Optimal. Leaf size=121 \[ -\frac {(A-B x) \sqrt {a+b x+c x^2}}{x}-\frac {(2 a B+A b) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {a}}+\frac {(2 A c+b B) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c}} \]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {812, 843, 621, 206, 724} \begin {gather*} -\frac {(A-B x) \sqrt {a+b x+c x^2}}{x}-\frac {(2 a B+A b) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {a}}+\frac {(2 A c+b B) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 621
Rule 724
Rule 812
Rule 843
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^2} \, dx &=-\frac {(A-B x) \sqrt {a+b x+c x^2}}{x}-\frac {1}{2} \int \frac {-A b-2 a B-(b B+2 A c) x}{x \sqrt {a+b x+c x^2}} \, dx\\ &=-\frac {(A-B x) \sqrt {a+b x+c x^2}}{x}-\frac {1}{2} (-A b-2 a B) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx-\frac {1}{2} (-b B-2 A c) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx\\ &=-\frac {(A-B x) \sqrt {a+b x+c x^2}}{x}-(A b+2 a B) \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x}{\sqrt {a+b x+c x^2}}\right )-(-b B-2 A c) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )\\ &=-\frac {(A-B x) \sqrt {a+b x+c x^2}}{x}-\frac {(A b+2 a B) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {a}}+\frac {(b B+2 A c) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.17, size = 118, normalized size = 0.98 \begin {gather*} \frac {(B x-A) \sqrt {a+x (b+c x)}}{x}-\frac {(2 a B+A b) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )}{2 \sqrt {a}}+\frac {(2 A c+b B) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{2 \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.54, size = 117, normalized size = 0.97 \begin {gather*} \frac {(B x-A) \sqrt {a+b x+c x^2}}{x}+\frac {(-2 A c-b B) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{2 \sqrt {c}}+\frac {(-2 a B-A b) \tanh ^{-1}\left (\frac {\sqrt {a+b x+c x^2}-\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.91, size = 648, normalized size = 5.36 \begin {gather*} \left [\frac {{\left (2 \, B a + A b\right )} \sqrt {a} c x \log \left (-\frac {8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right ) + {\left (B a b + 2 \, A a c\right )} \sqrt {c} x \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (B a c x - A a c\right )} \sqrt {c x^{2} + b x + a}}{4 \, a c x}, \frac {{\left (2 \, B a + A b\right )} \sqrt {a} c x \log \left (-\frac {8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right ) - 2 \, {\left (B a b + 2 \, A a c\right )} \sqrt {-c} x \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 4 \, {\left (B a c x - A a c\right )} \sqrt {c x^{2} + b x + a}}{4 \, a c x}, \frac {2 \, {\left (2 \, B a + A b\right )} \sqrt {-a} c x \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{2} + a b x + a^{2}\right )}}\right ) + {\left (B a b + 2 \, A a c\right )} \sqrt {c} x \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (B a c x - A a c\right )} \sqrt {c x^{2} + b x + a}}{4 \, a c x}, \frac {{\left (2 \, B a + A b\right )} \sqrt {-a} c x \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{2} + a b x + a^{2}\right )}}\right ) - {\left (B a b + 2 \, A a c\right )} \sqrt {-c} x \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (B a c x - A a c\right )} \sqrt {c x^{2} + b x + a}}{2 \, a c x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.08, size = 207, normalized size = 1.71 \begin {gather*} -\frac {A b \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{2 \sqrt {a}}+A \sqrt {c}\, \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )-B \sqrt {a}\, \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )+\frac {B b \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 \sqrt {c}}+\frac {\sqrt {c \,x^{2}+b x +a}\, A c x}{a}+\frac {\sqrt {c \,x^{2}+b x +a}\, A b}{a}+\sqrt {c \,x^{2}+b x +a}\, B -\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A}{a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.56, size = 166, normalized size = 1.37 \begin {gather*} B\,\sqrt {c\,x^2+b\,x+a}-\frac {A\,\sqrt {c\,x^2+b\,x+a}}{x}-B\,\sqrt {a}\,\ln \left (\frac {b}{2}+\frac {a}{x}+\frac {\sqrt {a}\,\sqrt {c\,x^2+b\,x+a}}{x}\right )+A\,\sqrt {c}\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )+\frac {B\,b\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )}{2\,\sqrt {c}}-\frac {A\,b\,\ln \left (\frac {b}{2}+\frac {a}{x}+\frac {\sqrt {a}\,\sqrt {c\,x^2+b\,x+a}}{x}\right )}{2\,\sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x\right ) \sqrt {a + b x + c x^{2}}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________