Optimal. Leaf size=316 \[ -\frac {4 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}{e^6 (a+b x)}+\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}{3 e^6 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^5}{e^6 (a+b x)}+\frac {2 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^6 (a+b x)}-\frac {10 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)}{9 e^6 (a+b x)}+\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2}{7 e^6 (a+b x)} \]
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Rubi [A] time = 0.10, antiderivative size = 316, 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 = {646, 43} \[ \frac {2 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^6 (a+b x)}-\frac {10 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)}{9 e^6 (a+b x)}+\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2}{7 e^6 (a+b x)}-\frac {4 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}{e^6 (a+b x)}+\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}{3 e^6 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^5}{e^6 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5}{\sqrt {d+e x}} \, 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 \sqrt {d+e x}}+\frac {5 b^6 (b d-a e)^4 \sqrt {d+e x}}{e^5}-\frac {10 b^7 (b d-a e)^3 (d+e x)^{3/2}}{e^5}+\frac {10 b^8 (b d-a e)^2 (d+e x)^{5/2}}{e^5}-\frac {5 b^9 (b d-a e) (d+e x)^{7/2}}{e^5}+\frac {b^{10} (d+e x)^{9/2}}{e^5}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac {2 (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}+\frac {10 b (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x)}-\frac {4 b^2 (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}+\frac {20 b^3 (b d-a e)^2 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x)}-\frac {10 b^4 (b d-a e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^6 (a+b x)}+\frac {2 b^5 (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^6 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 234, normalized size = 0.74 \[ \frac {2 \sqrt {(a+b x)^2} \sqrt {d+e x} \left (693 a^5 e^5+1155 a^4 b e^4 (e x-2 d)+462 a^3 b^2 e^3 \left (8 d^2-4 d e x+3 e^2 x^2\right )+198 a^2 b^3 e^2 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+11 a b^4 e \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )+b^5 \left (-256 d^5+128 d^4 e x-96 d^3 e^2 x^2+80 d^2 e^3 x^3-70 d e^4 x^4+63 e^5 x^5\right )\right )}{693 e^6 (a+b x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 261, normalized size = 0.83 \[ \frac {2 \, {\left (63 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 1408 \, a b^{4} d^{4} e - 3168 \, a^{2} b^{3} d^{3} e^{2} + 3696 \, a^{3} b^{2} d^{2} e^{3} - 2310 \, a^{4} b d e^{4} + 693 \, a^{5} e^{5} - 35 \, {\left (2 \, b^{5} d e^{4} - 11 \, a b^{4} e^{5}\right )} x^{4} + 10 \, {\left (8 \, b^{5} d^{2} e^{3} - 44 \, a b^{4} d e^{4} + 99 \, a^{2} b^{3} e^{5}\right )} x^{3} - 6 \, {\left (16 \, b^{5} d^{3} e^{2} - 88 \, a b^{4} d^{2} e^{3} + 198 \, a^{2} b^{3} d e^{4} - 231 \, a^{3} b^{2} e^{5}\right )} x^{2} + {\left (128 \, b^{5} d^{4} e - 704 \, a b^{4} d^{3} e^{2} + 1584 \, a^{2} b^{3} d^{2} e^{3} - 1848 \, a^{3} b^{2} d e^{4} + 1155 \, a^{4} b e^{5}\right )} x\right )} \sqrt {e x + d}}{693 \, e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 334, normalized size = 1.06 \[ \frac {2}{693} \, {\left (1155 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{4} b