Optimal. Leaf size=360 \[ -\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x) (d+e x)^{13}}+\frac {b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^{12}}-\frac {15 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x) (d+e x)^{11}}+\frac {2 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^{10}}-\frac {5 b^4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^9}+\frac {3 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^8}-\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^7} \]
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Rubi [A]
time = 0.13, antiderivative size = 360, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {784, 21, 45}
\begin {gather*} -\frac {15 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{11 e^7 (a+b x) (d+e x)^{11}}+\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{2 e^7 (a+b x) (d+e x)^{12}}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{13 e^7 (a+b x) (d+e x)^{13}}-\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^7}+\frac {3 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{4 e^7 (a+b x) (d+e x)^8}-\frac {5 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{3 e^7 (a+b x) (d+e x)^9}+\frac {2 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) (d+e x)^{10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 45
Rule 784
Rubi steps
\begin {align*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{14}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x) \left (a b+b^2 x\right )^5}{(d+e x)^{14}} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^6}{(d+e x)^{14}} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(-b d+a e)^6}{e^6 (d+e x)^{14}}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^{13}}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^{12}}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^{11}}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)^{10}}-\frac {6 b^5 (b d-a e)}{e^6 (d+e x)^9}+\frac {b^6}{e^6 (d+e x)^8}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x) (d+e x)^{13}}+\frac {b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^{12}}-\frac {15 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x) (d+e x)^{11}}+\frac {2 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^{10}}-\frac {5 b^4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^9}+\frac {3 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^8}-\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^7}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 295, normalized size = 0.82 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (924 a^6 e^6+462 a^5 b e^5 (d+13 e x)+210 a^4 b^2 e^4 \left (d^2+13 d e x+78 e^2 x^2\right )+84 a^3 b^3 e^3 \left (d^3+13 d^2 e x+78 d e^2 x^2+286 e^3 x^3\right )+28 a^2 b^4 e^2 \left (d^4+13 d^3 e x+78 d^2 e^2 x^2+286 d e^3 x^3+715 e^4 x^4\right )+7 a b^5 e \left (d^5+13 d^4 e x+78 d^3 e^2 x^2+286 d^2 e^3 x^3+715 d e^4 x^4+1287 e^5 x^5\right )+b^6 \left (d^6+13 d^5 e x+78 d^4 e^2 x^2+286 d^3 e^3 x^3+715 d^2 e^4 x^4+1287 d e^5 x^5+1716 e^6 x^6\right )\right )}{12012 e^7 (a+b x) (d+e x)^{13}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 392, normalized size = 1.09
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{6} x^{6}}{7 e}-\frac {3 b^{5} \left (7 a e +b d \right ) x^{5}}{28 e^{2}}-\frac {5 b^{4} \left (28 a^{2} e^{2}+7 a b d e +b^{2} d^{2}\right ) x^{4}}{84 e^{3}}-\frac {b^{3} \left (84 a^{3} e^{3}+28 a^{2} b d \,e^{2}+7 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{42 e^{4}}-\frac {b^{2} \left (210 a^{4} e^{4}+84 a^{3} b d \,e^{3}+28 a^{2} b^{2} d^{2} e^{2}+7 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{2}}{154 e^{5}}-\frac {b \left (462 a^{5} e^{5}+210 a^{4} b d \,e^{4}+84 a^{3} b^{2} d^{2} e^{3}+28 a^{2} b^{3} d^{3} e^{2}+7 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) x}{924 e^{6}}-\frac {924 e^{6} a^{6}+462 d \,e^{5} a^{5} b +210 d^{2} e^{4} a^{4} b^{2}+84 d^{3} e^{3} a^{3} b^{3}+28 d^{4} e^{2} a^{2} b^{4}+7 d^{5} e a \,b^{5}+d^{6} b^{6}}{12012 e^{7}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{13}}\) | \(351\) |
