Optimal. Leaf size=52 \[ \frac {3}{2} a b \sqrt {1+x^2}+\frac {1}{2} b (a+b x) \sqrt {1+x^2}+\frac {1}{2} \left (2 a^2-b^2\right ) \sinh ^{-1}(x) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {757, 655, 221}
\begin {gather*} \frac {1}{2} \left (2 a^2-b^2\right ) \sinh ^{-1}(x)+\frac {3}{2} a b \sqrt {x^2+1}+\frac {1}{2} b \sqrt {x^2+1} (a+b x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 221
Rule 655
Rule 757
Rubi steps
\begin {align*} \int \frac {(a+b x)^2}{\sqrt {1+x^2}} \, dx &=\frac {1}{2} b (a+b x) \sqrt {1+x^2}+\frac {1}{2} \int \frac {2 a^2-b^2+3 a b x}{\sqrt {1+x^2}} \, dx\\ &=\frac {3}{2} a b \sqrt {1+x^2}+\frac {1}{2} b (a+b x) \sqrt {1+x^2}+\frac {1}{2} \left (2 a^2-b^2\right ) \int \frac {1}{\sqrt {1+x^2}} \, dx\\ &=\frac {3}{2} a b \sqrt {1+x^2}+\frac {1}{2} b (a+b x) \sqrt {1+x^2}+\frac {1}{2} \left (2 a^2-b^2\right ) \sinh ^{-1}(x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.16, size = 49, normalized size = 0.94 \begin {gather*} \frac {1}{2} b (4 a+b x) \sqrt {1+x^2}+\frac {1}{2} \left (2 a^2-b^2\right ) \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.45, size = 38, normalized size = 0.73
method | result | size |
risch | \(\frac {b \left (b x +4 a \right ) \sqrt {x^{2}+1}}{2}+\left (a^{2}-\frac {b^{2}}{2}\right ) \arcsinh \left (x \right )\) | \(31\) |
default | \(b^{2} \left (\frac {x \sqrt {x^{2}+1}}{2}-\frac {\arcsinh \left (x \right )}{2}\right )+2 a b \sqrt {x^{2}+1}+a^{2} \arcsinh \left (x \right )\) | \(38\) |
trager | \(\frac {b \left (b x +4 a \right ) \sqrt {x^{2}+1}}{2}+\frac {\left (2 a^{2}-b^{2}\right ) \ln \left (\sqrt {x^{2}+1}+x \right )}{2}\) | \(42\) |
meijerg | \(\frac {b^{2} \left (\sqrt {\pi }\, x \sqrt {x^{2}+1}-\sqrt {\pi }\, \arcsinh \left (x \right )\right )}{2 \sqrt {\pi }}+\frac {a b \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {x^{2}+1}\right )}{\sqrt {\pi }}+a^{2} \arcsinh \left (x \right )\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.48, size = 38, normalized size = 0.73 \begin {gather*} \frac {1}{2} \, \sqrt {x^{2} + 1} b^{2} x + a^{2} \operatorname {arsinh}\left (x\right ) - \frac {1}{2} \, b^{2} \operatorname {arsinh}\left (x\right ) + 2 \, \sqrt {x^{2} + 1} a b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 3.66, size = 45, normalized size = 0.87 \begin {gather*} -\frac {1}{2} \, {\left (2 \, a^{2} - b^{2}\right )} \log \left (-x + \sqrt {x^{2} + 1}\right ) + \frac {1}{2} \, {\left (b^{2} x + 4 \, a b\right )} \sqrt {x^{2} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.08, size = 42, normalized size = 0.81 \begin {gather*} a^{2} \operatorname {asinh}{\left (x \right )} + 2 a b \sqrt {x^{2} + 1} + \frac {b^{2} x \sqrt {x^{2} + 1}}{2} - \frac {b^{2} \operatorname {asinh}{\left (x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.95, size = 45, normalized size = 0.87 \begin {gather*} -\frac {1}{2} \, {\left (2 \, a^{2} - b^{2}\right )} \log \left (-x + \sqrt {x^{2} + 1}\right ) + \frac {1}{2} \, {\left (b^{2} x + 4 \, a b\right )} \sqrt {x^{2} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.03, size = 32, normalized size = 0.62 \begin {gather*} \left (\frac {x\,b^2}{2}+2\,a\,b\right )\,\sqrt {x^2+1}+\mathrm {asinh}\left (x\right )\,\left (a^2-\frac {b^2}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________