Optimal. Leaf size=103 \[ \frac{2 b^3}{3 a^4 x^{3/2}}-\frac{b^2}{2 a^3 x^2}+\frac{2 b^5}{a^6 \sqrt{x}}-\frac{b^4}{a^5 x}-\frac{2 b^6 \log \left (a+b \sqrt{x}\right )}{a^7}+\frac{b^6 \log (x)}{a^7}+\frac{2 b}{5 a^2 x^{5/2}}-\frac{1}{3 a x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0515842, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 44} \[ \frac{2 b^3}{3 a^4 x^{3/2}}-\frac{b^2}{2 a^3 x^2}+\frac{2 b^5}{a^6 \sqrt{x}}-\frac{b^4}{a^5 x}-\frac{2 b^6 \log \left (a+b \sqrt{x}\right )}{a^7}+\frac{b^6 \log (x)}{a^7}+\frac{2 b}{5 a^2 x^{5/2}}-\frac{1}{3 a x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \sqrt{x}\right ) x^4} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x^7 (a+b x)} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{1}{a x^7}-\frac{b}{a^2 x^6}+\frac{b^2}{a^3 x^5}-\frac{b^3}{a^4 x^4}+\frac{b^4}{a^5 x^3}-\frac{b^5}{a^6 x^2}+\frac{b^6}{a^7 x}-\frac{b^7}{a^7 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{1}{3 a x^3}+\frac{2 b}{5 a^2 x^{5/2}}-\frac{b^2}{2 a^3 x^2}+\frac{2 b^3}{3 a^4 x^{3/2}}-\frac{b^4}{a^5 x}+\frac{2 b^5}{a^6 \sqrt{x}}-\frac{2 b^6 \log \left (a+b \sqrt{x}\right )}{a^7}+\frac{b^6 \log (x)}{a^7}\\ \end{align*}
Mathematica [A] time = 0.0573427, size = 93, normalized size = 0.9 \[ \frac{\frac{a \left (20 a^2 b^3 x^{3/2}-15 a^3 b^2 x+12 a^4 b \sqrt{x}-10 a^5-30 a b^4 x^2+60 b^5 x^{5/2}\right )}{x^3}-60 b^6 \log \left (a+b \sqrt{x}\right )+30 b^6 \log (x)}{30 a^7} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.01, size = 88, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,a{x}^{3}}}+{\frac{2\,b}{5\,{a}^{2}}{x}^{-{\frac{5}{2}}}}-{\frac{{b}^{2}}{2\,{x}^{2}{a}^{3}}}+{\frac{2\,{b}^{3}}{3\,{a}^{4}}{x}^{-{\frac{3}{2}}}}-{\frac{{b}^{4}}{x{a}^{5}}}+{\frac{{b}^{6}\ln \left ( x \right ) }{{a}^{7}}}-2\,{\frac{{b}^{6}\ln \left ( a+b\sqrt{x} \right ) }{{a}^{7}}}+2\,{\frac{{b}^{5}}{{a}^{6}\sqrt{x}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.969261, size = 116, normalized size = 1.13 \begin{align*} -\frac{2 \, b^{6} \log \left (b \sqrt{x} + a\right )}{a^{7}} + \frac{b^{6} \log \left (x\right )}{a^{7}} + \frac{60 \, b^{5} x^{\frac{5}{2}} - 30 \, a b^{4} x^{2} + 20 \, a^{2} b^{3} x^{\frac{3}{2}} - 15 \, a^{3} b^{2} x + 12 \, a^{4} b \sqrt{x} - 10 \, a^{5}}{30 \, a^{6} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.28383, size = 225, normalized size = 2.18 \begin{align*} -\frac{60 \, b^{6} x^{3} \log \left (b \sqrt{x} + a\right ) - 60 \, b^{6} x^{3} \log \left (\sqrt{x}\right ) + 30 \, a^{2} b^{4} x^{2} + 15 \, a^{4} b^{2} x + 10 \, a^{6} - 4 \,{\left (15 \, a b^{5} x^{2} + 5 \, a^{3} b^{3} x + 3 \, a^{5} b\right )} \sqrt{x}}{30 \, a^{7} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 9.54432, size = 126, normalized size = 1.22 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{7}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{7 b x^{\frac{7}{2}}} & \text{for}\: a = 0 \\- \frac{1}{3 a x^{3}} & \text{for}\: b = 0 \\- \frac{1}{3 a x^{3}} + \frac{2 b}{5 a^{2} x^{\frac{5}{2}}} - \frac{b^{2}}{2 a^{3} x^{2}} + \frac{2 b^{3}}{3 a^{4} x^{\frac{3}{2}}} - \frac{b^{4}}{a^{5} x} + \frac{2 b^{5}}{a^{6} \sqrt{x}} + \frac{b^{6} \log{\left (x \right )}}{a^{7}} - \frac{2 b^{6} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{7}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.11855, size = 123, normalized size = 1.19 \begin{align*} -\frac{2 \, b^{6} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{a^{7}} + \frac{b^{6} \log \left ({\left | x \right |}\right )}{a^{7}} + \frac{60 \, a b^{5} x^{\frac{5}{2}} - 30 \, a^{2} b^{4} x^{2} + 20 \, a^{3} b^{3} x^{\frac{3}{2}} - 15 \, a^{4} b^{2} x + 12 \, a^{5} b \sqrt{x} - 10 \, a^{6}}{30 \, a^{7} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]