Optimal. Leaf size=118 \[ -\frac {2 (d+e x)}{3 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {8 e \left (c d^2+a e^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {652, 627}
\begin {gather*} \frac {8 e \left (a e^2+c d^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 (d+e x)}{3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 627
Rule 652
Rubi steps
\begin {align*} \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=-\frac {2 (d+e x)}{3 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {(4 e) \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 \left (c d^2-a e^2\right )}\\ &=-\frac {2 (d+e x)}{3 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {8 e \left (c d^2+a e^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 90, normalized size = 0.76 \begin {gather*} \frac {2 (d+e x) \left (3 a^2 e^4+6 a c d e^2 (d+2 e x)+c^2 d^2 \left (-d^2+4 d e x+8 e^2 x^2\right )\right )}{3 \left (c d^2-a e^2\right )^3 ((a e+c d x) (d+e x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(370\) vs.
\(2(110)=220\).
time = 0.65, size = 371, normalized size = 3.14
method | result | size |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (e x +d \right )^{2} \left (8 e^{2} x^{2} c^{2} d^{2}+12 a c d \,e^{3} x +4 c^{2} d^{3} e x +3 a^{2} e^{4}+6 a c \,d^{2} e^{2}-c^{2} d^{4}\right )}{3 \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}\) | \(146\) |
trager | \(-\frac {2 \left (8 e^{2} x^{2} c^{2} d^{2}+12 a c d \,e^{3} x +4 c^{2} d^{3} e x +3 a^{2} e^{4}+6 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{3 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (c d x +a e \right )^{2} \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )}\) | \(147\) |
default | \(e \left (-\frac {1}{3 c d e \left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}-\frac {\left (e^{2} a +c \,d^{2}\right ) \left (\frac {\frac {4}{3} c d e x +\frac {2}{3} e^{2} a +\frac {2}{3} c \,d^{2}}{\left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}+\frac {16 c d e \left (2 c d e x +e^{2} a +c \,d^{2}\right )}{3 \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right )^{2} \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{2 c d e}\right )+d \left (\frac {\frac {4}{3} c d e x +\frac {2}{3} e^{2} a +\frac {2}{3} c \,d^{2}}{\left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}+\frac {16 c d e \left (2 c d e x +e^{2} a +c \,d^{2}\right )}{3 \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right )^{2} \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}\right )\) | \(371\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 320 vs.
\(2 (112) = 224\).
time = 7.89, size = 320, normalized size = 2.71 \begin {gather*} \frac {2 \, {\left (4 \, c^{2} d^{3} x e - c^{2} d^{4} + 12 \, a c d x e^{3} + 3 \, a^{2} e^{4} + 2 \, {\left (4 \, c^{2} d^{2} x^{2} + 3 \, a c d^{2}\right )} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{3 \, {\left (c^{5} d^{9} x^{2} - a^{5} x e^{9} - {\left (2 \, a^{4} c d x^{2} + a^{5} d\right )} e^{8} - {\left (a^{3} c^{2} d^{2} x^{3} - a^{4} c d^{2} x\right )} e^{7} + {\left (5 \, a^{3} c^{2} d^{3} x^{2} + 3 \, a^{4} c d^{3}\right )} e^{6} + 3 \, {\left (a^{2} c^{3} d^{4} x^{3} + a^{3} c^{2} d^{4} x\right )} e^{5} - 3 \, {\left (a^{2} c^{3} d^{5} x^{2} + a^{3} c^{2} d^{5}\right )} e^{4} - {\left (3 \, a c^{4} d^{6} x^{3} + 5 \, a^{2} c^{3} d^{6} x\right )} e^{3} - {\left (a c^{4} d^{7} x^{2} - a^{2} c^{3} d^{7}\right )} e^{2} + {\left (c^{5} d^{8} x^{3} + 2 \, a c^{4} d^{8} x\right )} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d + e x}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.14, size = 120, normalized size = 1.02 \begin {gather*} -\frac {2\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (3\,a^2\,e^4+6\,a\,c\,d^2\,e^2+12\,a\,c\,d\,e^3\,x-c^2\,d^4+4\,c^2\,d^3\,e\,x+8\,c^2\,d^2\,e^2\,x^2\right )}{3\,{\left (a\,e+c\,d\,x\right )}^2\,{\left (a\,e^2-c\,d^2\right )}^3\,\left (d+e\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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