Optimal. Leaf size=147 \[ \frac {1}{a d \sqrt {a+c x^2}}-\frac {e (a e+c d x)}{a d \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {e^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d \left (c d^2+a e^2\right )^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2} d} \]
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Rubi [A]
time = 0.09, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {975, 272, 53,
65, 214, 755, 12, 739, 212} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {e (a e+c d x)}{a d \sqrt {a+c x^2} \left (a e^2+c d^2\right )}+\frac {e^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{d \left (a e^2+c d^2\right )^{3/2}}+\frac {1}{a d \sqrt {a+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 53
Rule 65
Rule 212
Rule 214
Rule 272
Rule 739
Rule 755
Rule 975
Rubi steps
\begin {align*} \int \frac {1}{x (d+e x) \left (a+c x^2\right )^{3/2}} \, dx &=\int \left (\frac {1}{d x \left (a+c x^2\right )^{3/2}}-\frac {e}{d (d+e x) \left (a+c x^2\right )^{3/2}}\right ) \, dx\\ &=\frac {\int \frac {1}{x \left (a+c x^2\right )^{3/2}} \, dx}{d}-\frac {e \int \frac {1}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx}{d}\\ &=-\frac {e (a e+c d x)}{a d \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {\text {Subst}\left (\int \frac {1}{x (a+c x)^{3/2}} \, dx,x,x^2\right )}{2 d}-\frac {e \int \frac {a e^2}{(d+e x) \sqrt {a+c x^2}} \, dx}{a d \left (c d^2+a e^2\right )}\\ &=\frac {1}{a d \sqrt {a+c x^2}}-\frac {e (a e+c d x)}{a d \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{2 a d}-\frac {e^3 \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{d \left (c d^2+a e^2\right )}\\ &=\frac {1}{a d \sqrt {a+c x^2}}-\frac {e (a e+c d x)}{a d \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {\text {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{a c d}+\frac {e^3 \text {Subst}\left (\int \frac {1}{c d^2+a e^2-x^2} \, dx,x,\frac {a e-c d x}{\sqrt {a+c x^2}}\right )}{d \left (c d^2+a e^2\right )}\\ &=\frac {1}{a d \sqrt {a+c x^2}}-\frac {e (a e+c d x)}{a d \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {e^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d \left (c d^2+a e^2\right )^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2} d}\\ \end {align*}
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Mathematica [A]
time = 0.75, size = 144, normalized size = 0.98 \begin {gather*} \frac {c (d-e x)}{a \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {2 e^3 \tan ^{-1}\left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{d \left (-c d^2-a e^2\right )^{3/2}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c} x-\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2} d} \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. \(362\) vs.
\(2(131)=262\).
time = 0.06, size = 363, normalized size = 2.47
method | result | size |
default | \(-\frac {\frac {e^{2}}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}+\frac {2 c d e \left (2 c \left (x +\frac {d}{e}\right )-\frac {2 c d}{e}\right )}{\left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {4 c \left (a \,e^{2}+c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}-\frac {e^{2} \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}}{d}+\frac {\frac {1}{a \sqrt {c \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}}{d}\) | \(363\) |
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. 302 vs.
\(2 (131) = 262\).
