Optimal. Leaf size=318 \[ \frac {2 (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^6 (a+b x) (d+e x)^{13/2}}-\frac {10 b (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^6 (a+b x) (d+e x)^{11/2}}+\frac {20 b^2 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^6 (a+b x) (d+e x)^{9/2}}-\frac {20 b^3 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x) (d+e x)^{7/2}}+\frac {2 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) (d+e x)^{5/2}}-\frac {2 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x) (d+e x)^{3/2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 318, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {660, 45}
\begin {gather*} \frac {20 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)^{9/2}}-\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{11 e^6 (a+b x) (d+e x)^{11/2}}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{13 e^6 (a+b x) (d+e x)^{13/2}}-\frac {2 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x) (d+e x)^{3/2}}+\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^6 (a+b x) (d+e x)^{5/2}}-\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^6 (a+b x) (d+e x)^{7/2}} \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)^{15/2}} \, 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)^{15/2}} \, 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)^{15/2}}+\frac {5 b^6 (b d-a e)^4}{e^5 (d+e x)^{13/2}}-\frac {10 b^7 (b d-a e)^3}{e^5 (d+e x)^{11/2}}+\frac {10 b^8 (b d-a e)^2}{e^5 (d+e x)^{9/2}}-\frac {5 b^9 (b d-a e)}{e^5 (d+e x)^{7/2}}+\frac {b^{10}}{e^5 (d+e x)^{5/2}}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {2 (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^6 (a+b x) (d+e x)^{13/2}}-\frac {10 b (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^6 (a+b x) (d+e x)^{11/2}}+\frac {20 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)^{9/2}}-\frac {20 b^3 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x) (d+e x)^{7/2}}+\frac {2 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) (d+e x)^{5/2}}-\frac {2 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x) (d+e x)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 235, normalized size = 0.74 \begin {gather*} -\frac {2 \sqrt {(a+b x)^2} \left (693 a^5 e^5+315 a^4 b e^4 (2 d+13 e x)+70 a^3 b^2 e^3 \left (8 d^2+52 d e x+143 e^2 x^2\right )+30 a^2 b^3 e^2 \left (16 d^3+104 d^2 e x+286 d e^2 x^2+429 e^3 x^3\right )+3 a b^4 e \left (128 d^4+832 d^3 e x+2288 d^2 e^2 x^2+3432 d e^3 x^3+3003 e^4 x^4\right )+b^5 \left (256 d^5+1664 d^4 e x+4576 d^3 e^2 x^2+6864 d^2 e^3 x^3+6006 d e^4 x^4+3003 e^5 x^5\right )\right )}{9009 e^6 (a+b x) (d+e x)^{13/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.68, size = 289, normalized size = 0.91
method | result | size |
gosper | \(-\frac {2 \left (3003 b^{5} e^{5} x^{5}+9009 a \,b^{4} e^{5} x^{4}+6006 b^{5} d \,e^{4} x^{4}+12870 a^{2} b^{3} e^{5} x^{3}+10296 a \,b^{4} d \,e^{4} x^{3}+6864 b^{5} d^{2} e^{3} x^{3}+10010 a^{3} b^{2} e^{5} x^{2}+8580 a^{2} b^{3} d \,e^{4} x^{2}+6864 a \,b^{4} d^{2} e^{3} x^{2}+4576 b^{5} d^{3} e^{2} x^{2}+4095 a^{4} b \,e^{5} x +3640 a^{3} b^{2} d \,e^{4} x +3120 a^{2} b^{3} d^{2} e^{3} x +2496 a \,b^{4} d^{3} e^{2} x +1664 b^{5} d^{4} e x +693 a^{5} e^{5}+630 a^{4} b d \,e^{4}+560 a^{3} b^{2} d^{2} e^{3}+480 a^{2} b^{3} d^{3} e^{2}+384 a \,b^{4} d^{4} e +256 b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{9009 \left (e x +d \right )^{\frac {13}{2}} e^{6} \left (b x +a \right )^{5}}\) | \(289\) |
