Optimal. Leaf size=63 \[ -\frac {\left (1-2 a^2\right ) \sinh ^{-1}(a+b x)}{2 b^3}-\frac {3 a \sqrt {(a+b x)^2+1}}{2 b^3}+\frac {x \sqrt {(a+b x)^2+1}}{2 b^2} \]
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Rubi [A] time = 0.04, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {371, 743, 641, 215} \[ -\frac {\left (1-2 a^2\right ) \sinh ^{-1}(a+b x)}{2 b^3}-\frac {3 a \sqrt {(a+b x)^2+1}}{2 b^3}+\frac {x \sqrt {(a+b x)^2+1}}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 215
Rule 371
Rule 641
Rule 743
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {1+(a+b x)^2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(-a+x)^2}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{b^3}\\ &=\frac {x \sqrt {1+(a+b x)^2}}{2 b^2}+\frac {\operatorname {Subst}\left (\int \frac {-1+2 a^2-3 a x}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{2 b^3}\\ &=-\frac {3 a \sqrt {1+(a+b x)^2}}{2 b^3}+\frac {x \sqrt {1+(a+b x)^2}}{2 b^2}-\frac {\left (1-2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{2 b^3}\\ &=-\frac {3 a \sqrt {1+(a+b x)^2}}{2 b^3}+\frac {x \sqrt {1+(a+b x)^2}}{2 b^2}-\frac {\left (1-2 a^2\right ) \sinh ^{-1}(a+b x)}{2 b^3}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 51, normalized size = 0.81 \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2+1} (b x-3 a)+\left (2 a^2-1\right ) \sinh ^{-1}(a+b x)}{2 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 70, normalized size = 1.11 \[ -\frac {{\left (2 \, a^{2} - 1\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (b x - 3 \, a\right )}}{2 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 70, normalized size = 1.11 \[ \frac {1}{2} \, \sqrt {{\left (b x + a\right )}^{2} + 1} {\left (\frac {x}{b^{2}} - \frac {3 \, a}{b^{3}}\right )} - \frac {{\left (2 \, a^{2} - 1\right )} \log \left (-a b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{2 \, b^{2} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 146, normalized size = 2.32 \[ \frac {a^{2} \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}\, b^{2}}+\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, x}{2 b^{2}}-\frac {\ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{2 \sqrt {b^{2}}\, b^{2}}-\frac {3 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a}{2 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.59, size = 135, normalized size = 2.14 \[ \frac {3 \, a^{2} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{2 \, b^{3}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} x}{2 \, b^{2}} - \frac {{\left (a^{2} + 1\right )} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{2 \, b^{3}} - \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a}{2 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^2}{\sqrt {{\left (a+b\,x\right )}^2+1}} \,d x \]
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 \[ \int \frac {x^{2}}{\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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