Optimal. Leaf size=162 \[ -\frac {2 \left (c d^2-b d e+a e^2\right )^2}{5 e^5 (d+e x)^{5/2}}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )}{3 e^5 (d+e x)^{3/2}}-\frac {2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )}{e^5 \sqrt {d+e x}}-\frac {4 c (2 c d-b e) \sqrt {d+e x}}{e^5}+\frac {2 c^2 (d+e x)^{3/2}}{3 e^5} \]
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Rubi [A]
time = 0.05, antiderivative size = 162, 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 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^5 \sqrt {d+e x}}+\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5 (d+e x)^{3/2}}-\frac {2 \left (a e^2-b d e+c d^2\right )^2}{5 e^5 (d+e x)^{5/2}}-\frac {4 c \sqrt {d+e x} (2 c d-b e)}{e^5}+\frac {2 c^2 (d+e x)^{3/2}}{3 e^5} \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 )^2}{(d+e x)^{7/2}} \, dx &=\int \left (\frac {\left (c d^2-b d e+a e^2\right )^2}{e^4 (d+e x)^{7/2}}+\frac {2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )}{e^4 (d+e x)^{5/2}}+\frac {6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{e^4 (d+e x)^{3/2}}-\frac {2 c (2 c d-b e)}{e^4 \sqrt {d+e x}}+\frac {c^2 \sqrt {d+e x}}{e^4}\right ) \, dx\\ &=-\frac {2 \left (c d^2-b d e+a e^2\right )^2}{5 e^5 (d+e x)^{5/2}}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )}{3 e^5 (d+e x)^{3/2}}-\frac {2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )}{e^5 \sqrt {d+e x}}-\frac {4 c (2 c d-b e) \sqrt {d+e x}}{e^5}+\frac {2 c^2 (d+e x)^{3/2}}{3 e^5}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 172, normalized size = 1.06 \begin {gather*} -\frac {2 \left (3 a^2 e^4+2 a b e^3 (2 d+5 e x)+b^2 e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )+2 a c e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )-6 b c e \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )+c^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 )}{15 e^5 (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.74, size = 192, normalized size = 1.19
method | result | size |
derivativedivides | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+4 b c e \sqrt {e x +d}-8 c^{2} d \sqrt {e x +d}-\frac {2 \left (2 a c \,e^{2}+b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}\right )}{\sqrt {e x +d}}-\frac {2 \left (a^{2} e^{4}-2 d \,e^{3} a b +2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 d^{3} e b c +c^{2} d^{4}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 \left (2 a b \,e^{3}-4 a d \,e^{2} c -2 b^{2} d \,e^{2}+6 d^{2} e b c -4 c^{2} d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{5}}\) | \(192\) |
default | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+4 b c e \sqrt {e x +d}-8 c^{2} d \sqrt {e x +d}-\frac {2 \left (2 a c \,e^{2}+b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}\right )}{\sqrt {e x +d}}-\frac {2 \left (a^{2} e^{4}-2 d \,e^{3} a b +2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 d^{3} e b c +c^{2} d^{4}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 \left (2 a b \,e^{3}-4 a d \,e^{2} c -2 b^{2} d \,e^{2}+6 d^{2} e b c -4 c^{2} d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{5}}\) | \(192\) |
gosper | \(-\frac {2 \left (-5 c^{2} x^{4} e^{4}-30 b c \,e^{4} x^{3}+40 c^{2} d \,e^{3} x^{3}+30 a c \,e^{4} x^{2}+15 b^{2} e^{4} x^{2}-180 b c d \,e^{3} x^{2}+240 c^{2} d^{2} e^{2} x^{2}+10 a b \,e^{4} x +40 a c d \,e^{3} x +20 b^{2} d \,e^{3} x -240 b c \,d^{2} e^{2} x +320 c^{2} d^{3} e x +3 a^{2} e^{4}+4 d \,e^{3} a b +16 a c \,d^{2} e^{2}+8 b^{2} d^{2} e^{2}-96 d^{3} e b c +128 c^{2} d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{5}}\) | \(194\) |
trager | \(-\frac {2 \left (-5 c^{2} x^{4} e^{4}-30 b c \,e^{4} x^{3}+40 c^{2} d \,e^{3} x^{3}+30 a c \,e^{4} x^{2}+15 b^{2} e^{4} x^{2}-180 b c d \,e^{3} x^{2}+240 c^{2} d^{2} e^{2} x^{2}+10 a b \,e^{4} x +40 a c d \,e^{3} x +20 b^{2} d \,e^{3} x -240 b c \,d^{2} e^{2} x +320 c^{2} d^{3} e x +3 a^{2} e^{4}+4 d \,e^{3} a b +16 a c \,d^{2} e^{2}+8 b^{2} d^{2} e^{2}-96 d^{3} e b c +128 c^{2} d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{5}}\) | \(194\) |
