Optimal. Leaf size=251 \[ \frac {x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 e}+\frac {\left (\left (5 c d^2-3 a e^2\right ) \left (3 c d^2+a e^2\right )-2 c d e \left (5 c d^2-a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 c^2 d^2 e^3}-\frac {\left (c d^2-a e^2\right ) \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right ) \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{16 c^{5/2} d^{5/2} e^{7/2}} \]
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Rubi [A]
time = 0.17, antiderivative size = 251, normalized size of antiderivative = 1.00, 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 = {865, 846, 793,
635, 212} \begin {gather*} -\frac {\left (c d^2-a e^2\right ) \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 )}{16 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}}{24 c^2 d^2 e^3}+\frac {x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 793
Rule 846
Rule 865
Rubi steps
\begin {align*} \int \frac {x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx &=\int \frac {x^2 (a e+c d x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\\ &=\frac {x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 e}+\frac {\int \frac {x \left (-2 a c d^2 e-\frac {1}{2} c d \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}{3 c d e}\\ &=\frac {x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 e}+\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}}{24 c^2 d^2 e^3}-\frac {\left (\left (c d^2-a e^2\right ) \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}{16 c^2 d^2 e^3}\\ &=\frac {x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 e}+\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}}{24 c^2 d^2 e^3}-\frac {\left (\left (c d^2-a e^2\right ) \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 )}{8 c^2 d^2 e^3}\\ &=\frac {x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 e}+\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}}{24 c^2 d^2 e^3}-\frac {\left (c d^2-a e^2\right ) \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 )}{16 c^{5/2} d^{5/2} e^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.78, size = 218, normalized size = 0.87 \begin {gather*} \frac {\sqrt {a e+c d x} \sqrt {d+e x} \left (-e \sqrt {a e+c d x} \sqrt {d+e x} \left (3 a^2 e^4+2 a c d e^2 (2 d-e x)+c^2 d^2 \left (-15 d^2+10 d e x-8 e^2 x^2\right )\right )-3 \sqrt {\frac {e}{c d}} \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 ) \log \left (-\sqrt {\frac {e}{c d}} \sqrt {a e+c d x}+\sqrt {d+e x}\right )\right )}{24 c^2 d^2 e^4 \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. \(511\) vs.
\(2(225)=450\).
time = 0.08, size = 512, normalized size = 2.04
method | result | size |
default | \(\frac {\frac {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{3 c d e}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {\left (2 c d e x +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{8 c d e \sqrt {c d e}}\right )}{2 c d e}}{e}-\frac {d \left (\frac {\left (2 c d e x +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{8 c d e \sqrt {c d e}}\right )}{e^{2}}+\frac {d^{2} \left (\sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+c d e \left (x +\frac {d}{e}\right )}{\sqrt {c d e}}+\sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{2 \sqrt {c d e}}\right )}{e^{3}}\) | \(512\) |
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 = 3.25, size = 517, normalized size = 2.06 \begin {gather*} \left [-\frac {{\left (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 {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 (10 \, c^{3} d^{4} x e^{2} - 15 \, c^{3} d^{5} e - 2 \, a c^{2} d^{2} x e^{4} + 3 \, a^{2} c d e^{5} - 4 \, {\left (2 \, c^{3} d^{3} x^{2} - a c^{2} d^{3}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-4\right )}}{96 \, c^{3} d^{3}}, \frac {{\left (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 {-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 (10 \, c^{3} d^{4} x e^{2} - 15 \, c^{3} d^{5} e - 2 \, a c^{2} d^{2} x e^{4} + 3 \, a^{2} c d e^{5} - 4 \, {\left (2 \, c^{3} d^{3} x^{2} - a c^{2} d^{3}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-4\right )}}{48 \, c^{3} d^{3}}\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^{2} \sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}{d + e x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.40, size = 217, normalized size = 0.86 \begin {gather*} \frac {1}{24} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} {\left (2 \, {\left (4 \, x e^{\left (-1\right )} - \frac {{\left (5 \, c^{2} d^{3} e - a c d e^{3}\right )} e^{\left (-3\right )}}{c^{2} d^{2}}\right )} x + \frac {{\left (15 \, c^{2} d^{4} - 4 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4}\right )} e^{\left (-3\right )}}{c^{2} d^{2}}\right )} + \frac {{\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 )} e^{\left (-\frac {7}{2}\right )} \log \left ({\left | -c d^{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 )} \sqrt {c d} e^{\frac {1}{2}} - a e^{2} \right |}\right )}{16 \, \sqrt {c d} c^{2} d^{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^2\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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