Optimal. Leaf size=195 \[ -\frac {d \left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{3/2} e^4}+\frac {\sqrt {a+c x^2} \left (11 c d^2-4 a e^2\right )}{6 c^2 e^3}-\frac {d^4 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^4 \sqrt {a e^2+c d^2}}-\frac {7 d \sqrt {a+c x^2} (d+e x)}{6 c e^3}+\frac {\sqrt {a+c x^2} (d+e x)^2}{3 c e^3} \]
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Rubi [A] time = 0.48, 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 = {1654, 844, 217, 206, 725} \[ \frac {\sqrt {a+c x^2} \left (11 c d^2-4 a e^2\right )}{6 c^2 e^3}-\frac {d \left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{3/2} e^4}-\frac {d^4 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^4 \sqrt {a e^2+c d^2}}-\frac {7 d \sqrt {a+c x^2} (d+e x)}{6 c e^3}+\frac {\sqrt {a+c x^2} (d+e x)^2}{3 c e^3} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 725
Rule 844
Rule 1654
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 \operatorname {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 ) \operatorname {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 = 0.23, size = 149, normalized size = 0.76 \[ \frac {-\frac {3 d \left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{3/2}}+\frac {e \sqrt {a+c x^2} \left (-4 a e^2+6 c d^2-3 c d e x+2 c e^2 x^2\right )}{c^2}-\frac {6 d^4 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{\sqrt {a e^2+c d^2}}}{6 e^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 5.99, size = 1060, normalized size = 5.44 \[ \left [\frac {6 \, \sqrt {c d^{2} + a e^{2}} c^{2} d^{4} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} - 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + 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 (6 \, c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - 4 \, a^{2} e^{5} + 2 \, {\left (c^{2} d^{2} e^{3} + a c e^{5}\right )} x^{2} - 3 \, {\left (c^{2} d^{3} e^{2} + a c d e^{4}\right )} x\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}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{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 (6 \, c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - 4 \, a^{2} e^{5} + 2 \, {\left (c^{2} d^{2} e^{3} + a c e^{5}\right )} x^{2} - 3 \, {\left (c^{2} d^{3} e^{2} + a c d e^{4}\right )} x\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 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} - 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + 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 (6 \, c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - 4 \, a^{2} e^{5} + 2 \, {\left (c^{2} d^{2} e^{3} + a c e^{5}\right )} x^{2} - 3 \, {\left (c^{2} d^{3} e^{2} + a c d e^{4}\right )} x\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}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{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 (6 \, c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - 4 \, a^{2} e^{5} + 2 \, {\left (c^{2} d^{2} e^{3} + a c e^{5}\right )} x^{2} - 3 \, {\left (c^{2} d^{3} e^{2} + a c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{6 \, {\left (c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 163, normalized size = 0.84 \[ \frac {2 \, d^{4} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right ) e^{\left (-4\right )}}{\sqrt {-c d^{2} - a e^{2}}} + \frac {1}{6} \, \sqrt {c x^{2} + a} {\left (x {\left (\frac {2 \, x e^{\left (-1\right )}}{c} - \frac {3 \, d e^{\left (-2\right )}}{c}\right )} + \frac {2 \, {\left (3 \, c^{2} d^{2} e^{7} - 2 \, a c e^{9}\right )} e^{\left (-10\right )}}{c^{3}}\right )} + \frac {{\left (2 \, c^{\frac {3}{2}} d^{3} - a \sqrt {c} d e^{2}\right )} e^{\left (-4\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{2 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 260, normalized size = 1.33 \[ \frac {\sqrt {c \,x^{2}+a}\, x^{2}}{3 c e}-\frac {d^{4} \ln \left (\frac {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, e^{5}}+\frac {a d \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 c^{\frac {3}{2}} e^{2}}-\frac {d^{3} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{\sqrt {c}\, e^{4}}-\frac {\sqrt {c \,x^{2}+a}\, d x}{2 c \,e^{2}}-\frac {2 \sqrt {c \,x^{2}+a}\, a}{3 c^{2} e}+\frac {\sqrt {c \,x^{2}+a}\, d^{2}}{c \,e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 171, normalized size = 0.88 \[ \frac {\sqrt {c x^{2} + a} x^{2}}{3 \, c e} - \frac {\sqrt {c x^{2} + a} d x}{2 \, c e^{2}} - \frac {d^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {c} e^{4}} + \frac {a d \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{2 \, c^{\frac {3}{2}} e^{2}} + \frac {d^{4} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | e x + d \right |}} - \frac {a e}{\sqrt {a c} {\left | e x + d \right |}}\right )}{\sqrt {a + \frac {c d^{2}}{e^{2}}} e^{5}} + \frac {\sqrt {c x^{2} + a} d^{2}}{c e^{3}} - \frac {2 \, \sqrt {c x^{2} + a} a}{3 \, c^{2} e} \]
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 \[ \int \frac {x^4}{\sqrt {c\,x^2+a}\,\left (d+e\,x\right )} \,d x \]
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 \[ \int \frac {x^{4}}{\sqrt {a + c x^{2}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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