Optimal. Leaf size=212 \[ -\frac {14 b^6 (d+e x)^{13/2} (b d-a e)}{13 e^8}+\frac {42 b^5 (d+e x)^{11/2} (b d-a e)^2}{11 e^8}-\frac {70 b^4 (d+e x)^{9/2} (b d-a e)^3}{9 e^8}+\frac {10 b^3 (d+e x)^{7/2} (b d-a e)^4}{e^8}-\frac {42 b^2 (d+e x)^{5/2} (b d-a e)^5}{5 e^8}+\frac {14 b (d+e x)^{3/2} (b d-a e)^6}{3 e^8}-\frac {2 \sqrt {d+e x} (b d-a e)^7}{e^8}+\frac {2 b^7 (d+e x)^{15/2}}{15 e^8} \]
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Rubi [A] time = 0.07, antiderivative size = 212, 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)^{13/2} (b d-a e)}{13 e^8}+\frac {42 b^5 (d+e x)^{11/2} (b d-a e)^2}{11 e^8}-\frac {70 b^4 (d+e x)^{9/2} (b d-a e)^3}{9 e^8}+\frac {10 b^3 (d+e x)^{7/2} (b d-a e)^4}{e^8}-\frac {42 b^2 (d+e x)^{5/2} (b d-a e)^5}{5 e^8}+\frac {14 b (d+e x)^{3/2} (b d-a e)^6}{3 e^8}-\frac {2 \sqrt {d+e x} (b d-a e)^7}{e^8}+\frac {2 b^7 (d+e x)^{15/2}}{15 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}{\sqrt {d+e x}} \, dx &=\int \frac {(a+b x)^7}{\sqrt {d+e x}} \, dx\\ &=\int \left (\frac {(-b d+a e)^7}{e^7 \sqrt {d+e x}}+\frac {7 b (b d-a e)^6 \sqrt {d+e x}}{e^7}-\frac {21 b^2 (b d-a e)^5 (d+e x)^{3/2}}{e^7}+\frac {35 b^3 (b d-a e)^4 (d+e x)^{5/2}}{e^7}-\frac {35 b^4 (b d-a e)^3 (d+e x)^{7/2}}{e^7}+\frac {21 b^5 (b d-a e)^2 (d+e x)^{9/2}}{e^7}-\frac {7 b^6 (b d-a e) (d+e x)^{11/2}}{e^7}+\frac {b^7 (d+e x)^{13/2}}{e^7}\right ) \, dx\\ &=-\frac {2 (b d-a e)^7 \sqrt {d+e x}}{e^8}+\frac {14 b (b d-a e)^6 (d+e x)^{3/2}}{3 e^8}-\frac {42 b^2 (b d-a e)^5 (d+e x)^{5/2}}{5 e^8}+\frac {10 b^3 (b d-a e)^4 (d+e x)^{7/2}}{e^8}-\frac {70 b^4 (b d-a e)^3 (d+e x)^{9/2}}{9 e^8}+\frac {42 b^5 (b d-a e)^2 (d+e x)^{11/2}}{11 e^8}-\frac {14 b^6 (b d-a e) (d+e x)^{13/2}}{13 e^8}+\frac {2 b^7 (d+e x)^{15/2}}{15 e^8}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 167, normalized size = 0.79 \begin {gather*} \frac {2 \sqrt {d+e x} \left (-3465 b^6 (d+e x)^6 (b d-a e)+12285 b^5 (d+e x)^5 (b d-a e)^2-25025 b^4 (d+e x)^4 (b d-a e)^3+32175 b^3 (d+e x)^3 (b d-a e)^4-27027 b^2 (d+e x)^2 (b d-a e)^5+15015 b (d+e x) (b d-a e)^6-6435 (b d-a e)^7+429 b^7 (d+e x)^7\right )}{6435 e^8} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.16, size = 582, normalized size = 2.75 \begin {gather*} \frac {2 \sqrt {d+e x} \left (6435 a^7 e^7+15015 a^6 b e^6 (d+e x)-45045 a^6 b d e^6+135135 a^5 b^2 d^2 e^5+27027 a^5 b^2 e^5 (d+e x)^2-90090 a^5 b^2 d e^5 (d+e x)-225225 a^4 b^3 d^3 e^4+225225 a^4 b^3 d^2 e^4 (d+e x)+32175 a^4 b^3 e^4 (d+e x)^3-135135 a^4 b^3 d e^4 (d+e x)^2+225225 a^3 b^4 d^4 e^3-300300 a^3 b^4 d^3 e^3 (d+e x)+270270 a^3 b^4 d^2 e^3 (d+e x)^2+25025 a^3 b^4 e^3 (d+e x)^4-128700 a^3 b^4 d e^3 (d+e x)^3-135135 a^2 b^5 d^5 e^2+225225 a^2 b^5 d^4 e^2 (d+e x)-270270 a^2 b^5 d^3 e^2 (d+e x)^2+193050 a^2 b^5 d^2 e^2 (d+e x)^3+12285 a^2 b^5 e^2 (d+e x)^5-75075 a^2 b^5 d e^2 (d+e x)^4+45045 a b^6 d^6 e-90090 a b^6 d^5 e (d+e x)+135135 a b^6 d^4 e (d+e x)^2-128700 a b^6 d^3 