Optimal. Leaf size=181 \[ \frac {2}{5 \left (c d^2-a e^2\right ) (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {4 c d}{5 \left (c d^2-a e^2\right )^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {16 c^2 d^2 \left (c d^2+a e^2+2 c d e x\right )}{5 \left (c d^2-a e^2\right )^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {672, 627}
\begin {gather*} -\frac {16 c^2 d^2 \left (a e^2+c d^2+2 c d e x\right )}{5 \left (c d^2-a e^2\right )^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {4 c d}{5 (d+e x) \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {2}{5 (d+e x)^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 627
Rule 672
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=\frac {2}{5 \left (c d^2-a e^2\right ) (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {(6 c d) \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{5 \left (c d^2-a e^2\right )}\\ &=\frac {2}{5 \left (c d^2-a e^2\right ) (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {4 c d}{5 \left (c d^2-a e^2\right )^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (8 c^2 d^2\right ) \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}{5 \left (c d^2-a e^2\right )^2}\\ &=\frac {2}{5 \left (c d^2-a e^2\right ) (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {4 c d}{5 \left (c d^2-a e^2\right )^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {16 c^2 d^2 \left (c d^2+a e^2+2 c d e x\right )}{5 \left (c d^2-a e^2\right )^4 \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 = 0.16, size = 123, normalized size = 0.68 \begin {gather*} -\frac {2 (a e+c d x)^5 \left (e^3-\frac {5 c d e^2 (d+e x)}{a e+c d x}+\frac {15 c^2 d^2 e (d+e x)^2}{(a e+c d x)^2}+\frac {5 c^3 d^3 (d+e x)^3}{(a e+c d x)^3}\right )}{5 \left (c d^2-a e^2\right )^4 ((a e+c d x) (d+e x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.72, size = 227, normalized size = 1.25
method | result | size |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (16 c^{3} d^{3} e^{3} x^{3}+8 a \,c^{2} d^{2} e^{4} x^{2}+40 c^{3} d^{4} e^{2} x^{2}-2 a^{2} c d \,e^{5} x +20 a \,c^{2} d^{3} e^{3} x +30 c^{3} d^{5} e x +e^{6} a^{3}-5 e^{4} d^{2} a^{2} c +15 d^{4} e^{2} c^{2} a +5 d^{6} c^{3}\right )}{5 \left (e x +d \right ) \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\) | \(216\) |
trager | \(-\frac {2 \left (16 c^{3} d^{3} e^{3} x^{3}+8 a \,c^{2} d^{2} e^{4} x^{2}+40 c^{3} d^{4} e^{2} x^{2}-2 a^{2} c d \,e^{5} x +20 a \,c^{2} d^{3} e^{3} x +30 c^{3} d^{5} e x +e^{6} a^{3}-5 e^{4} d^{2} a^{2} c +15 d^{4} e^{2} c^{2} a +5 d^{6} c^{3}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{5 \left (c d x +a e \right ) \left (e^{2} a -c \,d^{2}\right ) \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right ) \left (e x +d \right )^{3}}\) | \(219\) |
default | \(\frac {-\frac {2}{5 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}-\frac {6 c d e \left (-\frac {2}{3 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}+\frac {8 c d e \left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right )}{3 \left (e^{2} a -c \,d^{2}\right )^{3} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}\right )}{5 \left (e^{2} a -c \,d^{2}\right )}}{e^{2}}\) | \(227\) |
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. 503 vs.
\(2 (171) = 342\).
time = 17.33, size = 503, normalized size = 2.78 \begin {gather*} -\frac {2 \, {\left (30 \, c^{3} d^{5} x e + 5 \, c^{3} d^{6} - 2 \, a^{2} c d x e^{5} + a^{3} e^{6} + {\left (8 \, a c^{2} d^{2} x^{2} - 5 \, a^{2} c d^{2}\right )} e^{4} + 4 \, {\left (4 \, c^{3} d^{3} x^{3} + 5 \, a c^{2} d^{3} x\right )} e^{3} + 5 \, {\left (8 \, c^{3} d^{4} x^{2} + 3 \, a c^{2} d^{4}\right )} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{5 \, {\left (c^{5} d^{12} x + a^{5} x^{3} e^{12} + {\left (a^{4} c d x^{4} + 3 \, a^{5} d x^{2}\right )} e^{11} - {\left (a^{4} c d^{2} x^{3} - 3 \, a^{5} d^{2} x\right )} e^{10} - {\left (4 \, a^{3} c^{2} d^{3} x^{4} + 9 \, a^{4} c d^{3} x^{2} - a^{5} d^{3}\right )} e^{9} - {\left (6 \, a^{3} c^{2} d^{4} x^{3} + 11 \, a^{4} c d^{4} x\right )} e^{8} + 2 \, {\left (3 \, a^{2} c^{3} d^{5} x^{4} + 3 \, a^{3} c^{2} d^{5} x^{2} - 2 \, a^{4} c d^{5}\right )} e^{7} + 14 \, {\left (a^{2} c^{3} d^{6} x^{3} + a^{3} c^{2} d^{6} x\right )} e^{6} - 2 \, {\left (2 \, a c^{4} d^{7} x^{4} - 3 \, a^{2} c^{3} d^{7} x^{2} - 3 \, a^{3} c^{2} d^{7}\right )} e^{5} - {\left (11 \, a c^{4} d^{8} x^{3} + 6 \, a^{2} c^{3} d^{8} x\right )} e^{4} + {\left (c^{5} d^{9} x^{4} - 9 \, a c^{4} d^{9} x^{2} - 4 \, a^{2} c^{3} d^{9}\right )} e^{3} + {\left (3 \, c^{5} d^{10} x^{3} - a c^{4} d^{10} x\right )} e^{2} + {\left (3 \, c^{5} d^{11} x^{2} + a c^{4} d^{11}\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 {1}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{2}}\, dx \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. 3774 vs.
