Optimal. Leaf size=172 \[ \frac {4 a^3}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a^4}{2 b^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 a x (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {x^2 (a+b x)}{2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {6 a^2 (a+b x) \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 172, 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 = {660, 45}
\begin {gather*} \frac {6 a^2 (a+b x) \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 a x (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {x^2 (a+b x)}{2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a^4}{2 b^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {4 a^3}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 660
Rubi steps
\begin {align*} \int \frac {x^4}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {x^4}{\left (a b+b^2 x\right )^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \left (-\frac {3 a}{b^7}+\frac {x}{b^6}+\frac {a^4}{b^7 (a+b x)^3}-\frac {4 a^3}{b^7 (a+b x)^2}+\frac {6 a^2}{b^7 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {4 a^3}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a^4}{2 b^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 a x (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {x^2 (a+b x)}{2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {6 a^2 (a+b x) \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 83, normalized size = 0.48 \begin {gather*} \frac {7 a^4+2 a^3 b x-11 a^2 b^2 x^2-4 a b^3 x^3+b^4 x^4+12 a^2 (a+b x)^2 \log (a+b x)}{2 b^5 (a+b x) \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.52, size = 101, normalized size = 0.59
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\frac {1}{2} b \,x^{2}-3 a x \right )}{\left (b x +a \right ) b^{4}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 a^{3} x +\frac {7 a^{4}}{2 b}\right )}{\left (b x +a \right )^{3} b^{4}}+\frac {6 \sqrt {\left (b x +a \right )^{2}}\, a^{2} \ln \left (b x +a \right )}{\left (b x +a \right ) b^{5}}\) | \(98\) |
default | \(\frac {\left (b^{4} x^{4}+12 \ln \left (b x +a \right ) a^{2} b^{2} x^{2}-4 a \,b^{3} x^{3}+24 \ln \left (b x +a \right ) a^{3} b x -11 a^{2} b^{2} x^{2}+12 a^{4} \ln \left (b x +a \right )+2 a^{3} b x +7 a^{4}\right ) \left (b x +a \right )}{2 b^{5} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(101\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 131, normalized size = 0.76 \begin {gather*} \frac {x^{3}}{2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac {5 \, a x^{2}}{2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{3}} + \frac {6 \, a^{2} \log \left (x + \frac {a}{b}\right )}{b^{5}} - \frac {5 \, a^{3}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{5}} + \frac {12 \, a^{3} x}{b^{6} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {23 \, a^{4}}{2 \, b^{7} {\left (x + \frac {a}{b}\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.56, size = 95, normalized size = 0.55 \begin {gather*} \frac {b^{4} x^{4} - 4 \, a b^{3} x^{3} - 11 \, a^{2} b^{2} x^{2} + 2 \, a^{3} b x + 7 \, a^{4} + 12 \, {\left (a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}\right )} \log \left (b x + a\right )}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \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 {x^{4}}{\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.82, size = 89, normalized size = 0.52 \begin {gather*} \frac {6 \, a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{5} \mathrm {sgn}\left (b x + a\right )} + \frac {b^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) - 6 \, a b^{2} x \mathrm {sgn}\left (b x + a\right )}{2 \, b^{6}} + \frac {8 \, a^{3} b x + 7 \, a^{4}}{2 \, {\left (b x + a\right )}^{2} b^{5} \mathrm {sgn}\left (b x + a\right )} \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 {x^4}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________