Optimal. Leaf size=206 \[ \frac {2 (b d-a e)^7}{e^8 \sqrt {d+e x}}+\frac {14 b (b d-a e)^6 \sqrt {d+e x}}{e^8}-\frac {14 b^2 (b d-a e)^5 (d+e x)^{3/2}}{e^8}+\frac {14 b^3 (b d-a e)^4 (d+e x)^{5/2}}{e^8}-\frac {10 b^4 (b d-a e)^3 (d+e x)^{7/2}}{e^8}+\frac {14 b^5 (b d-a e)^2 (d+e x)^{9/2}}{3 e^8}-\frac {14 b^6 (b d-a e) (d+e x)^{11/2}}{11 e^8}+\frac {2 b^7 (d+e x)^{13/2}}{13 e^8} \]
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Rubi [A]
time = 0.05, antiderivative size = 206, 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, 45}
\begin {gather*} -\frac {14 b^6 (d+e x)^{11/2} (b d-a e)}{11 e^8}+\frac {14 b^5 (d+e x)^{9/2} (b d-a e)^2}{3 e^8}-\frac {10 b^4 (d+e x)^{7/2} (b d-a e)^3}{e^8}+\frac {14 b^3 (d+e x)^{5/2} (b d-a e)^4}{e^8}-\frac {14 b^2 (d+e x)^{3/2} (b d-a e)^5}{e^8}+\frac {14 b \sqrt {d+e x} (b d-a e)^6}{e^8}+\frac {2 (b d-a e)^7}{e^8 \sqrt {d+e x}}+\frac {2 b^7 (d+e x)^{13/2}}{13 e^8} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 45
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)^{3/2}} \, dx &=\int \frac {(a+b x)^7}{(d+e x)^{3/2}} \, dx\\ &=\int \left (\frac {(-b d+a e)^7}{e^7 (d+e x)^{3/2}}+\frac {7 b (b d-a e)^6}{e^7 \sqrt {d+e x}}-\frac {21 b^2 (b d-a e)^5 \sqrt {d+e x}}{e^7}+\frac {35 b^3 (b d-a e)^4 (d+e x)^{3/2}}{e^7}-\frac {35 b^4 (b d-a e)^3 (d+e x)^{5/2}}{e^7}+\frac {21 b^5 (b d-a e)^2 (d+e x)^{7/2}}{e^7}-\frac {7 b^6 (b d-a e) (d+e x)^{9/2}}{e^7}+\frac {b^7 (d+e x)^{11/2}}{e^7}\right ) \, dx\\ &=\frac {2 (b d-a e)^7}{e^8 \sqrt {d+e x}}+\frac {14 b (b d-a e)^6 \sqrt {d+e x}}{e^8}-\frac {14 b^2 (b d-a e)^5 (d+e x)^{3/2}}{e^8}+\frac {14 b^3 (b d-a e)^4 (d+e x)^{5/2}}{e^8}-\frac {10 b^4 (b d-a e)^3 (d+e x)^{7/2}}{e^8}+\frac {14 b^5 (b d-a e)^2 (d+e x)^{9/2}}{3 e^8}-\frac {14 b^6 (b d-a e) (d+e x)^{11/2}}{11 e^8}+\frac {2 b^7 (d+e x)^{13/2}}{13 e^8}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 374, normalized size = 1.82 \begin {gather*} \frac {-858 a^7 e^7+6006 a^6 b e^6 (2 d+e x)+6006 a^5 b^2 e^5 \left (-8 d^2-4 d e x+e^2 x^2\right )+6006 a^4 b^3 e^4 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+858 a^3 b^4 e^3 \left (-128 d^4-64 d^3 e x+16 d^2 e^2 x^2-8 d e^3 x^3+5 e^4 x^4\right )+286 a^2 b^5 e^2 \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )+26 a b^6 e \left (-1024 d^6-512 d^5 e x+128 d^4 e^2 x^2-64 d^3 e^3 x^3+40 d^2 e^4 x^4-28 d e^5 x^5+21 e^6 x^6\right )+2 b^7 \left (2048 d^7+1024 d^6 e x-256 d^5 e^2 x^2+128 d^4 e^3 x^3-80 d^3 e^4 x^4+56 d^2 e^5 x^5-42 d e^6 x^6+33 e^7 x^7\right )}{429 e^8 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(594\) vs.
