Optimal. Leaf size=255 \[ -\frac {d \left (-a^2 e^4+4 a c d^2 e^2+8 c^2 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{3/2} e^6}+\frac {\left (a+c x^2\right )^{3/2} \left (47 c d^2-8 a e^2\right )}{60 c^2 e^3}-\frac {d^4 \sqrt {a e^2+c d^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^6}+\frac {d \sqrt {a+c x^2} \left (8 c d^3-e x \left (4 c d^2-a e^2\right )\right )}{8 c e^5}-\frac {13 d \left (a+c x^2\right )^{3/2} (d+e x)}{20 c e^3}+\frac {\left (a+c x^2\right )^{3/2} (d+e x)^2}{5 c e^3} \]
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Rubi [A] time = 0.63, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1654, 815, 844, 217, 206, 725} \[ -\frac {d \left (-a^2 e^4+4 a c d^2 e^2+8 c^2 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{3/2} e^6}+\frac {\left (a+c x^2\right )^{3/2} \left (47 c d^2-8 a e^2\right )}{60 c^2 e^3}+\frac {d \sqrt {a+c x^2} \left (8 c d^3-e x \left (4 c d^2-a e^2\right )\right )}{8 c e^5}-\frac {d^4 \sqrt {a e^2+c d^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^6}-\frac {13 d \left (a+c x^2\right )^{3/2} (d+e x)}{20 c e^3}+\frac {\left (a+c x^2\right )^{3/2} (d+e x)^2}{5 c e^3} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 725
Rule 815
Rule 844
Rule 1654
Rubi steps
\begin {align*} \int \frac {x^4 \sqrt {a+c x^2}}{d+e x} \, dx &=\frac {(d+e x)^2 \left (a+c x^2\right )^{3/2}}{5 c e^3}+\frac {\int \frac {\sqrt {a+c x^2} \left (-2 a d^2 e^2-d e \left (3 c d^2+4 a e^2\right ) x-e^2 \left (11 c d^2+2 a e^2\right ) x^2-13 c d e^3 x^3\right )}{d+e x} \, dx}{5 c e^4}\\ &=-\frac {13 d (d+e x) \left (a+c x^2\right )^{3/2}}{20 c e^3}+\frac {(d+e x)^2 \left (a+c x^2\right )^{3/2}}{5 c e^3}+\frac {\int \frac {\sqrt {a+c x^2} \left (5 a c d^2 e^5+3 c d e^4 \left (9 c d^2-a e^2\right ) x+c e^5 \left (47 c d^2-8 a e^2\right ) x^2\right )}{d+e x} \, dx}{20 c^2 e^7}\\ &=\frac {\left (47 c d^2-8 a e^2\right ) \left (a+c x^2\right )^{3/2}}{60 c^2 e^3}-\frac {13 d (d+e x) \left (a+c x^2\right )^{3/2}}{20 c e^3}+\frac {(d+e x)^2 \left (a+c x^2\right )^{3/2}}{5 c e^3}+\frac {\int \frac {\left (15 a c^2 d^2 e^7-15 c^2 d e^6 \left (4 c d^2-a e^2\right ) x\right ) \sqrt {a+c x^2}}{d+e x} \, dx}{60 c^3 e^9}\\ &=\frac {d \left (8 c d^3-e \left (4 c d^2-a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 c e^5}+\frac {\left (47 c d^2-8 a e^2\right ) \left (a+c x^2\right )^{3/2}}{60 c^2 e^3}-\frac {13 d (d+e x) \left (a+c x^2\right )^{3/2}}{20 c e^3}+\frac {(d+e x)^2 \left (a+c x^2\right )^{3/2}}{5 c e^3}+\frac {\int \frac {15 a c^3 d^2 e^7 \left (4 c d^2+a e^2\right )-15 c^3 d e^6 \left (8 c^2 d^4+4 a c d^2 e^2-a^2 e^4\right ) x}{(d+e x) \sqrt {a+c x^2}} \, dx}{120 c^4 e^{11}}\\ &=\frac {d \left (8 c d^3-e \left (4 c d^2-a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 c e^5}+\frac {\left (47 c d^2-8 a e^2\right ) \left (a+c x^2\right )^{3/2}}{60 c^2 e^3}-\frac {13 d (d+e x) \left (a+c x^2\right )^{3/2}}{20 c e^3}+\frac {(d+e x)^2 \left (a+c x^2\right )^{3/2}}{5 c e^3}+\frac {\left (d^4 \left (c d^2+a e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{e^6}-\frac {\left (d \left (8 c^2 d^4+4 a c d^2 e^2-a^2 e^4\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{8 c e^6}\\ &=\frac {d \left (8 c d^3-e \left (4 c d^2-a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 c e^5}+\frac {\left (47 c d^2-8 a e^2\right ) \left (a+c x^2\right )^{3/2}}{60 c^2 e^3}-\frac {13 d (d+e x) \left (a+c x^2\right )^{3/2}}{20 c e^3}+\frac {(d+e x)^2 \left (a+c x^2\right )^{3/2}}{5 c e^3}-\frac {\left (d^4 \left (c d^2+a e^2\right )\right ) \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^6}-\frac {\left (d \left (8 c^2 d^4+4 a c d^2 e^2-a^2 e^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{8 c e^6}\\ &=\frac {d \left (8 c d^3-e \left (4 c d^2-a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 c e^5}+\frac {\left (47 c d^2-8 a e^2\right ) \left (a+c x^2\right )^{3/2}}{60 c^2 e^3}-\frac {13 d (d+e x) \left (a+c x^2\right )^{3/2}}{20 c e^3}+\frac {(d+e x)^2 \left (a+c x^2\right )^{3/2}}{5 c e^3}-\frac {d \left (8 c^2 d^4+4 a c d^2 e^2-a^2 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{3/2} e^6}-\frac {d^4 \sqrt {c d^2+a e^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^6}\\ \end {align*}
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Mathematica [A] time = 0.61, size = 259, normalized size = 1.02 \[ \frac {e \sqrt {a+c x^2} \left (-16 a^2 e^4+a c e^2 \left (40 d^2-15 d e x+8 e^2 x^2\right )+2 c^2 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )-120 c^{5/2} d^5 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )-120 c^2 d^4 \sqrt {a e^2+c d^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )+\frac {15 \sqrt {a} \sqrt {c} d e^2 \sqrt {a+c x^2} \left (a e^2-4 c d^2\right ) \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {\frac {c x^2}{a}+1}}}{120 c^2 e^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 13.72, size = 1104, normalized size = 4.33 \[ \left [\frac {120 \, \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 ) - 15 \, {\left (8 \, c^{2} d^{5} + 4 \, 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 (24 \, c^{2} e^{5} x^{4} - 30 \, c^{2} d e^{4} x^{3} + 120 \, c^{2} d^{4} e + 40 \, a c d^{2} e^{3} - 16 \, a^{2} e^{5} + 8 \, {\left (5 \, c^{2} d^{2} e^{3} + a c e^{5}\right )} x^{2} - 15 \, {\left (4 \, c^{2} d^{3} e^{2} + a c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{240 \, c^{2} e^{6}}, -\frac {240 \, \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 ) + 15 \, {\left (8 \, c^{2} d^{5} + 4 \, 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 (24 \, c^{2} e^{5} x^{4} - 30 \, c^{2} d e^{4} x^{3} + 120 \, c^{2} d^{4} e + 40 \, a c d^{2} e^{3} - 16 \, a^{2} e^{5} + 8 \, {\left (5 \, c^{2} d^{2} e^{3} + a c e^{5}\right )} x^{2} - 15 \, {\left (4 \, c^{2} d^{3} e^{2} + a c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{240 \, c^{2} e^{6}}, \frac {60 \, \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 ) + 15 \, {\left (8 \, c^{2} d^{5} + 4 \, 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 (24 \, c^{2} e^{5} x^{4} - 30 \, c^{2} d e^{4} x^{3} + 120 \, c^{2} d^{4} e + 40 \, a c d^{2} e^{3} - 16 \, a^{2} e^{5} + 8 \, {\left (5 \, c^{2} d^{2} e^{3} + a c e^{5}\right )} x^{2} - 15 \, {\left (4 \, c^{2} d^{3} e^{2} + a c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{120 \, c^{2} e^{6}}, -\frac {120 \, \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 ) - 15 \, {\left (8 \, c^{2} d^{5} + 4 \, 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 (24 \, c^{2} e^{5} x^{4} - 30 \, c^{2} d e^{4} x^{3} + 120 \, c^{2} d^{4} e + 40 \, a c d^{2} e^{3} - 16 \, a^{2} e^{5} + 8 \, {\left (5 \, c^{2} d^{2} e^{3} + a c e^{5}\right )} x^{2} - 15 \, {\left (4 \, c^{2} d^{3} e^{2} + a