Optimal. Leaf size=214 \[ -\frac {14 b^6 (d+e x)^{15/2} (b d-a e)}{15 e^8}+\frac {42 b^5 (d+e x)^{13/2} (b d-a e)^2}{13 e^8}-\frac {70 b^4 (d+e x)^{11/2} (b d-a e)^3}{11 e^8}+\frac {70 b^3 (d+e x)^{9/2} (b d-a e)^4}{9 e^8}-\frac {6 b^2 (d+e x)^{7/2} (b d-a e)^5}{e^8}+\frac {14 b (d+e x)^{5/2} (b d-a e)^6}{5 e^8}-\frac {2 (d+e x)^{3/2} (b d-a e)^7}{3 e^8}+\frac {2 b^7 (d+e x)^{17/2}}{17 e^8} \]
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Rubi [A] time = 0.07, antiderivative size = 214, 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)^{15/2} (b d-a e)}{15 e^8}+\frac {42 b^5 (d+e x)^{13/2} (b d-a e)^2}{13 e^8}-\frac {70 b^4 (d+e x)^{11/2} (b d-a e)^3}{11 e^8}+\frac {70 b^3 (d+e x)^{9/2} (b d-a e)^4}{9 e^8}-\frac {6 b^2 (d+e x)^{7/2} (b d-a e)^5}{e^8}+\frac {14 b (d+e x)^{5/2} (b d-a e)^6}{5 e^8}-\frac {2 (d+e x)^{3/2} (b d-a e)^7}{3 e^8}+\frac {2 b^7 (d+e x)^{17/2}}{17 e^8} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int (a+b x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^7 \sqrt {d+e x} \, dx\\ &=\int \left (\frac {(-b d+a e)^7 \sqrt {d+e x}}{e^7}+\frac {7 b (b d-a e)^6 (d+e x)^{3/2}}{e^7}-\frac {21 b^2 (b d-a e)^5 (d+e x)^{5/2}}{e^7}+\frac {35 b^3 (b d-a e)^4 (d+e x)^{7/2}}{e^7}-\frac {35 b^4 (b d-a e)^3 (d+e x)^{9/2}}{e^7}+\frac {21 b^5 (b d-a e)^2 (d+e x)^{11/2}}{e^7}-\frac {7 b^6 (b d-a e) (d+e x)^{13/2}}{e^7}+\frac {b^7 (d+e x)^{15/2}}{e^7}\right ) \, dx\\ &=-\frac {2 (b d-a e)^7 (d+e x)^{3/2}}{3 e^8}+\frac {14 b (b d-a e)^6 (d+e x)^{5/2}}{5 e^8}-\frac {6 b^2 (b d-a e)^5 (d+e x)^{7/2}}{e^8}+\frac {70 b^3 (b d-a e)^4 (d+e x)^{9/2}}{9 e^8}-\frac {70 b^4 (b d-a e)^3 (d+e x)^{11/2}}{11 e^8}+\frac {42 b^5 (b d-a e)^2 (d+e x)^{13/2}}{13 e^8}-\frac {14 b^6 (b d-a e) (d+e x)^{15/2}}{15 e^8}+\frac {2 b^7 (d+e x)^{17/2}}{17 e^8}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 167, normalized size = 0.78 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (-51051 b^6 (d+e x)^6 (b d-a e)+176715 b^5 (d+e x)^5 (b d-a e)^2-348075 b^4 (d+e x)^4 (b d-a e)^3+425425 b^3 (d+e x)^3 (b d-a e)^4-328185 b^2 (d+e x)^2 (b d-a e)^5+153153 b (d+e x) (b d-a e)^6-36465 (b d-a e)^7+6435 b^7 (d+e x)^7\right )}{109395 e^8} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.17, size = 582, normalized size = 2.72 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (36465 a^7 e^7+153153 a^6 b e^6 (d+e x)-255255 a^6 b d e^6+765765 a^5 b^2 d^2 e^5+328185 a^5 b^2 e^5 (d+e x)^2-918918 a^5 b^2 d e^5 (d+e x)-1276275 a^4 b^3 d^3 e^4+2297295 a^4 b^3 d^2 e^4 (d+e x)+425425 a^4 b^3 e^4 (d+e x)^3-1640925 a^4 b^3 d e^4 (d+e x)^2+1276275 a^3 b^4 d^4 e^3-3063060 a^3 b^4 d^3 e^3 (d+e x)+3281850 a^3 b^4 d^2 e^3 (d+e x)^2+348075 a^3 b^4 e^3 (d+e x)^4-1701700 a^3 b^4 d e^3 (d+e x)^3-765765 a^2 b^5 d^5 e^2+2297295 a^2 b^5 d^4 e^2 (d+e x)-3281850 a^2 b^5 d^3 e^2 (d+e x)^2+2552550 a^2 b^5 d^2 e^2 (d+e x)^3+176715 a^2 b^5 e^2 (d+e x)^5-1044225 a^2 b^5 d e^2 (d+e x)^4+255255 a b^6 d^6 e-918918 a b^6 d^5 e (d+e x)+1640925 a b^6 d^4 e (d+e x)^2-1701700 a b^6 d^3 e (d+e x)^3+1044225 a b^6 d^2 e (d+e x)^4+51051 a b^6 e (d+e x)^6-353430 a b^6 d e (d+e x)^5-36465 b^7 d^7+153153 b^7 