Optimal. Leaf size=360 \[ -\frac {15 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{11 e^7 (a+b x) (d+e x)^{11}}+\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{2 e^7 (a+b x) (d+e x)^{12}}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{13 e^7 (a+b x) (d+e x)^{13}}-\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^7}+\frac {3 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{4 e^7 (a+b x) (d+e x)^8}-\frac {5 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{3 e^7 (a+b x) (d+e x)^9}+\frac {2 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) (d+e x)^{10}} \]
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Rubi [A] time = 0.19, 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 = {770, 21, 43} \[ -\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^7}+\frac {3 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{4 e^7 (a+b x) (d+e x)^8}-\frac {5 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{3 e^7 (a+b x) (d+e x)^9}+\frac {2 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) (d+e x)^{10}}-\frac {15 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{11 e^7 (a+b x) (d+e x)^{11}}+\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{2 e^7 (a+b x) (d+e x)^{12}}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{13 e^7 (a+b x) (d+e x)^{13}} \]
Antiderivative was successfully verified.
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Rule 21
Rule 43
Rule 770
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 = 0.12, size = 295, normalized size = 0.82 \[ -\frac {\sqrt {(a+b x)^2} \left (924 a^6 e^6+462 a^5 b e^5 (d+13 e x)+210 a^4 b^2 e^4 \left (d^2+13 d e x+78 e^2 x^2\right )+84 a^3 b^3 e^3 \left (d^3+13 d^2 e x+78 d e^2 x^2+286 e^3 x^3\right )+28 a^2 b^4 e^2 \left (d^4+13 d^3 e x+78 d^2 e^2 x^2+286 d e^3 x^3+715 e^4 x^4\right )+7 a b^5 e \left (d^5+13 d^4 e x+78 d^3 e^2 x^2+286 d^2 e^3 x^3+715 d e^4 x^4+1287 e^5 x^5\right )+b^6 \left (d^6+13 d^5 e x+78 d^4 e^2 x^2+286 d^3 e^3 x^3+715 d^2 e^4 x^4+1287 d e^5 x^5+1716 e^6 x^6\right )\right )}{12012 e^7 (a+b x) (d+e x)^{13}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 485, normalized size = 1.35 \[ -\frac {1716 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 7 \, a b^{5} d^{5} e + 28 \, a^{2} b^{4} d^{4} e^{2} + 84 \, a^{3} b^{3} d^{3} e^{3} + 210 \, a^{4} b^{2} d^{2} e^{4} + 462 \, a^{5} b d e^{5} + 924 \, a^{6} e^{6} + 1287 \, {\left (b^{6} d e^{5} + 7 \, a b^{5} e^{6}\right )} x^{5} + 715 \, {\left (b^{6} d^{2} e^{4} + 7 \, a b^{5} d e^{5} + 28 \, a^{2} b^{4} e^{6}\right )} x^{4} + 286 \, {\left (b^{6} d^{3} e^{3} + 7 \, a b^{5} d^{2} e^{4} + 28 \, a^{2} b^{4} d e^{5} + 84 \, a^{3} b^{3} e^{6}\right )} x^{3} + 78 \, {\left (b^{6} d^{4} e^{2} + 7 \, a b^{5} d^{3} e^{3} + 28 \, a^{2} b^{4} d^{2} e^{4} + 84 \, a^{3} b^{3} d e^{5} + 210 \, a^{4} b^{2} e^{6}\right )} x^{2} + 13 \, {\left (b^{6} d^{5} e + 7 \, a b^{5} d^{4} e^{2} + 28 \, a^{2} b^{4} d^{3} e^{3} + 84 \, a^{3} b^{3} d^{2} e^{4} + 210 \, a^{4} b^{2} d e^{5} + 462 \, a^{5} b e^{6}\right )} x}{12012 \, {\left (e^{20} x^{13} + 13 \, d e^{19} x^{12} + 78 \, d^{2} e^{18} x^{11} + 286 \, d^{3} e^{17} x^{10} + 715 \, d^{4} e^{16} x^{9} + 1287 \, d^{5} e^{15} x^{8} + 1716 \, d^{6} e^{14} x^{7} + 1716 \, d^{7} e^{13} x^{6} + 1287 \, d^{8} e^{12} x^{5} + 