risch | \(\frac {2 c \left (c e x +6 b e -11 c d \right ) \sqrt {e x +d}}{3 e^{5}}-\frac {2 \left (30 a c \,e^{4} x^{2}+15 b^{2} e^{4} x^{2}-90 b c d \,e^{3} x^{2}+90 c^{2} d^{2} e^{2} x^{2}+10 a b \,e^{4} x +40 a c d \,e^{3} x +20 b^{2} d \,e^{3} x -150 b c \,d^{2} e^{2} x +160 c^{2} d^{3} e x +3 a^{2} e^{4}+4 d \,e^{3} a b +16 a c \,d^{2} e^{2}+8 b^{2} d^{2} e^{2}-66 d^{3} e b c +73 c^{2} d^{4}\right )}{15 e^{5} \sqrt {e x +d}\, \left (e^{2} x^{2}+2 d x e +d^{2}\right )}\) | \(206\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 192, normalized size = 1.19 \begin {gather*} \frac {2}{15} \, {\left (5 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} c^{2} - 6 \, {\left (2 \, c^{2} d - b c e\right )} \sqrt {x e + d}\right )} e^{\left (-4\right )} - \frac {{\left (3 \, c^{2} d^{4} - 6 \, b c d^{3} e - 6 \, a b d e^{3} + 15 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2} + 2 \, a c e^{2}\right )} {\left (x e + d\right )}^{2} + 3 \, {\left (b^{2} e^{2} + 2 \, a c e^{2}\right )} d^{2} + 3 \, a^{2} e^{4} - 10 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} + {\left (b^{2} e^{2} + 2 \, a c e^{2}\right )} d\right )} {\left (x e + d\right )}\right )} e^{\left (-4\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 = 2.98, size = 188, normalized size = 1.16 \begin {gather*} -\frac {2 \, {\left (128 \, c^{2} d^{4} - {\left (5 \, c^{2} x^{4} + 30 \, b c x^{3} - 10 \, a b x - 15 \, {\left (b^{2} + 2 \, a c\right )} x^{2} - 3 \, a^{2}\right )} e^{4} + 4 \, {\left (10 \, c^{2} d x^{3} - 45 \, b c d x^{2} + a b d + 5 \, {\left (b^{2} + 2 \, a c\right )} d x\right )} e^{3} + 8 \, {\left (30 \, c^{2} d^{2} x^{2} - 30 \, b c d^{2} x + {\left (b^{2} + 2 \, a c\right )} d^{2}\right )} e^{2} + 32 \, {\left (10 \, c^{2} d^{3} x - 3 \, b c d^{3}\right )} e\right )} \sqrt {x e + d}}{15 \, {\left (x^{3} e^{8} + 3 \, d x^{2} e^{7} + 3 \, d^{2} x e^{6} + d^{3} e^{5}\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. 1180 vs.
\(2 (156) = 312\).
time = 0.78, size = 1180, normalized size = 7.28 \begin {gather*} \begin {cases} - \frac {6 a^{2} e^{4}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {8 a b d e^{3}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {20 a b e^{4} x}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {32 a c d^{2} e^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {80 a c d e^{3} x}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {60 a c e^{4} x^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {16 b^{2} d^{2} e^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {40 b^{2} d e^{3} x}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {30 b^{2} e^{4} x^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {192 b c d^{3} e}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {480 b c d^{2} e^{2} x}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {360 b c d e^{3} x^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {60 b c e^{4} x^{3}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {256 c^{2} d^{4}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {640 c^{2} d^{3} e x}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {480 c^{2} d^{2} e^{2} x^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {80 c^{2} d e^{3} x^{3}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {10 c^{2} e^{4} x^{4}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {a^{2} x + a b x^{2} + \frac {2 a c x^{3}}{3} + \frac {b^{2} x^{3}}{3} + \frac {b c x^{4}}{2} + \frac {c^{2} x^{5}}{5}}{d^{\frac {7}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.64, size = 241, normalized size = 1.49 \begin {gather*} \frac {2}{3} \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} c^{2} e^{10} - 12 \, \sqrt {x e + d} c^{2} d e^{10} + 6 \, \sqrt {x e + d} b c e^{11}\right )} e^{\left (-15\right )} - \frac {2 \, {\left (90 \, {\left (x e + d\right )}^{2} c^{2} d^{2} - 20 \, {\left (x e + d\right )} c^{2} d^{3} + 3 \, c^{2} d^{4} - 90 \, {\left (x e + d\right )}^{2} b c d e + 30 \, {\left (x e + d\right )} b c d^{2} e - 6 \, b c d^{3} e + 15 \, {\left (x e + d\right )}^{2} b^{2} e^{2} + 30 \, {\left (x e + d\right )}^{2} a c e^{2} - 10 \, {\left (x e + d\right )} b^{2} d e^{2} - 20 \, {\left (x e + d\right )} a c d e^{2} + 3 \, b^{2} d^{2} e^{2} + 6 \, a c d^{2} e^{2} + 10 \, {\left (x e + d\right )} a b e^{3} - 6 \, a b d e^{3} + 3 \, a^{2} e^{4}\right )} e^{\left (-5\right )}}{15 \, {\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 = 0.90, size = 193, normalized size = 1.19 \begin {gather*} -\frac {2\,\left (3\,a^2\,e^4+4\,a\,b\,d\,e^3+10\,a\,b\,e^4\,x+16\,a\,c\,d^2\,e^2+40\,a\,c\,d\,e^3\,x+30\,a\,c\,e^4\,x^2+8\,b^2\,d^2\,e^2+20\,b^2\,d\,e^3\,x+15\,b^2\,e^4\,x^2-96\,b\,c\,d^3\,e-240\,b\,c\,d^2\,e^2\,x-180\,b\,c\,d\,e^3\,x^2-30\,b\,c\,e^4\,x^3+128\,c^2\,d^4+320\,c^2\,d^3\,e\,x+240\,c^2\,d^2\,e^2\,x^2+40\,c^2\,d\,e^3\,x^3-5\,c^2\,e^4\,x^4\right )}{15\,e^5\,{\left (d+e\,x\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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