Optimal. Leaf size=200 \[ \frac {(a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{10 (b d-a e) (d+e x)^{10}}+\frac {b (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{30 (b d-a e)^2 (d+e x)^9}+\frac {b^2 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{120 (b d-a e)^3 (d+e x)^8}+\frac {b^3 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{840 (b d-a e)^4 (d+e x)^7} \]
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Rubi [A]
time = 0.06, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {784, 21, 47, 37}
\begin {gather*} \frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6}{120 (d+e x)^8 (b d-a e)^3}+\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6}{30 (d+e x)^9 (b d-a e)^2}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6}{10 (d+e x)^{10} (b d-a e)}+\frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6}{840 (d+e x)^7 (b d-a e)^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 37
Rule 47
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)^{11}} \, 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)^{11}} \, 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)^{11}} \, dx}{a b+b^2 x}\\ &=\frac {(a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{10 (b d-a e) (d+e x)^{10}}+\frac {\left (3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^6}{(d+e x)^{10}} \, dx}{10 (b d-a e) \left (a b+b^2 x\right )}\\ &=\frac {(a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{10 (b d-a e) (d+e x)^{10}}+\frac {b (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{30 (b d-a e)^2 (d+e x)^9}+\frac {\left (b^3 \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^6}{(d+e x)^9} \, dx}{15 (b d-a e)^2 \left (a b+b^2 x\right )}\\ &=\frac {(a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{10 (b d-a e) (d+e x)^{10}}+\frac {b (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{30 (b d-a e)^2 (d+e x)^9}+\frac {b^2 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{120 (b d-a e)^3 (d+e x)^8}+\frac {\left (b^4 \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^6}{(d+e x)^8} \, dx}{120 (b d-a e)^3 \left (a b+b^2 x\right )}\\ &=\frac {(a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{10 (b d-a e) (d+e x)^{10}}+\frac {b (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{30 (b d-a e)^2 (d+e x)^9}+\frac {b^2 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{120 (b d-a e)^3 (d+e x)^8}+\frac {b^3 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{840 (b d-a e)^4 (d+e x)^7}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 295, normalized size = 1.48 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (84 a^6 e^6+56 a^5 b e^5 (d+10 e x)+35 a^4 b^2 e^4 \left (d^2+10 d e x+45 e^2 x^2\right )+20 a^3 b^3 e^3 \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+10 a^2 b^4 e^2 \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )+4 a b^5 e \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )+b^6 \left (d^6+10 d^5 e x+45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+252 d e^5 x^5+210 e^6 x^6\right )\right )}{840 e^7 (a+b x) (d+e x)^{10}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(391\) vs.
\(2(148)=296\).
time = 0.08, size = 392, normalized size = 1.96
