Integrand size = 28, antiderivative size = 398 \[ \int \frac {x^2}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx=\frac {d^2 (d+e x) \sqrt {1+\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}\right )}{e^3 \sqrt {a+b (d+e x)^3+c (d+e x)^6}}-\frac {d (d+e x)^2 \sqrt {1+\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},\frac {1}{2},\frac {5}{3},-\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}\right )}{e^3 \sqrt {a+b (d+e x)^3+c (d+e x)^6}}+\frac {\text {arctanh}\left (\frac {b+2 c (d+e x)^3}{2 \sqrt {c} \sqrt {a+b (d+e x)^3+c (d+e x)^6}}\right )}{3 \sqrt {c} e^3} \]
[Out]
Time = 0.43 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {1403, 1804, 1362, 440, 1399, 524, 1366, 635, 212} \[ \int \frac {x^2}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx=\frac {d^2 (d+e x) \sqrt {\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c (d+e x)^3}{\sqrt {b^2-4 a c}+b}+1} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}\right )}{e^3 \sqrt {a+b (d+e x)^3+c (d+e x)^6}}-\frac {d (d+e x)^2 \sqrt {\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c (d+e x)^3}{\sqrt {b^2-4 a c}+b}+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},\frac {1}{2},\frac {5}{3},-\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}\right )}{e^3 \sqrt {a+b (d+e x)^3+c (d+e x)^6}}+\frac {\text {arctanh}\left (\frac {b+2 c (d+e x)^3}{2 \sqrt {c} \sqrt {a+b (d+e x)^3+c (d+e x)^6}}\right )}{3 \sqrt {c} e^3} \]
[In]
[Out]
Rule 212
Rule 440
Rule 524
Rule 635
Rule 1362
Rule 1366
Rule 1399
Rule 1403
Rule 1804
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(-d+x)^2}{\sqrt {a+b x^3+c x^6}} \, dx,x,d+e x\right )}{e^3} \\ & = \frac {\text {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 {\text {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) \text {Subst}\left (\int \frac {x}{\sqrt {a+b x^3+c x^6}} \, dx,x,d+e x\right )}{e^3}+\frac {d^2 \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^3+c x^6}} \, dx,x,d+e x\right )}{e^3} \\ & = \frac {\text {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 ) \text {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 ) \text {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 \text {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*}
\[ \int \frac {x^2}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx=\int \frac {x^2}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx \]
[In]
[Out]
\[\int \frac {x^{2}}{\sqrt {a +b \left (e x +d \right )^{3}+c \left (e x +d \right )^{6}}}d x\]
[In]
[Out]
\[ \int \frac {x^2}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx=\int { \frac {x^{2}}{\sqrt {{\left (e x + d\right )}^{6} c + {\left (e x + d\right )}^{3} b + a}} \,d x } \]
[In]
[Out]
\[ \int \frac {x^2}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx=\int \frac {x^{2}}{\sqrt {a + b d^{3} + 3 b d^{2} e x + 3 b d e^{2} x^{2} + b e^{3} x^{3} + c d^{6} + 6 c d^{5} e x + 15 c d^{4} e^{2} x^{2} + 20 c d^{3} e^{3} x^{3} + 15 c d^{2} e^{4} x^{4} + 6 c d e^{5} x^{5} + c e^{6} x^{6}}}\, dx \]
[In]
[Out]
\[ \int \frac {x^2}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx=\int { \frac {x^{2}}{\sqrt {{\left (e x + d\right )}^{6} c + {\left (e x + d\right )}^{3} b + a}} \,d x } \]
[In]
[Out]
\[ \int \frac {x^2}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx=\int { \frac {x^{2}}{\sqrt {{\left (e x + d\right )}^{6} c + {\left (e x + d\right )}^{3} b + a}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x^2}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx=\int \frac {x^2}{\sqrt {a+b\,{\left (d+e\,x\right )}^3+c\,{\left (d+e\,x\right )}^6}} \,d x \]
[In]
[Out]