Optimal. Leaf size=137 \[ -\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d x}-\frac {\left (c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{2 \sqrt {a} d^{3/2} \sqrt {e}} \]
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Rubi [A]
time = 0.10, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {863, 820, 738,
212} \begin {gather*} -\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d x}-\frac {\left (c d^2-a e^2\right ) \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 )}{2 \sqrt {a} d^{3/2} \sqrt {e}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 738
Rule 820
Rule 863
Rubi steps
\begin {align*} \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^2 (d+e x)} \, dx &=\int \frac {a e+c d x}{x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d x}-\frac {\left (-2 a c d^2 e+a e \left (c d^2+a e^2\right )\right ) \int \frac {1}{x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2 a d e}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d x}+\frac {\left (-2 a c d^2 e+a e \left (c d^2+a e^2\right )\right ) \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 e}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d x}-\frac {\left (c d^2-a e^2\right ) \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 )}{2 \sqrt {a} d^{3/2} \sqrt {e}}\\ \end {align*}
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Mathematica [A]
time = 0.66, size = 141, normalized size = 1.03 \begin {gather*} \frac {\sqrt {(a e+c d x) (d+e x)} \left (-d+\frac {\left (-c d^2 x+a e^2 x\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \left (-e x+\sqrt {\frac {e}{c d}} \sqrt {a e+c d x} \sqrt {d+e x}\right )}{\sqrt {a} e}\right )}{\sqrt {a} \sqrt {c} \sqrt {\frac {e}{c d}} \sqrt {a e+c d x} \sqrt {d+e x}}\right )}{d^2 x} \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. \(707\) vs.
\(2(117)=234\).
time = 0.10, size = 708, normalized size = 5.17
method | result | size |
default | \(\frac {e \left (\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 {\left (a \,e^{2}-c \,d^{2}\right ) \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+c d e \left (x +\frac {d}{e}\right )}{\sqrt {c d e}}+\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 )}\right )}{2 \sqrt {c d e}}\right )}{d^{2}}+\frac {-\frac {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{a d e x}+\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (\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 ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{2 \sqrt {c d e}}-\frac {a d e \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 )}{\sqrt {a d e}}\right )}{2 a d e}+\frac {2 c \left (\frac {\left (2 c d e x +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{8 c d e \sqrt {c d e}}\right )}{a}}{d}-\frac {e \left (\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 ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{2 \sqrt {c d e}}-\frac {a d e \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 )}{\sqrt {a d e}}\right )}{d^{2}}\) | \(708\) |
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 [A]
time = 2.62, size = 356, normalized size = 2.60 \begin {gather*} \left [-\frac {{\left (4 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} a d e + {\left (c d^{2} x - a x e^{2}\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 )\right )} e^{\left (-1\right )}}{4 \, a d^{2} x}, -\frac {{\left (2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} a d e - {\left (c d^{2} x - a x e^{2}\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 )\right )} e^{\left (-1\right )}}{2 \, a d^{2} x}\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 {\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}{x^{2} \left (d + e x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.15, size = 229, normalized size = 1.67 \begin {gather*} \frac {{\left (c d^{2} - a e^{2}\right )} \arctan \left (-\frac {\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}}{\sqrt {-a d e}}\right )}{\sqrt {-a d e} d} - \frac {{\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} c d^{2} + 2 \, \sqrt {c d} a d e^{\frac {3}{2}} + {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} a e^{2}}{{\left (a d e - {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{2}\right )} d} \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 = -0.01 \begin {gather*} \int \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x^2\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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