Optimal. Leaf size=195 \[ \frac {\left (11 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{6 c^2 e^3}-\frac {7 d (d+e x) \sqrt {a+c x^2}}{6 c e^3}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^3}-\frac {d \left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{3/2} e^4}-\frac {d^4 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^4 \sqrt {c d^2+a e^2}} \]
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Rubi [A]
time = 0.31, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1668, 858, 223,
212, 739} \begin {gather*} -\frac {d \left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{3/2} e^4}+\frac {\sqrt {a+c x^2} \left (11 c d^2-4 a e^2\right )}{6 c^2 e^3}-\frac {d^4 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^4 \sqrt {a e^2+c d^2}}-\frac {7 d \sqrt {a+c x^2} (d+e x)}{6 c e^3}+\frac {\sqrt {a+c x^2} (d+e x)^2}{3 c e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 739
Rule 858
Rule 1668
Rubi steps
\begin {align*} \int \frac {x^4}{(d+e x) \sqrt {a+c x^2}} \, dx &=\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^3}+\frac {\int \frac {-2 a d^2 e^2-d e \left (c d^2+4 a e^2\right ) x-e^2 \left (5 c d^2+2 a e^2\right ) x^2-7 c d e^3 x^3}{(d+e x) \sqrt {a+c x^2}} \, dx}{3 c e^4}\\ &=-\frac {7 d (d+e x) \sqrt {a+c x^2}}{6 c e^3}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^3}+\frac {\int \frac {3 a c d^2 e^5+c d e^4 \left (5 c d^2-a e^2\right ) x+c e^5 \left (11 c d^2-4 a e^2\right ) x^2}{(d+e x) \sqrt {a+c x^2}} \, dx}{6 c^2 e^7}\\ &=\frac {\left (11 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{6 c^2 e^3}-\frac {7 d (d+e x) \sqrt {a+c x^2}}{6 c e^3}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^3}+\frac {\int \frac {3 a c^2 d^2 e^7-3 c^2 d e^6 \left (2 c d^2-a e^2\right ) x}{(d+e x) \sqrt {a+c x^2}} \, dx}{6 c^3 e^9}\\ &=\frac {\left (11 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{6 c^2 e^3}-\frac {7 d (d+e x) \sqrt {a+c x^2}}{6 c e^3}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^3}+\frac {d^4 \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{e^4}-\frac {\left (d \left (2 c d^2-a e^2\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{2 c e^4}\\ &=\frac {\left (11 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{6 c^2 e^3}-\frac {7 d (d+e x) \sqrt {a+c x^2}}{6 c e^3}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^3}-\frac {d^4 \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 )}{e^4}-\frac {\left (d \left (2 c d^2-a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{2 c e^4}\\ &=\frac {\left (11 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{6 c^2 e^3}-\frac {7 d (d+e x) \sqrt {a+c x^2}}{6 c e^3}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^3}-\frac {d \left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{3/2} e^4}-\frac {d^4 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^4 \sqrt {c d^2+a e^2}}\\ \end {align*}
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Mathematica [A]
time = 0.55, size = 161, normalized size = 0.83 \begin {gather*} \frac {\frac {e \sqrt {a+c x^2} \left (-4 a e^2+c \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )}{c^2}-\frac {12 d^4 \tan ^{-1}\left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\sqrt {-c d^2-a e^2}}+\frac {3 d \left (2 c d^2-a e^2\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{c^{3/2}}}{6 e^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 258, normalized size = 1.32
method | result | size |
risch | \(-\frac {\left (-2 c \,e^{2} x^{2}+3 c d e x +4 a \,e^{2}-6 c \,d^{2}\right ) \sqrt {c \,x^{2}+a}}{6 c^{2} e^{3}}+\frac {d \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right ) a}{2 c^{\frac {3}{2}} e^{2}}-\frac {d^{3} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e^{4} \sqrt {c}}-\frac {d^{4} \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{5} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}\) | \(229\) |