e (d+e x)^3+75075 a b^6 d^2 e (d+e x)^4+3465 a b^6 e (d+e x)^6-24570 a b^6 d e (d+e x)^5-6435 b^7 d^7+15015 b^7 d^6 (d+e x)-27027 b^7 d^5 (d+e x)^2+32175 b^7 d^4 (d+e x)^3-25025 b^7 d^3 (d+e x)^4+12285 b^7 d^2 (d+e x)^5+429 b^7 (d+e x)^7-3465 b^7 d (d+e x)^6\right )}{6435 e^8} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 463, normalized size = 2.18 \begin {gather*} \frac {2 \, {\left (429 \, b^{7} e^{7} x^{7} - 2048 \, b^{7} d^{7} + 15360 \, a b^{6} d^{6} e - 49920 \, a^{2} b^{5} d^{5} e^{2} + 91520 \, a^{3} b^{4} d^{4} e^{3} - 102960 \, a^{4} b^{3} d^{3} e^{4} + 72072 \, a^{5} b^{2} d^{2} e^{5} - 30030 \, a^{6} b d e^{6} + 6435 \, a^{7} e^{7} - 231 \, {\left (2 \, b^{7} d e^{6} - 15 \, a b^{6} e^{7}\right )} x^{6} + 63 \, {\left (8 \, b^{7} d^{2} e^{5} - 60 \, a b^{6} d e^{6} + 195 \, a^{2} b^{5} e^{7}\right )} x^{5} - 35 \, {\left (16 \, b^{7} d^{3} e^{4} - 120 \, a b^{6} d^{2} e^{5} + 390 \, a^{2} b^{5} d e^{6} - 715 \, a^{3} b^{4} e^{7}\right )} x^{4} + 5 \, {\left (128 \, b^{7} d^{4} e^{3} - 960 \, a b^{6} d^{3} e^{4} + 3120 \, a^{2} b^{5} d^{2} e^{5} - 5720 \, a^{3} b^{4} d e^{6} + 6435 \, a^{4} b^{3} e^{7}\right )} x^{3} - 3 \, {\left (256 \, b^{7} d^{5} e^{2} - 1920 \, a b^{6} d^{4} e^{3} + 6240 \, a^{2} b^{5} d^{3} e^{4} - 11440 \, a^{3} b^{4} d^{2} e^{5} + 12870 \, a^{4} b^{3} d e^{6} - 9009 \, a^{5} b^{2} e^{7}\right )} x^{2} + {\left (1024 \, b^{7} d^{6} e - 7680 \, a b^{6} d^{5} e^{2} + 24960 \, a^{2} b^{5} d^{4} e^{3} - 45760 \, a^{3} b^{4} d^{3} e^{4} + 51480 \, a^{4} b^{3} d^{2} e^{5} - 36036 \, a^{5} b^{2} d e^{6} + 15015 \, a^{6} b e^{7}\right )} x\right )} \sqrt {e x + d}}{6435 \, e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 505, normalized size = 2.38 \begin {gather*} \frac {2}{6435} \, {\left (15015 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{6} b e^{\left (-1\right )} + 9009 \, {\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^{5} b^{2} e^{\left (-2\right )} + 6435 \, {\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^{4} b^{3} e^{\left (-3\right )} + 715 \, {\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^{3} b^{4} e^{\left (-4\right )} + 195 \, {\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 )} a^{2} b^{5} e^{\left (-5\right )} + 15 \, {\left (231 \, {\left (x e + d\right )}^{\frac {13}{2}} - 1638 \, {\left (x e + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {x e + d} d^{6}\right )} a b^{6} e^{\left (-6\right )} + {\left (429 \, {\left (x e + d\right )}^{\frac {15}{2}} - 3465 \, {\left (x e + d\right )}^{\frac {13}{2}} d + 12285 \, {\left (x e + d\right )}^{\frac {11}{2}} d^{2} - 25025 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{3} + 32175 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{4} - 27027 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{5} + 15015 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{6} - 6435 \, \sqrt {x e + d} d^{7}\right )} b^{7} e^{\left (-7\right )} + 6435 \, \sqrt {x e + d} a^{7}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 498, normalized size = 2.35 \begin {gather*} \frac {2 \left (429 b^{7} x^{7} e^{7}+3465 a \,b^{6} e^{7} x^{6}-462 b^{7} d \,e^{6} x^{6}+12285 a^{2} b^{5} e^{7} x^{5}-3780 a \,b^{6} d \,e^{6} x^{5}+504 b^{7} d^{2} e^{5} x^{5}+25025 a^{3} b^{4} e^{7} x^{4}-13650 a^{2} b^{5} d \,e^{6} x^{4}+4200 a \,b^{6} d^{2} e^{5} x^{4}-560 b^{7} d^{3} e^{4} x^{4}+32175 a^{4} b^{3} e^{7} x^{3}-28600 a^{3} b^{4} d \,e^{6} x^{3}+15600 a^{2} b^{5} d^{2} e^{5} x^{3}-4800 a \,b^{6} d^{3} e^{4} x^{3}+640 b^{7} d^{4} e^{3} x^{3}+27027 a^{5} b^{2} e^{7} x^{2}-38610 a^{4} b^{3} d \,e^{6} x^{2}+34320 a^{3} b^{4} d^{2} e^{5} x^{2}-18720 a^{2} b^{5} d^{3} e^{4} x^{2}+5760 a \,b^{6} d^{4} e^{3} x^{2}-768 b^{7} d^{5} e^{2} x^{2}+15015 a^{6} b \,e^{7} x -36036 a^{5} b^{2} d \,e^{6} x +51480 a^{4} b^{3} d^{2} e^{5} x -45760 a^{3} b^{4} d^{3} e^{4} x +24960 a^{2} b^{5} d^{4} e^{3} x -7680 a \,b^{6} d^{5} e^{2} x +1024 b^{7} d^{6} e x +6435 a^{7} e^{7}-30030 a^{6} b d \,e^{6}+72072 a^{5} b^{2} d^{2} e^{5}-102960 a^{4} b^{3} d^{3} e^{4}+91520 a^{3} b^{4} d^{4} e^{3}-49920 a^{2} b^{5} d^{5} e^{2}+15360 a \,b^{6} d^{6} e -2048 b^{7} d^{7}\right ) \sqrt {e x +d}}{6435 e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.71, size = 456, normalized size = 2.15 \begin {gather*} \frac {2 \, {\left (429 \, {\left (e x + d\right )}^{\frac {15}{2}} b^{7} - 3465 \, {\left (b^{7} d - a b^{6} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 12285 \, {\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 25025 \, {\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 {9}{2}} + 32175 \, {\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 {7}{2}} - 27027 \, {\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 {5}{2}} + 15015 \, {\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 {3}{2}} - 6435 \, {\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 )} \sqrt {e x + d}\right )}}{6435 \, e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.07, size = 187, normalized size = 0.88 \begin {gather*} \frac {2\,b^7\,{\left (d+e\,x\right )}^{15/2}}{15\,e^8}-\frac {\left (14\,b^7\,d-14\,a\,b^6\,e\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^8}+\frac {2\,{\left (a\,e-b\,d\right )}^7\,\sqrt {d+e\,x}}{e^8}+\frac {42\,b^2\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{5/2}}{5\,e^8}+\frac {10\,b^3\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{7/2}}{e^8}+\frac {70\,b^4\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^8}+\frac {42\,b^5\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{11/2}}{11\,e^8}+\frac {14\,b\,{\left (a\,e-b\,d\right )}^6\,{\left (d+e\,x\right )}^{3/2}}{3\,e^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 140.05, size = 1217, normalized size = 5.74
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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