Optimal. Leaf size=230 \[ \frac {\left (d+e x^2\right )^3 \left (a+b \text {sech}^{-1}(c x)\right )}{6 e}-\frac {b d^3 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{6 e}+\frac {b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2} \left (3 c^2 d+2 e\right )}{18 c^6}-\frac {b e^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{30 c^6}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (3 c^4 d^2+3 c^2 d e+e^2\right )}{6 c^6} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.25, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6299, 517, 446, 88, 63, 208} \[ \frac {\left (d+e x^2\right )^3 \left (a+b \text {sech}^{-1}(c x)\right )}{6 e}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (3 c^4 d^2+3 c^2 d e+e^2\right )}{6 c^6}-\frac {b d^3 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{6 e}+\frac {b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2} \left (3 c^2 d+2 e\right )}{18 c^6}-\frac {b e^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{30 c^6} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 88
Rule 208
Rule 446
Rule 517
Rule 6299
Rubi steps
\begin {align*} \int x \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx &=\frac {\left (d+e x^2\right )^3 \left (a+b \text {sech}^{-1}(c x)\right )}{6 e}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\left (d+e x^2\right )^3}{x \sqrt {1-c x} \sqrt {1+c x}} \, dx}{6 e}\\ &=\frac {\left (d+e x^2\right )^3 \left (a+b \text {sech}^{-1}(c x)\right )}{6 e}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\left (d+e x^2\right )^3}{x \sqrt {1-c^2 x^2}} \, dx}{6 e}\\ &=\frac {\left (d+e x^2\right )^3 \left (a+b \text {sech}^{-1}(c x)\right )}{6 e}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {(d+e x)^3}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{12 e}\\ &=\frac {\left (d+e x^2\right )^3 \left (a+b \text {sech}^{-1}(c x)\right )}{6 e}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \left (\frac {e \left (3 c^4 d^2+3 c^2 d e+e^2\right )}{c^4 \sqrt {1-c^2 x}}+\frac {d^3}{x \sqrt {1-c^2 x}}-\frac {e^2 \left (3 c^2 d+2 e\right ) \sqrt {1-c^2 x}}{c^4}+\frac {e^3 \left (1-c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )}{12 e}\\ &=-\frac {b \left (3 c^4 d^2+3 c^2 d e+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{6 c^6}+\frac {b e \left (3 c^2 d+2 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{18 c^6}-\frac {b e^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \left (1-c^2 x^2\right )^{5/2}}{30 c^6}+\frac {\left (d+e x^2\right )^3 \left (a+b \text {sech}^{-1}(c x)\right )}{6 e}+\frac {\left (b d^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{12 e}\\ &=-\frac {b \left (3 c^4 d^2+3 c^2 d e+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{6 c^6}+\frac {b e \left (3 c^2 d+2 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{18 c^6}-\frac {b e^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \left (1-c^2 x^2\right )^{5/2}}{30 c^6}+\frac {\left (d+e x^2\right )^3 \left (a+b \text {sech}^{-1}(c x)\right )}{6 e}-\frac {\left (b d^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-c^2 x^2}\right )}{6 c^2 e}\\ &=-\frac {b \left (3 c^4 d^2+3 c^2 d e+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{6 c^6}+\frac {b e \left (3 c^2 d+2 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{18 c^6}-\frac {b e^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \left (1-c^2 x^2\right )^{5/2}}{30 c^6}+\frac {\left (d+e x^2\right )^3 \left (a+b \text {sech}^{-1}(c x)\right )}{6 e}-\frac {b d^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{6 e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.32, size = 139, normalized size = 0.60 \[ \frac {1}{6} a x^2 \left (3 d^2+3 d e x^2+e^2 x^4\right )-\frac {b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (3 c^4 \left (15 d^2+5 d e x^2+e^2 x^4\right )+2 c^2 e \left (15 d+2 e x^2\right )+8 e^2\right )}{90 c^6}+\frac {1}{6} b x^2 \text {sech}^{-1}(c x) \left (3 d^2+3 d e x^2+e^2 x^4\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.00, size = 192, normalized size = 0.83 \[ \frac {15 \, a c^{5} e^{2} x^{6} + 45 \, a c^{5} d e x^{4} + 45 \, a c^{5} d^{2} x^{2} + 15 \, {\left (b c^{5} e^{2} x^{6} + 3 \, b c^{5} d e x^{4} + 3 \, b c^{5} d^{2} x^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (3 \, b c^{4} e^{2} x^{5} + {\left (15 \, b c^{4} d e + 4 \, b c^{2} e^{2}\right )} x^{3} + {\left (45 \, b c^{4} d^{2} + 30 \, b c^{2} d e + 8 \, b e^{2}\right )} x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{90 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 180, normalized size = 0.78 \[ \frac {\frac {a \left (\frac {1}{6} c^{6} e^{2} x^{6}+\frac {1}{2} c^{6} d e \,x^{4}+\frac {1}{2} c^{6} d^{2} x^{2}\right )}{c^{4}}+\frac {b \left (\frac {\mathrm {arcsech}\left (c x \right ) e^{2} c^{6} x^{6}}{6}+\frac {\mathrm {arcsech}\left (c x \right ) c^{6} d e \,x^{4}}{2}+\frac {\mathrm {arcsech}\left (c x \right ) c^{6} x^{2} d^{2}}{2}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (3 c^{4} e^{2} x^{4}+15 c^{4} d e \,x^{2}+45 d^{2} c^{4}+4 c^{2} e^{2} x^{2}+30 c^{2} d e +8 e^{2}\right )}{90}\right )}{c^{4}}}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 185, normalized size = 0.80 \[ \frac {1}{6} \, a e^{2} x^{6} + \frac {1}{2} \, a d e x^{4} + \frac {1}{2} \, a d^{2} x^{2} + \frac {1}{2} \, {\left (x^{2} \operatorname {arsech}\left (c x\right ) - \frac {x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c}\right )} b d^{2} + \frac {1}{6} \, {\left (3 \, x^{4} \operatorname {arsech}\left (c x\right ) + \frac {c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} - 3 \, x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{3}}\right )} b d e + \frac {1}{90} \, {\left (15 \, x^{6} \operatorname {arsech}\left (c x\right ) - \frac {3 \, c^{4} x^{5} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} - 10 \, c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{5}}\right )} b e^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x\,{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 6.16, size = 252, normalized size = 1.10 \[ \begin {cases} \frac {a d^{2} x^{2}}{2} + \frac {a d e x^{4}}{2} + \frac {a e^{2} x^{6}}{6} + \frac {b d^{2} x^{2} \operatorname {asech}{\left (c x \right )}}{2} + \frac {b d e x^{4} \operatorname {asech}{\left (c x \right )}}{2} + \frac {b e^{2} x^{6} \operatorname {asech}{\left (c x \right )}}{6} - \frac {b d^{2} \sqrt {- c^{2} x^{2} + 1}}{2 c^{2}} - \frac {b d e x^{2} \sqrt {- c^{2} x^{2} + 1}}{6 c^{2}} - \frac {b e^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{30 c^{2}} - \frac {b d e \sqrt {- c^{2} x^{2} + 1}}{3 c^{4}} - \frac {2 b e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{45 c^{4}} - \frac {4 b e^{2} \sqrt {- c^{2} x^{2} + 1}}{45 c^{6}} & \text {for}\: c \neq 0 \\\left (a + \infty b\right ) \left (\frac {d^{2} x^{2}}{2} + \frac {d e x^{4}}{2} + \frac {e^{2} x^{6}}{6}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________