default | \(-\frac {2 \left (3003 b^{5} e^{5} x^{5}+9009 a \,b^{4} e^{5} x^{4}+6006 b^{5} d \,e^{4} x^{4}+12870 a^{2} b^{3} e^{5} x^{3}+10296 a \,b^{4} d \,e^{4} x^{3}+6864 b^{5} d^{2} e^{3} x^{3}+10010 a^{3} b^{2} e^{5} x^{2}+8580 a^{2} b^{3} d \,e^{4} x^{2}+6864 a \,b^{4} d^{2} e^{3} x^{2}+4576 b^{5} d^{3} e^{2} x^{2}+4095 a^{4} b \,e^{5} x +3640 a^{3} b^{2} d \,e^{4} x +3120 a^{2} b^{3} d^{2} e^{3} x +2496 a \,b^{4} d^{3} e^{2} x +1664 b^{5} d^{4} e x +693 a^{5} e^{5}+630 a^{4} b d \,e^{4}+560 a^{3} b^{2} d^{2} e^{3}+480 a^{2} b^{3} d^{3} e^{2}+384 a \,b^{4} d^{4} e +256 b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{9009 \left (e x +d \right )^{\frac {13}{2}} e^{6} \left (b x +a \right )^{5}}\) | \(289\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 305, normalized size = 0.96 \begin {gather*} -\frac {2 \, {\left (3003 \, b^{5} x^{5} e^{5} + 256 \, b^{5} d^{5} + 384 \, a b^{4} d^{4} e + 480 \, a^{2} b^{3} d^{3} e^{2} + 560 \, a^{3} b^{2} d^{2} e^{3} + 630 \, a^{4} b d e^{4} + 693 \, a^{5} e^{5} + 3003 \, {\left (2 \, b^{5} d e^{4} + 3 \, a b^{4} e^{5}\right )} x^{4} + 858 \, {\left (8 \, b^{5} d^{2} e^{3} + 12 \, a b^{4} d e^{4} + 15 \, a^{2} b^{3} e^{5}\right )} x^{3} + 286 \, {\left (16 \, b^{5} d^{3} e^{2} + 24 \, a b^{4} d^{2} e^{3} + 30 \, a^{2} b^{3} d e^{4} + 35 \, a^{3} b^{2} e^{5}\right )} x^{2} + 13 \, {\left (128 \, b^{5} d^{4} e + 192 \, a b^{4} d^{3} e^{2} + 240 \, a^{2} b^{3} d^{2} e^{3} + 280 \, a^{3} b^{2} d e^{4} + 315 \, a^{4} b e^{5}\right )} x\right )}}{9009 \, {\left (x^{6} e^{12} + 6 \, d x^{5} e^{11} + 15 \, d^{2} x^{4} e^{10} + 20 \, d^{3} x^{3} e^{9} + 15 \, d^{4} x^{2} e^{8} + 6 \, d^{5} x e^{7} + d^{6} e^{6}\right )} \sqrt {x e + d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.77, size = 309, normalized size = 0.97 \begin {gather*} -\frac {2 \, {\left (256 \, b^{5} d^{5} + {\left (3003 \, b^{5} x^{5} + 9009 \, a b^{4} x^{4} + 12870 \, a^{2} b^{3} x^{3} + 10010 \, a^{3} b^{2} x^{2} + 4095 \, a^{4} b x + 693 \, a^{5}\right )} e^{5} + 2 \, {\left (3003 \, b^{5} d x^{4} + 5148 \, a b^{4} d x^{3} + 4290 \, a^{2} b^{3} d x^{2} + 1820 \, a^{3} b^{2} d x + 315 \, a^{4} b d\right )} e^{4} + 16 \, {\left (429 \, b^{5} d^{2} x^{3} + 429 \, a b^{4} d^{2} x^{2} + 195 \, a^{2} b^{3} d^{2} x + 35 \, a^{3} b^{2} d^{2}\right )} e^{3} + 32 \, {\left (143 \, b^{5} d^{3} x^{2} + 78 \, a b^{4} d^{3} x + 15 \, a^{2} b^{3} d^{3}\right )} e^{2} + 128 \, {\left (13 \, b^{5} d^{4} x + 3 \, a b^{4} d^{4}\right )} e\right )} \sqrt {x e + d}}{9009 \, {\left (x^{7} e^{13} + 7 \, d x^{6} e^{12} + 21 \, d^{2} x^{5} e^{11} + 35 \, d^{3} x^{4} e^{10} + 35 \, d^{4} x^{3} e^{9} + 21 \, d^{5} x^{2} e^{8} + 7 \, d^{6} x e^{7} + d^{7} 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: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.86, size = 447, normalized size = 1.41 \begin {gather*} -\frac {2 \, {\left (3003 \, {\left (x e + d\right )}^{5} b^{5} \mathrm {sgn}\left (b x + a\right ) - 9009 \, {\left (x e + d\right )}^{4} b^{5} d \mathrm {sgn}\left (b x + a\right ) + 12870 \, {\left (x e + d\right )}^{3} b^{5} d^{2} \mathrm {sgn}\left (b x + a\right ) - 10010 \, {\left (x e + d\right )}^{2} b^{5} d^{3} \mathrm {sgn}\left (b x + a\right ) + 4095 \, {\left (x e + d\right )} b^{5} d^{4} \mathrm {sgn}\left (b x + a\right ) - 693 \, b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) + 9009 \, {\left (x e + d\right )}^{4} a b^{4} e \mathrm {sgn}\left (b x + a\right ) - 25740 \, {\left (x e + d\right )}^{3} a b^{4} d e \mathrm {sgn}\left (b x + a\right ) + 30030 \, {\left (x e + d\right )}^{2} a b^{4} d^{2} e \mathrm {sgn}\left (b x + a\right ) - 16380 \, {\left (x e + d\right )} a b^{4} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 3465 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 12870 \, {\left (x e + d\right )}^{3} a^{2} b^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 30030 \, {\left (x e + d\right )}^{2} a^{2} b^{3} d e^{2} \mathrm {sgn}\left (b x + a\right ) + 24570 \, {\left (x e + d\right )} a^{2} b^{3} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 6930 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 10010 \, {\left (x e + d\right )}^{2} a^{3} b^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 16380 \, {\left (x e + d\right )} a^{3} b^{2} d e^{3} \mathrm {sgn}\left (b x + a\right ) + 6930 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 4095 \, {\left (x e + d\right )} a^{4} b e^{4} \mathrm {sgn}\left (b x + a\right ) - 3465 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) + 693 \, a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-6\right )}}{9009 \, {\left (x e + d\right )}^{\frac {13}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.77, size = 472, normalized size = 1.48 \begin {gather*} -\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {1386\,a^5\,e^5+1260\,a^4\,b\,d\,e^4+1120\,a^3\,b^2\,d^2\,e^3+960\,a^2\,b^3\,d^3\,e^2+768\,a\,b^4\,d^4\,e+512\,b^5\,d^5}{9009\,b\,e^{12}}+\frac {2\,b^4\,x^5}{3\,e^7}+\frac {2\,b^3\,x^4\,\left (3\,a\,e+2\,b\,d\right )}{3\,e^8}+\frac {x\,\left (8190\,a^4\,b\,e^5+7280\,a^3\,b^2\,d\,e^4+6240\,a^2\,b^3\,d^2\,e^3+4992\,a\,b^4\,d^3\,e^2+3328\,b^5\,d^4\,e\right )}{9009\,b\,e^{12}}+\frac {4\,b^2\,x^3\,\left (15\,a^2\,e^2+12\,a\,b\,d\,e+8\,b^2\,d^2\right )}{21\,e^9}+\frac {4\,b\,x^2\,\left (35\,a^3\,e^3+30\,a^2\,b\,d\,e^2+24\,a\,b^2\,d^2\,e+16\,b^3\,d^3\right )}{63\,e^{10}}\right )}{x^7\,\sqrt {d+e\,x}+\frac {a\,d^6\,\sqrt {d+e\,x}}{b\,e^6}+\frac {x^6\,\left (a\,e+6\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e}+\frac {3\,d\,x^5\,\left (2\,a\,e+5\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^2}+\frac {d^5\,x\,\left (6\,a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^6}+\frac {5\,d^2\,x^4\,\left (3\,a\,e+4\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^3}+\frac {5\,d^3\,x^3\,\left (4\,a\,e+3\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^4}+\frac {3\,d^4\,x^2\,\left (5\,a\,e+2\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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