Optimal. Leaf size=111 \[ \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 \left (c d^2-a e^2\right ) (d+e x)^6}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{35 \left (c d^2-a e^2\right )^2 (d+e x)^5} \]
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Rubi [A]
time = 0.03, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {672, 664}
\begin {gather*} \frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{35 (d+e x)^5 \left (c d^2-a e^2\right )^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 (d+e x)^6 \left (c d^2-a e^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 664
Rule 672
Rubi steps
\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^6} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 \left (c d^2-a e^2\right ) (d+e x)^6}+\frac {(2 c d) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^5} \, dx}{7 \left (c d^2-a e^2\right )}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 \left (c d^2-a e^2\right ) (d+e x)^6}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{35 \left (c d^2-a e^2\right )^2 (d+e x)^5}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 61, normalized size = 0.55 \begin {gather*} \frac {2 ((a e+c d x) (d+e x))^{5/2} \left (-5 a e^2+c d (7 d+2 e x)\right )}{35 \left (c d^2-a e^2\right )^2 (d+e x)^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.79, size = 131, normalized size = 1.18
method | result | size |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-2 c d e x +5 e^{2} a -7 c \,d^{2}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{35 \left (e x +d \right )^{5} \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}\) | \(90\) |
default | \(\frac {-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{7 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{6}}+\frac {4 c d e \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{35 \left (e^{2} a -c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )^{5}}}{e^{6}}\) | \(131\) |
trager | \(-\frac {2 \left (-2 c^{3} d^{3} e \,x^{3}+d^{2} e^{2} c^{2} a \,x^{2}-7 c^{3} d^{4} x^{2}+8 a^{2} c d \,e^{3} x -14 d^{3} e \,c^{2} a x +5 e^{4} a^{3}-7 d^{2} e^{2} a^{2} c \right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{35 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (e x +d \right )^{4}}\) | \(143\) |
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. 275 vs.
\(2 (105) = 210\).
time = 7.71, size = 275, normalized size = 2.48 \begin {gather*} \frac {2 \, {\left (7 \, c^{3} d^{4} x^{2} - 8 \, a^{2} c d x e^{3} - 5 \, a^{3} e^{4} - {\left (a c^{2} d^{2} x^{2} - 7 \, a^{2} c d^{2}\right )} e^{2} + 2 \, {\left (c^{3} d^{3} x^{3} + 7 \, a c^{2} d^{3} x\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{35 \, {\left (4 \, c^{2} d^{7} x e + c^{2} d^{8} + a^{2} x^{4} e^{8} + 4 \, a^{2} d x^{3} e^{7} - 2 \, {\left (a c d^{2} x^{4} - 3 \, a^{2} d^{2} x^{2}\right )} e^{6} - 4 \, {\left (2 \, a c d^{3} x^{3} - a^{2} d^{3} x\right )} e^{5} + {\left (c^{2} d^{4} x^{4} - 12 \, a c d^{4} x^{2} + a^{2} d^{4}\right )} e^{4} + 4 \, {\left (c^{2} d^{5} x^{3} - 2 \, a c d^{5} x\right )} e^{3} + 2 \, {\left (3 \, c^{2} d^{6} x^{2} - a c d^{6}\right )} e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.77, size = 1477, normalized size = 13.31 \begin {gather*} \frac {\left (\frac {d\,\left (\frac {4\,c^3\,d^4}{7\,\left (a\,e^2-c\,d^2\right )\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )}-\frac {2\,c^2\,d^2\,\left (5\,a\,e^2-c\,d^2\right )}{7\,\left (a\,e^2-c\,d^2\right )\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )}\right )}{e}+\frac {4\,a^2\,c\,d\,e^4+2\,a\,c^2\,d^3\,e^2-2\,c^3\,d^5}{7\,e\,\left (a\,e^2-c\,d^2\right )\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^3}-\frac {\left (\frac {14\,c^3\,d^4-34\,a\,c^2\,d^2\,e^2}{35\,e\,\left (a\,e^2-c\,d^2\right )\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}+\frac {4\,c^3\,d^4}{7\,e\,\left (a\,e^2-c\,d^2\right )\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^2}-\frac {\left (\frac {d\,\left (\frac {56\,c^4\,d^5-72\,a\,c^3\,d^3\,e^2}{35\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}+\frac {8\,c^4\,d^5}{35\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}\right )}{e}+\frac {8\,a\,c^2\,d^2\,e\,\left (8\,a\,e^2-7\,c\,d^2\right )}{35\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^2}+\frac {\left (\frac {d\,\left (\frac {16\,c^5\,d^6}{105\,e\,{\left (a\,e^2-c\,d^2\right )}^4}-\frac {8\,c^4\,d^4\,\left (19\,a\,e^2-15\,c\,d^2\right )}{105\,e\,{\left (a\,e^2-c\,d^2\right )}^4}\right )}{e}+\frac {8\,c^3\,d^3\,\left (9\,a^2\,e^4+a\,c\,d^2\,e^2-8\,c^2\,d^4\right )}{105\,e^2\,{\left (a\,e^2-c\,d^2\right )}^4}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x}-\frac {\left (\frac {2\,a^2\,e^3}{7\,a\,e^3-7\,c\,d^2\,e}+\frac {d\,\left (\frac {2\,c^2\,d^3}{7\,a\,e^3-7\,c\,d^2\,e}-\frac {4\,a\,c\,d\,e^2}{7\,a\,e^3-7\,c\,d^2\,e}\right )}{e}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^4}-\frac {\left (\frac {28\,c^4\,d^5-36\,a\,c^3\,d^3\,e^2}{35\,e^2\,{\left (a\,e^2-c\,d^2\right )}^3}+\frac {8\,c^4\,d^5}{35\,e^2\,{\left (a\,e^2-c\,d^2\right )}^3}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x}-\frac {\left (\frac {d\,\left (\frac {16\,c^5\,d^6}{105\,e\,{\left (a\,e^2-c\,d^2\right )}^4}-\frac {16\,c^4\,d^4\,\left (11\,a\,e^2-9\,c\,d^2\right )}{105\,e\,{\left (a\,e^2-c\,d^2\right )}^4}\right )}{e}+\frac {16\,a\,c^3\,d^3\,\left (10\,a\,e^2-9\,c\,d^2\right )}{105\,{\left (a\,e^2-c\,d^2\right )}^4}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x}-\frac {\left (\frac {d\,\left (\frac {16\,c^3\,d^4-24\,a\,c^2\,d^2\,e^2}{7\,\left (a\,e^2-c\,d^2\right )\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )}+\frac {4\,c^3\,d^4}{7\,\left (a\,e^2-c\,d^2\right )\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )}\right )}{e}+\frac {4\,a\,c\,d\,e\,\left (5\,a\,e^2-4\,c\,d^2\right )}{7\,\left (a\,e^2-c\,d^2\right )\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^3}+\frac {\left (\frac {d\,\left (\frac {8\,c^4\,d^5}{35\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}-\frac {4\,c^3\,d^3\,\left (13\,a\,e^2-9\,c\,d^2\right )}{35\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}\right )}{e}+\frac {24\,a^2\,c^2\,d^2\,e^4+4\,a\,c^3\,d^4\,e^2-20\,c^4\,d^6}{35\,e\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^2}-\frac {8\,c^3\,d^3\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{105\,e^2\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (d+e\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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