c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{120 \, c^{2} e^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 252, normalized size = 0.99 \[ \frac {2 \, {\left (c d^{6} + a d^{4} e^{2}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right ) e^{\left (-6\right )}}{\sqrt {-c d^{2} - a e^{2}}} + \frac {1}{120} \, \sqrt {c x^{2} + a} {\left ({\left (2 \, {\left (3 \, {\left (4 \, x e^{\left (-1\right )} - 5 \, d e^{\left (-2\right )}\right )} x + \frac {4 \, {\left (5 \, c^{3} d^{2} e^{18} + a c^{2} e^{20}\right )} e^{\left (-21\right )}}{c^{3}}\right )} x - \frac {15 \, {\left (4 \, c^{3} d^{3} e^{17} + a c^{2} d e^{19}\right )} e^{\left (-21\right )}}{c^{3}}\right )} x + \frac {8 \, {\left (15 \, c^{3} d^{4} e^{16} + 5 \, a c^{2} d^{2} e^{18} - 2 \, a^{2} c e^{20}\right )} e^{\left (-21\right )}}{c^{3}}\right )} + \frac {{\left (8 \, c^{\frac {5}{2}} d^{5} + 4 \, a c^{\frac {3}{2}} d^{3} e^{2} - a^{2} \sqrt {c} d e^{4}\right )} e^{\left (-6\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{8 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 560, normalized size = 2.20 \[ -\frac {a \,d^{4} \ln \left (\frac {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, e^{5}}-\frac {c \,d^{6} \ln \left (\frac {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, e^{7}}+\frac {a^{2} d \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{8 c^{\frac {3}{2}} e^{2}}-\frac {a \,d^{3} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}\, e^{4}}-\frac {\sqrt {c}\, d^{5} \ln \left (\frac {-\frac {c d}{e}+\left (x +\frac {d}{e}\right ) c}{\sqrt {c}}+\sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\right )}{e^{6}}+\frac {\sqrt {c \,x^{2}+a}\, a d x}{8 c \,e^{2}}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} x^{2}}{5 c e}-\frac {\sqrt {c \,x^{2}+a}\, d^{3} x}{2 e^{4}}-\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} d x}{4 c \,e^{2}}+\frac {\sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, d^{4}}{e^{5}}-\frac {2 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a}{15 c^{2} e}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} d^{2}}{3 c \,e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 249, normalized size = 0.98 \[ \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} x^{2}}{5 \, c e} - \frac {\sqrt {c x^{2} + a} d^{3} x}{2 \, e^{4}} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} d x}{4 \, c e^{2}} + \frac {\sqrt {c x^{2} + a} a d x}{8 \, c e^{2}} - \frac {\sqrt {c} d^{5} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{e^{6}} - \frac {a d^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {c} e^{4}} + \frac {a^{2} d \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{8 \, c^{\frac {3}{2}} e^{2}} + \frac {\sqrt {a + \frac {c d^{2}}{e^{2}}} d^{4} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | e x + d \right |}} - \frac {a e}{\sqrt {a c} {\left | e x + d \right |}}\right )}{e^{5}} + \frac {\sqrt {c x^{2} + a} d^{4}}{e^{5}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} d^{2}}{3 \, c e^{3}} - \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a}{15 \, c^{2} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^4\,\sqrt {c\,x^2+a}}{d+e\,x} \,d x \]
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 \[ \int \frac {x^{4} \sqrt {a + c x^{2}}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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