Optimal. Leaf size=211 \[ -\frac {\left (8 c d^3-e \left (4 c d^2-a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 c e^4}-\frac {7 d \left (a+c x^2\right )^{3/2}}{12 c e^2}+\frac {(d+e x) \left (a+c x^2\right )^{3/2}}{4 c e^2}+\frac {\left (8 c^2 d^4+4 a c d^2 e^2-a^2 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{3/2} e^5}+\frac {d^3 \sqrt {c d^2+a e^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^5} \]
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Rubi [A]
time = 0.24, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps
used = 8, 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 {\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^5}+\frac {d^3 \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^5}-\frac {\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^4}-\frac {7 d \left (a+c x^2\right )^{3/2}}{12 c e^2}+\frac {\left (a+c x^2\right )^{3/2} (d+e x)}{4 c e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 739
Rule 829
Rule 858
Rule 1668
Rubi steps
\begin {align*} \int \frac {x^3 \sqrt {a+c x^2}}{d+e x} \, dx &=\frac {(d+e x) \left (a+c x^2\right )^{3/2}}{4 c e^2}+\frac {\int \frac {\sqrt {a+c x^2} \left (-a d e^2-e \left (3 c d^2+a e^2\right ) x-7 c d e^2 x^2\right )}{d+e x} \, dx}{4 c e^3}\\ &=-\frac {7 d \left (a+c x^2\right )^{3/2}}{12 c e^2}+\frac {(d+e x) \left (a+c x^2\right )^{3/2}}{4 c e^2}+\frac {\int \frac {\left (-3 a c d e^4+3 c e^3 \left (4 c d^2-a e^2\right ) x\right ) \sqrt {a+c x^2}}{d+e x} \, dx}{12 c^2 e^5}\\ &=-\frac {\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^4}-\frac {7 d \left (a+c x^2\right )^{3/2}}{12 c e^2}+\frac {(d+e x) \left (a+c x^2\right )^{3/2}}{4 c e^2}+\frac {\int \frac {-3 a c^2 d e^4 \left (4 c d^2+a e^2\right )+3 c^2 e^3 \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}{24 c^3 e^7}\\ &=-\frac {\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^4}-\frac {7 d \left (a+c x^2\right )^{3/2}}{12 c e^2}+\frac {(d+e x) \left (a+c x^2\right )^{3/2}}{4 c e^2}-\frac {\left (d^3 \left (c d^2+a e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{e^5}+\frac {\left (8 c^2 d^4+4 a c d^2 e^2-a^2 e^4\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{8 c e^5}\\ &=-\frac {\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^4}-\frac {7 d \left (a+c x^2\right )^{3/2}}{12 c e^2}+\frac {(d+e x) \left (a+c x^2\right )^{3/2}}{4 c e^2}+\frac {\left (d^3 \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^5}+\frac {\left (8 c^2 d^4+4 a c d^2 e^2-a^2 e^4\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{8 c e^5}\\ &=-\frac {\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^4}-\frac {7 d \left (a+c x^2\right )^{3/2}}{12 c e^2}+\frac {(d+e x) \left (a+c x^2\right )^{3/2}}{4 c e^2}+\frac {\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^5}+\frac {d^3 \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^5}\\ \end {align*}
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Mathematica [A]
time = 0.48, size = 199, normalized size = 0.94 \begin {gather*} \frac {\sqrt {c} e \sqrt {a+c x^2} \left (a e^2 (-8 d+3 e x)+c \left (-24 d^3+12 d^2 e x-8 d e^2 x^2+6 e^3 x^3\right )\right )-48 c^{3/2} d^3 \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 )-3 \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 )}{24 c^{3/2} e^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(386\) vs.
\(2(185)=370\).