\(2 (171) = 342\).
time = 2.17, size = 3774, normalized size = 20.85 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.52, size = 1005, normalized size = 5.55 \begin {gather*} \frac {\left (\frac {16\,c^3\,d^4\,e}{15\,{\left (a\,e^2-c\,d^2\right )}^5}-\frac {8\,c^2\,d^2\,e\,\left (c\,d^2+a\,e^2\right )}{15\,{\left (a\,e^2-c\,d^2\right )}^5}\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 {e^2\,\left (10\,c^2\,d^3-18\,a\,c\,d\,e^2\right )}{5\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a^2\,e^5-6\,a\,c\,d^2\,e^3+3\,c^2\,d^4\,e\right )}+\frac {8\,c^2\,d^3\,e^2}{5\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a^2\,e^5-6\,a\,c\,d^2\,e^3+3\,c^2\,d^4\,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 {2\,e^2\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^3\,\left (5\,a^2\,e^5-10\,a\,c\,d^2\,e^3+5\,c^2\,d^4\,e\right )}-\frac {\left (x\,\left (\frac {32\,a\,c^5\,d^6\,e^4}{15\,{\left (a\,e^2-c\,d^2\right )}^4\,\left (a^2\,c\,d\,e^5-2\,a\,c^2\,d^3\,e^3+c^3\,d^5\,e\right )}-\frac {\left (c\,d^2+a\,e^2\right )\,\left (\frac {16\,c^5\,d^5\,e^3\,\left (c\,d^2+a\,e^2\right )}{15\,{\left (a\,e^2-c\,d^2\right )}^4\,\left (a^2\,c\,d\,e^5-2\,a\,c^2\,d^3\,e^3+c^3\,d^5\,e\right )}+\frac {8\,c^5\,d^5\,e^3\,\left (3\,a\,e^2-11\,c\,d^2\right )}{15\,{\left (a\,e^2-c\,d^2\right )}^4\,\left (a^2\,c\,d\,e^5-2\,a\,c^2\,d^3\,e^3+c^3\,d^5\,e\right )}\right )}{c\,d\,e}+\frac {2\,c^2\,d^2\,e^2\,\left (58\,a^2\,c^2\,d^2\,e^4-104\,a\,c^3\,d^4\,e^2+30\,c^4\,d^6\right )}{15\,{\left (a\,e^2-c\,d^2\right )}^4\,\left (a^2\,c\,d\,e^5-2\,a\,c^2\,d^3\,e^3+c^3\,d^5\,e\right )}+\frac {4\,c^4\,d^4\,e^2\,\left (c\,d^2+a\,e^2\right )\,\left (3\,a\,e^2-11\,c\,d^2\right )}{15\,{\left (a\,e^2-c\,d^2\right )}^4\,\left (a^2\,c\,d\,e^5-2\,a\,c^2\,d^3\,e^3+c^3\,d^5\,e\right )}\right )-\frac {a\,\left (\frac {16\,c^5\,d^5\,e^3\,\left (c\,d^2+a\,e^2\right )}{15\,{\left (a\,e^2-c\,d^2\right )}^4\,\left (a^2\,c\,d\,e^5-2\,a\,c^2\,d^3\,e^3+c^3\,d^5\,e\right )}+\frac {8\,c^5\,d^5\,e^3\,\left (3\,a\,e^2-11\,c\,d^2\right )}{15\,{\left (a\,e^2-c\,d^2\right )}^4\,\left (a^2\,c\,d\,e^5-2\,a\,c^2\,d^3\,e^3+c^3\,d^5\,e\right )}\right )}{c}+\frac {c\,d\,e\,\left (c\,d^2+a\,e^2\right )\,\left (58\,a^2\,c^2\,d^2\,e^4-104\,a\,c^3\,d^4\,e^2+30\,c^4\,d^6\right )}{15\,{\left (a\,e^2-c\,d^2\right )}^4\,\left (a^2\,c\,d\,e^5-2\,a\,c^2\,d^3\,e^3+c^3\,d^5\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{\left (a\,e+c\,d\,x\right )\,\left (d+e\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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