Optimal. Leaf size=398 \[ \frac {d^2 (d+e x) \sqrt {\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c (d+e x)^3}{\sqrt {b^2-4 a c}+b}+1} F_1\left (\frac {1}{3};\frac {1}{2},\frac {1}{2};\frac {4}{3};-\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}\right )}{e^3 \sqrt {a+b (d+e x)^3+c (d+e x)^6}}-\frac {d (d+e x)^2 \sqrt {\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c (d+e x)^3}{\sqrt {b^2-4 a c}+b}+1} F_1\left (\frac {2}{3};\frac {1}{2},\frac {1}{2};\frac {5}{3};-\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}\right )}{e^3 \sqrt {a+b (d+e x)^3+c (d+e x)^6}}+\frac {\tanh ^{-1}\left (\frac {b+2 c (d+e x)^3}{2 \sqrt {c} \sqrt {a+b (d+e x)^3+c (d+e x)^6}}\right )}{3 \sqrt {c} e^3} \]
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Rubi [A] time = 0.69, antiderivative size = 398, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {1389, 1790, 1348, 429, 1385, 510, 1352, 621, 206} \[ \frac {d^2 (d+e x) \sqrt {\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c (d+e x)^3}{\sqrt {b^2-4 a c}+b}+1} F_1\left (\frac {1}{3};\frac {1}{2},\frac {1}{2};\frac {4}{3};-\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}\right )}{e^3 \sqrt {a+b (d+e x)^3+c (d+e x)^6}}-\frac {d (d+e x)^2 \sqrt {\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c (d+e x)^3}{\sqrt {b^2-4 a c}+b}+1} F_1\left (\frac {2}{3};\frac {1}{2},\frac {1}{2};\frac {5}{3};-\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}\right )}{e^3 \sqrt {a+b (d+e x)^3+c (d+e x)^6}}+\frac {\tanh ^{-1}\left (\frac {b+2 c (d+e x)^3}{2 \sqrt {c} \sqrt {a+b (d+e x)^3+c (d+e x)^6}}\right )}{3 \sqrt {c} e^3} \]
Antiderivative was successfully verified.
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Rule 206
Rule 429
Rule 510
Rule 621
Rule 1348
Rule 1352
Rule 1385
Rule 1389
Rule 1790
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(-d+x)^2}{\sqrt {a+b x^3+c x^6}} \, dx,x,d+e x\right )}{e^3}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {d^2}{\sqrt {a+b x^3+c x^6}}-\frac {2 d x}{\sqrt {a+b x^3+c x^6}}+\frac {x^2}{\sqrt {a+b x^3+c x^6}}\right ) \, dx,x,d+e x\right )}{e^3}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+b x^3+c x^6}} \, dx,x,d+e x\right )}{e^3}-\frac {(2 d) \operatorname {Subst}\left (\int \frac {x}{\sqrt {a+b x^3+c x^6}} \, dx,x,d+e x\right )}{e^3}+\frac {d^2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^3+c x^6}} \, dx,x,d+e x\right )}{e^3}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,(d+e x)^3\right )}{3 e^3}-\frac {\left (2 d \sqrt {1+\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}}} \, dx,x,d+e x\right )}{e^3 \sqrt {a+b (d+e x)^3+c (d+e x)^6}}+\frac {\left (d^2 \sqrt {1+\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}}} \, dx,x,d+e x\right )}{e^3 \sqrt {a+b (d+e x)^3+c (d+e x)^6}}\\ &=\frac {d^2 (d+e x) \sqrt {1+\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}} F_1\left (\frac {1}{3};\frac {1}{2},\frac {1}{2};\frac {4}{3};-\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}\right )}{e^3 \sqrt {a+b (d+e x)^3+c (d+e x)^6}}-\frac {d (d+e x)^2 \sqrt {1+\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}} F_1\left (\frac {2}{3};\frac {1}{2},\frac {1}{2};\frac {5}{3};-\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}\right )}{e^3 \sqrt {a+b (d+e x)^3+c (d+e x)^6}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c (d+e x)^3}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}}\right )}{3 e^3}\\ &=\frac {d^2 (d+e x) \sqrt {1+\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}} F_1\left (\frac {1}{3};\frac {1}{2},\frac {1}{2};\frac {4}{3};-\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}\right )}{e^3 \sqrt {a+b (d+e x)^3+c (d+e x)^6}}-\frac {d (d+e x)^2 \sqrt {1+\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}} F_1\left (\frac {2}{3};\frac {1}{2},\frac {1}{2};\frac {5}{3};-\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}\right )}{e^3 \sqrt {a+b (d+e x)^3+c (d+e x)^6}}+\frac {\tanh ^{-1}\left (\frac {b+2 c (d+e x)^3}{2 \sqrt {c} \sqrt {a+b (d+e x)^3+c (d+e x)^6}}\right )}{3 \sqrt {c} e^3}\\ \end {align*}
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Mathematica [F] time = 0.44, size = 0, normalized size = 0.00 \[ \int \frac {x^2}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 11.32, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2}}{\sqrt {c e^{6} x^{6} + 6 \, c d e^{5} x^{5} + 15 \, c d^{2} e^{4} x^{4} + c d^{6} + {\left (20 \, c d^{3} + b\right )} e^{3} x^{3} + 3 \, {\left (5 \, c d^{4} + b d\right )} e^{2} x^{2} + b d^{3} + 3 \, {\left (2 \, c d^{5} + b d^{2}\right )} e x + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {{\left (e x + d\right )}^{6} c + {\left (e x + d\right )}^{3} b + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {a +\left (e x +d \right )^{3} b +\left (e x +d \right )^{6} c}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {{\left (e x + d\right )}^{6} c + {\left (e x + d\right )}^{3} b + a}}\,{d x} \]
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 \[ \int \frac {x^2}{\sqrt {a+b\,{\left (d+e\,x\right )}^3+c\,{\left (d+e\,x\right )}^6}} \,d x \]
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 \[ \int \frac {x^{2}}{\sqrt {a + b d^{3} + 3 b d^{2} e x + 3 b d e^{2} x^{2} + b e^{3} x^{3} + c d^{6} + 6 c d^{5} e x + 15 c d^{4} e^{2} x^{2} + 20 c d^{3} e^{3} x^{3} + 15 c d^{2} e^{4} x^{4} + 6 c d e^{5} x^{5} + c e^{6} x^{6}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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