Optimal. Leaf size=271 \[ -\frac {3 \left (3 c d^2+a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^2 d^2 e^3}-\frac {2 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e^3 \left (c d^2-a e^2\right ) (d+e x)}+\frac {(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 c d e^3}+\frac {3 \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right ) \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 c^{5/2} d^{5/2} e^{7/2}} \]
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Rubi [A]
time = 0.22, antiderivative size = 298, normalized size of antiderivative = 1.10, number of steps
used = 5, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {863, 832, 793,
635, 212} \begin {gather*} \frac {3 \left (a^2 e^4+2 a c d^2 e^2+5 c^2 d^4\right ) \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{5/2} d^{5/2} e^{7/2}}-\frac {\left (\left (5 c d^2-3 a e^2\right ) \left (a e^2+3 c d^2\right )-2 c d e x \left (5 c d^2-a e^2\right )\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c^2 d^2 e^3 \left (c d^2-a e^2\right )}-\frac {2 d x^2 \left (c d x \left (c d^2-a e^2\right )+a e \left (c d^2-a e^2\right )\right )}{e \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 793
Rule 832
Rule 863
Rubi steps
\begin {align*} \int \frac {x^3}{(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\int \frac {x^3 (a e+c d x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx\\ &=-\frac {2 d x^2 \left (a e \left (c d^2-a e^2\right )+c d \left (c d^2-a e^2\right ) x\right )}{e \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 \int \frac {x \left (2 a c d^2 e \left (c d^2-a e^2\right )+\frac {1}{2} c d \left (c d^2-a e^2\right ) \left (5 c d^2-a e^2\right ) x\right )}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{c d e \left (c d^2-a e^2\right )^2}\\ &=-\frac {2 d x^2 \left (a e \left (c d^2-a e^2\right )+c d \left (c d^2-a e^2\right ) x\right )}{e \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (\left (5 c d^2-3 a e^2\right ) \left (3 c d^2+a e^2\right )-2 c d e \left (5 c d^2-a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^2 d^2 e^3 \left (c d^2-a e^2\right )}+\frac {\left (3 \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right )\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 c^2 d^2 e^3}\\ &=-\frac {2 d x^2 \left (a e \left (c d^2-a e^2\right )+c d \left (c d^2-a e^2\right ) x\right )}{e \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (\left (5 c d^2-3 a e^2\right ) \left (3 c d^2+a e^2\right )-2 c d e \left (5 c d^2-a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^2 d^2 e^3 \left (c d^2-a e^2\right )}+\frac {\left (3 \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right )\right ) \text {Subst}\left (\int \frac {1}{4 c d e-x^2} \, dx,x,\frac {c d^2+a e^2+2 c d e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 c^2 d^2 e^3}\\ &=-\frac {2 d x^2 \left (a e \left (c d^2-a e^2\right )+c d \left (c d^2-a e^2\right ) x\right )}{e \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (\left (5 c d^2-3 a e^2\right ) \left (3 c d^2+a e^2\right )-2 c d e \left (5 c d^2-a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^2 d^2 e^3 \left (c d^2-a e^2\right )}+\frac {3 \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right ) \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 c^{5/2} d^{5/2} e^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.48, size = 277, normalized size = 1.02 \begin {gather*} \frac {\sqrt {c} \sqrt {d} \sqrt {e} \left (3 a^3 e^5 (d+e x)+a^2 c d e^3 \left (4 d^2+5 d e x+e^2 x^2\right )+c^3 d^4 x \left (-15 d^2-5 d e x+2 e^2 x^2\right )-a c^2 d^2 e \left (15 d^3+d^2 e x-4 d e^2 x^2+2 e^3 x^3\right )\right )+3 \left (5 c^3 d^6-3 a c^2 d^4 e^2-a^2 c d^2 e^4-a^3 e^6\right ) \sqrt {a e+c d x} \sqrt {d+e x} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{4 c^{5/2} d^{5/2} e^{7/2} \left (c d^2-a e^2\right ) \sqrt {(a e+c d x) (d+e x)}} \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. \(513\) vs.
