Optimal. Leaf size=70 \[ -\frac {b^2 \left (a+b \sqrt {x}\right )^6}{84 a^3 x^3}+\frac {b \left (a+b \sqrt {x}\right )^6}{14 a^2 x^{7/2}}-\frac {\left (a+b \sqrt {x}\right )^6}{4 a x^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {266, 45, 37} \[ -\frac {b^2 \left (a+b \sqrt {x}\right )^6}{84 a^3 x^3}+\frac {b \left (a+b \sqrt {x}\right )^6}{14 a^2 x^{7/2}}-\frac {\left (a+b \sqrt {x}\right )^6}{4 a x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 45
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b \sqrt {x}\right )^5}{x^5} \, dx &=2 \operatorname {Subst}\left (\int \frac {(a+b x)^5}{x^9} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\left (a+b \sqrt {x}\right )^6}{4 a x^4}-\frac {b \operatorname {Subst}\left (\int \frac {(a+b x)^5}{x^8} \, dx,x,\sqrt {x}\right )}{2 a}\\ &=-\frac {\left (a+b \sqrt {x}\right )^6}{4 a x^4}+\frac {b \left (a+b \sqrt {x}\right )^6}{14 a^2 x^{7/2}}+\frac {b^2 \operatorname {Subst}\left (\int \frac {(a+b x)^5}{x^7} \, dx,x,\sqrt {x}\right )}{14 a^2}\\ &=-\frac {\left (a+b \sqrt {x}\right )^6}{4 a x^4}+\frac {b \left (a+b \sqrt {x}\right )^6}{14 a^2 x^{7/2}}-\frac {b^2 \left (a+b \sqrt {x}\right )^6}{84 a^3 x^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 65, normalized size = 0.93 \[ -\frac {21 a^5+120 a^4 b \sqrt {x}+280 a^3 b^2 x+336 a^2 b^3 x^{3/2}+210 a b^4 x^2+56 b^5 x^{5/2}}{84 x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.95, size = 58, normalized size = 0.83 \[ -\frac {210 \, a b^{4} x^{2} + 280 \, a^{3} b^{2} x + 21 \, a^{5} + 8 \, {\left (7 \, b^{5} x^{2} + 42 \, a^{2} b^{3} x + 15 \, a^{4} b\right )} \sqrt {x}}{84 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.21, size = 57, normalized size = 0.81 \[ -\frac {56 \, b^{5} x^{\frac {5}{2}} + 210 \, a b^{4} x^{2} + 336 \, a^{2} b^{3} x^{\frac {3}{2}} + 280 \, a^{3} b^{2} x + 120 \, a^{4} b \sqrt {x} + 21 \, a^{5}}{84 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 58, normalized size = 0.83 \[ -\frac {2 b^{5}}{3 x^{\frac {3}{2}}}-\frac {5 a \,b^{4}}{2 x^{2}}-\frac {4 a^{2} b^{3}}{x^{\frac {5}{2}}}-\frac {10 a^{3} b^{2}}{3 x^{3}}-\frac {10 a^{4} b}{7 x^{\frac {7}{2}}}-\frac {a^{5}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.86, size = 57, normalized size = 0.81 \[ -\frac {56 \, b^{5} x^{\frac {5}{2}} + 210 \, a b^{4} x^{2} + 336 \, a^{2} b^{3} x^{\frac {3}{2}} + 280 \, a^{3} b^{2} x + 120 \, a^{4} b \sqrt {x} + 21 \, a^{5}}{84 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.10, size = 57, normalized size = 0.81 \[ -\frac {21\,a^5+56\,b^5\,x^{5/2}+280\,a^3\,b^2\,x+210\,a\,b^4\,x^2+120\,a^4\,b\,\sqrt {x}+336\,a^2\,b^3\,x^{3/2}}{84\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.64, size = 73, normalized size = 1.04 \[ - \frac {a^{5}}{4 x^{4}} - \frac {10 a^{4} b}{7 x^{\frac {7}{2}}} - \frac {10 a^{3} b^{2}}{3 x^{3}} - \frac {4 a^{2} b^{3}}{x^{\frac {5}{2}}} - \frac {5 a b^{4}}{2 x^{2}} - \frac {2 b^{5}}{3 x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________