Optimal. Leaf size=172 \[ -\frac{a^4}{2 b^5 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{4 a^3}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 a x (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^2 (a+b x)}{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{6 a^2 (a+b x) \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0727534, antiderivative size = 172, normalized size of antiderivative = 1., 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} \[ -\frac{a^4}{2 b^5 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{4 a^3}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 a x (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^2 (a+b x)}{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{6 a^2 (a+b x) \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 646
Rule 43
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 = 0.0266041, size = 83, normalized size = 0.48 \[ \frac{-11 a^2 b^2 x^2+2 a^3 b x+12 a^2 (a+b x)^2 \log (a+b x)+7 a^4-4 a b^3 x^3+b^4 x^4}{2 b^5 (a+b x) \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.229, size = 101, normalized size = 0.6 \begin{align*}{\frac{ \left ({b}^{4}{x}^{4}+12\,\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{2}-4\,a{b}^{3}{x}^{3}+24\,\ln \left ( bx+a \right ) x{a}^{3}b-11\,{x}^{2}{a}^{2}{b}^{2}+12\,{a}^{4}\ln \left ( bx+a \right ) +2\,x{a}^{3}b+7\,{a}^{4} \right ) \left ( bx+a \right ) }{2\,{b}^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.62069, size = 223, normalized size = 1.3 \begin{align*} \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 )}{{\left (b^{2}\right )}^{\frac{3}{2}} b^{2}} + \frac{9 \, a^{4}}{{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{12 \, a^{3} x}{{\left (b^{2}\right )}^{\frac{5}{2}} b{\left (x + \frac{a}{b}\right )}^{2}} - \frac{5 \, a^{3}}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{5}} + \frac{5 \, a^{4}}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}} b^{4}{\left (x + \frac{a}{b}\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.58875, size = 200, normalized size = 1.16 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]