Optimal. Leaf size=208 \[ -\frac {14 b^6 (d+e x)^{9/2} (b d-a e)}{9 e^8}+\frac {6 b^5 (d+e x)^{7/2} (b d-a e)^2}{e^8}-\frac {14 b^4 (d+e x)^{5/2} (b d-a e)^3}{e^8}+\frac {70 b^3 (d+e x)^{3/2} (b d-a e)^4}{3 e^8}-\frac {42 b^2 \sqrt {d+e x} (b d-a e)^5}{e^8}-\frac {14 b (b d-a e)^6}{e^8 \sqrt {d+e x}}+\frac {2 (b d-a e)^7}{3 e^8 (d+e x)^{3/2}}+\frac {2 b^7 (d+e x)^{11/2}}{11 e^8} \]
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Rubi [A] time = 0.08, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {27, 43} \begin {gather*} -\frac {14 b^6 (d+e x)^{9/2} (b d-a e)}{9 e^8}+\frac {6 b^5 (d+e x)^{7/2} (b d-a e)^2}{e^8}-\frac {14 b^4 (d+e x)^{5/2} (b d-a e)^3}{e^8}+\frac {70 b^3 (d+e x)^{3/2} (b d-a e)^4}{3 e^8}-\frac {42 b^2 \sqrt {d+e x} (b d-a e)^5}{e^8}-\frac {14 b (b d-a e)^6}{e^8 \sqrt {d+e x}}+\frac {2 (b d-a e)^7}{3 e^8 (d+e x)^{3/2}}+\frac {2 b^7 (d+e x)^{11/2}}{11 e^8} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{5/2}} \, dx &=\int \frac {(a+b x)^7}{(d+e x)^{5/2}} \, dx\\ &=\int \left (\frac {(-b d+a e)^7}{e^7 (d+e x)^{5/2}}+\frac {7 b (b d-a e)^6}{e^7 (d+e x)^{3/2}}-\frac {21 b^2 (b d-a e)^5}{e^7 \sqrt {d+e x}}+\frac {35 b^3 (b d-a e)^4 \sqrt {d+e x}}{e^7}-\frac {35 b^4 (b d-a e)^3 (d+e x)^{3/2}}{e^7}+\frac {21 b^5 (b d-a e)^2 (d+e x)^{5/2}}{e^7}-\frac {7 b^6 (b d-a e) (d+e x)^{7/2}}{e^7}+\frac {b^7 (d+e x)^{9/2}}{e^7}\right ) \, dx\\ &=\frac {2 (b d-a e)^7}{3 e^8 (d+e x)^{3/2}}-\frac {14 b (b d-a e)^6}{e^8 \sqrt {d+e x}}-\frac {42 b^2 (b d-a e)^5 \sqrt {d+e x}}{e^8}+\frac {70 b^3 (b d-a e)^4 (d+e x)^{3/2}}{3 e^8}-\frac {14 b^4 (b d-a e)^3 (d+e x)^{5/2}}{e^8}+\frac {6 b^5 (b d-a e)^2 (d+e x)^{7/2}}{e^8}-\frac {14 b^6 (b d-a e) (d+e x)^{9/2}}{9 e^8}+\frac {2 b^7 (d+e x)^{11/2}}{11 e^8}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 167, normalized size = 0.80 \begin {gather*} \frac {2 \left (-77 b^6 (d+e x)^6 (b d-a e)+297 b^5 (d+e x)^5 (b d-a e)^2-693 b^4 (d+e x)^4 (b d-a e)^3+1155 b^3 (d+e x)^3 (b d-a e)^4-2079 b^2 (d+e x)^2 (b d-a e)^5-693 b (d+e x) (b d-a e)^6+33 (b d-a e)^7+9 b^7 (d+e x)^7\right )}{99 e^8 (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.12, size = 582, normalized size = 2.80 \begin {gather*} \frac {2 \left (-33 a^7 e^7-693 a^6 b e^6 (d+e x)+231 a^6 b d e^6-693 a^5 b^2 d^2 e^5+2079 a^5 b^2 e^5 (d+e x)^2+4158 a^5 b^2 d e^5 (d+e x)+1155 a^4 b^3 d^3 e^4-10395 a^4 b^3 d^2 e^4 (d+e x)+1155 a^4 b^3 e^4 (d+e x)^3-10395 a^4 b^3 d e^4 (d+e x)^2-1155 a^3 b^4 d^4 e^3+13860 a^3 b^4 d^3 e^3 (d+e x)+20790 a^3 b^4 d^2 e^3 (d+e x)^2+693 a^3 b^4 e^3 (d+e x)^4-4620 a^3 b^4 d e^3 (d+e x)^3+693 a^2 b^5 d^5 e^2-10395 a^2 b^5 d^4 e^2 (d+e x)-20790 a^2 b^5 d^3 e^2 (d+e x)^2+6930 a^2 b^5 d^2 e^2 (d+e x)^3+297 a^2 b^5 e^2 (d+e x)^5-2079 a^2 b^5 d e^2 (d+e x)^4-231 a b^6 