Optimal. Leaf size=219 \[ -\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{5 a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{x^2 (a+b x)}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{b^5 x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{5 a b^4 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0528697, antiderivative size = 219, 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} \[ -\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{5 a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{x^2 (a+b x)}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{b^5 x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{5 a b^4 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]
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 )^{5/2}}{x^5} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^5}{x^5} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (b^{10}+\frac{a^5 b^5}{x^5}+\frac{5 a^4 b^6}{x^4}+\frac{10 a^3 b^7}{x^3}+\frac{10 a^2 b^8}{x^2}+\frac{5 a b^9}{x}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{5 a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{x^2 (a+b x)}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{b^5 x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{5 a b^4 \sqrt{a^2+2 a b x+b^2 x^2} \log (x)}{a+b x}\\ \end{align*}
Mathematica [A] time = 0.019687, size = 79, normalized size = 0.36 \[ -\frac{\sqrt{(a+b x)^2} \left (60 a^3 b^2 x^2+120 a^2 b^3 x^3+20 a^4 b x+3 a^5-60 a b^4 x^4 \log (x)-12 b^5 x^5\right )}{12 x^4 (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.225, size = 76, normalized size = 0.4 \begin{align*}{\frac{60\,a{b}^{4}\ln \left ( x \right ){x}^{4}+12\,{b}^{5}{x}^{5}-120\,{a}^{2}{b}^{3}{x}^{3}-60\,{a}^{3}{b}^{2}{x}^{2}-20\,{a}^{4}bx-3\,{a}^{5}}{12\, \left ( bx+a \right ) ^{5}{x}^{4}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{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 = 1.74022, size = 136, normalized size = 0.62 \begin{align*} \frac{12 \, b^{5} x^{5} + 60 \, a b^{4} x^{4} \log \left (x\right ) - 120 \, a^{2} b^{3} x^{3} - 60 \, a^{3} b^{2} x^{2} - 20 \, a^{4} b x - 3 \, a^{5}}{12 \, x^{4}} \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{5}{2}}}{x^{5}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.47272, size = 123, normalized size = 0.56 \begin{align*} b^{5} x \mathrm{sgn}\left (b x + a\right ) + 5 \, a b^{4} \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (b x + a\right ) - \frac{120 \, a^{2} b^{3} x^{3} \mathrm{sgn}\left (b x + a\right ) + 60 \, a^{3} b^{2} x^{2} \mathrm{sgn}\left (b x + a\right ) + 20 \, a^{4} b x \mathrm{sgn}\left (b x + a\right ) + 3 \, a^{5} \mathrm{sgn}\left (b x + a\right )}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]