Optimal. Leaf size=115 \[ -\frac {\left (1-4 a^2\right ) e^{2 \sinh ^{-1}(a+b x)}}{16 b^3}-\frac {\left (1-4 a^2\right ) \sinh ^{-1}(a+b x)}{8 b^3}-\frac {a e^{-\sinh ^{-1}(a+b x)}}{2 b^3}-\frac {a e^{3 \sinh ^{-1}(a+b x)}}{6 b^3}-\frac {e^{-2 \sinh ^{-1}(a+b x)}}{16 b^3}+\frac {e^{4 \sinh ^{-1}(a+b x)}}{32 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5898, 2282, 12, 1628} \[ -\frac {\left (1-4 a^2\right ) e^{2 \sinh ^{-1}(a+b x)}}{16 b^3}-\frac {\left (1-4 a^2\right ) \sinh ^{-1}(a+b x)}{8 b^3}-\frac {a e^{-\sinh ^{-1}(a+b x)}}{2 b^3}-\frac {a e^{3 \sinh ^{-1}(a+b x)}}{6 b^3}-\frac {e^{-2 \sinh ^{-1}(a+b x)}}{16 b^3}+\frac {e^{4 \sinh ^{-1}(a+b x)}}{32 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 1628
Rule 2282
Rule 5898
Rubi steps
\begin {align*} \int e^{\sinh ^{-1}(a+b x)} x^2 \, dx &=\frac {\operatorname {Subst}\left (\int e^x \cosh (x) \left (-\frac {a}{b}+\frac {\sinh (x)}{b}\right )^2 \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+2 a x-x^2\right )^2 \left (1+x^2\right )}{8 b^2 x^3} \, dx,x,e^{\sinh ^{-1}(a+b x)}\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+2 a x-x^2\right )^2 \left (1+x^2\right )}{x^3} \, dx,x,e^{\sinh ^{-1}(a+b x)}\right )}{8 b^3}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{x^3}+\frac {4 a}{x^2}+\frac {-1+4 a^2}{x}+\left (-1+4 a^2\right ) x-4 a x^2+x^3\right ) \, dx,x,e^{\sinh ^{-1}(a+b x)}\right )}{8 b^3}\\ &=-\frac {e^{-2 \sinh ^{-1}(a+b x)}}{16 b^3}-\frac {a e^{-\sinh ^{-1}(a+b x)}}{2 b^3}-\frac {\left (1-4 a^2\right ) e^{2 \sinh ^{-1}(a+b x)}}{16 b^3}-\frac {a e^{3 \sinh ^{-1}(a+b x)}}{6 b^3}+\frac {e^{4 \sinh ^{-1}(a+b x)}}{32 b^3}-\frac {\left (1-4 a^2\right ) \sinh ^{-1}(a+b x)}{8 b^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 102, normalized size = 0.89 \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2+1} \left (2 a^3-2 a^2 b x+a \left (2 b^2 x^2-13\right )+6 b^3 x^3+3 b x\right )+8 a b^3 x^3+3 (2 a-1) (2 a+1) \sinh ^{-1}(a+b x)+6 b^4 x^4}{24 b^3} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.63, size = 117, normalized size = 1.02 \[ \frac {6 \, b^{4} x^{4} + 8 \, a b^{3} x^{3} - 3 \, {\left (4 \, a^{2} - 1\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + {\left (6 \, b^{3} x^{3} + 2 \, a b^{2} x^{2} + 2 \, a^{3} - {\left (2 \, a^{2} - 3\right )} b x - 13 \, a\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{24 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.61, size = 140, normalized size = 1.22 \[ \frac {1}{4} \, b x^{4} + \frac {1}{3} \, a x^{3} + \frac {1}{24} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left ({\left (2 \, {\left (3 \, x + \frac {a}{b}\right )} x - \frac {2 \, a^{2} b^{3} - 3 \, b^{3}}{b^{5}}\right )} x + \frac {2 \, a^{3} b^{2} - 13 \, a b^{2}}{b^{5}}\right )} - \frac {{\left (4 \, a^{2} - 1\right )} \log \left (-a b - {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} {\left | b \right |}\right )}{8 \, b^{2} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 264, normalized size = 2.30 \[ \frac {x \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{4 b^{2}}-\frac {5 a \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{12 b^{3}}+\frac {a^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, x}{2 b^{2}}+\frac {a^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 b^{3}}+\frac {a^{2} \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{2 b^{2} \sqrt {b^{2}}}-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, x}{8 b^{2}}-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a}{8 b^{3}}-\frac {\ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{8 b^{2} \sqrt {b^{2}}}+\frac {b \,x^{4}}{4}+\frac {x^{3} a}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.69, size = 273, normalized size = 2.37 \[ \frac {1}{4} \, b x^{4} + \frac {1}{3} \, a x^{3} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} x}{4 \, b^{2}} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a}{12 \, b^{3}} - \frac {{\left (5 \, a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} a^{2} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{8 \, b^{5}} + \frac {{\left (5 \, a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} x}{8 \, b^{4}} + \frac {{\left (5 \, a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} {\left (a^{2} + 1\right )} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{8 \, b^{5}} + \frac {{\left (5 \, a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a}{8 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,\left (a+\sqrt {{\left (a+b\,x\right )}^2+1}+b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \left (a + b x + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________