715 \, d^{9} e^{11} x^{4} + 286 \, d^{10} e^{10} x^{3} + 78 \, d^{11} e^{9} x^{2} + 13 \, d^{12} e^{8} x + d^{13} e^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 520, normalized size = 1.44 \[ -\frac {{\left (1716 \, b^{6} x^{6} e^{6} \mathrm {sgn}\left (b x + a\right ) + 1287 \, b^{6} d x^{5} e^{5} \mathrm {sgn}\left (b x + a\right ) + 715 \, b^{6} d^{2} x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 286 \, b^{6} d^{3} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 78 \, b^{6} d^{4} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 13 \, b^{6} d^{5} x e \mathrm {sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) + 9009 \, a b^{5} x^{5} e^{6} \mathrm {sgn}\left (b x + a\right ) + 5005 \, a b^{5} d x^{4} e^{5} \mathrm {sgn}\left (b x + a\right ) + 2002 \, a b^{5} d^{2} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 546 \, a b^{5} d^{3} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 91 \, a b^{5} d^{4} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 7 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 20020 \, a^{2} b^{4} x^{4} e^{6} \mathrm {sgn}\left (b x + a\right ) + 8008 \, a^{2} b^{4} d x^{3} e^{5} \mathrm {sgn}\left (b x + a\right ) + 2184 \, a^{2} b^{4} d^{2} x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 364 \, a^{2} b^{4} d^{3} x e^{3} \mathrm {sgn}\left (b x + a\right ) + 28 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 24024 \, a^{3} b^{3} x^{3} e^{6} \mathrm {sgn}\left (b x + a\right ) + 6552 \, a^{3} b^{3} d x^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 1092 \, a^{3} b^{3} d^{2} x e^{4} \mathrm {sgn}\left (b x + a\right ) + 84 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 16380 \, a^{4} b^{2} x^{2} e^{6} \mathrm {sgn}\left (b x + a\right ) + 2730 \, a^{4} b^{2} d x e^{5} \mathrm {sgn}\left (b x + a\right ) + 210 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 6006 \, a^{5} b x e^{6} \mathrm {sgn}\left (b x + a\right ) + 462 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + 924 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{12012 \, {\left (x e + d\right )}^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 392, normalized size = 1.09 \[ -\frac {\left (1716 b^{6} e^{6} x^{6}+9009 a \,b^{5} e^{6} x^{5}+1287 b^{6} d \,e^{5} x^{5}+20020 a^{2} b^{4} e^{6} x^{4}+5005 a \,b^{5} d \,e^{5} x^{4}+715 b^{6} d^{2} e^{4} x^{4}+24024 a^{3} b^{3} e^{6} x^{3}+8008 a^{2} b^{4} d \,e^{5} x^{3}+2002 a \,b^{5} d^{2} e^{4} x^{3}+286 b^{6} d^{3} e^{3} x^{3}+16380 a^{4} b^{2} e^{6} x^{2}+6552 a^{3} b^{3} d \,e^{5} x^{2}+2184 a^{2} b^{4} d^{2} e^{4} x^{2}+546 a \,b^{5} d^{3} e^{3} x^{2}+78 b^{6} d^{4} e^{2} x^{2}+6006 a^{5} b \,e^{6} x +2730 a^{4} b^{2} d \,e^{5} x +1092 a^{3} b^{3} d^{2} e^{4} x +364 a^{2} b^{4} d^{3} e^{3} x +91 a \,b^{5} d^{4} e^{2} x +13 b^{6} d^{5} e x +924 a^{6} e^{6}+462 a^{5} b d \,e^{5}+210 a^{4} b^{2} d^{2} e^{4}+84 a^{3} b^{3} d^{3} e^{3}+28 a^{2} b^{4} d^{4} e^{2}+7 a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{12012 \left (e x +d \right )^{13} \left (b x +a \right )^{5} e^{7}} \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ \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} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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