5 d (d+e x)^4\right )}{640 \sqrt {d+e x} (b d-a e)^6 (-a e-b (d+e x)+b d)^5}+\frac {693 \sqrt {b} e^5 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{128 (a e-b d)^{13/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 2314, normalized size = 9.68
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 571, normalized size = 2.39 \begin {gather*} -\frac {693 \, b \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{5}}{128 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \sqrt {-b^{2} d + a b e}} - \frac {2 \, e^{5}}{{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \sqrt {x e + d}} - \frac {2185 \, {\left (x e + d\right )}^{\frac {9}{2}} b^{5} e^{5} - 9770 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{5} d e^{5} + 16768 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{5} d^{2} e^{5} - 13270 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{5} d^{3} e^{5} + 4215 \, \sqrt {x e + d} b^{5} d^{4} e^{5} + 9770 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{4} e^{6} - 33536 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{4} d e^{6} + 39810 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{4} d^{2} e^{6} - 16860 \, \sqrt {x e + d} a b^{4} d^{3} e^{6} + 16768 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{3} e^{7} - 39810 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{3} d e^{7} + 25290 \, \sqrt {x e + d} a^{2} b^{3} d^{2} e^{7} + 13270 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b^{2} e^{8} - 16860 \, \sqrt {x e + d} a^{3} b^{2} d e^{8} + 4215 \, \sqrt {x e + d} a^{4} b e^{9}}{640 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\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 [B] time = 0.07, size = 641, normalized size = 2.68 \begin {gather*} -\frac {843 \sqrt {e x +d}\, a^{4} b \,e^{9}}{128 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{5}}+\frac {843 \sqrt {e x +d}\, a^{3} b^{2} d \,e^{8}}{32 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{5}}-\frac {2529 \sqrt {e x +d}\, a^{2} b^{3} d^{2} e^{7}}{64 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{5}}+\frac {843 \sqrt {e x +d}\, a \,b^{4} d^{3} e^{6}}{32 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{5}}-\frac {843 \sqrt {e x +d}\, b^{5} d^{4} e^{5}}{128 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{5}}-\frac {1327 \left (e x +d \right )^{\frac {3}{2}} a^{3} b^{2} e^{8}}{64 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{5}}+\frac {3981 \left (e x +d \right )^{\frac {3}{2}} a^{2} b^{3} d \,e^{7}}{64 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{5}}-\frac {3981 \left (e x +d \right )^{\frac {3}{2}} a \,b^{4} d^{2} e^{6}}{64 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{5}}+\frac {1327 \left (e x +d \right )^{\frac {3}{2}} b^{5} d^{3} e^{5}}{64 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{5}}-\frac {131 \left (e x +d \right )^{\frac {5}{2}} a^{2} b^{3} e^{7}}{5 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{5}}+\frac {262 \left (e x +d \right )^{\frac {5}{2}} a \,b^{4} d \,e^{6}}{5 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{5}}-\frac {131 \left (e x +d \right )^{\frac {5}{2}} b^{5} d^{2} e^{5}}{5 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{5}}-\frac {977 \left (e x +d \right )^{\frac {7}{2}} a \,b^{4} e^{6}}{64 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{5}}+\frac {977 \left (e x +d \right )^{\frac {7}{2}} b^{5} d \,e^{5}}{64 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{5}}-\frac {437 \left (e x +d \right )^{\frac {9}{2}} b^{5} e^{5}}{128 \left (a e -b d \right )^{6} \left (b e x +a e \right )^{5}}-\frac {693 b \,e^{5} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \left (a e -b d \right )^{6} \sqrt {\left (a e -b d \right ) b}}-\frac {2 e^{5}}{\left (a e -b d \right )^{6} \sqrt {e x +d}} \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 = 1.10, size = 515, normalized size = 2.15 \begin {gather*} -\frac {\frac {2\,e^5}{a\,e-b\,d}+\frac {2607\,b^2\,e^5\,{\left (d+e\,x\right )}^2}{64\,{\left (a\,e-b\,d\right )}^3}+\frac {231\,b^3\,e^5\,{\left (d+e\,x\right )}^3}{5\,{\left (a\,e-b\,d\right )}^4}+\frac {1617\,b^4\,e^5\,{\left (d+e\,x\right )}^4}{64\,{\left (a\,e-b\,d\right )}^5}+\frac {693\,b^5\,e^5\,{\left (d+e\,x\right )}^5}{128\,{\left (a\,e-b\,d\right )}^6}+\frac {2123\,b\,e^5\,\left (d+e\,x\right )}{128\,{\left (a\,e-b\,d\right )}^2}}{\sqrt {d+e\,x}\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )-{\left (d+e\,x\right )}^{5/2}\,\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 )+{\left (d+e\,x\right )}^{3/2}\,\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 )+b^5\,{\left (d+e\,x\right )}^{11/2}-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^{9/2}+{\left (d+e\,x\right )}^{7/2}\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )}-\frac {693\,\sqrt {b}\,e^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6\right )}{{\left (a\,e-b\,d\right )}^{13/2}}\right )}{128\,{\left (a\,e-b\,d\right )}^{13/2}} \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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