time = 0.09, size = 387, normalized size = 1.83
method | result | size |
default | \(\frac {\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4 c}-\frac {a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4 c}}{e}-\frac {d \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{3 c \,e^{2}}+\frac {d^{2} \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{e^{3}}-\frac {d^{3} \left (\sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}-\frac {\sqrt {c}\, d \ln \left (\frac {-\frac {c d}{e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\right )}{e}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}\right )}{e^{4}}\) | \(387\) |
risch | \(-\frac {\left (-6 c \,e^{3} x^{3}+8 d \,e^{2} c \,x^{2}-3 a \,e^{3} x -12 c \,d^{2} e x +8 a d \,e^{2}+24 c \,d^{3}\right ) \sqrt {c \,x^{2}+a}}{24 c \,e^{4}}-\frac {\ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right ) a^{2}}{8 c^{\frac {3}{2}} e}+\frac {\ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right ) a \,d^{2}}{2 \sqrt {c}\, e^{3}}+\frac {\sqrt {c}\, \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right ) d^{4}}{e^{5}}+\frac {d^{3} \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right ) a}{e^{4} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}+\frac {c \,d^{5} \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{6} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}\) | \(406\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 200, normalized size = 0.95 \begin {gather*} \sqrt {c} d^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-5\right )} - \sqrt {c d^{2} e^{\left (-2\right )} + a} d^{3} \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 (-4\right )} + \frac {1}{2} \, \sqrt {c x^{2} + a} d^{2} x e^{\left (-3\right )} + \frac {a d^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-3\right )}}{2 \, \sqrt {c}} - \sqrt {c x^{2} + a} d^{3} e^{\left (-4\right )} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} x e^{\left (-1\right )}}{4 \, c} - \frac {\sqrt {c x^{2} + a} a x e^{\left (-1\right )}}{8 \, c} - \frac {a^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-1\right )}}{8 \, c^{\frac {3}{2}}} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} d e^{\left (-2\right )}}{3 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 10.54, size = 920, normalized size = 4.36 \begin {gather*} \left [\frac {{\left (24 \, \sqrt {c d^{2} + a e^{2}} c^{2} d^{3} \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 (8 \, c^{2} d^{4} + 4 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (12 \, c^{2} d^{2} x e^{2} - 24 \, c^{2} d^{3} e + 3 \, {\left (2 \, c^{2} x^{3} + a c x\right )} e^{4} - 8 \, {\left (c^{2} d x^{2} + a c d\right )} e^{3}\right )} \sqrt {c x^{2} + a}\right )} e^{\left (-5\right )}}{48 \, c^{2}}, -\frac {{\left (48 \, \sqrt {-c d^{2} - a e^{2}} c^{2} d^{3} \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 (8 \, c^{2} d^{4} + 4 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - 2 \, {\left (12 \, c^{2} d^{2} x e^{2} - 24 \, c^{2} d^{3} e + 3 \, {\left (2 \, c^{2} x^{3} + a c x\right )} e^{4} - 8 \, {\left (c^{2} d x^{2} + a c d\right )} e^{3}\right )} \sqrt {c x^{2} + a}\right )} e^{\left (-5\right )}}{48 \, c^{2}}, \frac {{\left (12 \, \sqrt {c d^{2} + a e^{2}} c^{2} d^{3} \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 (8 \, c^{2} d^{4} + 4 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (12 \, c^{2} d^{2} x e^{2} - 24 \, c^{2} d^{3} e + 3 \, {\left (2 \, c^{2} x^{3} + a c x\right )} e^{4} - 8 \, {\left (c^{2} d x^{2} + a c d\right )} e^{3}\right )} \sqrt {c x^{2} + a}\right )} e^{\left (-5\right )}}{24 \, c^{2}}, -\frac {{\left (24 \, \sqrt {-c d^{2} - a e^{2}} c^{2} d^{3} \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 (8 \, c^{2} d^{4} + 4 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (12 \, c^{2} d^{2} x e^{2} - 24 \, c^{2} d^{3} e + 3 \, {\left (2 \, c^{2} x^{3} + a c x\right )} e^{4} - 8 \, {\left (c^{2} d x^{2} + a c d\right )} e^{3}\right )} \sqrt {c x^{2} + a}\right )} e^{\left (-5\right )}}{24 \, c^{2}}\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^{3} \sqrt {a + c x^{2}}}{d + e x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.98, size = 201, normalized size = 0.95 \begin {gather*} -\frac {2 \, {\left (c d^{5} + a d^{3} 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 (-5\right )}}{\sqrt {-c d^{2} - a e^{2}}} + \frac {1}{24} \, \sqrt {c x^{2} + a} {\left ({\left (2 \, {\left (3 \, x e^{\left (-1\right )} - 4 \, d e^{\left (-2\right )}\right )} x + \frac {3 \, {\left (4 \, c^{2} d^{2} e^{12} + a c e^{14}\right )} e^{\left (-15\right )}}{c^{2}}\right )} x - \frac {8 \, {\left (3 \, c^{2} d^{3} e^{11} + a c d e^{13}\right )} e^{\left (-15\right )}}{c^{2}}\right )} - \frac {{\left (8 \, c^{2} d^{4} + 4 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} e^{\left (-5\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{8 \, c^{\frac {3}{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 {x^3\,\sqrt {c\,x^2+a}}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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