Optimal. Leaf size=276 \[ -\frac {3 c}{2 a^2 d \sqrt {a+c x^2}}+\frac {e^2}{a d^3 \sqrt {a+c x^2}}-\frac {1}{2 a d x^2 \sqrt {a+c x^2}}+\frac {e}{a d^2 x \sqrt {a+c x^2}}+\frac {2 c e x}{a^2 d^2 \sqrt {a+c x^2}}-\frac {e^3 (a e+c d x)}{a d^3 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {e^5 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^3 \left (c d^2+a e^2\right )^{3/2}}+\frac {3 c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 a^{5/2} d}-\frac {e^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2} d^3} \]
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Rubi [A]
time = 0.15, antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 12, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {975, 272, 44,
53, 65, 214, 277, 197, 755, 12, 739, 212} \begin {gather*} -\frac {e^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2} d^3}+\frac {3 c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 a^{5/2} d}+\frac {2 c e x}{a^2 d^2 \sqrt {a+c x^2}}-\frac {3 c}{2 a^2 d \sqrt {a+c x^2}}+\frac {e^2}{a d^3 \sqrt {a+c x^2}}+\frac {e}{a d^2 x \sqrt {a+c x^2}}+\frac {e^5 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{d^3 \left (a e^2+c d^2\right )^{3/2}}-\frac {e^3 (a e+c d x)}{a d^3 \sqrt {a+c x^2} \left (a e^2+c d^2\right )}-\frac {1}{2 a d x^2 \sqrt {a+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 44
Rule 53
Rule 65
Rule 197
Rule 212
Rule 214
Rule 272
Rule 277
Rule 739
Rule 755
Rule 975
Rubi steps
\begin {align*} \int \frac {1}{x^3 (d+e x) \left (a+c x^2\right )^{3/2}} \, dx &=\int \left (\frac {1}{d x^3 \left (a+c x^2\right )^{3/2}}-\frac {e}{d^2 x^2 \left (a+c x^2\right )^{3/2}}+\frac {e^2}{d^3 x \left (a+c x^2\right )^{3/2}}-\frac {e^3}{d^3 (d+e x) \left (a+c x^2\right )^{3/2}}\right ) \, dx\\ &=\frac {\int \frac {1}{x^3 \left (a+c x^2\right )^{3/2}} \, dx}{d}-\frac {e \int \frac {1}{x^2 \left (a+c x^2\right )^{3/2}} \, dx}{d^2}+\frac {e^2 \int \frac {1}{x \left (a+c x^2\right )^{3/2}} \, dx}{d^3}-\frac {e^3 \int \frac {1}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx}{d^3}\\ &=\frac {e}{a d^2 x \sqrt {a+c x^2}}-\frac {e^3 (a e+c d x)}{a d^3 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {\text {Subst}\left (\int \frac {1}{x^2 (a+c x)^{3/2}} \, dx,x,x^2\right )}{2 d}+\frac {(2 c e) \int \frac {1}{\left (a+c x^2\right )^{3/2}} \, dx}{a d^2}+\frac {e^2 \text {Subst}\left (\int \frac {1}{x (a+c x)^{3/2}} \, dx,x,x^2\right )}{2 d^3}-\frac {e^3 \int \frac {a e^2}{(d+e x) \sqrt {a+c x^2}} \, dx}{a d^3 \left (c d^2+a e^2\right )}\\ &=\frac {e^2}{a d^3 \sqrt {a+c x^2}}+\frac {1}{a d x^2 \sqrt {a+c x^2}}+\frac {e}{a d^2 x \sqrt {a+c x^2}}+\frac {2 c e x}{a^2 d^2 \sqrt {a+c x^2}}-\frac {e^3 (a e+c d x)}{a d^3 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {3 \text {Subst}\left (\int \frac {1}{x^2 \sqrt {a+c x}} \, dx,x,x^2\right )}{2 a d}+\frac {e^2 \text {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{2 a d^3}-\frac {e^5 \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{d^3 \left (c d^2+a e^2\right )}\\ &=\frac {e^2}{a d^3 \sqrt {a+c x^2}}+\frac {1}{a d x^2 \sqrt {a+c x^2}}+\frac {e}{a d^2 x \sqrt {a+c x^2}}+\frac {2 c e x}{a^2 d^2 \sqrt {a+c x^2}}-\frac {e^3 (a e+c d x)}{a d^3 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {3 \sqrt {a+c x^2}}{2 a^2 d x^2}-\frac {(3 c) \text {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{4 a^2 d}+\frac {e^2 \text {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{a c d^3}+\frac {e^5 \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^3 \left (c d^2+a e^2\right )}\\ &=\frac {e^2}{a d^3 \sqrt {a+c x^2}}+\frac {1}{a d x^2 \sqrt {a+c x^2}}+\frac {e}{a d^2 x \sqrt {a+c x^2}}+\frac {2 c e x}{a^2 d^2 \sqrt {a+c x^2}}-\frac {e^3 (a e+c d x)}{a d^3 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {3 \sqrt {a+c x^2}}{2 a^2 d x^2}+\frac {e^5 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^3 \left (c d^2+a e^2\right )^{3/2}}-\frac {e^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2} d^3}-\frac {3 \text {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{2 a^2 d}\\ &=\frac {e^2}{a d^3 \sqrt {a+c x^2}}+\frac {1}{a d x^2 \sqrt {a+c x^2}}+\frac {e}{a d^2 x \sqrt {a+c x^2}}+\frac {2 c e x}{a^2 d^2 \sqrt {a+c x^2}}-\frac {e^3 (a e+c d x)}{a d^3 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {3 \sqrt {a+c x^2}}{2 a^2 d x^2}+\frac {e^5 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^3 \left (c d^2+a e^2\right )^{3/2}}+\frac {3 c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 a^{5/2} d}-\frac {e^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2} d^3}\\ \end {align*}
