Optimal. Leaf size=125 \[ -\frac {2 (b d-a e)^4}{3 e^5 (d+e x)^{3/2}}+\frac {8 b (b d-a e)^3}{e^5 \sqrt {d+e x}}+\frac {12 b^2 (b d-a e)^2 \sqrt {d+e x}}{e^5}-\frac {8 b^3 (b d-a e) (d+e x)^{3/2}}{3 e^5}+\frac {2 b^4 (d+e x)^{5/2}}{5 e^5} \]
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Rubi [A]
time = 0.03, antiderivative size = 125, 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, 45}
\begin {gather*} -\frac {8 b^3 (d+e x)^{3/2} (b d-a e)}{3 e^5}+\frac {12 b^2 \sqrt {d+e x} (b d-a e)^2}{e^5}+\frac {8 b (b d-a e)^3}{e^5 \sqrt {d+e x}}-\frac {2 (b d-a e)^4}{3 e^5 (d+e x)^{3/2}}+\frac {2 b^4 (d+e x)^{5/2}}{5 e^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 45
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{5/2}} \, dx &=\int \frac {(a+b x)^4}{(d+e x)^{5/2}} \, dx\\ &=\int \left (\frac {(-b d+a e)^4}{e^4 (d+e x)^{5/2}}-\frac {4 b (b d-a e)^3}{e^4 (d+e x)^{3/2}}+\frac {6 b^2 (b d-a e)^2}{e^4 \sqrt {d+e x}}-\frac {4 b^3 (b d-a e) \sqrt {d+e x}}{e^4}+\frac {b^4 (d+e x)^{3/2}}{e^4}\right ) \, dx\\ &=-\frac {2 (b d-a e)^4}{3 e^5 (d+e x)^{3/2}}+\frac {8 b (b d-a e)^3}{e^5 \sqrt {d+e x}}+\frac {12 b^2 (b d-a e)^2 \sqrt {d+e x}}{e^5}-\frac {8 b^3 (b d-a e) (d+e x)^{3/2}}{3 e^5}+\frac {2 b^4 (d+e x)^{5/2}}{5 e^5}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 153, normalized size = 1.22 \begin {gather*} \frac {2 \left (-5 a^4 e^4-20 a^3 b e^3 (2 d+3 e x)+30 a^2 b^2 e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )+20 a b^3 e \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )+b^4 \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )\right )}{15 e^5 (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.67, size = 198, normalized size = 1.58
method | result | size |
risch | \(\frac {2 b^{2} \left (3 b^{2} x^{2} e^{2}+20 a b \,e^{2} x -14 b^{2} d e x +90 a^{2} e^{2}-160 a b d e +73 b^{2} d^{2}\right ) \sqrt {e x +d}}{15 e^{5}}-\frac {2 \left (12 b e x +a e +11 b d \right ) \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{3 e^{5} \left (e x +d \right )^{\frac {3}{2}}}\) | \(128\) |
gosper | \(-\frac {2 \left (-3 b^{4} x^{4} e^{4}-20 a \,b^{3} e^{4} x^{3}+8 b^{4} d \,e^{3} x^{3}-90 a^{2} b^{2} e^{4} x^{2}+120 a \,b^{3} d \,e^{3} x^{2}-48 b^{4} d^{2} e^{2} x^{2}+60 a^{3} b \,e^{4} x -360 a^{2} b^{2} d \,e^{3} x +480 a \,b^{3} d^{2} e^{2} x -192 b^{4} d^{3} e x +5 e^{4} a^{4}+40 a^{3} b d \,e^{3}-240 a^{2} b^{2} d^{2} e^{2}+320 a \,b^{3} d^{3} e -128 b^{4} d^{4}\right )}{15 \left (e x +d \right )^{\frac {3}{2}} e^{5}}\) | \(186\) |
trager | \(-\frac {2 \left (-3 b^{4} x^{4} e^{4}-20 a \,b^{3} e^{4} x^{3}+8 b^{4} d \,e^{3} x^{3}-90 a^{2} b^{2} e^{4} x^{2}+120 a \,b^{3} d \,e^{3} x^{2}-48 b^{4} d^{2} e^{2} x^{2}+60 a^{3} b \,e^{4} x -360 a^{2} b^{2} d \,e^{3} x +480 a \,b^{3} d^{2} e^{2} x -192 b^{4} d^{3} e x +5 e^{4} a^{4}+40 a^{3} b d \,e^{3}-240 a^{2} b^{2} d^{2} e^{2}+320 a \,b^{3} d^{3} e -128 b^{4} d^{4}\right )}{15 \left (e x +d \right )^{\frac {3}{2}} e^{5}}\) | \(186\) |
derivativedivides | \(\frac {\frac {2 b^{4} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {8 a \,b^{3} e \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {8 b^{4} d \left (e x +d \right )^{\frac {3}{2}}}{3}+12 a^{2} b^{2} e^{2} \sqrt {e x +d}-24 a \,b^{3} d e \sqrt {e x +d}+12 b^{4} d^{2} \sqrt {e x +d}-\frac {2 \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {8 b \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{\sqrt {e x +d}}}{e^{5}}\) | \(198\) |