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{6} x^{6}}{4 e}-\frac {3 b^{5} \left (4 a e +b d \right ) x^{5}}{10 e^{2}}-\frac {b^{4} \left (10 a^{2} e^{2}+4 a b d e +b^{2} d^{2}\right ) x^{4}}{4 e^{3}}-\frac {b^{3} \left (20 a^{3} e^{3}+10 a^{2} b d \,e^{2}+4 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{7 e^{4}}-\frac {3 b^{2} \left (35 a^{4} e^{4}+20 a^{3} b d \,e^{3}+10 a^{2} b^{2} d^{2} e^{2}+4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{2}}{56 e^{5}}-\frac {b \left (56 a^{5} e^{5}+35 a^{4} b d \,e^{4}+20 a^{3} b^{2} d^{2} e^{3}+10 a^{2} b^{3} d^{3} e^{2}+4 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) x}{84 e^{6}}-\frac {84 e^{6} a^{6}+56 d \,e^{5} a^{5} b +35 d^{2} e^{4} a^{4} b^{2}+20 d^{3} e^{3} a^{3} b^{3}+10 d^{4} e^{2} a^{2} b^{4}+4 d^{5} e a \,b^{5}+d^{6} b^{6}}{840 e^{7}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{10}}\) | \(351\) |
gosper | \(-\frac {\left (210 b^{6} e^{6} x^{6}+1008 a \,b^{5} e^{6} x^{5}+252 b^{6} d \,e^{5} x^{5}+2100 a^{2} b^{4} e^{6} x^{4}+840 a \,b^{5} d \,e^{5} x^{4}+210 b^{6} d^{2} e^{4} x^{4}+2400 a^{3} b^{3} e^{6} x^{3}+1200 a^{2} b^{4} d \,e^{5} x^{3}+480 a \,b^{5} d^{2} e^{4} x^{3}+120 b^{6} d^{3} e^{3} x^{3}+1575 a^{4} b^{2} e^{6} x^{2}+900 a^{3} b^{3} d \,e^{5} x^{2}+450 a^{2} b^{4} d^{2} e^{4} x^{2}+180 a \,b^{5} d^{3} e^{3} x^{2}+45 b^{6} d^{4} e^{2} x^{2}+560 a^{5} b \,e^{6} x +350 a^{4} b^{2} d \,e^{5} x +200 a^{3} b^{3} d^{2} e^{4} x +100 a^{2} b^{4} d^{3} e^{3} x +40 a \,b^{5} d^{4} e^{2} x +10 b^{6} d^{5} e x +84 e^{6} a^{6}+56 d \,e^{5} a^{5} b +35 d^{2} e^{4} a^{4} b^{2}+20 d^{3} e^{3} a^{3} b^{3}+10 d^{4} e^{2} a^{2} b^{4}+4 d^{5} e a \,b^{5}+d^{6} b^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{840 e^{7} \left (e x +d \right )^{10} \left (b x +a \right )^{5}}\) | \(392\) |
default | \(-\frac {\left (210 b^{6} e^{6} x^{6}+1008 a \,b^{5} e^{6} x^{5}+252 b^{6} d \,e^{5} x^{5}+2100 a^{2} b^{4} e^{6} x^{4}+840 a \,b^{5} d \,e^{5} x^{4}+210 b^{6} d^{2} e^{4} x^{4}+2400 a^{3} b^{3} e^{6} x^{3}+1200 a^{2} b^{4} d \,e^{5} x^{3}+480 a \,b^{5} d^{2} e^{4} x^{3}+120 b^{6} d^{3} e^{3} x^{3}+1575 a^{4} b^{2} e^{6} x^{2}+900 a^{3} b^{3} d \,e^{5} x^{2}+450 a^{2} b^{4} d^{2} e^{4} x^{2}+180 a \,b^{5} d^{3} e^{3} x^{2}+45 b^{6} d^{4} e^{2} x^{2}+560 a^{5} b \,e^{6} x +350 a^{4} b^{2} d \,e^{5} x +200 a^{3} b^{3} d^{2} e^{4} x +100 a^{2} b^{4} d^{3} e^{3} x +40 a \,b^{5} d^{4} e^{2} x +10 b^{6} d^{5} e x +84 e^{6} a^{6}+56 d \,e^{5} a^{5} b +35 d^{2} e^{4} a^{4} b^{2}+20 d^{3} e^{3} a^{3} b^{3}+10 d^{4} e^{2} a^{2} b^{4}+4 d^{5} e a \,b^{5}+d^{6} b^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{840 e^{7} \left (e x +d \right )^{10} \left (b x +a \right )^{5}}\) | \(392\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 417 vs.
\(2 (156) = 312\).
time = 2.52, size = 417, normalized size = 2.08 \begin {gather*} -\frac {b^{6} d^{6} + {\left (210 \, b^{6} x^{6} + 1008 \, a b^{5} x^{5} + 2100 \, a^{2} b^{4} x^{4} + 2400 \, a^{3} b^{3} x^{3} + 1575 \, a^{4} b^{2} x^{2} + 560 \, a^{5} b x + 84 \, a^{6}\right )} e^{6} + 2 \, {\left (126 \, b^{6} d x^{5} + 420 \, a b^{5} d x^{4} + 600 \, a^{2} b^{4} d x^{3} + 450 \, a^{3} b^{3} d x^{2} + 175 \, a^{4} b^{2} d x + 28 \, a^{5} b d\right )} e^{5} + 5 \, {\left (42 \, b^{6} d^{2} x^{4} + 96 \, a b^{5} d^{2} x^{3} + 90 \, a^{2} b^{4} d^{2} x^{2} + 40 \, a^{3} b^{3} d^{2} x + 7 \, a^{4} b^{2} d^{2}\right )} e^{4} + 20 \, {\left (6 \, b^{6} d^{3} x^{3} + 9 \, a b^{5} d^{3} x^{2} + 5 \, a^{2} b^{4} d^{3} x + a^{3} b^{3} d^{3}\right )} e^{3} + 5 \, {\left (9 \, b^{6} d^{4} x^{2} + 8 \, a b^{5} d^{4} x + 2 \, a^{2} b^{4} d^{4}\right )} e^{2} + 2 \, {\left (5 \, b^{6} d^{5} x + 2 \, a b^{5} d^{5}\right )} e}{840 \, {\left (x^{10} e^{17} + 10 \, d x^{9} e^{16} + 45 \, d^{2} x^{8} e^{15} + 120 \, d^{3} x^{7} e^{14} + 210 \, d^{4} x^{6} e^{13} + 252 \, d^{5} x^{5} e^{12} + 210 \, d^{6} x^{4} e^{11} + 120 \, d^{7} x^{3} e^{10} + 45 \, d^{8} x^{2} e^{9} + 10 \, d^{9} x e^{8} + d^{10} e^{7}\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. 520 vs.
