Optimal. Leaf size=210 \[ -\frac {2 b^6 (d+e x)^{7/2} (b d-a e)}{e^8}+\frac {42 b^5 (d+e x)^{5/2} (b d-a e)^2}{5 e^8}-\frac {70 b^4 (d+e x)^{3/2} (b d-a e)^3}{3 e^8}+\frac {70 b^3 \sqrt {d+e x} (b d-a e)^4}{e^8}+\frac {42 b^2 (b d-a e)^5}{e^8 \sqrt {d+e x}}-\frac {14 b (b d-a e)^6}{3 e^8 (d+e x)^{3/2}}+\frac {2 (b d-a e)^7}{5 e^8 (d+e x)^{5/2}}+\frac {2 b^7 (d+e x)^{9/2}}{9 e^8} \]
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Rubi [A] time = 0.08, antiderivative size = 210, 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 {2 b^6 (d+e x)^{7/2} (b d-a e)}{e^8}+\frac {42 b^5 (d+e x)^{5/2} (b d-a e)^2}{5 e^8}-\frac {70 b^4 (d+e x)^{3/2} (b d-a e)^3}{3 e^8}+\frac {70 b^3 \sqrt {d+e x} (b d-a e)^4}{e^8}+\frac {42 b^2 (b d-a e)^5}{e^8 \sqrt {d+e x}}-\frac {14 b (b d-a e)^6}{3 e^8 (d+e x)^{3/2}}+\frac {2 (b d-a e)^7}{5 e^8 (d+e x)^{5/2}}+\frac {2 b^7 (d+e x)^{9/2}}{9 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)^{7/2}} \, dx &=\int \frac {(a+b x)^7}{(d+e x)^{7/2}} \, dx\\ &=\int \left (\frac {(-b d+a e)^7}{e^7 (d+e x)^{7/2}}+\frac {7 b (b d-a e)^6}{e^7 (d+e x)^{5/2}}-\frac {21 b^2 (b d-a e)^5}{e^7 (d+e x)^{3/2}}+\frac {35 b^3 (b d-a e)^4}{e^7 \sqrt {d+e x}}-\frac {35 b^4 (b d-a e)^3 \sqrt {d+e x}}{e^7}+\frac {21 b^5 (b d-a e)^2 (d+e x)^{3/2}}{e^7}-\frac {7 b^6 (b d-a e) (d+e x)^{5/2}}{e^7}+\frac {b^7 (d+e x)^{7/2}}{e^7}\right ) \, dx\\ &=\frac {2 (b d-a e)^7}{5 e^8 (d+e x)^{5/2}}-\frac {14 b (b d-a e)^6}{3 e^8 (d+e x)^{3/2}}+\frac {42 b^2 (b d-a e)^5}{e^8 \sqrt {d+e x}}+\frac {70 b^3 (b d-a e)^4 \sqrt {d+e x}}{e^8}-\frac {70 b^4 (b d-a e)^3 (d+e x)^{3/2}}{3 e^8}+\frac {42 b^5 (b d-a e)^2 (d+e x)^{5/2}}{5 e^8}-\frac {2 b^6 (b d-a e) (d+e x)^{7/2}}{e^8}+\frac {2 b^7 (d+e x)^{9/2}}{9 e^8}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 167, normalized size = 0.80 \begin {gather*} \frac {2 \left (-45 b^6 (d+e x)^6 (b d-a e)+189 b^5 (d+e x)^5 (b d-a e)^2-525 b^4 (d+e x)^4 (b d-a e)^3+1575 b^3 (d+e x)^3 (b d-a e)^4+945 b^2 (d+e x)^2 (b d-a e)^5-105 b (d+e x) (b d-a e)^6+9 (b d-a e)^7+5 b^7 (d+e x)^7\right )}{45 e^8 (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.12, size = 582, normalized size = 2.77 \begin {gather*} \frac {2 \left (-9 a^7 e^7-105 a^6 b e^6 (d+e x)+63 a^6 b d e^6-189 a^5 b^2 d^2 e^5-945 a^5 b^2 e^5 (d+e x)^2+630 a^5 b^2 d e^5 (d+e x)+315 a^4 b^3 d^3 e^4-1575 a^4 b^3 d^2 e^4 (d+e x)+1575 a^4 b^3 e^4 (d+e x)^3+4725 a^4 b^3 d e^4 (d+e x)^2-315 a^3 b^4 d^4 e^3+2100 a^3 b^4 d^3 e^3 (d+e x)-9450 a^3 b^4 d^2 e^3 (d+e x)^2+525 a^3 b^4 e^3 (d+e x)^4-6300 a^3 b^4 d e^3 (d+e x)^3+189 a^2 b^5 d^5 e^2-1575 a^2 b^5 d^4 e^2 (d+e x)+9450 a^2 b^5 d^3 e^2 (d+e x)^2+9450 a^2 b^5 d^2 e^2 (d+e x)^3+189 a^2 b^5 e^2 (d+e x)^5-1575 a^2 b^5 d e^2 (d+e x)^4-63 a b^6 d^6 e+630 a b^6 d^5 e (d+e x)-4725 a b^6 d^4 e (d+e x)^2-6300 a b^6 d^3 e (d+e x)^3+1575 a b^6 d^2 e (d+e x)^4+45 a b^6 e (d+e x)^6-378 a b^6 d e (d+e x)^5+9 b^7 d^7-105 b^7 d^6 (d+e x)+945 b^7 d^5 (d+e x)^2+1575 b^7 d^4 (d+e x)^3-525 b^7 d^3 (d+e x)^4+189 b^7 d^2 (d+e x)^5+5 b^7 (d+e x)^7-45 b^7 d (d+e x)^6\right )}{45 e^8 (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 496, normalized size = 2.36 \begin {gather*} \frac {2 \, {\left (5 \, b^{7} e^{7} x^{7} + 2048 \, b^{7} d^{7} - 9216 \, a b^{6} d^{6} e + 16128 \, a^{2} b^{5} d^{5} e^{2} - 13440 \, a^{3} b^{4} d^{4} e^{3} + 5040 \, a^{4} b^{3} d^{3} e^{4} - 504 \, a^{5} b^{2} d^{2} e^{5} - 42 \, a^{6} b d e^{6} - 9 \, a^{7} e^{7} - 5 \, {\left (2 \, b^{7} d e^{6} - 9 \, a b^{6} e^{7}\right )} x^{6} + 3 \, {\left (8 \, b^{7} d^{2} e^{5} - 36 \, a b^{6} d e^{6} + 63 \, a^{2} b^{5} e^{7}\right )} x^{5} - 5 \, {\left (16 \, b^{7} d^{3} e^{4} - 72 \, a b^{6} d^{2} e^{5} + 126 \, a^{2} b^{5} d e^{6} - 105 \, a^{3} b^{4} e^{7}\right )} x^{4} + 5 \, {\left (128 \, b^{7} d^{4} e^{3} - 576 \, a b^{6} d^{3} e^{4} + 1008 \, a^{2} b^{5} d^{2} e^{5} - 840 \, a^{3} b^{4} d e^{6} + 315 \, a^{4} b^{3} e^{7}\right )} x^{3} + 15 \, {\left (256 \, b^{7} d^{5} e^{2} - 1152 \, a b^{6} d^{4} e^{3} + 2016 \, a^{2} b^{5} d^{3} e^{4} - 1680 \, a^{3} b^{4} d^{2} e^{5} + 630 \, a^{4} b^{3} d e^{6} - 63 \, a^{5} b^{2} e^{7}\right )} x^{2} + 5 \, {\left (1024 \, b^{7} d^{6} e - 4608 \, a b^{6} d^{5} e^{2} + 8064 \, a^{2} b^{5} d^{4} e^{3} - 6720 \, a^{3} b^{4} d^{3} e^{4} + 2520 \, a^{4} b^{3} d^{2} e^{5} - 252 \, a^{5} b^{2} d e^{6} - 21 \, a^{6} b e^{7}\right )} x\right )} \sqrt {e x + d}}{45 \, {\left (e^{11} x^{3} + 3 \, d e^{10} x^{2} + 3 \, d^{2} e^{9} x + d^{3} e^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 608, normalized size = 2.90 \begin {gather*} \frac {2}{45} \, {\left (5 \, {\left (x e + d\right )}^{\frac {9}{2}} b^{7} e^{64} - 45 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{7} d e^{64} + 189 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{7} d^{2} e^{64} - 525 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{7} d^{3} e^{64} + 1575 \, \sqrt {x e + d} b^{7} d^{4} e^{64} + 45 