d^6 (d+e x)-328185 b^7 d^5 (d+e x)^2+425425 b^7 d^4 (d+e x)^3-348075 b^7 d^3 (d+e x)^4+176715 b^7 d^2 (d+e x)^5+6435 b^7 (d+e x)^7-51051 b^7 d (d+e x)^6\right )}{109395 e^8} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 568, normalized size = 2.65 \begin {gather*} \frac {2 \, {\left (6435 \, b^{7} e^{8} x^{8} - 2048 \, b^{7} d^{8} + 17408 \, a b^{6} d^{7} e - 65280 \, a^{2} b^{5} d^{6} e^{2} + 141440 \, a^{3} b^{4} d^{5} e^{3} - 194480 \, a^{4} b^{3} d^{4} e^{4} + 175032 \, a^{5} b^{2} d^{3} e^{5} - 102102 \, a^{6} b d^{2} e^{6} + 36465 \, a^{7} d e^{7} + 429 \, {\left (b^{7} d e^{7} + 119 \, a b^{6} e^{8}\right )} x^{7} - 231 \, {\left (2 \, b^{7} d^{2} e^{6} - 17 \, a b^{6} d e^{7} - 765 \, a^{2} b^{5} e^{8}\right )} x^{6} + 63 \, {\left (8 \, b^{7} d^{3} e^{5} - 68 \, a b^{6} d^{2} e^{6} + 255 \, a^{2} b^{5} d e^{7} + 5525 \, a^{3} b^{4} e^{8}\right )} x^{5} - 35 \, {\left (16 \, b^{7} d^{4} e^{4} - 136 \, a b^{6} d^{3} e^{5} + 510 \, a^{2} b^{5} d^{2} e^{6} - 1105 \, a^{3} b^{4} d e^{7} - 12155 \, a^{4} b^{3} e^{8}\right )} x^{4} + 5 \, {\left (128 \, b^{7} d^{5} e^{3} - 1088 \, a b^{6} d^{4} e^{4} + 4080 \, a^{2} b^{5} d^{3} e^{5} - 8840 \, a^{3} b^{4} d^{2} e^{6} + 12155 \, a^{4} b^{3} d e^{7} + 65637 \, a^{5} b^{2} e^{8}\right )} x^{3} - 3 \, {\left (256 \, b^{7} d^{6} e^{2} - 2176 \, a b^{6} d^{5} e^{3} + 8160 \, a^{2} b^{5} d^{4} e^{4} - 17680 \, a^{3} b^{4} d^{3} e^{5} + 24310 \, a^{4} b^{3} d^{2} e^{6} - 21879 \, a^{5} b^{2} d e^{7} - 51051 \, a^{6} b e^{8}\right )} x^{2} + {\left (1024 \, b^{7} d^{7} e - 8704 \, a b^{6} d^{6} e^{2} + 32640 \, a^{2} b^{5} d^{5} e^{3} - 70720 \, a^{3} b^{4} d^{4} e^{4} + 97240 \, a^{4} b^{3} d^{3} e^{5} - 87516 \, a^{5} b^{2} d^{2} e^{6} + 51051 \, a^{6} b d e^{7} + 36465 \, a^{7} e^{8}\right )} x\right )} \sqrt {e x + d}}{109395 \, e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 1119, normalized size = 5.23
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 498, normalized size = 2.33 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (6435 b^{7} x^{7} e^{7}+51051 a \,b^{6} e^{7} x^{6}-6006 b^{7} d \,e^{6} x^{6}+176715 a^{2} b^{5} e^{7} x^{5}-47124 a \,b^{6} d \,e^{6} x^{5}+5544 b^{7} d^{2} e^{5} x^{5}+348075 a^{3} b^{4} e^{7} x^{4}-160650 a^{2} b^{5} d \,e^{6} x^{4}+42840 a \,b^{6} d^{2} e^{5} x^{4}-5040 b^{7} d^{3} e^{4} x^{4}+425425 a^{4} b^{3} e^{7} x^{3}-309400 a^{3} b^{4} d \,e^{6} x^{3}+142800 a^{2} b^{5} d^{2} e^{5} x^{3}-38080 a \,b^{6} d^{3} e^{4} x^{3}+4480 b^{7} d^{4} e^{3} x^{3}+328185 a^{5} b^{2} e^{7} x^{2}-364650 a^{4} b^{3} d \,e^{6} x^{2}+265200 a^{3} b^{4} d^{2} e^{5} x^{2}-122400 a^{2} b^{5} d^{3} e^{4} x^{2}+32640 a \,b^{6} d^{4} e^{3} x^{2}-3840 b^{7} d^{5} e^{2} x^{2}+153153 a^{6} b \,e^{7} x -262548 a^{5} b^{2} d \,e^{6} x +291720 a^{4} b^{3} d^{2} e^{5} x -212160 a^{3} b^{4} d^{3} e^{4} x +97920 a^{2} b^{5} d^{4} e^{3} x -26112 a \,b^{6} d^{5} e^{2} x +3072 b^{7} d^{6} e x +36465 a^{7} e^{7}-102102 a^{6} b d \,e^{6}+175032 a^{5} b^{2} d^{2} e^{5}-194480 a^{4} b^{3} d^{3} e^{4}+141440 a^{3} b^{4} d^{4} e^{3}-65280 a^{2} b^{5} d^{5} e^{2}+17408 a \,b^{6} d^{6} e -2048 