Optimal. Leaf size=278 \[ -\frac {2 \left (c d^2-b d e+a e^2\right )^3}{5 e^7 (d+e x)^{5/2}}+\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{e^7 (d+e x)^{3/2}}-\frac {6 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^7 \sqrt {d+e x}}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) \sqrt {d+e x}}{e^7}+\frac {2 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{3/2}}{e^7}-\frac {6 c^2 (2 c d-b e) (d+e x)^{5/2}}{5 e^7}+\frac {2 c^3 (d+e x)^{7/2}}{7 e^7} \]
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Rubi [A]
time = 0.08, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {712}
\begin {gather*} \frac {2 c (d+e x)^{3/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}-\frac {2 \sqrt {d+e x} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7}-\frac {6 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 \sqrt {d+e x}}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 (d+e x)^{3/2}}-\frac {2 \left (a e^2-b d e+c d^2\right )^3}{5 e^7 (d+e x)^{5/2}}-\frac {6 c^2 (d+e x)^{5/2} (2 c d-b e)}{5 e^7}+\frac {2 c^3 (d+e x)^{7/2}}{7 e^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 712
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{7/2}} \, dx &=\int \left (\frac {\left (c d^2-b d e+a e^2\right )^3}{e^6 (d+e x)^{7/2}}+\frac {3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^6 (d+e x)^{5/2}}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right )}{e^6 (d+e x)^{3/2}}+\frac {(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right )}{e^6 \sqrt {d+e x}}+\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) \sqrt {d+e x}}{e^6}-\frac {3 c^2 (2 c d-b e) (d+e x)^{3/2}}{e^6}+\frac {c^3 (d+e x)^{5/2}}{e^6}\right ) \, dx\\ &=-\frac {2 \left (c d^2-b d e+a e^2\right )^3}{5 e^7 (d+e x)^{5/2}}+\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{e^7 (d+e x)^{3/2}}-\frac {6 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^7 \sqrt {d+e x}}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) \sqrt {d+e x}}{e^7}+\frac {2 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{3/2}}{e^7}-\frac {6 c^2 (2 c d-b e) (d+e x)^{5/2}}{5 e^7}+\frac {2 c^3 (d+e x)^{7/2}}{7 e^7}\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 391, normalized size = 1.41 \begin {gather*} -\frac {2 \left (c^3 \left (1024 d^6+2560 d^5 e x+1920 d^4 e^2 x^2+320 d^3 e^3 x^3-40 d^2 e^4 x^4+12 d e^5 x^5-5 e^6 x^6\right )+7 e^3 \left (a^3 e^3+a^2 b e^2 (2 d+5 e x)+a b^2 e \left (8 d^2+20 d e x+15 e^2 x^2\right )-b^3 \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )\right )+7 c e^2 \left (a^2 e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )-6 a b e \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )+b^2 \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )\right )-7 c^2 e \left (a e \left (-128 d^4-320 d^3 e x-240 d^2 e^2 x^2-40 d e^3 x^3+5 e^4 x^4\right )+b \left (256 d^5+640 d^4 e x+480 d^3 e^2 x^2+80 d^2 e^3 x^3-10 d e^4 x^4+3 e^5 x^5\right )\right )\right )}{35 e^7 (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.76, size = 506, normalized size = 1.82 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 430, normalized size = 1.55 \begin {gather*} \frac {2}{35} \, {\left ({\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{3} - 21 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (x e + d\right )}^{\frac {5}{2}} + 35 