Optimal. Leaf size=116 \[ -\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b \sqrt {x}}}\right )}{4 a^{7/2}}+\frac {5 b^2 \sqrt {a x+b \sqrt {x}}}{4 a^3}-\frac {5 b \sqrt {x} \sqrt {a x+b \sqrt {x}}}{6 a^2}+\frac {2 x \sqrt {a x+b \sqrt {x}}}{3 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {2018, 670, 640, 620, 206} \[ \frac {5 b^2 \sqrt {a x+b \sqrt {x}}}{4 a^3}-\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b \sqrt {x}}}\right )}{4 a^{7/2}}-\frac {5 b \sqrt {x} \sqrt {a x+b \sqrt {x}}}{6 a^2}+\frac {2 x \sqrt {a x+b \sqrt {x}}}{3 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 620
Rule 640
Rule 670
Rule 2018
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {b \sqrt {x}+a x}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {2 x \sqrt {b \sqrt {x}+a x}}{3 a}-\frac {(5 b) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{3 a}\\ &=-\frac {5 b \sqrt {x} \sqrt {b \sqrt {x}+a x}}{6 a^2}+\frac {2 x \sqrt {b \sqrt {x}+a x}}{3 a}+\frac {\left (5 b^2\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{4 a^2}\\ &=\frac {5 b^2 \sqrt {b \sqrt {x}+a x}}{4 a^3}-\frac {5 b \sqrt {x} \sqrt {b \sqrt {x}+a x}}{6 a^2}+\frac {2 x \sqrt {b \sqrt {x}+a x}}{3 a}-\frac {\left (5 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{8 a^3}\\ &=\frac {5 b^2 \sqrt {b \sqrt {x}+a x}}{4 a^3}-\frac {5 b \sqrt {x} \sqrt {b \sqrt {x}+a x}}{6 a^2}+\frac {2 x \sqrt {b \sqrt {x}+a x}}{3 a}-\frac {\left (5 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{4 a^3}\\ &=\frac {5 b^2 \sqrt {b \sqrt {x}+a x}}{4 a^3}-\frac {5 b \sqrt {x} \sqrt {b \sqrt {x}+a x}}{6 a^2}+\frac {2 x \sqrt {b \sqrt {x}+a x}}{3 a}-\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{4 a^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 129, normalized size = 1.11 \[ \frac {5 b^4 \left (\frac {a \sqrt {x}}{b}+1\right ) \left (\frac {16 a^3 x^{3/2}}{15 b^3}-\frac {4 a^2 x}{3 b^2}+\frac {2 a \sqrt {x}}{b}-\frac {2 \sqrt {a} \sqrt [4]{x} \sinh ^{-1}\left (\frac {\sqrt {a} \sqrt [4]{x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {\frac {a \sqrt {x}}{b}+1}}\right )}{8 a^4 \sqrt {\sqrt {x} \left (a \sqrt {x}+b\right )}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.31, size = 83, normalized size = 0.72 \[ \frac {1}{12} \, \sqrt {a x + b \sqrt {x}} {\left (2 \, \sqrt {x} {\left (\frac {4 \, \sqrt {x}}{a} - \frac {5 \, b}{a^{2}}\right )} + \frac {15 \, b^{2}}{a^{3}}\right )} + \frac {5 \, b^{3} \log \left ({\left | -2 \, \sqrt {a} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} - b \right |}\right )}{8 \, a^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.05, size = 181, normalized size = 1.56 \[ -\frac {\sqrt {a x +b \sqrt {x}}\, \left (24 a \,b^{3} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-9 a \,b^{3} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+36 \sqrt {a x +b \sqrt {x}}\, a^{\frac {5}{2}} b \sqrt {x}-48 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {3}{2}} b^{2}+18 \sqrt {a x +b \sqrt {x}}\, a^{\frac {3}{2}} b^{2}-16 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {5}{2}}\right )}{24 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {a x + b \sqrt {x}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x}{\sqrt {a\,x+b\,\sqrt {x}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {a x + b \sqrt {x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________