default | \(\frac {\frac {x^{2} \sqrt {c \,x^{2}+a}}{3 c}-\frac {2 a \sqrt {c \,x^{2}+a}}{3 c^{2}}}{e}-\frac {d \left (\frac {x \sqrt {c \,x^{2}+a}}{2 c}-\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 c^{\frac {3}{2}}}\right )}{e^{2}}+\frac {d^{2} \sqrt {c \,x^{2}+a}}{e^{3} c}-\frac {d^{3} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e^{4} \sqrt {c}}-\frac {d^{4} \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{5} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}\) | \(258\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 166, normalized size = 0.85 \begin {gather*} \frac {d^{4} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | x e + d \right |}} - \frac {a e}{\sqrt {a c} {\left | x e + d \right |}}\right ) e^{\left (-5\right )}}{\sqrt {c d^{2} e^{\left (-2\right )} + a}} - \frac {d^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-4\right )}}{\sqrt {c}} + \frac {\sqrt {c x^{2} + a} x^{2} e^{\left (-1\right )}}{3 \, c} - \frac {\sqrt {c x^{2} + a} d x e^{\left (-2\right )}}{2 \, c} + \frac {a d \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-2\right )}}{2 \, c^{\frac {3}{2}}} + \frac {\sqrt {c x^{2} + a} d^{2} e^{\left (-3\right )}}{c} - \frac {2 \, \sqrt {c x^{2} + a} a e^{\left (-1\right )}}{3 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 6.81, size = 1021, normalized size = 5.24 \begin {gather*} \left [\frac {6 \, \sqrt {c d^{2} + a e^{2}} c^{2} d^{4} \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 ) - 3 \, {\left (2 \, c^{2} d^{5} + a c d^{3} e^{2} - a^{2} d e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - 2 \, {\left (3 \, c^{2} d^{3} x e^{2} - 6 \, c^{2} d^{4} e + 3 \, a c d x e^{4} - 2 \, {\left (a c x^{2} - 2 \, a^{2}\right )} e^{5} - 2 \, {\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + a}}{12 \, {\left (c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )}}, \frac {12 \, \sqrt {-c d^{2} - a e^{2}} c^{2} d^{4} \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 ) - 3 \, {\left (2 \, c^{2} d^{5} + a c d^{3} e^{2} - a^{2} d e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - 2 \, {\left (3 \, c^{2} d^{3} x e^{2} - 6 \, c^{2} d^{4} e + 3 \, a c d x e^{4} - 2 \, {\left (a c x^{2} - 2 \, a^{2}\right )} e^{5} - 2 \, {\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + a}}{12 \, {\left (c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )}}, \frac {3 \, \sqrt {c d^{2} + a e^{2}} c^{2} d^{4} \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 ) + 3 \, {\left (2 \, c^{2} d^{5} + a c d^{3} e^{2} - a^{2} d e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (3 \, c^{2} d^{3} x e^{2} - 6 \, c^{2} d^{4} e + 3 \, a c d x e^{4} - 2 \, {\left (a c x^{2} - 2 \, a^{2}\right )} e^{5} - 2 \, {\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + a}}{6 \, {\left (c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )}}, \frac {6 \, \sqrt {-c d^{2} - a e^{2}} c^{2} d^{4} \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 ) + 3 \, {\left (2 \, c^{2} d^{5} + a c d^{3} e^{2} - a^{2} d e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (3 \, c^{2} d^{3} x e^{2} - 6 \, c^{2} d^{4} e + 3 \, a c d x e^{4} - 2 \, {\left (a c x^{2} - 2 \, a^{2}\right )} e^{5} - 2 \, {\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + a}}{6 \, {\left (c^{3} d^{2} e^{4} + a c^{2} e^{6}\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 {x^{4}}{\sqrt {a + c 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 = 1.42, size = 163, normalized size = 0.84 \begin {gather*} \frac {2 \, d^{4} \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^{\left (-4\right )}}{\sqrt {-c d^{2} - a e^{2}}} + \frac {1}{6} \, \sqrt {c x^{2} + a} {\left (x {\left (\frac {2 \, x e^{\left (-1\right )}}{c} - \frac {3 \, d e^{\left (-2\right )}}{c}\right )} + \frac {2 \, {\left (3 \, c^{2} d^{2} e^{7} - 2 \, a c e^{9}\right )} e^{\left (-10\right )}}{c^{3}}\right )} + \frac {{\left (2 \, c^{\frac {3}{2}} d^{3} - a \sqrt {c} d e^{2}\right )} e^{\left (-4\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{2 \, c^{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 = -0.01 \begin {gather*} \int \frac {x^4}{\sqrt {c\,x^2+a}\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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