Optimal. Leaf size=92 \[ \frac {5 a^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 \sqrt {b}}+\frac {5}{8} a^2 \sqrt {x} \sqrt {a+b x}+\frac {5}{12} a \sqrt {x} (a+b x)^{3/2}+\frac {1}{3} \sqrt {x} (a+b x)^{5/2} \]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {50, 63, 217, 206} \begin {gather*} \frac {5}{8} a^2 \sqrt {x} \sqrt {a+b x}+\frac {5 a^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 \sqrt {b}}+\frac {5}{12} a \sqrt {x} (a+b x)^{3/2}+\frac {1}{3} \sqrt {x} (a+b x)^{5/2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2}}{\sqrt {x}} \, dx &=\frac {1}{3} \sqrt {x} (a+b x)^{5/2}+\frac {1}{6} (5 a) \int \frac {(a+b x)^{3/2}}{\sqrt {x}} \, dx\\ &=\frac {5}{12} a \sqrt {x} (a+b x)^{3/2}+\frac {1}{3} \sqrt {x} (a+b x)^{5/2}+\frac {1}{8} \left (5 a^2\right ) \int \frac {\sqrt {a+b x}}{\sqrt {x}} \, dx\\ &=\frac {5}{8} a^2 \sqrt {x} \sqrt {a+b x}+\frac {5}{12} a \sqrt {x} (a+b x)^{3/2}+\frac {1}{3} \sqrt {x} (a+b x)^{5/2}+\frac {1}{16} \left (5 a^3\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx\\ &=\frac {5}{8} a^2 \sqrt {x} \sqrt {a+b x}+\frac {5}{12} a \sqrt {x} (a+b x)^{3/2}+\frac {1}{3} \sqrt {x} (a+b x)^{5/2}+\frac {1}{8} \left (5 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {5}{8} a^2 \sqrt {x} \sqrt {a+b x}+\frac {5}{12} a \sqrt {x} (a+b x)^{3/2}+\frac {1}{3} \sqrt {x} (a+b x)^{5/2}+\frac {1}{8} \left (5 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )\\ &=\frac {5}{8} a^2 \sqrt {x} \sqrt {a+b x}+\frac {5}{12} a \sqrt {x} (a+b x)^{3/2}+\frac {1}{3} \sqrt {x} (a+b x)^{5/2}+\frac {5 a^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 \sqrt {b}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 80, normalized size = 0.87 \begin {gather*} \frac {1}{24} \sqrt {a+b x} \left (\frac {15 a^{5/2} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {b} \sqrt {\frac {b x}{a}+1}}+\sqrt {x} \left (33 a^2+26 a b x+8 b^2 x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.11, size = 79, normalized size = 0.86 \begin {gather*} \frac {1}{24} \sqrt {a+b x} \left (33 a^2 \sqrt {x}+26 a b x^{3/2}+8 b^2 x^{5/2}\right )-\frac {5 a^3 \log \left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right )}{8 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.51, size = 141, normalized size = 1.53 \begin {gather*} \left [\frac {15 \, a^{3} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (8 \, b^{3} x^{2} + 26 \, a b^{2} x + 33 \, a^{2} b\right )} \sqrt {b x + a} \sqrt {x}}{48 \, b}, -\frac {15 \, a^{3} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (8 \, b^{3} x^{2} + 26 \, a b^{2} x + 33 \, a^{2} b\right )} \sqrt {b x + a} \sqrt {x}}{24 \, b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 93, normalized size = 1.01 \begin {gather*} \frac {5 \sqrt {\left (b x +a \right ) x}\, a^{3} \ln \left (\frac {b x +\frac {a}{2}}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{16 \sqrt {b x +a}\, \sqrt {b}\, \sqrt {x}}+\frac {5 \sqrt {b x +a}\, a^{2} \sqrt {x}}{8}+\frac {5 \left (b x +a \right )^{\frac {3}{2}} a \sqrt {x}}{12}+\frac {\left (b x +a \right )^{\frac {5}{2}} \sqrt {x}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 2.99, size = 141, normalized size = 1.53 \begin {gather*} -\frac {5 \, a^{3} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right )}{16 \, \sqrt {b}} - \frac {\frac {15 \, \sqrt {b x + a} a^{3} b^{2}}{\sqrt {x}} - \frac {40 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} b}{x^{\frac {3}{2}}} + \frac {33 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3}}{x^{\frac {5}{2}}}}{24 \, {\left (b^{3} - \frac {3 \, {\left (b x + a\right )} b^{2}}{x} + \frac {3 \, {\left (b x + a\right )}^{2} b}{x^{2}} - \frac {{\left (b x + a\right )}^{3}}{x^{3}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{5/2}}{\sqrt {x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 6.23, size = 102, normalized size = 1.11 \begin {gather*} \frac {11 a^{\frac {5}{2}} \sqrt {x} \sqrt {1 + \frac {b x}{a}}}{8} + \frac {13 a^{\frac {3}{2}} b x^{\frac {3}{2}} \sqrt {1 + \frac {b x}{a}}}{12} + \frac {\sqrt {a} b^{2} x^{\frac {5}{2}} \sqrt {1 + \frac {b x}{a}}}{3} + \frac {5 a^{3} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{8 \sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________