\(2 (156) = 312\).
time = 0.88, size = 520, normalized size = 2.60 \begin {gather*} -\frac {{\left (210 \, b^{6} x^{6} e^{6} \mathrm {sgn}\left (b x + a\right ) + 252 \, b^{6} d x^{5} e^{5} \mathrm {sgn}\left (b x + a\right ) + 210 \, b^{6} d^{2} x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 120 \, b^{6} d^{3} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 45 \, b^{6} d^{4} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, b^{6} d^{5} x e \mathrm {sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) + 1008 \, a b^{5} x^{5} e^{6} \mathrm {sgn}\left (b x + a\right ) + 840 \, a b^{5} d x^{4} e^{5} \mathrm {sgn}\left (b x + a\right ) + 480 \, a b^{5} d^{2} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 180 \, a b^{5} d^{3} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 40 \, a b^{5} d^{4} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 2100 \, a^{2} b^{4} x^{4} e^{6} \mathrm {sgn}\left (b x + a\right ) + 1200 \, a^{2} b^{4} d x^{3} e^{5} \mathrm {sgn}\left (b x + a\right ) + 450 \, a^{2} b^{4} d^{2} x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 100 \, a^{2} b^{4} d^{3} x e^{3} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 2400 \, a^{3} b^{3} x^{3} e^{6} \mathrm {sgn}\left (b x + a\right ) + 900 \, a^{3} b^{3} d x^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 200 \, a^{3} b^{3} d^{2} x e^{4} \mathrm {sgn}\left (b x + a\right ) + 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 1575 \, a^{4} b^{2} x^{2} e^{6} \mathrm {sgn}\left (b x + a\right ) + 350 \, a^{4} b^{2} d x e^{5} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 560 \, a^{5} b x e^{6} \mathrm {sgn}\left (b x + a\right ) + 56 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + 84 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{840 \, {\left (x e + d\right )}^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.53, size = 1010, normalized size = 5.05 \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}{9\,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}{9\,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}{9\,e^7}-\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{9\,e^3}-\frac {b^5\,\left (6\,a\,e-b\,d\right )}{9\,e^3}\right )}{e}+\frac {b^4\,\left (15\,a^2\,e^2-6\,a\,b\,d\,e+b^2\,d^2\right )}{9\,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 )}^9}-\frac {\left (\frac {15\,a^2\,b^4\,e^2-24\,a\,b^5\,d\,e+10\,b^6\,d^2}{6\,e^7}+\frac {d\,\left (\frac {b^6\,d}{6\,e^6}-\frac {b^5\,\left (3\,a\,e-2\,b\,d\right )}{3\,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 )}^6}-\frac {\left (\frac {a^6}{10\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {3\,a\,b^5}{5\,e}-\frac {b^6\,d}{10\,e^2}\right )}{e}-\frac {3\,a^2\,b^4}{2\,e}\right )}{e}+\frac {2\,a^3\,b^3}{e}\right )}{e}-\frac {3\,a^4\,b^2}{2\,e}\right )}{e}+\frac {3\,a^5\,b}{5\,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 {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}{8\,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}{8\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{8\,e^4}-\frac {b^5\,\left (3\,a\,e-b\,d\right )}{4\,e^4}\right )}{e}+\frac {3\,b^4\,\left (5\,a^2\,e^2-4\,a\,b\,d\,e+b^2\,d^2\right )}{8\,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 )}^8}+\frac {\left (\frac {5\,b^6\,d-6\,a\,b^5\,e}{5\,e^7}+\frac {b^6\,d}{5\,e^7}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5}+\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}{7\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{7\,e^5}-\frac {3\,b^5\,\left (2\,a\,e-b\,d\right )}{7\,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 )}{7\,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 )}^7}-\frac {b^6\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,e^7\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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