Optimal. Leaf size=271 \[ -\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6+a c^2 d^4 e^2+7 a^2 c d^2 e^4-3 a^3 e^6+c d e \left (3 c d^2-a e^2\right ) \left (c d^2+3 a e^2\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\tanh ^{-1}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{a^{3/2} d^{5/2} e^{3/2}} \]
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Rubi [A]
time = 0.22, antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {865, 836, 12,
738, 212} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{a^{3/2} d^{5/2} e^{3/2}}+\frac {2 \left (-3 a^3 e^6+7 a^2 c d^2 e^4+a c^2 d^4 e^2+c d e x \left (3 c d^2-a e^2\right ) \left (3 a e^2+c d^2\right )+3 c^3 d^6\right )}{3 a d^2 e \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 e (a e+c d x)}{3 d \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 12
Rule 212
Rule 738
Rule 836
Rule 865
Rubi steps
\begin {align*} \int \frac {1}{x (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 &=\int \frac {a e+c d x}{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 e (a e+c d x)}{3 d \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 {2 \int \frac {-\frac {3}{2} a e \left (c d^2-a e^2\right )^2+2 a c d e^2 \left (c d^2-a e^2\right ) x}{x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 a d e \left (c d^2-a e^2\right )^2}\\ &=-\frac {2 e (a e+c d x)}{3 d \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 {2 \left (3 c^3 d^6+a c^2 d^4 e^2+7 a^2 c d^2 e^4-3 a^3 e^6+c d e \left (3 c d^2-a e^2\right ) \left (c d^2+3 a e^2\right ) x\right )}{3 a d^2 e \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}}+\frac {4 \int \frac {3 a e \left (c d^2-a e^2\right )^4}{4 x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{3 a^2 d^2 e^2 \left (c d^2-a e^2\right )^4}\\ &=-\frac {2 e (a e+c d x)}{3 d \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 {2 \left (3 c^3 d^6+a c^2 d^4 e^2+7 a^2 c d^2 e^4-3 a^3 e^6+c d e \left (3 c d^2-a e^2\right ) \left (c d^2+3 a e^2\right ) x\right )}{3 a d^2 e \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}}+\frac {\int \frac {1}{x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{a d^2 e}\\ &=-\frac {2 e (a e+c d x)}{3 d \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 {2 \left (3 c^3 d^6+a c^2 d^4 e^2+7 a^2 c d^2 e^4-3 a^3 e^6+c d e \left (3 c d^2-a e^2\right ) \left (c d^2+3 a e^2\right ) x\right )}{3 a d^2 e \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}}-\frac {2 \text {Subst}\left (\int \frac {1}{4 a d e-x^2} \, dx,x,\frac {2 a d e-\left (-c d^2-a e^2\right ) x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{a d^2 e}\\ &=-\frac {2 e (a e+c d x)}{3 d \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 {2 \left (3 c^3 d^6+a c^2 d^4 e^2+7 a^2 c d^2 e^4-3 a^3 e^6+c d e \left (3 c d^2-a e^2\right ) \left (c d^2+3 a e^2\right ) x\right )}{3 a d^2 e \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}}-\frac {\tanh ^{-1}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{a^{3/2} d^{5/2} e^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.35, size = 217, normalized size = 0.80 \begin {gather*} \frac {2 \left (\frac {\sqrt {a} \sqrt {d} \sqrt {e} (a e+c d x) \left (-3 c^3 d^5 (d+e x)^2+a^3 e^6 (4 d+3 e x)-a c^2 d^3 e^3 x (9 d+8 e x)+a^2 c d e^4 \left (-9 d^2-4 d e x+3 e^2 x^2\right )\right )}{\left (-c d^2+a e^2\right )^3}-3 (a e+c d x)^{3/2} (d+e x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {e} \sqrt {d+e x}}{\sqrt {d} \sqrt {a e+c d x}}\right )\right )}{3 a^{3/2} d^{5/2} e^{3/2} ((a e+c d x) (d+e x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 360, normalized size = 1.33
method | result | size |
default | \(-\frac {-\frac {2}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}+\frac {8 c d e \left (2 c d e \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right )}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}}{d}+\frac {\frac {1}{a d e \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d e x +a \,e^{2}+c \,d^{2}\right )}{a d e \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {\ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{x}\right )}{a d e \sqrt {a d e}}}{d}\) | \(360\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 729 vs.
\(2 (243) = 486\).