time = 2.93, size = 1284, normalized size = 8.73 \begin {gather*} \left [\frac {{\left (a^{2} c x^{2} + a^{3}\right )} \sqrt {c d^{2} + a e^{2}} e^{3} \log \left (-\frac {2 \, c^{2} d^{2} x^{2} - 2 \, a c d x e + a c d^{2} - 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a} + {\left (a c x^{2} + 2 \, a^{2}\right )} e^{2}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) + {\left (c^{3} d^{4} x^{2} + a c^{2} d^{4} + {\left (a^{2} c x^{2} + a^{3}\right )} e^{4} + 2 \, {\left (a c^{2} d^{2} x^{2} + a^{2} c d^{2}\right )} e^{2}\right )} \sqrt {a} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (a c^{2} d^{3} x e - a c^{2} d^{4} + a^{2} c d x e^{3} - a^{2} c d^{2} e^{2}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (a^{2} c^{3} d^{5} x^{2} + a^{3} c^{2} d^{5} + {\left (a^{4} c d x^{2} + a^{5} d\right )} e^{4} + 2 \, {\left (a^{3} c^{2} d^{3} x^{2} + a^{4} c d^{3}\right )} e^{2}\right )}}, -\frac {2 \, {\left (a^{2} c x^{2} + a^{3}\right )} \sqrt {-c d^{2} - a e^{2}} \arctan \left (-\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{c^{2} d^{2} x^{2} + a c d^{2} + {\left (a c x^{2} + a^{2}\right )} e^{2}}\right ) e^{3} - {\left (c^{3} d^{4} x^{2} + a c^{2} d^{4} + {\left (a^{2} c x^{2} + a^{3}\right )} e^{4} + 2 \, {\left (a c^{2} d^{2} x^{2} + a^{2} c d^{2}\right )} e^{2}\right )} \sqrt {a} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (a c^{2} d^{3} x e - a c^{2} d^{4} + a^{2} c d x e^{3} - a^{2} c d^{2} e^{2}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (a^{2} c^{3} d^{5} x^{2} + a^{3} c^{2} d^{5} + {\left (a^{4} c d x^{2} + a^{5} d\right )} e^{4} + 2 \, {\left (a^{3} c^{2} d^{3} x^{2} + a^{4} c d^{3}\right )} e^{2}\right )}}, \frac {{\left (a^{2} c x^{2} + a^{3}\right )} \sqrt {c d^{2} + a e^{2}} e^{3} \log \left (-\frac {2 \, c^{2} d^{2} x^{2} - 2 \, a c d x e + a c d^{2} - 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a} + {\left (a c x^{2} + 2 \, a^{2}\right )} e^{2}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) + 2 \, {\left (c^{3} d^{4} x^{2} + a c^{2} d^{4} + {\left (a^{2} c x^{2} + a^{3}\right )} e^{4} + 2 \, {\left (a c^{2} d^{2} x^{2} + a^{2} c d^{2}\right )} e^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - 2 \, {\left (a c^{2} d^{3} x e - a c^{2} d^{4} + a^{2} c d x e^{3} - a^{2} c d^{2} e^{2}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (a^{2} c^{3} d^{5} x^{2} + a^{3} c^{2} d^{5} + {\left (a^{4} c d x^{2} + a^{5} d\right )} e^{4} + 2 \, {\left (a^{3} c^{2} d^{3} x^{2} + a^{4} c d^{3}\right )} e^{2}\right )}}, -\frac {{\left (a^{2} c x^{2} + a^{3}\right )} \sqrt {-c d^{2} - a e^{2}} \arctan \left (-\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{c^{2} d^{2} x^{2} + a c d^{2} + {\left (a c x^{2} + a^{2}\right )} e^{2}}\right ) e^{3} - {\left (c^{3} d^{4} x^{2} + a c^{2} d^{4} + {\left (a^{2} c x^{2} + a^{3}\right )} e^{4} + 2 \, {\left (a c^{2} d^{2} x^{2} + a^{2} c d^{2}\right )} e^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) + {\left (a c^{2} d^{3} x e - a c^{2} d^{4} + a^{2} c d x e^{3} - a^{2} c d^{2} e^{2}\right )} \sqrt {c x^{2} + a}}{a^{2} c^{3} d^{5} x^{2} + a^{3} c^{2} d^{5} + {\left (a^{4} c d x^{2} + a^{5} d\right )} e^{4} + 2 \, {\left (a^{3} c^{2} d^{3} x^{2} + a^{4} c d^{3}\right )} e^{2}}\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 (a + c x^{2}\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(-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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x\,{\left (c\,x^2+a\right )}^{3/2}\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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