d^6 e+4158 a b^6 d^5 e (d+e x)+10395 a b^6 d^4 e (d+e x)^2-4620 a b^6 d^3 e (d+e x)^3+2079 a b^6 d^2 e (d+e x)^4+77 a b^6 e (d+e x)^6-594 a b^6 d e (d+e x)^5+33 b^7 d^7-693 b^7 d^6 (d+e x)-2079 b^7 d^5 (d+e x)^2+1155 b^7 d^4 (d+e x)^3-693 b^7 d^3 (d+e x)^4+297 b^7 d^2 (d+e x)^5+9 b^7 (d+e x)^7-77 b^7 d (d+e x)^6\right )}{99 e^8 (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 484, normalized size = 2.33 \begin {gather*} \frac {2 \, {\left (9 \, b^{7} e^{7} x^{7} - 2048 \, b^{7} d^{7} + 11264 \, a b^{6} d^{6} e - 25344 \, a^{2} b^{5} d^{5} e^{2} + 29568 \, a^{3} b^{4} d^{4} e^{3} - 18480 \, a^{4} b^{3} d^{3} e^{4} + 5544 \, a^{5} b^{2} d^{2} e^{5} - 462 \, a^{6} b d e^{6} - 33 \, a^{7} e^{7} - 7 \, {\left (2 \, b^{7} d e^{6} - 11 \, a b^{6} e^{7}\right )} x^{6} + 3 \, {\left (8 \, b^{7} d^{2} e^{5} - 44 \, a b^{6} d e^{6} + 99 \, a^{2} b^{5} e^{7}\right )} x^{5} - 3 \, {\left (16 \, b^{7} d^{3} e^{4} - 88 \, a b^{6} d^{2} e^{5} + 198 \, a^{2} b^{5} d e^{6} - 231 \, a^{3} b^{4} e^{7}\right )} x^{4} + {\left (128 \, b^{7} d^{4} e^{3} - 704 \, a b^{6} d^{3} e^{4} + 1584 \, a^{2} b^{5} d^{2} e^{5} - 1848 \, a^{3} b^{4} d e^{6} + 1155 \, a^{4} b^{3} e^{7}\right )} x^{3} - 3 \, {\left (256 \, b^{7} d^{5} e^{2} - 1408 \, a b^{6} d^{4} e^{3} + 3168 \, a^{2} b^{5} d^{3} e^{4} - 3696 \, a^{3} b^{4} d^{2} e^{5} + 2310 \, a^{4} b^{3} d e^{6} - 693 \, a^{5} b^{2} e^{7}\right )} x^{2} - 3 \, {\left (1024 \, b^{7} d^{6} e - 5632 \, a b^{6} d^{5} e^{2} + 12672 \, a^{2} b^{5} d^{4} e^{3} - 14784 \, a^{3} b^{4} d^{3} e^{4} + 9240 \, a^{4} b^{3} d^{2} e^{5} - 2772 \, a^{5} b^{2} d e^{6} + 231 \, a^{6} b e^{7}\right )} x\right )} \sqrt {e x + d}}{99 \, {\left (e^{10} x^{2} + 2 \, d e^{9} x + d^{2} e^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.34, size = 609, normalized size = 2.93 \begin {gather*} \frac {2}{99} \, {\left (9 \, {\left (x e + d\right )}^{\frac {11}{2}} b^{7} e^{80} - 77 \, {\left (x e + d\right )}^{\frac {9}{2}} b^{7} d e^{80} + 297 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{7} d^{2} e^{80} - 693 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{7} d^{3} e^{80} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{7} d^{4} e^{80} - 2079 \, \sqrt {x e + d} b^{7} d^{5} e^{80} + 77 \, {\left (x e + d\right )}^{\frac {9}{2}} a b^{6} e^{81} - 594 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{6} d e^{81} + 2079 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{6} d^{2} e^{81} - 4620 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{6} d^{3} e^{81} + 10395 \, \sqrt {x e + d} a b^{6} d^{4} e^{81} + 297 \, {\left (x e + d\right )}^{\frac {7}{2}} a^{2} b^{5} e^{82} - 2079 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{5} d e^{82} + 6930 