\(2(184)=368\).
time = 0.05, size = 595, normalized size = 2.89 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 455 vs.
\(2 (191) = 382\).
time = 0.31, size = 455, normalized size = 2.21 \begin {gather*} \frac {2}{429} \, {\left ({\left (33 \, {\left (x e + d\right )}^{\frac {13}{2}} b^{7} - 273 \, {\left (b^{7} d - a b^{6} e\right )} {\left (x e + d\right )}^{\frac {11}{2}} + 1001 \, {\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )} {\left (x e + d\right )}^{\frac {9}{2}} - 2145 \, {\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 (x e + d\right )}^{\frac {7}{2}} + 3003 \, {\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 (x e + d\right )}^{\frac {5}{2}} - 3003 \, {\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 (x e + d\right )}^{\frac {3}{2}} + 3003 \, {\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 )} \sqrt {x e + d}\right )} e^{\left (-7\right )} + \frac {429 \, {\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 )} e^{\left (-7\right )}}{\sqrt {x e + d}}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 436 vs.
\(2 (191) = 382\).
time = 2.80, size = 436, normalized size = 2.12 \begin {gather*} \frac {2 \, {\left (2048 \, b^{7} d^{7} + {\left (33 \, b^{7} x^{7} + 273 \, a b^{6} x^{6} + 1001 \, a^{2} b^{5} x^{5} + 2145 \, a^{3} b^{4} x^{4} + 3003 \, a^{4} b^{3} x^{3} + 3003 \, a^{5} b^{2} x^{2} + 3003 \, a^{6} b x - 429 \, a^{7}\right )} e^{7} - 2 \, {\left (21 \, b^{7} d x^{6} + 182 \, a b^{6} d x^{5} + 715 \, a^{2} b^{5} d x^{4} + 1716 \, a^{3} b^{4} d x^{3} + 3003 \, a^{4} b^{3} d x^{2} + 6006 \, a^{5} b^{2} d x - 3003 \, a^{6} b d\right )} e^{6} + 8 \, {\left (7 \, b^{7} d^{2} x^{5} + 65 \, a b^{6} d^{2} x^{4} + 286 \, a^{2} b^{5} d^{2} x^{3} + 858 \, a^{3} b^{4} d^{2} x^{2} + 3003 \, a^{4} b^{3} d^{2} x - 3003 \, a^{5} b^{2} d^{2}\right )} e^{5} - 16 \, {\left (5 \, b^{7} d^{3} x^{4} + 52 \, a b^{6} d^{3} x^{3} + 286 \, a^{2} b^{5} d^{3} x^{2} + 1716 \, a^{3} b^{4} d^{3} x - 3003 \, a^{4} b^{3} d^{3}\right )} e^{4} + 128 \, {\left (b^{7} d^{4} x^{3} + 13 \, a b^{6} d^{4} x^{2} + 143 \, a^{2} b^{5} d^{4} x - 429 \, a^{3} b^{4} d^{4}\right )} e^{3} - 256 \, {\left (b^{7} d^{5} x^{2} + 26 \, a b^{6} d^{5} x - 143 \, a^{2} b^{5} d^{5}\right )} e^{2} + 1024 \, {\left (b^{7} d^{6} x - 13 \, a b^{6} d^{6}\right )} e\right )} \sqrt {x e + d}}{429 \, {\left (x e^{9} + d e^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 439 vs.