time = 8.95, size = 1483, normalized size = 5.47 \begin {gather*} \left [\frac {3 \, {\left (c^{4} d^{9} x - a^{4} x^{2} e^{9} - {\left (a^{3} c d x^{3} + 2 \, a^{4} d x\right )} e^{8} + {\left (a^{3} c d^{2} x^{2} - a^{4} d^{2}\right )} e^{7} + {\left (3 \, a^{2} c^{2} d^{3} x^{3} + 5 \, a^{3} c d^{3} x\right )} e^{6} + 3 \, {\left (a^{2} c^{2} d^{4} x^{2} + a^{3} c d^{4}\right )} e^{5} - 3 \, {\left (a c^{3} d^{5} x^{3} + a^{2} c^{2} d^{5} x\right )} e^{4} - {\left (5 \, a c^{3} d^{6} x^{2} + 3 \, a^{2} c^{2} d^{6}\right )} e^{3} + {\left (c^{4} d^{7} x^{3} - a c^{3} d^{7} x\right )} e^{2} + {\left (2 \, c^{4} d^{8} x^{2} + a c^{3} d^{8}\right )} e\right )} \sqrt {a d} e^{\frac {1}{2}} \log \left (\frac {c^{2} d^{4} x^{2} + 8 \, a c d^{3} x e + a^{2} x^{2} e^{4} + 8 \, a^{2} d x e^{3} - 4 \, {\left (c d^{2} x + a x e^{2} + 2 \, a d e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {a d} e^{\frac {1}{2}} + 2 \, {\left (3 \, a c d^{2} x^{2} + 4 \, a^{2} d^{2}\right )} e^{2}}{x^{2}}\right ) + 4 \, {\left (3 \, a c^{3} d^{6} x^{2} e^{3} + 6 \, a c^{3} d^{7} x e^{2} + 3 \, a c^{3} d^{8} e + 9 \, a^{2} c^{2} d^{5} x e^{4} + 4 \, a^{3} c d^{3} x e^{6} - 3 \, a^{4} d x e^{8} - {\left (3 \, a^{3} c d^{2} x^{2} + 4 \, a^{4} d^{2}\right )} e^{7} + {\left (8 \, a^{2} c^{2} d^{4} x^{2} + 9 \, a^{3} c d^{4}\right )} e^{5}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{6 \, {\left (a^{2} c^{4} d^{12} x e^{2} - a^{6} d^{3} x^{2} e^{11} - {\left (a^{5} c d^{4} x^{3} + 2 \, a^{6} d^{4} x\right )} e^{10} + {\left (a^{5} c d^{5} x^{2} - a^{6} d^{5}\right )} e^{9} + {\left (3 \, a^{4} c^{2} d^{6} x^{3} + 5 \, a^{5} c d^{6} x\right )} e^{8} + 3 \, {\left (a^{4} c^{2} d^{7} x^{2} + a^{5} c d^{7}\right )} e^{7} - 3 \, {\left (a^{3} c^{3} d^{8} x^{3} + a^{4} c^{2} d^{8} x\right )} e^{6} - {\left (5 \, a^{3} c^{3} d^{9} x^{2} + 3 \, a^{4} c^{2} d^{9}\right )} e^{5} + {\left (a^{2} c^{4} d^{10} x^{3} - a^{3} c^{3} d^{10} x\right )} e^{4} + {\left (2 \, a^{2} c^{4} d^{11} x^{2} + a^{3} c^{3} d^{11}\right )} e^{3}\right )}}, \frac {3 \, {\left (c^{4} d^{9} x - a^{4} x^{2} e^{9} - {\left (a^{3} c d x^{3} + 2 \, a^{4} d x\right )} e^{8} + {\left (a^{3} c d^{2} x^{2} - a^{4} d^{2}\right )} e^{7} + {\left (3 \, a^{2} c^{2} d^{3} x^{3} + 5 \, a^{3} c d^{3} x\right )} e^{6} + 3 \, {\left (a^{2} c^{2} d^{4} x^{2} + a^{3} c d^{4}\right )} e^{5} - 3 \, {\left (a c^{3} d^{5} x^{3} + a^{2} c^{2} d^{5} x\right )} e^{4} - {\left (5 \, a c^{3} d^{6} x^{2} + 3 \, a^{2} c^{2} d^{6}\right )} e^{3} + {\left (c^{4} d^{7} x^{3} - a c^{3} d^{7} x\right )} e^{2} + {\left (2 \, c^{4} d^{8} x^{2} + a c^{3} d^{8}\right )} e\right )} \sqrt {-a d e} \arctan \left (\frac {{\left (c d^{2} x + a x e^{2} + 2 \, a d e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {-a d e}}{2 \, {\left (a c d^{3} x e + a^{2} d x e^{3} + {\left (a c d^{2} x^{2} + a^{2} d^{2}\right )} e^{2}\right )}}\right ) + 2 \, {\left (3 \, a c^{3} d^{6} x^{2} e^{3} + 6 \, a c^{3} d^{7} x e^{2} + 3 \, a c^{3} d^{8} e + 9 \, a^{2} c^{2} d^{5} x e^{4} + 4 \, a^{3} c d^{3} x e^{6} - 3 \, a^{4} d x e^{8} - {\left (3 \, a^{3} c d^{2} x^{2} + 4 \, a^{4} d^{2}\right )} e^{7} + {\left (8 \, a^{2} c^{2} d^{4} x^{2} + 9 \, a^{3} c d^{4}\right )} e^{5}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{3 \, {\left (a^{2} c^{4} d^{12} x e^{2} - a^{6} d^{3} x^{2} e^{11} - {\left (a^{5} c d^{4} x^{3} + 2 \, a^{6} d^{4} x\right )} e^{10} + {\left (a^{5} c d^{5} x^{2} - a^{6} d^{5}\right )} e^{9} + {\left (3 \, a^{4} c^{2} d^{6} x^{3} + 5 \, a^{5} c d^{6} x\right )} e^{8} + 3 \, {\left (a^{4} c^{2} d^{7} x^{2} + a^{5} c d^{7}\right )} e^{7} - 3 \, {\left (a^{3} c^{3} d^{8} x^{3} + a^{4} c^{2} d^{8} x\right )} e^{6} - {\left (5 \, a^{3} c^{3} d^{9} x^{2} + 3 \, a^{4} c^{2} d^{9}\right )} e^{5} + {\left (a^{2} c^{4} d^{10} x^{3} - a^{3} c^{3} d^{10} x\right )} e^{4} + {\left (2 \, a^{2} c^{4} d^{11} x^{2} + a^{3} c^{3} d^{11}\right )} e^{3}\right )}}\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}{x \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, 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 [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x\,\left (d+e\,x\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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