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2} + a c^{2} e^{2}\right )} {\left (x e + d\right )}^{\frac {3}{2}} - 35 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e - b^{3} e^{3} - 6 \, a b c e^{3} + 12 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d\right )} \sqrt {x e + d}\right )} e^{\left (-6\right )} - \frac {7 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d^{4} - 3 \, a^{2} b d e^{5} - {\left (b^{3} e^{3} + 6 \, a b c e^{3}\right )} d^{3} + a^{3} e^{6} + 15 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + a b^{2} e^{4} + a^{2} c e^{4} + 6 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d^{2} - {\left (b^{3} e^{3} + 6 \, a b c e^{3}\right )} d\right )} {\left (x e + d\right )}^{2} + 3 \, {\left (a b^{2} e^{4} + a^{2} c e^{4}\right )} d^{2} - 5 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d^{3} - a^{2} b e^{5} - {\left (b^{3} e^{3} + 6 \, a b c e^{3}\right )} d^{2} + 2 \, {\left (a b^{2} e^{4} + a^{2} c e^{4}\right )} d\right )} {\left (x e + d\right )}\right )} e^{\left (-6\right )}}{{\left (x e + d\right )}^{\frac {5}{2}}}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.69, size = 410, normalized size = 1.47 \begin {gather*} -\frac {2 \, {\left (1024 \, c^{3} d^{6} - {\left (5 \, c^{3} x^{6} + 21 \, b c^{2} x^{5} + 35 \, {\left (b^{2} c + a c^{2}\right )} x^{4} - 35 \, a^{2} b x + 35 \, {\left (b^{3} + 6 \, a b c\right )} x^{3} - 7 \, a^{3} - 105 \, {\left (a b^{2} + a^{2} c\right )} x^{2}\right )} e^{6} + 2 \, {\left (6 \, c^{3} d x^{5} + 35 \, b c^{2} d x^{4} + 140 \, {\left (b^{2} c + a c^{2}\right )} d x^{3} + 7 \, a^{2} b d - 105 \, {\left (b^{3} + 6 \, a b c\right )} d x^{2} + 70 \, {\left (a b^{2} + a^{2} c\right )} d x\right )} e^{5} - 8 \, {\left (5 \, c^{3} d^{2} x^{4} + 70 \, b c^{2} d^{2} x^{3} - 210 \, {\left (b^{2} c + a c^{2}\right )} d^{2} x^{2} + 35 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} x - 7 \, {\left (a b^{2} + a^{2} c\right )} d^{2}\right )} e^{4} + 16 \, {\left (20 \, c^{3} d^{3} x^{3} - 210 \, b c^{2} d^{3} x^{2} + 140 \, {\left (b^{2} c + a c^{2}\right )} d^{3} x - 7 \, {\left (b^{3} + 6 \, a b c\right )} d^{3}\right )} e^{3} + 128 \, {\left (15 \, c^{3} d^{4} x^{2} - 35 \, b c^{2} d^{4} x + 7 \, {\left (b^{2} c + a c^{2}\right )} d^{4}\right )} e^{2} + 256 \, {\left (10 \, c^{3} d^{5} x - 7 \, b c^{2} d^{5}\right )} e\right )} \sqrt {x e + d}}{35 \, {\left (x^{3} e^{10} + 3 \, d x^{2} e^{9} + 3 \, d^{2} x e^{8} + d^{3} e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 114.93, size = 304, normalized size = 1.09 \begin {gather*} \frac {2 c^{3} \left (d + e x\right )^{\frac {7}{2}}}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (6 b c^{2} e - 12 c^{3} d\right )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (6 a c^{2} e^{2} + 6 b^{2} c e^{2} - 30 b c^{2} d e + 30 c^{3} d^{2}\right )}{3 e^{7}} + \frac {\sqrt {d + e x} \left (12 a b c e^{3} - 24 a c^{2} d e^{2} + 2 b^{3} e^{3} - 24 b^{2} c d e^{2} + 60 b c^{2} d^{2} e - 40 c^{3} d^{3}\right )}{e^{7}} - \frac {6 \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{e^{7} \sqrt {d + e x}} - \frac {2 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{e^{7} \left (d + e x\right )^{\frac {3}{2}}} - \frac {2 \left (a e^{2} - b d e + c d^{2}\right )^{3}}{5 e^{7} \left (d + e x\right )^{\frac {5}{2}}} \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. 609 vs.
