Optimal. Leaf size=308 \[ \frac {(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^6 (a+b x) (d+e x)^{11}}-\frac {b (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^6 (a+b x) (d+e x)^{10}}+\frac {10 b^2 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^6 (a+b x) (d+e x)^9}-\frac {5 b^3 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^6 (a+b x) (d+e x)^8}+\frac {5 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x) (d+e x)^7}-\frac {b^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^6 (a+b x) (d+e x)^6} \]
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Rubi [A]
time = 0.10, antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {660, 45}
\begin {gather*} \frac {10 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{9 e^6 (a+b x) (d+e x)^9}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{2 e^6 (a+b x) (d+e x)^{10}}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{11 e^6 (a+b x) (d+e x)^{11}}-\frac {b^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^6 (a+b x) (d+e x)^6}+\frac {5 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{7 e^6 (a+b x) (d+e x)^7}-\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{4 e^6 (a+b x) (d+e x)^8} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 660
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5}{(d+e x)^{12}} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^5 (b d-a e)^5}{e^5 (d+e x)^{12}}+\frac {5 b^6 (b d-a e)^4}{e^5 (d+e x)^{11}}-\frac {10 b^7 (b d-a e)^3}{e^5 (d+e x)^{10}}+\frac {10 b^8 (b d-a e)^2}{e^5 (d+e x)^9}-\frac {5 b^9 (b d-a e)}{e^5 (d+e x)^8}+\frac {b^{10}}{e^5 (d+e x)^7}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^6 (a+b x) (d+e x)^{11}}-\frac {b (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^6 (a+b x) (d+e x)^{10}}+\frac {10 b^2 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^6 (a+b x) (d+e x)^9}-\frac {5 b^3 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^6 (a+b x) (d+e x)^8}+\frac {5 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x) (d+e x)^7}-\frac {b^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^6 (a+b x) (d+e x)^6}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 223, normalized size = 0.72 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (252 a^5 e^5+126 a^4 b e^4 (d+11 e x)+56 a^3 b^2 e^3 \left (d^2+11 d e x+55 e^2 x^2\right )+21 a^2 b^3 e^2 \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+6 a b^4 e \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )+b^5 \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )\right )}{2772 e^6 (a+b x) (d+e x)^{11}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.70, size = 288, normalized size = 0.94
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{5} x^{5}}{6 e}-\frac {5 b^{4} \left (6 a e +b d \right ) x^{4}}{42 e^{2}}-\frac {5 b^{3} \left (21 a^{2} e^{2}+6 a b d e +b^{2} d^{2}\right ) x^{3}}{84 e^{3}}-\frac {5 b^{2} \left (56 e^{3} a^{3}+21 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{2}}{252 e^{4}}-\frac {b \left (126 e^{4} a^{4}+56 a^{3} b d \,e^{3}+21 a^{2} b^{2} d^{2} e^{2}+6 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x}{252 e^{5}}-\frac {252 a^{5} e^{5}+126 a^{4} b d \,e^{4}+56 a^{3} b^{2} d^{2} e^{3}+21 a^{2} b^{3} d^{3} e^{2}+6 a \,b^{4} d^{4} e +b^{5} d^{5}}{2772 e^{6}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{11}}\) | \(262\) |
gosper | \(-\frac {\left (462 b^{5} e^{5} x^{5}+1980 a \,b^{4} e^{5} x^{4}+330 b^{5} d \,e^{4} x^{4}+3465 a^{2} b^{3} e^{5} x^{3}+990 a \,b^{4} d \,e^{4} x^{3}+165 b^{5} d^{2} e^{3} x^{3}+3080 a^{3} b^{2} e^{5} x^{2}+1155 a^{2} b^{3} d \,e^{4} x^{2}+330 a \,b^{4} d^{2} e^{3} x^{2}+55 b^{5} d^{3} e^{2} x^{2}+1386 a^{4} b \,e^{5} x +616 a^{3} b^{2} d \,e^{4} x +231 a^{2} b^{3} d^{2} e^{3} x +66 a \,b^{4} d^{3} e^{2} x +11 b^{5} d^{4} e x +252 a^{5} e^{5}+126 a^{4} b d \,e^{4}+56 a^{3} b^{2} d^{2} e^{3}+21 a^{2} b^{3} d^{3} e^{2}+6 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{2772 e^{6} \left (e x +d \right )^{11} \left (b x +a \right )^{5}}\) | \(288\) |
default | \(-\frac {\left (462 b^{5} e^{5} x^{5}+1980 a \,b^{4} e^{5} x^{4}+330 b^{5} d \,e^{4} x^{4}+3465 a^{2} b^{3} e^{5} x^{3}+990 a \,b^{4} d \,e^{4} x^{3}+165 b^{5} d^{2} e^{3} x^{3}+3080 a^{3} b^{2} e^{5} x^{2}+1155 a^{2} b^{3} d \,e^{4} x^{2}+330 a \,b^{4} d^{2} e^{3} x^{2}+55 b^{5} d^{3} e^{2} x^{2}+1386 a^{4} b \,e^{5} x +616 a^{3} b^{2} d \,e^{4} x +231 a^{2} b^{3} d^{2} e^{3} x +66 a \,b^{4} d^{3} e^{2} x +11 b^{5} d^{4} e x +252 a^{5} e^{5}+126 a^{4} b d \,e^{4}+56 a^{3} b^{2} d^{2} e^{3}+21 a^{2} b^{3} d^{3} e^{2}+6 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{2772 e^{6} \left (e x +d \right )^{11} \left (b x +a \right )^{5}}\) | \(288\) |
