Optimal. Leaf size=145 \[ -\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac{3 a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac{3 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{b^3 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0344629, antiderivative size = 145, 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} \[ -\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac{3 a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac{3 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{b^3 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 646
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^4} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^3}{x^4} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (\frac{a^3 b^3}{x^4}+\frac{3 a^2 b^4}{x^3}+\frac{3 a b^5}{x^2}+\frac{b^6}{x}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac{3 a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac{3 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2} \log (x)}{a+b x}\\ \end{align*}
Mathematica [A] time = 0.0188272, size = 57, normalized size = 0.39 \[ -\frac{\sqrt{(a+b x)^2} \left (a \left (2 a^2+9 a b x+18 b^2 x^2\right )-6 b^3 x^3 \log (x)\right )}{6 x^3 (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.223, size = 54, normalized size = 0.4 \begin{align*}{\frac{6\,{b}^{3}\ln \left ( x \right ){x}^{3}-18\,a{b}^{2}{x}^{2}-9\,b{a}^{2}x-2\,{a}^{3}}{6\, \left ( bx+a \right ) ^{3}{x}^{3}} \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.01924, size = 85, normalized size = 0.59 \begin{align*} \frac{6 \, b^{3} x^{3} \log \left (x\right ) - 18 \, a b^{2} x^{2} - 9 \, a^{2} b x - 2 \, a^{3}}{6 \, x^{3}} \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{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.26428, size = 80, normalized size = 0.55 \begin{align*} b^{3} \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (b x + a\right ) - \frac{18 \, a b^{2} x^{2} \mathrm{sgn}\left (b x + a\right ) + 9 \, a^{2} b x \mathrm{sgn}\left (b x + a\right ) + 2 \, a^{3} \mathrm{sgn}\left (b x + a\right )}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]