\(2 (264) = 528\).
time = 1.09, size = 609, normalized size = 2.19 \begin {gather*} \frac {2}{35} \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{3} e^{42} - 42 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{3} d e^{42} + 175 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{2} e^{42} - 700 \, \sqrt {x e + d} c^{3} d^{3} e^{42} + 21 \, {\left (x e + d\right )}^{\frac {5}{2}} b c^{2} e^{43} - 175 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{2} d e^{43} + 1050 \, \sqrt {x e + d} b c^{2} d^{2} e^{43} + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c e^{44} + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{2} e^{44} - 420 \, \sqrt {x e + d} b^{2} c d e^{44} - 420 \, \sqrt {x e + d} a c^{2} d e^{44} + 35 \, \sqrt {x e + d} b^{3} e^{45} + 210 \, \sqrt {x e + d} a b c e^{45}\right )} e^{\left (-49\right )} - \frac {2 \, {\left (75 \, {\left (x e + d\right )}^{2} c^{3} d^{4} - 10 \, {\left (x e + d\right )} c^{3} d^{5} + c^{3} d^{6} - 150 \, {\left (x e + d\right )}^{2} b c^{2} d^{3} e + 25 \, {\left (x e + d\right )} b c^{2} d^{4} e - 3 \, b c^{2} d^{5} e + 90 \, {\left (x e + d\right )}^{2} b^{2} c d^{2} e^{2} + 90 \, {\left (x e + d\right )}^{2} a c^{2} d^{2} e^{2} - 20 \, {\left (x e + d\right )} b^{2} c d^{3} e^{2} - 20 \, {\left (x e + d\right )} a c^{2} d^{3} e^{2} + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - 15 \, {\left (x e + d\right )}^{2} b^{3} d e^{3} - 90 \, {\left (x e + d\right )}^{2} a b c d e^{3} + 5 \, {\left (x e + d\right )} b^{3} d^{2} e^{3} + 30 \, {\left (x e + d\right )} a b c d^{2} e^{3} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 15 \, {\left (x e + d\right )}^{2} a b^{2} e^{4} + 15 \, {\left (x e + d\right )}^{2} a^{2} c e^{4} - 10 \, {\left (x e + d\right )} a b^{2} d e^{4} - 10 \, {\left (x e + d\right )} a^{2} c d e^{4} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} + 5 \, {\left (x e + d\right )} a^{2} b e^{5} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}\right )} e^{\left (-7\right )}}{5 \, {\left (x e + d\right )}^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.89, size = 445, normalized size = 1.60 \begin {gather*} \frac {2\,c^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^7}+\frac {{\left (d+e\,x\right )}^{3/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2+6\,a\,c^2\,e^2\right )}{3\,e^7}-\frac {{\left (d+e\,x\right )}^2\,\left (6\,a^2\,c\,e^4+6\,a\,b^2\,e^4-36\,a\,b\,c\,d\,e^3+36\,a\,c^2\,d^2\,e^2-6\,b^3\,d\,e^3+36\,b^2\,c\,d^2\,e^2-60\,b\,c^2\,d^3\,e+30\,c^3\,d^4\right )-\left (d+e\,x\right )\,\left (-2\,a^2\,b\,e^5+4\,a^2\,c\,d\,e^4+4\,a\,b^2\,d\,e^4-12\,a\,b\,c\,d^2\,e^3+8\,a\,c^2\,d^3\,e^2-2\,b^3\,d^2\,e^3+8\,b^2\,c\,d^3\,e^2-10\,b\,c^2\,d^4\,e+4\,c^3\,d^5\right )+\frac {2\,a^3\,e^6}{5}+\frac {2\,c^3\,d^6}{5}-\frac {2\,b^3\,d^3\,e^3}{5}+\frac {6\,a\,b^2\,d^2\,e^4}{5}+\frac {6\,a\,c^2\,d^4\,e^2}{5}+\frac {6\,a^2\,c\,d^2\,e^4}{5}+\frac {6\,b^2\,c\,d^4\,e^2}{5}-\frac {6\,a^2\,b\,d\,e^5}{5}-\frac {6\,b\,c^2\,d^5\,e}{5}-\frac {12\,a\,b\,c\,d^3\,e^3}{5}}{e^7\,{\left (d+e\,x\right )}^{5/2}}+\frac {2\,\left (b\,e-2\,c\,d\right )\,\sqrt {d+e\,x}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{e^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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