Optimal. Leaf size=95 \[ \frac {\sqrt {a+b x+c x^2}}{d^3 \left (b^2-4 a c\right ) (b+2 c x)^2}+\frac {\tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{2 \sqrt {c} d^3 \left (b^2-4 a c\right )^{3/2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {693, 688, 205} \begin {gather*} \frac {\sqrt {a+b x+c x^2}}{d^3 \left (b^2-4 a c\right ) (b+2 c x)^2}+\frac {\tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{2 \sqrt {c} d^3 \left (b^2-4 a c\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 688
Rule 693
Rubi steps
\begin {align*} \int \frac {1}{(b d+2 c d x)^3 \sqrt {a+b x+c x^2}} \, dx &=\frac {\sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2}+\frac {\int \frac {1}{(b d+2 c d x) \sqrt {a+b x+c x^2}} \, dx}{2 \left (b^2-4 a c\right ) d^2}\\ &=\frac {\sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2}+\frac {(2 c) \operatorname {Subst}\left (\int \frac {1}{2 b^2 c d-8 a c^2 d+8 c^2 d x^2} \, dx,x,\sqrt {a+b x+c x^2}\right )}{\left (b^2-4 a c\right ) d^2}\\ &=\frac {\sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2}+\frac {\tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{2 \sqrt {c} \left (b^2-4 a c\right )^{3/2} d^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.28, size = 107, normalized size = 1.13 \begin {gather*} \frac {\sqrt {a+x (b+c x)} \left (\frac {2 \left (b^2-4 a c\right )}{(b+2 c x)^2}+\frac {\tanh ^{-1}\left (2 \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}}\right )}{\sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}}}\right )}{2 d^3 \left (b^2-4 a c\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.96, size = 124, normalized size = 1.31 \begin {gather*} \frac {\sqrt {a+b x+c x^2}}{d^3 \left (b^2-4 a c\right ) (b+2 c x)^2}-\frac {\tan ^{-1}\left (-\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+\frac {2 c x}{\sqrt {b^2-4 a c}}+\frac {b}{\sqrt {b^2-4 a c}}\right )}{\sqrt {c} d^3 \left (b^2-4 a c\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.55, size = 440, normalized size = 4.63 \begin {gather*} \left [\frac {{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \sqrt {-b^{2} c + 4 \, a c^{2}} \log \left (-\frac {4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c + 4 \, \sqrt {-b^{2} c + 4 \, a c^{2}} \sqrt {c x^{2} + b x + a}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) + 4 \, {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {c x^{2} + b x + a}}{4 \, {\left (4 \, {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{3} x^{2} + 4 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{3} x + {\left (b^{6} c - 8 \, a b^{4} c^{2} + 16 \, a^{2} b^{2} c^{3}\right )} d^{3}\right )}}, -\frac {{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \sqrt {b^{2} c - 4 \, a c^{2}} \arctan \left (\frac {\sqrt {b^{2} c - 4 \, a c^{2}} \sqrt {c x^{2} + b x + a}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {c x^{2} + b x + a}}{2 \, {\left (4 \, {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{3} x^{2} + 4 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{3} x + {\left (b^{6} c - 8 \, a b^{4} c^{2} + 16 \, a^{2} b^{2} c^{3}\right )} d^{3}\right )}}\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 = 0.06, size = 174, normalized size = 1.83 \begin {gather*} \frac {\ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{2 \left (4 a c -b^{2}\right ) \sqrt {\frac {4 a c -b^{2}}{c}}\, c \,d^{3}}-\frac {\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}}{4 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{2} c^{2} d^{3}} \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 [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (b\,d+2\,c\,d\,x\right )}^3\,\sqrt {c\,x^2+b\,x+a}} \,d x \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*} \frac {\int \frac {1}{b^{3} \sqrt {a + b x + c x^{2}} + 6 b^{2} c x \sqrt {a + b x + c x^{2}} + 12 b c^{2} x^{2} \sqrt {a + b x + c x^{2}} + 8 c^{3} x^{3} \sqrt {a + b x + c x^{2}}}\, dx}{d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________