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{6} e^{65} - 378 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{6} d e^{65} + 1575 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{6} d^{2} e^{65} - 6300 \, \sqrt {x e + d} a b^{6} d^{3} e^{65} + 189 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{5} e^{66} - 1575 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{5} d e^{66} + 9450 \, \sqrt {x e + d} a^{2} b^{5} d^{2} e^{66} + 525 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b^{4} e^{67} - 6300 \, \sqrt {x e + d} a^{3} b^{4} d e^{67} + 1575 \, \sqrt {x e + d} a^{4} b^{3} e^{68}\right )} e^{\left (-72\right )} + \frac {2 \, {\left (315 \, {\left (x e + d\right )}^{2} b^{7} d^{5} - 35 \, {\left (x e + d\right )} b^{7} d^{6} + 3 \, b^{7} d^{7} - 1575 \, {\left (x e + d\right )}^{2} a b^{6} d^{4} e + 210 \, {\left (x e + d\right )} a b^{6} d^{5} e - 21 \, a b^{6} d^{6} e + 3150 \, {\left (x e + d\right )}^{2} a^{2} b^{5} d^{3} e^{2} - 525 \, {\left (x e + d\right )} a^{2} b^{5} d^{4} e^{2} + 63 \, a^{2} b^{5} d^{5} e^{2} - 3150 \, {\left (x e + d\right )}^{2} a^{3} b^{4} d^{2} e^{3} + 700 \, {\left (x e + d\right )} a^{3} b^{4} d^{3} e^{3} - 105 \, a^{3} b^{4} d^{4} e^{3} + 1575 \, {\left (x e + d\right )}^{2} a^{4} b^{3} d e^{4} - 525 \, {\left (x e + d\right )} a^{4} b^{3} d^{2} e^{4} + 105 \, a^{4} b^{3} d^{3} e^{4} - 315 \, {\left (x e + d\right )}^{2} a^{5} b^{2} e^{5} + 210 \, {\left (x e + d\right )} a^{5} b^{2} d e^{5} - 63 \, a^{5} b^{2} d^{2} e^{5} - 35 \, {\left (x e + d\right )} a^{6} b e^{6} + 21 \, a^{6} b d e^{6} - 3 \, a^{7} e^{7}\right )} e^{\left (-8\right )}}{15 \, {\left (x e + d\right )}^{\frac {5}{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 = 2.37 \begin {gather*} -\frac {2 \left (-5 b^{7} x^{7} e^{7}-45 a \,b^{6} e^{7} x^{6}+10 b^{7} d \,e^{6} x^{6}-189 a^{2} b^{5} e^{7} x^{5}+108 a \,b^{6} d \,e^{6} x^{5}-24 b^{7} d^{2} e^{5} x^{5}-525 a^{3} b^{4} e^{7} x^{4}+630 a^{2} b^{5} d \,e^{6} x^{4}-360 a \,b^{6} d^{2} e^{5} x^{4}+80 b^{7} d^{3} e^{4} x^{4}-1575 a^{4} b^{3} e^{7} x^{3}+4200 a^{3} b^{4} d \,e^{6} x^{3}-5040 a^{2} b^{5} d^{2} e^{5} x^{3}+2880 a \,b^{6} d^{3} e^{4} x^{3}-640 b^{7} d^{4} e^{3} x^{3}+945 a^{5} b^{2} e^{7} x^{2}-9450 a^{4} b^{3} d \,e^{6} x^{2}+25200 a^{3} b^{4} d^{2} e^{5} x^{2}-30240 a^{2} b^{5} d^{3} e^{4} x^{2}+17280 a \,b^{6} d^{4} e^{3} x^{2}-3840 b^{7} d^{5} e^{2} x^{2}+105 a^{6} b \,e^{7} x +1260 a^{5} b^{2} d \,e^{6} x -12600 