Optimal. Leaf size=181 \[ -\frac {12 b^5 (d+e x)^{7/2} (b d-a e)}{7 e^7}+\frac {6 b^4 (d+e x)^{5/2} (b d-a e)^2}{e^7}-\frac {40 b^3 (d+e x)^{3/2} (b d-a e)^3}{3 e^7}+\frac {30 b^2 \sqrt {d+e x} (b d-a e)^4}{e^7}+\frac {12 b (b d-a e)^5}{e^7 \sqrt {d+e x}}-\frac {2 (b d-a e)^6}{3 e^7 (d+e x)^{3/2}}+\frac {2 b^6 (d+e x)^{9/2}}{9 e^7} \]
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Rubi [A] time = 0.06, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {27, 43} \[ -\frac {12 b^5 (d+e x)^{7/2} (b d-a e)}{7 e^7}+\frac {6 b^4 (d+e x)^{5/2} (b d-a e)^2}{e^7}-\frac {40 b^3 (d+e x)^{3/2} (b d-a e)^3}{3 e^7}+\frac {30 b^2 \sqrt {d+e x} (b d-a e)^4}{e^7}+\frac {12 b (b d-a e)^5}{e^7 \sqrt {d+e x}}-\frac {2 (b d-a e)^6}{3 e^7 (d+e x)^{3/2}}+\frac {2 b^6 (d+e x)^{9/2}}{9 e^7} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{5/2}} \, dx &=\int \frac {(a+b x)^6}{(d+e x)^{5/2}} \, dx\\ &=\int \left (\frac {(-b d+a e)^6}{e^6 (d+e x)^{5/2}}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^{3/2}}+\frac {15 b^2 (b d-a e)^4}{e^6 \sqrt {d+e x}}-\frac {20 b^3 (b d-a e)^3 \sqrt {d+e x}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{3/2}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{5/2}}{e^6}+\frac {b^6 (d+e x)^{7/2}}{e^6}\right ) \, dx\\ &=-\frac {2 (b d-a e)^6}{3 e^7 (d+e x)^{3/2}}+\frac {12 b (b d-a e)^5}{e^7 \sqrt {d+e x}}+\frac {30 b^2 (b d-a e)^4 \sqrt {d+e x}}{e^7}-\frac {40 b^3 (b d-a e)^3 (d+e x)^{3/2}}{3 e^7}+\frac {6 b^4 (b d-a e)^2 (d+e x)^{5/2}}{e^7}-\frac {12 b^5 (b d-a e) (d+e x)^{7/2}}{7 e^7}+\frac {2 b^6 (d+e x)^{9/2}}{9 e^7}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 145, normalized size = 0.80 \[ \frac {2 \left (-54 b^5 (d+e x)^5 (b d-a e)+189 b^4 (d+e x)^4 (b d-a e)^2-420 b^3 (d+e x)^3 (b d-a e)^3+945 b^2 (d+e x)^2 (b d-a e)^4+378 b (d+e x) (b d-a e)^5-21 (b d-a e)^6+7 b^6 (d+e x)^6\right )}{63 e^7 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.86, size = 377, normalized size = 2.08 \[ \frac {2 \, {\left (7 \, b^{6} e^{6} x^{6} + 1024 \, b^{6} d^{6} - 4608 \, a b^{5} d^{5} e + 8064 \, a^{2} b^{4} d^{4} e^{2} - 6720 \, a^{3} b^{3} d^{3} e^{3} + 2520 \, a^{4} b^{2} d^{2} e^{4} - 252 \, a^{5} b d e^{5} - 21 \, a^{6} e^{6} - 6 \, {\left (2 \, b^{6} d e^{5} - 9 \, a b^{5} e^{6}\right )} x^{5} + 3 \, {\left (8 \, b^{6} d^{2} e^{4} - 36 \, a b^{5} d e^{5} + 63 \, a^{2} b^{4} e^{6}\right )} x^{4} - 4 \, {\left (16 \, b^{6} d^{3} e^{3} - 72 \, a b^{5} d^{2} e^{4} + 126 \, a^{2} b^{4} d e^{5} - 105 \, a^{3} b^{3} e^{6}\right )} x^{3} + 3 \, {\left (128 \, b^{6} d^{4} e^{2} - 576 \, a b^{5} d^{3} e^{3} + 1008 \, a^{2} b^{4} d^{2} e^{4} - 840 \, a^{3} b^{3} d e^{5} + 315 \, a^{4} b^{2} e^{6}\right )} x^{2} + 6 \, {\left (256 \, b^{6} d^{5} e - 1152 \, a b^{5} d^{4} e^{2} + 2016 \, a^{2} b^{4} d^{3} e^{3} - 1680 \, a^{3} b^{3} d^{2} e^{4} + 630 \, a^{4} b^{2} d e^{5} - 63 \, a^{5} b e^{6}\right )} x\right )} \sqrt {e x + d}}{63 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 462, normalized size = 2.55 \[ \frac {2}{63} \, {\left (7 \, {\left (x e + d\right )}^{\frac {9}{2}} b^{6} e^{56} - 54 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{6} d e^{56} + 189 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{6} d^{2} e^{56} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{6} d^{3} e^{56} + 945 \, \sqrt {x e + d} b^{6} d^{4} e^{56} + 54 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{5} e^{57} - 378 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{5} d e^{57} + 1260 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{5} d^{2} e^{57} - 3780 \, \sqrt {x e + d} a b^{5} d^{3} e^{57} + 189 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{4} e^{58} - 1260 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{4} d e^{58} + 5670 \, \sqrt {x e + d} a^{2} b^{4} d^{2} e^{58} + 420 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b^{3} e^{59} - 3780 \, \sqrt {x e + d} a^{3} b^{3} d e^{59} + 945 \, \sqrt {x e + d} a^{4} b^{2} e^{60}\right )} e^{\left (-63\right )} + \frac {2 \, {\left (18 \, {\left (x e + d\right )} b^{6} d^{5} - b^{6} d^{6} - 90 \, {\left (x e + d\right )} a b^{5} d^{4} e + 6 \, a b^{5} d^{5} e + 180 \, {\left (x e + d\right )} a^{2} b^{4} d^{3} e^{2} - 15 \, a^{2} b^{4} d^{4} e^{2} - 180 \, {\left (x e + d\right )} a^{3} b^{3} d^{2} e^{3} + 20 \, a^{3} b^{3} d^{3} e^{3} + 90 \, {\left (x e + d\right )} a^{4} b^{2} d e^{4} - 15 \, a^{4} b^{2} d^{2} e^{4} - 18 \, {\left (x e + d\right )} a^{5} b e^{5} + 6 \, a^{5} b d e^{5} - a^{6} e^{6}\right )} e^{\left (-7\right )}}{3 \, {\left (x e + d\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 377, normalized size = 2.08 \[ -\frac {2 \left (-7 b^{6} e^{6} x^{6}-54 a \,b^{5} e^{6} x^{5}+12 b^{6} d \,e^{5} x^{5}-189 a^{2} b^{4} e^{6} x^{4}+108 a \,b^{5} d \,e^{5} x^{4}-24 b^{6} d^{2} e^{4} x^{4}-420 a^{3} b^{3} e^{6} x^{3}+504 a^{2} b^{4} d \,e^{5} x^{3}-288 a \,b^{5} d^{2} e^{4} x^{3}+64 b^{6} d^{3} e^{3} x^{3}-945 a^{4} b^{2} e^{6} x^{2}+2520 a^{3} b^{3} d \,e^{5} x^{2}-3024 a^{2} b^{4} d^{2} e^{4} x^{2}+1728 a \,b^{5} d^{3} e^{3} x^{2}-384 b^{6} d^{4} e^{2} x^{2}+378 a^{5} b \,e^{6} x -3780 a^{4} b^{2} d \,e^{5} x +10080 a^{3} b^{3} d^{2} e^{4} x -12096 a^{2} b^{4} d^{3} e^{3} x +6912 a \,b^{5} d^{4} e^{2} x -1536 b^{6} d^{5} e x +21 a^{6} e^{6}+252 a^{5} b d \,e^{5}-2520 a^{4} b^{2} d^{2} e^{4}+6720 a^{3} b^{3} d^{3} e^{3}-8064 a^{2} b^{4} d^{4} e^{2}+4608 a \,b^{5} d^{5} e -1024 b^{6} d^{6}\right )}{63 \left (e x +d \right )^{\frac {3}{2}} e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.19, size = 356, normalized size = 1.97 \[ \frac {2 \, {\left (\frac {7 \, {\left (e x + d\right )}^{\frac {9}{2}} b^{6} - 54 \, {\left (b^{6} d - a b^{5} e\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 189 \, {\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 420 \, {\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 945 \, {\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} \sqrt {e x + d}}{e^{6}} - \frac {21 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6} - 18 \, {\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )} {\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{6}}\right )}}{63 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.56, size = 281, normalized size = 1.55 \[ \frac {2\,b^6\,{\left (d+e\,x\right )}^{9/2}}{9\,e^7}-\frac {\left (12\,b^6\,d-12\,a\,b^5\,e\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}+\frac {\left (d+e\,x\right )\,\left (-12\,a^5\,b\,e^5+60\,a^4\,b^2\,d\,e^4-120\,a^3\,b^3\,d^2\,e^3+120\,a^2\,b^4\,d^3\,e^2-60\,a\,b^5\,d^4\,e+12\,b^6\,d^5\right )-\frac {2\,a^6\,e^6}{3}-\frac {2\,b^6\,d^6}{3}-10\,a^2\,b^4\,d^4\,e^2+\frac {40\,a^3\,b^3\,d^3\,e^3}{3}-10\,a^4\,b^2\,d^2\,e^4+4\,a\,b^5\,d^5\,e+4\,a^5\,b\,d\,e^5}{e^7\,{\left (d+e\,x\right )}^{3/2}}+\frac {30\,b^2\,{\left (a\,e-b\,d\right )}^4\,\sqrt {d+e\,x}}{e^7}+\frac {40\,b^3\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{3/2}}{3\,e^7}+\frac {6\,b^4\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{5/2}}{e^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 83.16, size = 270, normalized size = 1.49 \[ \frac {2 b^{6} \left (d + e x\right )^{\frac {9}{2}}}{9 e^{7}} - \frac {12 b \left (a e - b d\right )^{5}}{e^{7} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (12 a b^{5} e - 12 b^{6} d\right )}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (30 a^{2} b^{4} e^{2} - 60 a b^{5} d e + 30 b^{6} d^{2}\right )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (40 a^{3} b^{3} e^{3} - 120 a^{2} b^{4} d e^{2} + 120 a b^{5} d^{2} e - 40 b^{6} d^{3}\right )}{3 e^{7}} + \frac {\sqrt {d + e x} \left (30 a^{4} b^{2} e^{4} - 120 a^{3} b^{3} d e^{3} + 180 a^{2} b^{4} d^{2} e^{2} - 120 a b^{5} d^{3} e + 30 b^{6} d^{4}\right )}{e^{7}} - \frac {2 \left (a e - b d\right )^{6}}{3 e^{7} \left (d + e x\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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