Optimal. Leaf size=69 \[ \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (A b-a B)}{2 b^2}+\frac {B \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {640, 609} \begin {gather*} \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (A b-a B)}{2 b^2}+\frac {B \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 609
Rule 640
Rubi steps
\begin {align*} \int (A+B x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx &=\frac {B \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b^2}+\frac {\left (2 A b^2-2 a b B\right ) \int \sqrt {a^2+2 a b x+b^2 x^2} \, dx}{2 b^2}\\ &=\frac {(A b-a B) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 b^2}+\frac {B \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 45, normalized size = 0.65 \begin {gather*} \frac {x \sqrt {(a+b x)^2} (3 a (2 A+B x)+b x (3 A+2 B x))}{6 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.59, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.39, size = 24, normalized size = 0.35 \begin {gather*} \frac {1}{3} \, B b x^{3} + A a x + \frac {1}{2} \, {\left (B a + A b\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.22, size = 74, normalized size = 1.07 \begin {gather*} \frac {1}{3} \, B b x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, B a x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, A b x^{2} \mathrm {sgn}\left (b x + a\right ) + A a x \mathrm {sgn}\left (b x + a\right ) - \frac {{\left (B a^{3} - 3 \, A a^{2} b\right )} \mathrm {sgn}\left (b x + a\right )}{6 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 42, normalized size = 0.61 \begin {gather*} \frac {\left (2 B b \,x^{2}+3 A b x +3 B a x +6 A a \right ) \sqrt {\left (b x +a \right )^{2}}\, x}{6 b x +6 a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.63, size = 125, normalized size = 1.81 \begin {gather*} \frac {1}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A x - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B a x}{2 \, b} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B a^{2}}{2 \, b^{2}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A a}{2 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B}{3 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.25, size = 77, normalized size = 1.12 \begin {gather*} \frac {A\,\sqrt {{\left (a+b\,x\right )}^2}\,\left (a+b\,x\right )}{2\,b}+\frac {B\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{24\,b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.10, size = 26, normalized size = 0.38 \begin {gather*} A a x + \frac {B b x^{3}}{3} + x^{2} \left (\frac {A b}{2} + \frac {B a}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________