gosper | \(-\frac {\left (1716 b^{6} e^{6} x^{6}+9009 a \,b^{5} e^{6} x^{5}+1287 b^{6} d \,e^{5} x^{5}+20020 a^{2} b^{4} e^{6} x^{4}+5005 a \,b^{5} d \,e^{5} x^{4}+715 b^{6} d^{2} e^{4} x^{4}+24024 a^{3} b^{3} e^{6} x^{3}+8008 a^{2} b^{4} d \,e^{5} x^{3}+2002 a \,b^{5} d^{2} e^{4} x^{3}+286 b^{6} d^{3} e^{3} x^{3}+16380 a^{4} b^{2} e^{6} x^{2}+6552 a^{3} b^{3} d \,e^{5} x^{2}+2184 a^{2} b^{4} d^{2} e^{4} x^{2}+546 a \,b^{5} d^{3} e^{3} x^{2}+78 b^{6} d^{4} e^{2} x^{2}+6006 a^{5} b \,e^{6} x +2730 a^{4} b^{2} d \,e^{5} x +1092 a^{3} b^{3} d^{2} e^{4} x +364 a^{2} b^{4} d^{3} e^{3} x +91 a \,b^{5} d^{4} e^{2} x +13 b^{6} d^{5} e x +924 e^{6} a^{6}+462 d \,e^{5} a^{5} b +210 d^{2} e^{4} a^{4} b^{2}+84 d^{3} e^{3} a^{3} b^{3}+28 d^{4} e^{2} a^{2} b^{4}+7 d^{5} e a \,b^{5}+d^{6} b^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{12012 e^{7} \left (e x +d \right )^{13} \left (b x +a \right )^{5}}\) | \(392\) |
default | \(-\frac {\left (1716 b^{6} e^{6} x^{6}+9009 a \,b^{5} e^{6} x^{5}+1287 b^{6} d \,e^{5} x^{5}+20020 a^{2} b^{4} e^{6} x^{4}+5005 a \,b^{5} d \,e^{5} x^{4}+715 b^{6} d^{2} e^{4} x^{4}+24024 a^{3} b^{3} e^{6} x^{3}+8008 a^{2} b^{4} d \,e^{5} x^{3}+2002 a \,b^{5} d^{2} e^{4} x^{3}+286 b^{6} d^{3} e^{3} x^{3}+16380 a^{4} b^{2} e^{6} x^{2}+6552 a^{3} b^{3} d \,e^{5} x^{2}+2184 a^{2} b^{4} d^{2} e^{4} x^{2}+546 a \,b^{5} d^{3} e^{3} x^{2}+78 b^{6} d^{4} e^{2} x^{2}+6006 a^{5} b \,e^{6} x +2730 a^{4} b^{2} d \,e^{5} x +1092 a^{3} b^{3} d^{2} e^{4} x +364 a^{2} b^{4} d^{3} e^{3} x +91 a \,b^{5} d^{4} e^{2} x +13 b^{6} d^{5} e x +924 e^{6} a^{6}+462 d \,e^{5} a^{5} b +210 d^{2} e^{4} a^{4} b^{2}+84 d^{3} e^{3} a^{3} b^{3}+28 d^{4} e^{2} a^{2} b^{4}+7 d^{5} e a \,b^{5}+d^{6} b^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{12012 e^{7} \left (e x +d \right )^{13} \left (b x +a \right )^{5}}\) | \(392\) |
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 [A]
time = 2.84, size = 444, normalized size = 1.23 \begin {gather*} -\frac {b^{6} d^{6} + {\left (1716 \, b^{6} x^{6} + 9009 \, a b^{5} x^{5} + 20020 \, a^{2} b^{4} x^{4} + 24024 \, a^{3} b^{3} x^{3} + 16380 \, a^{4} b^{2} x^{2} + 6006 \, a^{5} b x + 924 \, a^{6}\right )} e^{6} + {\left (1287 \, b^{6} d x^{5} + 5005 \, a b^{5} d x^{4} + 8008 \, a^{2} b^{4} d x^{3} + 6552 \, a^{3} b^{3} d x^{2} + 2730 \, a^{4} b^{2} d x + 462 \, a^{5} b d\right )} e^{5} + {\left (715 \, b^{6} d^{2} x^{4} + 2002 \, a b^{5} d^{2} x^{3} + 2184 \, a^{2} b^{4} d^{2} x^{2} + 1092 \, a^{3} b^{3} d^{2} x + 210 \, a^{4} b^{2} d^{2}\right )} e^{4} + 2 \, {\left (143 \, b^{6} d^{3} x^{3} + 273 \, a b^{5} d^{3} x^{2} + 182 \, a^{2} b^{4} d^{3} x + 42 \, a^{3} b^{3} d^{3}\right )} e^{3} + {\left (78 \, b^{6} d^{4} x^{2} + 91 \, a b^{5} d^{4} x + 28 \, a^{2} b^{4} d^{4}\right )} e^{2} + {\left (13 \, b^{6} d^{5} x + 7 \, a b^{5} d^{5}\right )} e}{12012 \, {\left (x^{13} e^{20} + 13 \, d x^{12} e^{19} + 78 \, d^{2} x^{11} e^{18} + 286 \, d^{3} x^{10} e^{17} + 715 \, d^{4} x^{9} e^{16} + 1287 \, d^{5} x^{8} e^{15} + 1716 \, d^{6} x^{7} e^{14} + 1716 \, d^{7} x^{6} e^{13} + 1287 \, d^{8} x^{5} e^{12} + 715 \, d^{9} x^{4} e^{11} + 286 \, d^{10} x^{3} e^{10} + 78 \, d^{11} x^{2} e^{9} + 13 \, d^{12} x e^{8} + d^{13} e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.84, size = 520, normalized size = 1.44 \begin {gather*} -\frac {{\left (1716 \, b^{6} x^{6} e^{6} \mathrm {sgn}\left (b x + a\right ) + 1287 \, b^{6} d x^{5} e^{5} \mathrm {sgn}\left (b x + a\right ) + 715 \, b^{6} d^{2} x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 286 \, b^{6} d^{3} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 78 \, b^{6} d^{4} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 13 \, b^{6} d^{5} x e \mathrm {sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) + 9009 \, a