Optimal. Leaf size=34 \[ -\frac {1}{4 b (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.00, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {607} \begin {gather*} -\frac {1}{4 b (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 607
Rubi steps
\begin {align*} \int \frac {1}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=-\frac {1}{4 b (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 23, normalized size = 0.68 \begin {gather*} -\frac {a+b x}{4 b \left ((a+b x)^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [B] time = 0.00, size = 197, normalized size = 5.79 \begin {gather*} \frac {-2 \left (b^5 \left (-x^4\right )-a^4 b\right )-2 \sqrt {b^2} \sqrt {a^2+2 a b x+b^2 x^2} \left (-a^3+a^2 b x-a b^2 x^2+b^3 x^3\right )}{b x^4 \sqrt {a^2+2 a b x+b^2 x^2} \left (-8 a^3 b^5-24 a^2 b^6 x-24 a b^7 x^2-8 b^8 x^3\right )+b \sqrt {b^2} x^4 \left (8 a^4 b^4+32 a^3 b^5 x+48 a^2 b^6 x^2+32 a b^7 x^3+8 b^8 x^4\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.40, size = 46, normalized size = 1.35 \begin {gather*} -\frac {1}{4 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 20, normalized size = 0.59 \begin {gather*} -\frac {b x +a}{4 \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.96, size = 14, normalized size = 0.41 \begin {gather*} -\frac {1}{4 \, b^{5} {\left (x + \frac {a}{b}\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.62, size = 30, normalized size = 0.88 \begin {gather*} -\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,b\,{\left (a+b\,x\right )}^5} \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 {1}{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________