Optimal. Leaf size=143 \[ \frac {3 a^2 b x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {3 a b^2 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac {b^3 x^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {a^3 \log (x) \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x} \]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {646, 43} \begin {gather*} \frac {3 a^2 b x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {3 a b^2 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac {b^3 x^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {a^3 \log (x) \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 646
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^3}{x} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (3 a^2 b^4+\frac {a^3 b^3}{x}+3 a b^5 x+b^6 x^2\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {3 a^2 b x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {3 a b^2 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac {b^3 x^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {a^3 \sqrt {a^2+2 a b x+b^2 x^2} \log (x)}{a+b x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 52, normalized size = 0.36 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (6 a^3 \log (x)+b x \left (18 a^2+9 a b x+2 b^2 x^2\right )\right )}{6 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.35, size = 238, normalized size = 1.66 \begin {gather*} \frac {1}{12} \sqrt {a^2+2 a b x+b^2 x^2} \left (11 a^2+7 a b x+2 b^2 x^2\right )+\frac {1}{12} \left (-18 a^2 \sqrt {b^2} x-9 a b \sqrt {b^2} x^2-2 \left (b^2\right )^{3/2} x^3\right )+\frac {1}{2} a^3 \log \left (\sqrt {a^2+2 a b x+b^2 x^2}-a-\sqrt {b^2} x\right )-\frac {a^3 \left (\sqrt {b^2}+b\right ) \log \left (\sqrt {a^2+2 a b x+b^2 x^2}+a-\sqrt {b^2} x\right )}{2 b}-\frac {a^3 \sqrt {b^2} \log \left (b \sqrt {a^2+2 a b x+b^2 x^2}-a b-\sqrt {b^2} b x\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.40, size = 31, normalized size = 0.22 \begin {gather*} \frac {1}{3} \, b^{3} x^{3} + \frac {3}{2} \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3} \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 56, normalized size = 0.39 \begin {gather*} \frac {1}{3} \, b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, a b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b x \mathrm {sgn}\left (b x + a\right ) + a^{3} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (b x + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 51, normalized size = 0.36 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \left (2 b^{3} x^{3}+9 a \,b^{2} x^{2}+6 a^{3} \ln \relax (x )+18 a^{2} b x \right )}{6 \left (b x +a \right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.36, size = 134, normalized size = 0.94 \begin {gather*} \left (-1\right )^{2 \, b^{2} x + 2 \, a b} a^{3} \log \left (2 \, b^{2} x + 2 \, a b\right ) - \left (-1\right )^{2 \, a b x + 2 \, a^{2}} a^{3} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) + \frac {1}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a b x + \frac {3}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} + \frac {1}{3} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} \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 {{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{x} \,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*} \int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________