b^{5} x^{5} e^{6} \mathrm {sgn}\left (b x + a\right ) + 5005 \, a b^{5} d x^{4} e^{5} \mathrm {sgn}\left (b x + a\right ) + 2002 \, a b^{5} d^{2} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 546 \, a b^{5} d^{3} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 91 \, a b^{5} d^{4} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 7 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 20020 \, a^{2} b^{4} x^{4} e^{6} \mathrm {sgn}\left (b x + a\right ) + 8008 \, a^{2} b^{4} d x^{3} e^{5} \mathrm {sgn}\left (b x + a\right ) + 2184 \, a^{2} b^{4} d^{2} x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 364 \, a^{2} b^{4} d^{3} x e^{3} \mathrm {sgn}\left (b x + a\right ) + 28 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 24024 \, a^{3} b^{3} x^{3} e^{6} \mathrm {sgn}\left (b x + a\right ) + 6552 \, a^{3} b^{3} d x^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 1092 \, a^{3} b^{3} d^{2} x e^{4} \mathrm {sgn}\left (b x + a\right ) + 84 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 16380 \, a^{4} b^{2} x^{2} e^{6} \mathrm {sgn}\left (b x + a\right ) + 2730 \, a^{4} b^{2} d x e^{5} \mathrm {sgn}\left (b x + a\right ) + 210 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 6006 \, a^{5} b x e^{6} \mathrm {sgn}\left (b x + a\right ) + 462 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + 924 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{12012 \, {\left (x e + d\right )}^{13}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.39, size = 1010, normalized size = 2.81 \begin {gather*} \frac {\left (\frac {-6\,a^5\,b\,e^5+15\,a^4\,b^2\,d\,e^4-20\,a^3\,b^3\,d^2\,e^3+15\,a^2\,b^4\,d^3\,e^2-6\,a\,b^5\,d^4\,e+b^6\,d^5}{12\,e^7}+\frac {d\,\left (\frac {15\,a^4\,b^2\,e^5-20\,a^3\,b^3\,d\,e^4+15\,a^2\,b^4\,d^2\,e^3-6\,a\,b^5\,d^3\,e^2+b^6\,d^4\,e}{12\,e^7}-\frac {d\,\left (\frac {20\,a^3\,b^3\,e^5-15\,a^2\,b^4\,d\,e^4+6\,a\,b^5\,d^2\,e^3-b^6\,d^3\,e^2}{12\,e^7}-\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{12\,e^3}-\frac {b^5\,\left (6\,a\,e-b\,d\right )}{12\,e^3}\right )}{e}+\frac {b^4\,\left (15\,a^2\,e^2-6\,a\,b\,d\,e+b^2\,d^2\right )}{12\,e^4}\right )}{e}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{12}}-\frac {\left (\frac {15\,a^2\,b^4\,e^2-24\,a\,b^5\,d\,e+10\,b^6\,d^2}{9\,e^7}+\frac {d\,\left (\frac {b^6\,d}{9\,e^6}-\frac {2\,b^5\,\left (3\,a\,e-2\,b\,d\right )}{9\,e^6}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^9}-\frac {\left (\frac {a^6}{13\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {6\,a\,b^5}{13\,e}-\frac {b^6\,d}{13\,e^2}\right )}{e}-\frac {15\,a^2\,b^4}{13\,e}\right )}{e}+\frac {20\,a^3\,b^3}{13\,e}\right )}{e}-\frac {15\,a^4\,b^2}{13\,e}\right )}{e}+\frac {6\,a^5\,b}{13\,e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{13}}-\frac {\left (\frac {15\,a^4\,b^2\,e^4-40\,a^3\,b^3\,d\,e^3+45\,a^2\,b^4\,d^2\,e^2-24\,a\,b^5\,d^3\,e+5\,b^6\,d^4}{11\,e^7}+\frac {d\,\left (\frac {-20\,a^3\,b^3\,e^4+30\,a^2\,b^4\,d\,e^3-18\,a\,b^5\,d^2\,e^2+4\,b^6\,d^3\,e}{11\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{11\,e^4}-\frac {2\,b^5\,\left (3\,a\,e-b\,d\right )}{11\,e^4}\right )}{e}+\frac {3\,b^4\,\left (5\,a^2\,e^2-4\,a\,b\,d\,e+b^2\,d^2\right )}{11\,e^5}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{11}}+\frac {\left (\frac {5\,b^6\,d-6\,a\,b^5\,e}{8\,e^7}+\frac {b^6\,d}{8\,e^7}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^8}+\frac {\left (\frac {-20\,a^3\,b^3\,e^3+45\,a^2\,b^4\,d\,e^2-36\,a\,b^5\,d^2\,e+10\,b^6\,d^3}{10\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{10\,e^5}-\frac {3\,b^5\,\left (2\,a\,e-b\,d\right )}{10\,e^5}\right )}{e}+\frac {3\,b^4\,\left (5\,a^2\,e^2-6\,a\,b\,d\,e+2\,b^2\,d^2\right )}{10\,e^6}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{10}}-\frac {b^6\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{7\,e^7\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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