Optimal. Leaf size=251 \[ -\frac {\left (c d^2-a e^2\right ) \left (a^2 e^4+2 a c d^2 e^2+5 c^2 d^4\right ) \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 c^{5/2} d^{5/2} e^{7/2}}+\frac {\left (\left (5 c d^2-3 a e^2\right ) \left (a e^2+3 c d^2\right )-2 c d e x \left (5 c d^2-a e^2\right )\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{24 c^2 d^2 e^3}+\frac {x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 e} \]
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Rubi [A] time = 0.26, 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 = {851, 832, 779, 621, 206} \[ -\frac {\left (c d^2-a e^2\right ) \left (a^2 e^4+2 a c d^2 e^2+5 c^2 d^4\right ) \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 c^{5/2} d^{5/2} e^{7/2}}+\frac {\left (\left (5 c d^2-3 a e^2\right ) \left (a e^2+3 c d^2\right )-2 c d e x \left (5 c d^2-a e^2\right )\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{24 c^2 d^2 e^3}+\frac {x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 e} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 779
Rule 832
Rule 851
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 ) \operatorname {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 = 0.82, size = 245, normalized size = 0.98 \[ \frac {\sqrt {(d+e x) (a e+c d x)} \left (\sqrt {c} \sqrt {d} \sqrt {e} \left (-3 a^2 e^4+2 a c d e^2 (e x-2 d)+c^2 d^2 \left (15 d^2-10 d e x+8 e^2 x^2\right )\right )-\frac {3 \sqrt {c d} \sqrt {c d^2-a e^2} \left (a^2 e^4+2 a c d^2 e^2+5 c^2 d^4\right ) \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d} \sqrt {c d^2-a e^2}}\right )}{\sqrt {a e+c d x} \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}}\right )}{24 c^{5/2} d^{5/2} e^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.09, size = 536, normalized size = 2.14 \[ \left [-\frac {3 \, {\left (5 \, c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} - a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) - 4 \, {\left (8 \, c^{3} d^{3} e^{3} x^{2} + 15 \, c^{3} d^{5} e - 4 \, a c^{2} d^{3} e^{3} - 3 \, a^{2} c d e^{5} - 2 \, {\left (5 \, c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{96 \, c^{3} d^{3} e^{4}}, \frac {3 \, {\left (5 \, c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} - a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) + 2 \, {\left (8 \, c^{3} d^{3} e^{3} x^{2} + 15 \, c^{3} d^{5} e - 4 \, a c^{2} d^{3} e^{3} - 3 \, a^{2} c d e^{5} - 2 \, {\left (5 \, c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{48 \, c^{3} d^{3} e^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 713, normalized size = 2.84 \[ \frac {a^{3} e^{3} \ln \left (\frac {c d e x +\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}}{\sqrt {c d e}}+\sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\right )}{16 \sqrt {c d e}\, c^{2} d^{2}}+\frac {a^{2} e \ln \left (\frac {c d e x +\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}}{\sqrt {c d e}}+\sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\right )}{16 \sqrt {c d e}\, c}+\frac {a \,d^{2} \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+\left (x +\frac {d}{e}\right ) c d e}{\sqrt {c d e}}+\sqrt {\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{2 \sqrt {c d e}\, e}-\frac {5 a \,d^{2} \ln \left (\frac {c d e x +\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}}{\sqrt {c d e}}+\sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\right )}{16 \sqrt {c d e}\, e}-\frac {c \,d^{4} \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+\left (x +\frac {d}{e}\right ) c d e}{\sqrt {c d e}}+\sqrt {\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{2 \sqrt {c d e}\, e^{3}}+\frac {3 c \,d^{4} \ln \left (\frac {c d e x +\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}}{\sqrt {c d e}}+\sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\right )}{16 \sqrt {c d e}\, e^{3}}-\frac {\sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, a x}{4 c d}-\frac {3 \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, d x}{4 e^{2}}-\frac {\sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, a^{2} e}{8 c^{2} d^{2}}-\frac {\sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, a}{2 c e}-\frac {3 \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, d^{2}}{8 e^{3}}+\frac {\sqrt {\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\, d^{2}}{e^{3}}+\frac {\left (c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x \right )^{\frac {3}{2}}}{3 c d \,e^{2}} \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {x^2\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x} \,d x \]
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 \[ \int \frac {x^{2} \sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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