Optimal. Leaf size=255 \[ \frac {d \left (8 c d^3-e \left (4 c d^2-a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 c e^5}+\frac {\left (47 c d^2-8 a e^2\right ) \left (a+c x^2\right )^{3/2}}{60 c^2 e^3}-\frac {13 d (d+e x) \left (a+c x^2\right )^{3/2}}{20 c e^3}+\frac {(d+e x)^2 \left (a+c x^2\right )^{3/2}}{5 c e^3}-\frac {d \left (8 c^2 d^4+4 a c d^2 e^2-a^2 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{3/2} e^6}-\frac {d^4 \sqrt {c d^2+a e^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^6} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.39, 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 = {1668, 829, 858,
223, 212, 739} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 223
Rule 739
Rule 829
Rule 858
Rule 1668
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 ) \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^6}-\frac {\left (d \left (8 c^2 d^4+4 a c d^2 e^2-a^2 e^4\right )\right ) \text {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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.63, size = 230, normalized size = 0.90 \begin {gather*} \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 )+240 c^2 d^4 \sqrt {-c d^2-a e^2} \tan ^{-1}\left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )+15 \sqrt {c} d \left (8 c^2 d^4+4 a c d^2 e^2-a^2 e^4\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{120 c^2 e^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.09, size = 428, normalized size = 1.68
method | result | size |
default | \(\frac {\frac {x^{2} \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{5 c}-\frac {2 a \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{15 c^{2}}}{e}-\frac {d \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4 c}-\frac {a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4 c}\right )}{e^{2}}+\frac {d^{2} \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{3 e^{3} c}-\frac {d^{3} \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{e^{4}}+\frac {d^{4} \left (\sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}-\frac {\sqrt {c}\, d \ln \left (\frac {-\frac {c d}{e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\right )}{e}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}\right )}{e^{5}}\) | \(428\) |
risch | \(-\frac {\left (-24 e^{4} c^{2} x^{4}+30 d \,e^{3} c^{2} x^{3}-8 a c \,e^{4} x^{2}-40 c^{2} d^{2} e^{2} x^{2}+15 a c d \,e^{3} x +60 c^{2} d^{3} e x +16 a^{2} e^{4}-40 a c \,d^{2} e^{2}-120 c^{2} d^{4}\right ) \sqrt {c \,x^{2}+a}}{120 c^{2} e^{5}}+\frac {d \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right ) a^{2}}{8 c^{\frac {3}{2}} e^{2}}-\frac {d^{3} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right ) a}{2 \sqrt {c}\, e^{4}}-\frac {\sqrt {c}\, d^{5} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e^{6}}-\frac {d^{4} \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right ) a}{e^{5} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}-\frac {c \,d^{6} \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{7} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}\) | \(455\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.34, size = 240, normalized size = 0.94 \begin {gather*} -\sqrt {c} d^{5} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-6\right )} + \sqrt {c d^{2} e^{\left (-2\right )} + a} 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 )} - \frac {1}{2} \, \sqrt {c x^{2} + a} d^{3} x e^{\left (-4\right )} - \frac {a d^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-4\right )}}{2 \, \sqrt {c}} + \sqrt {c x^{2} + a} d^{4} e^{\left (-5\right )} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} x^{2} e^{\left (-1\right )}}{5 \, c} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} d x e^{\left (-2\right )}}{4 \, c} + \frac {\sqrt {c x^{2} + a} a d x e^{\left (-2\right )}}{8 \, c} + \frac {a^{2} d \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-2\right )}}{8 \, c^{\frac {3}{2}}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} d^{2} e^{\left (-3\right )}}{3 \, c} - \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a e^{\left (-1\right )}}{15 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 13.04, size = 1045, normalized size = 4.10 \begin {gather*} \left [\frac {{\left (120 \, \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 ) - 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 (60 \, c^{2} d^{3} x e^{2} - 120 \, c^{2} d^{4} e - 8 \, {\left (3 \, c^{2} x^{4} + a c x^{2} - 2 \, a^{2}\right )} e^{5} + 15 \, {\left (2 \, c^{2} d x^{3} + a c d x\right )} e^{4} - 40 \, {\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + a}\right )} e^{\left (-6\right )}}{240 \, c^{2}}, \frac {{\left (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}}{c^{2} d^{2} x^{2} + a c d^{2} + {\left (a c x^{2} + a^{2}\right )} e^{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 (60 \, c^{2} d^{3} x e^{2} - 120 \, c^{2} d^{4} e - 8 \, {\left (3 \, c^{2} x^{4} + a c x^{2} - 2 \, a^{2}\right )} e^{5} + 15 \, {\left (2 \, c^{2} d x^{3} + a c d x\right )} e^{4} - 40 \, {\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + a}\right )} e^{\left (-6\right )}}{240 \, c^{2}}, \frac {{\left (60 \, \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 ) + 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 (60 \, c^{2} d^{3} x e^{2} - 120 \, c^{2} d^{4} e - 8 \, {\left (3 \, c^{2} x^{4} + a c x^{2} - 2 \, a^{2}\right )} e^{5} + 15 \, {\left (2 \, c^{2} d x^{3} + a c d x\right )} e^{4} - 40 \, {\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + a}\right )} e^{\left (-6\right )}}{120 \, c^{2}}, \frac {{\left (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}}{c^{2} d^{2} x^{2} + a c d^{2} + {\left (a c x^{2} + a^{2}\right )} e^{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 (60 \, c^{2} d^{3} x e^{2} - 120 \, c^{2} d^{4} e - 8 \, {\left (3 \, c^{2} x^{4} + a c x^{2} - 2 \, a^{2}\right )} e^{5} + 15 \, {\left (2 \, c^{2} d x^{3} + a c d x\right )} e^{4} - 40 \, {\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + a}\right )} e^{\left (-6\right )}}{120 \, c^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} \sqrt {a + c x^{2}}}{d + e x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.15, size = 252, normalized size = 0.99 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^4\,\sqrt {c\,x^2+a}}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________