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Mathematica [A]
time = 0.81, size = 217, normalized size = 0.79 \begin {gather*} -\frac {\frac {d \left (c^2 d^2 x^2 (3 d-4 e x)+a^2 e^2 (d-2 e x)+a c \left (d^3-2 d^2 e x+d e^2 x^2-2 e^3 x^3\right )\right )}{a^2 \left (c d^2+a e^2\right ) x^2 \sqrt {a+c x^2}}+\frac {4 e^5 \tan ^{-1}\left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\left (-c d^2-a e^2\right )^{3/2}}+\frac {2 \left (3 c d^2-2 a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x-\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{5/2}}}{2 d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 479, normalized size = 1.74
method | result | size |
risch | \(-\frac {\sqrt {c \,x^{2}+a}\, \left (-2 e x +d \right )}{2 a^{2} d^{2} x^{2}}-\frac {c \,e^{4} \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{d^{3} \left (\sqrt {-a c}\, e +c d \right ) \left (\sqrt {-a c}\, e -c d \right ) \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right ) e^{2}}{a^{\frac {3}{2}} d^{3}}+\frac {3 \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right ) c}{2 a^{\frac {5}{2}} d}-\frac {c^{2} \sqrt {c \left (x +\frac {\sqrt {-a c}}{c}\right )^{2}-2 \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )}}{2 a^{2} \left (\sqrt {-a c}\, e -c d \right ) \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )}-\frac {c^{2} \sqrt {c \left (x -\frac {\sqrt {-a c}}{c}\right )^{2}+2 \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )}}{2 a^{2} \left (\sqrt {-a c}\, e +c d \right ) \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )}\) | \(419\) |
default | \(-\frac {e^{2} \left (\frac {e^{2}}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}+\frac {2 c d e \left (2 c \left (x +\frac {d}{e}\right )-\frac {2 c d}{e}\right )}{\left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {4 c \left (a \,e^{2}+c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}-\frac {e^{2} \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}\right )}{d^{3}}+\frac {-\frac {1}{2 a \,x^{2} \sqrt {c \,x^{2}+a}}-\frac {3 c \left (\frac {1}{a \sqrt {c \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}}{d}-\frac {e \left (-\frac {1}{a x \sqrt {c \,x^{2}+a}}-\frac {2 c x}{a^{2} \sqrt {c \,x^{2}+a}}\right )}{d^{2}}+\frac {e^{2} \left (\frac {1}{a \sqrt {c \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{d^{3}}\) | \(479\) |
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 = 3.32, size = 1961, normalized size = 7.11 \begin {gather*} \text {Too large to display} \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^{3} \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 [A]
time = 1.78, size = 358, normalized size = 1.30 \begin {gather*} \frac {\frac {{\left (a^{2} c^{3} d^{2} e + a^{3} c^{2} e^{3}\right )} x}{a^{4} c^{2} d^{4} + 2 \, a^{5} c d^{2} e^{2} + a^{6} e^{4}} - \frac {a^{2} c^{3} d^{3} + a^{3} c^{2} d e^{2}}{a^{4} c^{2} d^{4} + 2 \, a^{5} c d^{2} e^{2} + a^{6} e^{4}}}{\sqrt {c x^{2} + a}} - \frac {2 \, \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right ) e^{5}}{{\left (c d^{5} + a d^{3} e^{2}\right )} \sqrt {-c d^{2} - a e^{2}}} - \frac {{\left (3 \, c d^{2} - 2 \, a e^{2}\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2} d^{3}} + \frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c d - 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a \sqrt {c} e + {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a c d + 2 \, a^{2} \sqrt {c} e}{{\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{2} a^{2} d^{2}} \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^3\,{\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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