a^{4} b^{3} d^{2} e^{5} x +33600 a^{3} b^{4} d^{3} e^{4} x -40320 a^{2} b^{5} d^{4} e^{3} x +23040 a \,b^{6} d^{5} e^{2} x -5120 b^{7} d^{6} e x +9 a^{7} e^{7}+42 a^{6} b d \,e^{6}+504 a^{5} b^{2} d^{2} e^{5}-5040 a^{4} b^{3} d^{3} e^{4}+13440 a^{3} b^{4} d^{4} e^{3}-16128 a^{2} b^{5} d^{5} e^{2}+9216 a \,b^{6} d^{6} e -2048 b^{7} d^{7}\right )}{45 \left (e x +d \right )^{\frac {5}{2}} e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.55, size = 463, normalized size = 2.20 \begin {gather*} \frac {2 \, {\left (\frac {5 \, {\left (e x + d\right )}^{\frac {9}{2}} b^{7} - 45 \, {\left (b^{7} d - a b^{6} e\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 189 \, {\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 525 \, {\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 {3}{2}} + 1575 \, {\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 )} \sqrt {e x + d}}{e^{7}} + \frac {3 \, {\left (3 \, b^{7} d^{7} - 21 \, a b^{6} d^{6} e + 63 \, a^{2} b^{5} d^{5} e^{2} - 105 \, a^{3} b^{4} d^{4} e^{3} + 105 \, a^{4} b^{3} d^{3} e^{4} - 63 \, a^{5} b^{2} d^{2} e^{5} + 21 \, a^{6} b d e^{6} - 3 \, a^{7} e^{7} + 315 \, {\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 )} {\left (e x + d\right )}^{2} - 35 \, {\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 {5}{2}} e^{7}}\right )}}{45 \, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 388, normalized size = 1.85 \begin {gather*} \frac {2\,b^7\,{\left (d+e\,x\right )}^{9/2}}{9\,e^8}-\frac {\left (14\,b^7\,d-14\,a\,b^6\,e\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^8}+\frac {{\left (d+e\,x\right )}^2\,\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 )-\left (d+e\,x\right )\,\left (\frac {14\,a^6\,b\,e^6}{3}-28\,a^5\,b^2\,d\,e^5+70\,a^4\,b^3\,d^2\,e^4-\frac {280\,a^3\,b^4\,d^3\,e^3}{3}+70\,a^2\,b^5\,d^4\,e^2-28\,a\,b^6\,d^5\,e+\frac {14\,b^7\,d^6}{3}\right )-\frac {2\,a^7\,e^7}{5}+\frac {2\,b^7\,d^7}{5}+\frac {42\,a^2\,b^5\,d^5\,e^2}{5}-14\,a^3\,b^4\,d^4\,e^3+14\,a^4\,b^3\,d^3\,e^4-\frac {42\,a^5\,b^2\,d^2\,e^5}{5}-\frac {14\,a\,b^6\,d^6\,e}{5}+\frac {14\,a^6\,b\,d\,e^6}{5}}{e^8\,{\left (d+e\,x\right )}^{5/2}}+\frac {70\,b^3\,{\left (a\,e-b\,d\right )}^4\,\sqrt {d+e\,x}}{e^8}+\frac {70\,b^4\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{3/2}}{3\,e^8}+\frac {42\,b^5\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^8} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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