\(2(243)=486\).
time = 0.09, size = 514, normalized size = 1.90
method | result | size |
default | \(\frac {\frac {x \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{2 c d e}-\frac {3 \left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{c d e}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{2 c d e \sqrt {c d e}}\right )}{4 c d e}-\frac {a \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{2 c \sqrt {c d e}}}{e}-\frac {d \left (\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{c d e}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{2 c d e \sqrt {c d e}}\right )}{e^{2}}+\frac {d^{2} \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{e^{3} \sqrt {c d e}}+\frac {2 d^{3} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{e^{4} \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\) | \(514\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.70, size = 729, normalized size = 2.69 \begin {gather*} \left [\frac {3 \, {\left (5 \, c^{3} d^{6} x e + 5 \, c^{3} d^{7} - 3 \, a c^{2} d^{4} x e^{3} - 3 \, a c^{2} d^{5} e^{2} - a^{2} c d^{2} x e^{5} - a^{2} c d^{3} e^{4} - a^{3} x e^{7} - a^{3} d e^{6}\right )} \sqrt {c d} e^{\frac {1}{2}} \log \left (8 \, c^{2} d^{3} x e + c^{2} d^{4} + 8 \, a c d x e^{3} + a^{2} e^{4} + 4 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (2 \, c d x e + c d^{2} + a e^{2}\right )} \sqrt {c d} e^{\frac {1}{2}} + 2 \, {\left (4 \, c^{2} d^{2} x^{2} + 3 \, a c d^{2}\right )} e^{2}\right ) - 4 \, {\left (5 \, c^{3} d^{5} x e^{2} + 15 \, c^{3} d^{6} e - 2 \, a c^{2} d^{3} x e^{4} - 3 \, a^{2} c d x e^{6} + {\left (2 \, a c^{2} d^{2} x^{2} - 3 \, a^{2} c d^{2}\right )} e^{5} - 2 \, {\left (c^{3} d^{4} x^{2} + 2 \, a c^{2} d^{4}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{16 \, {\left (c^{4} d^{5} x e^{5} + c^{4} d^{6} e^{4} - a c^{3} d^{3} x e^{7} - a c^{3} d^{4} e^{6}\right )}}, -\frac {3 \, {\left (5 \, c^{3} d^{6} x e + 5 \, c^{3} d^{7} - 3 \, a c^{2} d^{4} x e^{3} - 3 \, a c^{2} d^{5} e^{2} - a^{2} c d^{2} x e^{5} - a^{2} c d^{3} e^{4} - a^{3} x e^{7} - a^{3} d e^{6}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (2 \, c d x e + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{3} x e + a c d x e^{3} + {\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{2}\right )}}\right ) + 2 \, {\left (5 \, c^{3} d^{5} x e^{2} + 15 \, c^{3} d^{6} e - 2 \, a c^{2} d^{3} x e^{4} - 3 \, a^{2} c d x e^{6} + {\left (2 \, a c^{2} d^{2} x^{2} - 3 \, a^{2} c d^{2}\right )} e^{5} - 2 \, {\left (c^{3} d^{4} x^{2} + 2 \, a c^{2} d^{4}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{8 \, {\left (c^{4} d^{5} x e^{5} + c^{4} d^{6} e^{4} - a c^{3} d^{3} x e^{7} - a c^{3} d^{4} 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^{3}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \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 = 1.17, size = 249, normalized size = 0.92 \begin {gather*} -\frac {2 \, d^{3} e^{\left (-3\right )}}{\sqrt {c d} d e^{\frac {1}{2}} + {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} e} + \frac {1}{4} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} {\left (\frac {2 \, x e^{\left (-2\right )}}{c d} - \frac {{\left (7 \, c d^{2} e^{5} + 3 \, a e^{7}\right )} e^{\left (-8\right )}}{c^{2} d^{2}}\right )} - \frac {3 \, {\left (5 \, \sqrt {c d} c^{2} d^{4} e^{\frac {1}{2}} + 2 \, \sqrt {c d} a c d^{2} e^{\frac {5}{2}} + \sqrt {c d} a^{2} e^{\frac {9}{2}}\right )} e^{\left (-4\right )} \log \left ({\left | -\sqrt {c d} c d^{2} e^{\frac {1}{2}} - 2 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} c d e - \sqrt {c d} a e^{\frac {5}{2}} \right |}\right )}{8 \, c^{3} d^{3}} \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}{\left (d+e\,x\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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