b^{7} d^{7}\right )}{109395 e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.60, size = 456, normalized size = 2.13 \begin {gather*} \frac {2 \, {\left (6435 \, {\left (e x + d\right )}^{\frac {17}{2}} b^{7} - 51051 \, {\left (b^{7} d - a b^{6} e\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 176715 \, {\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 348075 \, {\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 {11}{2}} + 425425 \, {\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 {9}{2}} - 328185 \, {\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 )}^{\frac {7}{2}} + 153153 \, {\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 )}^{\frac {5}{2}} - 36465 \, {\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}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{109395 \, e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.05, size = 187, normalized size = 0.87 \begin {gather*} \frac {2\,b^7\,{\left (d+e\,x\right )}^{17/2}}{17\,e^8}-\frac {\left (14\,b^7\,d-14\,a\,b^6\,e\right )\,{\left (d+e\,x\right )}^{15/2}}{15\,e^8}+\frac {2\,{\left (a\,e-b\,d\right )}^7\,{\left (d+e\,x\right )}^{3/2}}{3\,e^8}+\frac {6\,b^2\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{7/2}}{e^8}+\frac {70\,b^3\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{9/2}}{9\,e^8}+\frac {70\,b^4\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{11/2}}{11\,e^8}+\frac {42\,b^5\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{13/2}}{13\,e^8}+\frac {14\,b\,{\left (a\,e-b\,d\right )}^6\,{\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 [B] time = 10.16, size = 544, normalized size = 2.54 \begin {gather*} \frac {2 \left (\frac {b^{7} \left (d + e x\right )^{\frac {17}{2}}}{17 e^{7}} + \frac {\left (d + e x\right )^{\frac {15}{2}} \left (7 a b^{6} e - 7 b^{7} d\right )}{15 e^{7}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \left (21 a^{2} b^{5} e^{2} - 42 a b^{6} d e + 21 b^{7} d^{2}\right )}{13 e^{7}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \left (35 a^{3} b^{4} e^{3} - 105 a^{2} b^{5} d e^{2} + 105 a b^{6} d^{2} e - 35 b^{7} d^{3}\right )}{11 e^{7}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (35 a^{4} b^{3} e^{4} - 140 a^{3} b^{4} d e^{3} + 210 a^{2} b^{5} d^{2} e^{2} - 140 a b^{6} d^{3} e + 35 b^{7} d^{4}\right )}{9 e^{7}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (21 a^{5} b^{2} e^{5} - 105 a^{4} b^{3} d e^{4} + 210 a^{3} b^{4} d^{2} e^{3} - 210 a^{2} b^{5} d^{3} e^{2} + 105 a b^{6} d^{4} e - 21 b^{7} d^{5}\right )}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (7 a^{6} b e^{6} - 42 a^{5} b^{2} d e^{5} + 105 a^{4} b^{3} d^{2} e^{4} - 140 a^{3} b^{4} d^{3} e^{3} + 105 a^{2} b^{5} d^{4} e^{2} - 42 a b^{6} d^{5} e + 7 b^{7} d^{6}\right )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a^{7} e^{7} - 7 a^{6} b d e^{6} + 21 a^{5} b^{2} d^{2} e^{5} - 35 a^{4} b^{3} d^{3} e^{4} + 35 a^{3} b^{4} d^{4} e^{3} - 21 a^{2} b^{5} d^{5} e^{2} + 7 a b^{6} d^{6} e - b^{7} d^{7}\right )}{3 e^{7}}\right )}{e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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