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 = 3.17, size = 336, normalized size = 1.09 \begin {gather*} -\frac {b^{5} d^{5} + {\left (462 \, b^{5} x^{5} + 1980 \, a b^{4} x^{4} + 3465 \, a^{2} b^{3} x^{3} + 3080 \, a^{3} b^{2} x^{2} + 1386 \, a^{4} b x + 252 \, a^{5}\right )} e^{5} + {\left (330 \, b^{5} d x^{4} + 990 \, a b^{4} d x^{3} + 1155 \, a^{2} b^{3} d x^{2} + 616 \, a^{3} b^{2} d x + 126 \, a^{4} b d\right )} e^{4} + {\left (165 \, b^{5} d^{2} x^{3} + 330 \, a b^{4} d^{2} x^{2} + 231 \, a^{2} b^{3} d^{2} x + 56 \, a^{3} b^{2} d^{2}\right )} e^{3} + {\left (55 \, b^{5} d^{3} x^{2} + 66 \, a b^{4} d^{3} x + 21 \, a^{2} b^{3} d^{3}\right )} e^{2} + {\left (11 \, b^{5} d^{4} x + 6 \, a b^{4} d^{4}\right )} e}{2772 \, {\left (x^{11} e^{17} + 11 \, d x^{10} e^{16} + 55 \, d^{2} x^{9} e^{15} + 165 \, d^{3} x^{8} e^{14} + 330 \, d^{4} x^{7} e^{13} + 462 \, d^{5} x^{6} e^{12} + 462 \, d^{6} x^{5} e^{11} + 330 \, d^{7} x^{4} e^{10} + 165 \, d^{8} x^{3} e^{9} + 55 \, d^{9} x^{2} e^{8} + 11 \, d^{10} x e^{7} + d^{11} e^{6}\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 = 1.86, size = 381, normalized size = 1.24 \begin {gather*} -\frac {{\left (462 \, b^{5} x^{5} e^{5} \mathrm {sgn}\left (b x + a\right ) + 330 \, b^{5} d x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 165 \, b^{5} d^{2} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 55 \, b^{5} d^{3} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 11 \, b^{5} d^{4} x e \mathrm {sgn}\left (b x + a\right ) + b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) + 1980 \, a b^{4} x^{4} e^{5} \mathrm {sgn}\left (b x + a\right ) + 990 \, a b^{4} d x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 330 \, a b^{4} d^{2} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 66 \, a b^{4} d^{3} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 3465 \, a^{2} b^{3} x^{3} e^{5} \mathrm {sgn}\left (b x + a\right ) + 1155 \, a^{2} b^{3} d x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 231 \, a^{2} b^{3} d^{2} x e^{3} \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 3080 \, a^{3} b^{2} x^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 616 \, a^{3} b^{2} d x e^{4} \mathrm {sgn}\left (b x + a\right ) + 56 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 1386 \, a^{4} b x e^{5} \mathrm {sgn}\left (b x + a\right ) + 126 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) + 252 \, a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-6\right )}}{2772 \, {\left (x e + d\right )}^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.77, size = 687, normalized size = 2.23 \begin {gather*} \frac {\left (\frac {4\,b^5\,d-5\,a\,b^4\,e}{7\,e^6}+\frac {b^5\,d}{7\,e^6}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^7}-\frac {\left (\frac {5\,a^4\,b\,e^4-10\,a^3\,b^2\,d\,e^3+10\,a^2\,b^3\,d^2\,e^2-5\,a\,b^4\,d^3\,e+b^5\,d^4}{10\,e^6}+\frac {d\,\left (\frac {-10\,a^3\,b^2\,e^4+10\,a^2\,b^3\,d\,e^3-5\,a\,b^4\,d^2\,e^2+b^5\,d^3\,e}{10\,e^6}+\frac {d\,\left (\frac {d\,\left (\frac {b^5\,d}{10\,e^3}-\frac {b^4\,\left (5\,a\,e-b\,d\right )}{10\,e^3}\right )}{e}+\frac {b^3\,\left (10\,a^2\,e^2-5\,a\,b\,d\,e+b^2\,d^2\right )}{10\,e^4}\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 )}^{10}}-\frac {\left (\frac {10\,a^2\,b^3\,e^2-15\,a\,b^4\,d\,e+6\,b^5\,d^2}{8\,e^6}+\frac {d\,\left (\frac {b^5\,d}{8\,e^5}-\frac {b^4\,\left (5\,a\,e-3\,b\,d\right )}{8\,e^5}\right )}{e}\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 {a^5}{11\,e}-\frac {d\,\left (\frac {5\,a^4\,b}{11\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {5\,a\,b^4}{11\,e}-\frac {b^5\,d}{11\,e^2}\right )}{e}-\frac {10\,a^2\,b^3}{11\,e}\right )}{e}+\frac {10\,a^3\,b^2}{11\,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 )}^{11}}+\frac {\left (\frac {-10\,a^3\,b^2\,e^3+20\,a^2\,b^3\,d\,e^2-15\,a\,b^4\,d^2\,e+4\,b^5\,d^3}{9\,e^6}+\frac {d\,\left (\frac {d\,\left (\frac {b^5\,d}{9\,e^4}-\frac {b^4\,\left (5\,a\,e-2\,b\,d\right )}{9\,e^4}\right )}{e}+\frac {b^3\,\left (10\,a^2\,e^2-10\,a\,b\,d\,e+3\,b^2\,d^2\right )}{9\,e^5}\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 {b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{6\,e^6\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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