e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) + 462 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a^{3} b^{2} e^{\left (-2\right )} \mathrm {sgn}\left (b x + a\right ) + 198 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a^{2} b^{3} e^{\left (-3\right )} \mathrm {sgn}\left (b x + a\right ) + 11 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} a b^{4} e^{\left (-4\right )} \mathrm {sgn}\left (b x + a\right ) + {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} b^{5} e^{\left (-5\right )} \mathrm {sgn}\left (b x + a\right ) + 693 \, \sqrt {x e + d} a^{5} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 289, normalized size = 0.91 \[ \frac {2 \sqrt {e x +d}\, \left (63 b^{5} e^{5} x^{5}+385 a \,b^{4} e^{5} x^{4}-70 b^{5} d \,e^{4} x^{4}+990 a^{2} b^{3} e^{5} x^{3}-440 a \,b^{4} d \,e^{4} x^{3}+80 b^{5} d^{2} e^{3} x^{3}+1386 a^{3} b^{2} e^{5} x^{2}-1188 a^{2} b^{3} d \,e^{4} x^{2}+528 a \,b^{4} d^{2} e^{3} x^{2}-96 b^{5} d^{3} e^{2} x^{2}+1155 a^{4} b \,e^{5} x -1848 a^{3} b^{2} d \,e^{4} x +1584 a^{2} b^{3} d^{2} e^{3} x -704 a \,b^{4} d^{3} e^{2} x +128 b^{5} d^{4} e x +693 a^{5} e^{5}-2310 a^{4} b d \,e^{4}+3696 a^{3} b^{2} d^{2} e^{3}-3168 a^{2} b^{3} d^{3} e^{2}+1408 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{693 \left (b x +a \right )^{5} e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 338, normalized size = 1.07 \[ \frac {2 \, {\left (63 \, b^{5} e^{6} x^{6} - 256 \, b^{5} d^{6} + 1408 \, a b^{4} d^{5} e - 3168 \, a^{2} b^{3} d^{4} e^{2} + 3696 \, a^{3} b^{2} d^{3} e^{3} - 2310 \, a^{4} b d^{2} e^{4} + 693 \, a^{5} d e^{5} - 7 \, {\left (b^{5} d e^{5} - 55 \, a b^{4} e^{6}\right )} x^{5} + 5 \, {\left (2 \, b^{5} d^{2} e^{4} - 11 \, a b^{4} d e^{5} + 198 \, a^{2} b^{3} e^{6}\right )} x^{4} - 2 \, {\left (8 \, b^{5} d^{3} e^{3} - 44 \, a b^{4} d^{2} e^{4} + 99 \, a^{2} b^{3} d e^{5} - 693 \, a^{3} b^{2} e^{6}\right )} x^{3} + {\left (32 \, b^{5} d^{4} e^{2} - 176 \, a b^{4} d^{3} e^{3} + 396 \, a^{2} b^{3} d^{2} e^{4} - 462 \, a^{3} b^{2} d e^{5} + 1155 \, a^{4} b e^{6}\right )} x^{2} - {\left (128 \, b^{5} d^{5} e - 704 \, a b^{4} d^{4} e^{2} + 1584 \, a^{2} b^{3} d^{3} e^{3} - 1848 \, a^{3} b^{2} d^{2} e^{4} + 1155 \, a^{4} b d e^{5} - 693 \, a^{5} e^{6}\right )} x\right )}}{693 \, \sqrt {e x + d} e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 375, normalized size = 1.19 \[ \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {2\,b^4\,x^6}{11}-\frac {-1386\,a^5\,d\,e^5+4620\,a^4\,b\,d^2\,e^4-7392\,a^3\,b^2\,d^3\,e^3+6336\,a^2\,b^3\,d^4\,e^2-2816\,a\,b^4\,d^5\,e+512\,b^5\,d^6}{693\,b\,e^6}+\frac {2\,b^3\,x^5\,\left (55\,a\,e-b\,d\right )}{99\,e}+\frac {x\,\left (1386\,a^5\,e^6-2310\,a^4\,b\,d\,e^5+3696\,a^3\,b^2\,d^2\,e^4-3168\,a^2\,b^3\,d^3\,e^3+1408\,a\,b^4\,d^4\,e^2-256\,b^5\,d^5\,e\right )}{693\,b\,e^6}+\frac {x^2\,\left (2310\,a^4\,b\,e^6-924\,a^3\,b^2\,d\,e^5+792\,a^2\,b^3\,d^2\,e^4-352\,a\,b^4\,d^3\,e^3+64\,b^5\,d^4\,e^2\right )}{693\,b\,e^6}+\frac {10\,b^2\,x^4\,\left (198\,a^2\,e^2-11\,a\,b\,d\,e+2\,b^2\,d^2\right )}{693\,e^2}+\frac {4\,b\,x^3\,\left (693\,a^3\,e^3-99\,a^2\,b\,d\,e^2+44\,a\,b^2\,d^2\,e-8\,b^3\,d^3\right )}{693\,e^3}\right )}{x\,\sqrt {d+e\,x}+\frac {a\,\sqrt {d+e\,x}}{b}} \]
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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