default | \(\frac {\frac {2 b^{4} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {8 a \,b^{3} e \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {8 b^{4} d \left (e x +d \right )^{\frac {3}{2}}}{3}+12 a^{2} b^{2} e^{2} \sqrt {e x +d}-24 a \,b^{3} d e \sqrt {e x +d}+12 b^{4} d^{2} \sqrt {e x +d}-\frac {2 \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {8 b \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{\sqrt {e x +d}}}{e^{5}}\) | \(198\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 187, normalized size = 1.50 \begin {gather*} \frac {2}{15} \, {\left ({\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{4} - 20 \, {\left (b^{4} d - a b^{3} e\right )} {\left (x e + d\right )}^{\frac {3}{2}} + 90 \, {\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} \sqrt {x e + d}\right )} e^{\left (-4\right )} - \frac {5 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4} - 12 \, {\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )} {\left (x e + d\right )}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{\frac {3}{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 = 2.05, size = 183, normalized size = 1.46 \begin {gather*} \frac {2 \, {\left (128 \, b^{4} d^{4} + {\left (3 \, b^{4} x^{4} + 20 \, a b^{3} x^{3} + 90 \, a^{2} b^{2} x^{2} - 60 \, a^{3} b x - 5 \, a^{4}\right )} e^{4} - 8 \, {\left (b^{4} d x^{3} + 15 \, a b^{3} d x^{2} - 45 \, a^{2} b^{2} d x + 5 \, a^{3} b d\right )} e^{3} + 48 \, {\left (b^{4} d^{2} x^{2} - 10 \, a b^{3} d^{2} x + 5 \, a^{2} b^{2} d^{2}\right )} e^{2} + 64 \, {\left (3 \, b^{4} d^{3} x - 5 \, a b^{3} d^{3}\right )} e\right )} \sqrt {x e + d}}{15 \, {\left (x^{2} e^{7} + 2 \, d x e^{6} + d^{2} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 18.44, size = 136, normalized size = 1.09 \begin {gather*} \frac {2 b^{4} \left (d + e x\right )^{\frac {5}{2}}}{5 e^{5}} - \frac {8 b \left (a e - b d\right )^{3}}{e^{5} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (8 a b^{3} e - 8 b^{4} d\right )}{3 e^{5}} + \frac {\sqrt {d + e x} \left (12 a^{2} b^{2} e^{2} - 24 a b^{3} d e + 12 b^{4} d^{2}\right )}{e^{5}} - \frac {2 \left (a e - b d\right )^{4}}{3 e^{5} \left (d + e x\right )^{\frac {3}{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. 229 vs.
\(2 (113) = 226\).
time = 1.13, size = 229, normalized size = 1.83 \begin {gather*} \frac {2}{15} \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{4} e^{20} - 20 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{4} d e^{20} + 90 \, \sqrt {x e + d} b^{4} d^{2} e^{20} + 20 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{3} e^{21} - 180 \, \sqrt {x e + d} a b^{3} d e^{21} + 90 \, \sqrt {x e + d} a^{2} b^{2} e^{22}\right )} e^{\left (-25\right )} + \frac {2 \, {\left (12 \, {\left (x e + d\right )} b^{4} d^{3} - b^{4} d^{4} - 36 \, {\left (x e + d\right )} a b^{3} d^{2} e + 4 \, a b^{3} d^{3} e + 36 \, {\left (x e + d\right )} a^{2} b^{2} d e^{2} - 6 \, a^{2} b^{2} d^{2} e^{2} - 12 \, {\left (x e + d\right )} a^{3} b e^{3} + 4 \, a^{3} b d e^{3} - a^{4} e^{4}\right )} e^{\left (-5\right )}}{3 \, {\left (x e + d\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 175, normalized size = 1.40 \begin {gather*} \frac {\left (d+e\,x\right )\,\left (-8\,a^3\,b\,e^3+24\,a^2\,b^2\,d\,e^2-24\,a\,b^3\,d^2\,e+8\,b^4\,d^3\right )-\frac {2\,a^4\,e^4}{3}-\frac {2\,b^4\,d^4}{3}-4\,a^2\,b^2\,d^2\,e^2+\frac {8\,a\,b^3\,d^3\,e}{3}+\frac {8\,a^3\,b\,d\,e^3}{3}}{e^5\,{\left (d+e\,x\right )}^{3/2}}+\frac {2\,b^4\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5}-\frac {\left (8\,b^4\,d-8\,a\,b^3\,e\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,e^5}+\frac {12\,b^2\,{\left (a\,e-b\,d\right )}^2\,\sqrt {d+e\,x}}{e^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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