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{5} d^{2} e^{82} - 20790 \, \sqrt {x e + d} a^{2} b^{5} d^{3} e^{82} + 693 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{3} b^{4} e^{83} - 4620 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b^{4} d e^{83} + 20790 \, \sqrt {x e + d} a^{3} b^{4} d^{2} e^{83} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{4} b^{3} e^{84} - 10395 \, \sqrt {x e + d} a^{4} b^{3} d e^{84} + 2079 \, \sqrt {x e + d} a^{5} b^{2} e^{85}\right )} e^{\left (-88\right )} - \frac {2 \, {\left (21 \, {\left (x e + d\right )} b^{7} d^{6} - b^{7} d^{7} - 126 \, {\left (x e + d\right )} a b^{6} d^{5} e + 7 \, a b^{6} d^{6} e + 315 \, {\left (x e + d\right )} a^{2} b^{5} d^{4} e^{2} - 21 \, a^{2} b^{5} d^{5} e^{2} - 420 \, {\left (x e + d\right )} a^{3} b^{4} d^{3} e^{3} + 35 \, a^{3} b^{4} d^{4} e^{3} + 315 \, {\left (x e + d\right )} a^{4} b^{3} d^{2} e^{4} - 35 \, a^{4} b^{3} d^{3} e^{4} - 126 \, {\left (x e + d\right )} a^{5} b^{2} d e^{5} + 21 \, a^{5} b^{2} d^{2} e^{5} + 21 \, {\left (x e + d\right )} a^{6} b e^{6} - 7 \, a^{6} b d e^{6} + a^{7} e^{7}\right )} e^{\left (-8\right )}}{3 \, {\left (x e + d\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 498, normalized size = 2.39 \begin {gather*} -\frac {2 \left (-9 b^{7} x^{7} e^{7}-77 a \,b^{6} e^{7} x^{6}+14 b^{7} d \,e^{6} x^{6}-297 a^{2} b^{5} e^{7} x^{5}+132 a \,b^{6} d \,e^{6} x^{5}-24 b^{7} d^{2} e^{5} x^{5}-693 a^{3} b^{4} e^{7} x^{4}+594 a^{2} b^{5} d \,e^{6} x^{4}-264 a \,b^{6} d^{2} e^{5} x^{4}+48 b^{7} d^{3} e^{4} x^{4}-1155 a^{4} b^{3} e^{7} x^{3}+1848 a^{3} b^{4} d \,e^{6} x^{3}-1584 a^{2} b^{5} d^{2} e^{5} x^{3}+704 a \,b^{6} d^{3} e^{4} x^{3}-128 b^{7} d^{4} e^{3} x^{3}-2079 a^{5} b^{2} e^{7} x^{2}+6930 a^{4} b^{3} d \,e^{6} x^{2}-11088 a^{3} b^{4} d^{2} e^{5} x^{2}+9504 a^{2} b^{5} d^{3} e^{4} x^{2}-4224 a \,b^{6} d^{4} e^{3} x^{2}+768 b^{7} d^{5} e^{2} x^{2}+693 a^{6} b \,e^{7} x -8316 a^{5} b^{2} d \,e^{6} x +27720 a^{4} b^{3} d^{2} e^{5} x -44352 a^{3} b^{4} d^{3} e^{4} x +38016 a^{2} b^{5} d^{4} e^{3} x -16896 a \,b^{6} d^{5} e^{2} x +3072 b^{7} d^{6} e x +33 a^{7} e^{7}+462 a^{6} b d \,e^{6}-5544 a^{5} b^{2} d^{2} e^{5}+18480 a^{4} b^{3} d^{3} e^{4}-29568 a^{3} b^{4} d^{4} e^{3}+25344 a^{2} b^{5} d^{5} e^{2}-11264 a \,b^{6} d^{6} e +2048 b^{7} d^{7}\right )}{99 \left (e x +d \right )^{\frac {3}{2}} e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.58, size = 462, normalized size = 2.22 \begin {gather*} \frac {2 \, {\left (\frac {9 \, {\left (e x + d\right )}^{\frac {11}{2}} b^{7} - 77 \, {\left (b^{7} d - a b^{6} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 297 \, {\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 693 \, {\left (b^{7} d^{3} - 3 \, a b^{6} d^{2} e + 3 \, a^{2} b^{5} d