Optimal. Leaf size=143 \[ -\frac {2 d^2 (c d-b e)^2}{5 e^5 (d+e x)^{5/2}}+\frac {4 d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^{3/2}}-\frac {2 \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )}{e^5 \sqrt {d+e x}}-\frac {4 c (2 c d-b e) \sqrt {d+e x}}{e^5}+\frac {2 c^2 (d+e x)^{3/2}}{3 e^5} \]
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Rubi [A]
time = 0.04, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {712}
\begin {gather*} -\frac {2 \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{e^5 \sqrt {d+e x}}-\frac {2 d^2 (c d-b e)^2}{5 e^5 (d+e x)^{5/2}}-\frac {4 c \sqrt {d+e x} (2 c d-b e)}{e^5}+\frac {4 d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^{3/2}}+\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 (b x+c x^2\right )^2}{(d+e x)^{7/2}} \, dx &=\int \left (\frac {d^2 (c d-b e)^2}{e^4 (d+e x)^{7/2}}+\frac {2 d (c d-b e) (-2 c d+b e)}{e^4 (d+e x)^{5/2}}+\frac {6 c^2 d^2-6 b c d e+b^2 e^2}{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 d^2 (c d-b e)^2}{5 e^5 (d+e x)^{5/2}}+\frac {4 d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^{3/2}}-\frac {2 \left (6 c^2 d^2-6 b c d e+b^2 e^2\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 = 0.08, size = 123, normalized size = 0.86 \begin {gather*} -\frac {2 \left (b^2 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 = 0.46, size = 140, normalized size = 0.98
method | result | size |
derivativedivides | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+4 b c e \sqrt {e x +d}-8 c^{2} d \sqrt {e x +d}-\frac {2 \left (b^{2} e^{2}-6 b c d e +6 d^{2} c^{2}\right )}{\sqrt {e x +d}}+\frac {4 d \left (b^{2} e^{2}-3 b c d e +2 d^{2} c^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 d^{2} \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}}{e^{5}}\) | \(140\) |
default | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+4 b c e \sqrt {e x +d}-8 c^{2} d \sqrt {e x +d}-\frac {2 \left (b^{2} e^{2}-6 b c d e +6 d^{2} c^{2}\right )}{\sqrt {e x +d}}+\frac {4 d \left (b^{2} e^{2}-3 b c d e +2 d^{2} c^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 d^{2} \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}}{e^{5}}\) | \(140\) |
gosper | \(-\frac {2 \left (-5 c^{2} x^{4} e^{4}-30 b c \,e^{4} x^{3}+40 c^{2} d \,e^{3} x^{3}+15 b^{2} e^{4} x^{2}-180 b c d \,e^{3} x^{2}+240 c^{2} d^{2} e^{2} x^{2}+20 b^{2} d \,e^{3} x -240 b c \,d^{2} e^{2} x +320 c^{2} d^{3} e x +8 d^{2} e^{2} b^{2}-96 b c \,d^{3} e +128 c^{2} d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{5}}\) | \(141\) |
trager | \(-\frac {2 \left (-5 c^{2} x^{4} e^{4}-30 b c \,e^{4} x^{3}+40 c^{2} d \,e^{3} x^{3}+15 b^{2} e^{4} x^{2}-180 b c d \,e^{3} x^{2}+240 c^{2} d^{2} e^{2} x^{2}+20 b^{2} d \,e^{3} x -240 b c \,d^{2} e^{2} x +320 c^{2} d^{3} e x +8 d^{2} e^{2} b^{2}-96 b c \,d^{3} e +128 c^{2} d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{5}}\) | \(141\) |
risch | \(\frac {2 c \left (c e x +6 b e -11 c d \right ) \sqrt {e x +d}}{3 e^{5}}-\frac {2 \left (15 b^{2} e^{4} x^{2}-90 b c d \,e^{3} x^{2}+90 c^{2} d^{2} e^{2} x^{2}+20 b^{2} d \,e^{3} x -150 b c \,d^{2} e^{2} x +160 c^{2} d^{3} e x +8 d^{2} e^{2} b^{2}-66 b c \,d^{3} e +73 c^{2} d^{4}\right )}{15 e^{5} \sqrt {e x +d}\, \left (x^{2} e^{2}+2 d x e +d^{2}\right )}\) | \(153\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 150, normalized size = 1.05 \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 + 3 \, b^{2} d^{2} e^{2} + 15 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )} {\left (x e + d\right )}^{2} - 10 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\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 = 1.50, size = 157, normalized size = 1.10 \begin {gather*} -\frac {2 \, {\left (128 \, c^{2} d^{4} - 5 \, {\left (c^{2} x^{4} + 6 \, b c x^{3} - 3 \, b^{2} x^{2}\right )} e^{4} + 20 \, {\left (2 \, c^{2} d x^{3} - 9 \, b c d x^{2} + b^{2} d x\right )} e^{3} + 8 \, {\left (30 \, c^{2} d^{2} x^{2} - 30 \, b c d^{2} x + b^{2} 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. 787 vs.
\(2 (138) = 276\).
time = 0.71, size = 787, normalized size = 5.50 \begin {gather*} \begin {cases} - \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 {\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 = 1.09, size = 179, normalized size = 1.25 \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} - 10 \, {\left (x e + d\right )} b^{2} d e^{2} + 3 \, b^{2} d^{2} e^{2}\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 = 0.24, size = 140, normalized size = 0.98 \begin {gather*} -\frac {2\,\left (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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