\(2 (192) = 384\).
time = 38.51, size = 439, normalized size = 2.13 \begin {gather*} \frac {2 b^{7} \left (d + e x\right )^{\frac {13}{2}}}{13 e^{8}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (14 a b^{6} e - 14 b^{7} d\right )}{11 e^{8}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (42 a^{2} b^{5} e^{2} - 84 a b^{6} d e + 42 b^{7} d^{2}\right )}{9 e^{8}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \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 )}{7 e^{8}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \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 )}{5 e^{8}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \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 )}{3 e^{8}} + \frac {\sqrt {d + e x} \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 )}{e^{8}} - \frac {2 \left (a e - b d\right )^{7}}{e^{8} \sqrt {d + e x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 625 vs.
\(2 (191) = 382\).
time = 1.50, size = 625, normalized size = 3.03 \begin {gather*} \frac {2}{429} \, {\left (33 \, {\left (x e + d\right )}^{\frac {13}{2}} b^{7} e^{96} - 273 \, {\left (x e + d\right )}^{\frac {11}{2}} b^{7} d e^{96} + 1001 \, {\left (x e + d\right )}^{\frac {9}{2}} b^{7} d^{2} e^{96} - 2145 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{7} d^{3} e^{96} + 3003 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{7} d^{4} e^{96} - 3003 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{7} d^{5} e^{96} + 3003 \, \sqrt {x e + d} b^{7} d^{6} e^{96} + 273 \, {\left (x e + d\right )}^{\frac {11}{2}} a b^{6} e^{97} - 2002 \, {\left (x e + d\right )}^{\frac {9}{2}} a b^{6} d e^{97} + 6435 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{6} d^{2} e^{97} - 12012 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{6} d^{3} e^{97} + 15015 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{6} d^{4} e^{97} - 18018 \, \sqrt {x e + d} a b^{6} d^{5} e^{97} + 1001 \, {\left (x e + d\right )}^{\frac {9}{2}} a^{2} b^{5} e^{98} - 6435 \, {\left (x e + d\right )}^{\frac {7}{2}} a^{2} b^{5} d e^{98} + 18018 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{5} d^{2} e^{98} - 30030 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{5} d^{3} e^{98} + 45045 \, \sqrt {x e + d} a^{2} b^{5} d^{4} e^{98} + 2145 \, {\left (x e + d\right )}^{\frac {7}{2}} a^{3} b^{4} e^{99} - 12012 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{3} b^{4} d e^{99} + 30030 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b^{4} d^{2} e^{99} - 60060 \, \sqrt {x e + d} a^{3} b^{4} d^{3} e^{99} + 3003 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{4} b^{3} e^{100} - 15015 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{4} b^{3} d e^{100} + 45045 \, \sqrt {x e + d} a^{4} b^{3} d^{2} e^{100} + 3003 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{5} b^{2} e^{101} - 18018 \, \sqrt {x e + d} a^{5} b^{2} d e^{101} + 3003 \, \sqrt {x e + d} a^{6} b e^{102}\right )} e^{\left (-104\right )} + \frac {2 \, {\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 )} e^{\left (-8\right )}}{\sqrt {x e + d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.07, size = 270, normalized size = 1.31 \begin {gather*} \frac {2\,b^7\,{\left (d+e\,x\right )}^{13/2}}{13\,e^8}-\frac {\left (14\,b^7\,d-14\,a\,b^6\,e\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^8}-\frac {2\,a^7\,e^7-14\,a^6\,b\,d\,e^6+42\,a^5\,b^2\,d^2\,e^5-70\,a^4\,b^3\,d^3\,e^4+70\,a^3\,b^4\,d^4\,e^3-42\,a^2\,b^5\,d^5\,e^2+14\,a\,b^6\,d^6\,e-2\,b^7\,d^7}{e^8\,\sqrt {d+e\,x}}+\frac {14\,b^2\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{3/2}}{e^8}+\frac {14\,b^3\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{5/2}}{e^8}+\frac {10\,b^4\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{7/2}}{e^8}+\frac {14\,b^5\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{9/2}}{3\,e^8}+\frac {14\,b\,{\left (a\,e-b\,d\right )}^6\,\sqrt {d+e\,x}}{e^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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