e^{2} - a^{3} b^{4} e^{3}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 1155 \, {\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 2079 \, {\left (b^{7} d^{5} - 5 \, a b^{6} d^{4} e + 10 \, a^{2} b^{5} d^{3} e^{2} - 10 \, a^{3} b^{4} d^{2} e^{3} + 5 \, a^{4} b^{3} d e^{4} - a^{5} b^{2} e^{5}\right )} \sqrt {e x + d}}{e^{7}} + \frac {33 \, {\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7} - 21 \, {\left (b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}\right )} {\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{7}}\right )}}{99 \, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.05, size = 335, normalized size = 1.61 \begin {gather*} \frac {2\,b^7\,{\left (d+e\,x\right )}^{11/2}}{11\,e^8}-\frac {\left (14\,b^7\,d-14\,a\,b^6\,e\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^8}-\frac {\left (d+e\,x\right )\,\left (14\,a^6\,b\,e^6-84\,a^5\,b^2\,d\,e^5+210\,a^4\,b^3\,d^2\,e^4-280\,a^3\,b^4\,d^3\,e^3+210\,a^2\,b^5\,d^4\,e^2-84\,a\,b^6\,d^5\,e+14\,b^7\,d^6\right )+\frac {2\,a^7\,e^7}{3}-\frac {2\,b^7\,d^7}{3}-14\,a^2\,b^5\,d^5\,e^2+\frac {70\,a^3\,b^4\,d^4\,e^3}{3}-\frac {70\,a^4\,b^3\,d^3\,e^4}{3}+14\,a^5\,b^2\,d^2\,e^5+\frac {14\,a\,b^6\,d^6\,e}{3}-\frac {14\,a^6\,b\,d\,e^6}{3}}{e^8\,{\left (d+e\,x\right )}^{3/2}}+\frac {42\,b^2\,{\left (a\,e-b\,d\right )}^5\,\sqrt {d+e\,x}}{e^8}+\frac {70\,b^3\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{3/2}}{3\,e^8}+\frac {14\,b^4\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{5/2}}{e^8}+\frac {6\,b^5\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{7/2}}{e^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 113.69, size = 360, normalized size = 1.73 \begin {gather*} \frac {2 b^{7} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{8}} - \frac {14 b \left (a e - b d\right )^{6}}{e^{8} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (14 a b^{6} e - 14 b^{7} d\right )}{9 e^{8}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (42 a^{2} b^{5} e^{2} - 84 a b^{6} d e + 42 b^{7} d^{2}\right )}{7 e^{8}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (70 a^{3} b^{4} e^{3} - 210 a^{2} b^{5} d e^{2} + 210 a b^{6} d^{2} e - 70 b^{7} d^{3}\right )}{5 e^{8}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (70 a^{4} b^{3} e^{4} - 280 a^{3} b^{4} d e^{3} + 420 a^{2} b^{5} d^{2} e^{2} - 280 a b^{6} d^{3} e + 70 b^{7} d^{4}\right )}{3 e^{8}} + \frac {\sqrt {d + e x} \left (42 a^{5} b^{2} e^{5} - 210 a^{4} b^{3} d e^{4} + 420 a^{3} b^{4} d^{2} e^{3} - 420 a^{2} b^{5} d^{3} e^{2} + 210 a b^{6} d^{4} e - 42 b^{7} d^{5}\right )}{e^{8}} - \frac {2 \left (a e - b d\right )^{7}}{3 e^{8} \left (d + e x\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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