Optimal. Leaf size=96 \[ \frac {15 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 b^{7/2}}-\frac {15 a \sqrt {x} \sqrt {a+b x}}{4 b^3}+\frac {5 x^{3/2} \sqrt {a+b x}}{2 b^2}-\frac {2 x^{5/2}}{b \sqrt {a+b x}} \]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {47, 50, 63, 217, 206} \begin {gather*} \frac {15 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 b^{7/2}}+\frac {5 x^{3/2} \sqrt {a+b x}}{2 b^2}-\frac {15 a \sqrt {x} \sqrt {a+b x}}{4 b^3}-\frac {2 x^{5/2}}{b \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
Rule 50
Rule 63
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {x^{5/2}}{(a+b x)^{3/2}} \, dx &=-\frac {2 x^{5/2}}{b \sqrt {a+b x}}+\frac {5 \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx}{b}\\ &=-\frac {2 x^{5/2}}{b \sqrt {a+b x}}+\frac {5 x^{3/2} \sqrt {a+b x}}{2 b^2}-\frac {(15 a) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{4 b^2}\\ &=-\frac {2 x^{5/2}}{b \sqrt {a+b x}}-\frac {15 a \sqrt {x} \sqrt {a+b x}}{4 b^3}+\frac {5 x^{3/2} \sqrt {a+b x}}{2 b^2}+\frac {\left (15 a^2\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{8 b^3}\\ &=-\frac {2 x^{5/2}}{b \sqrt {a+b x}}-\frac {15 a \sqrt {x} \sqrt {a+b x}}{4 b^3}+\frac {5 x^{3/2} \sqrt {a+b x}}{2 b^2}+\frac {\left (15 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{4 b^3}\\ &=-\frac {2 x^{5/2}}{b \sqrt {a+b x}}-\frac {15 a \sqrt {x} \sqrt {a+b x}}{4 b^3}+\frac {5 x^{3/2} \sqrt {a+b x}}{2 b^2}+\frac {\left (15 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{4 b^3}\\ &=-\frac {2 x^{5/2}}{b \sqrt {a+b x}}-\frac {15 a \sqrt {x} \sqrt {a+b x}}{4 b^3}+\frac {5 x^{3/2} \sqrt {a+b x}}{2 b^2}+\frac {15 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 b^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 50, normalized size = 0.52 \begin {gather*} \frac {2 x^{7/2} \sqrt {\frac {b x}{a}+1} \, _2F_1\left (\frac {3}{2},\frac {7}{2};\frac {9}{2};-\frac {b x}{a}\right )}{7 a \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.14, size = 82, normalized size = 0.85 \begin {gather*} \frac {-15 a^2 \sqrt {x}-5 a b x^{3/2}+2 b^2 x^{5/2}}{4 b^3 \sqrt {a+b x}}-\frac {15 a^2 \log \left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right )}{4 b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.41, size = 175, normalized size = 1.82 \begin {gather*} \left [\frac {15 \, {\left (a^{2} b x + a^{3}\right )} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (2 \, b^{3} x^{2} - 5 \, a b^{2} x - 15 \, a^{2} b\right )} \sqrt {b x + a} \sqrt {x}}{8 \, {\left (b^{5} x + a b^{4}\right )}}, -\frac {15 \, {\left (a^{2} b x + a^{3}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (2 \, b^{3} x^{2} - 5 \, a b^{2} x - 15 \, a^{2} b\right )} \sqrt {b x + a} \sqrt {x}}{4 \, {\left (b^{5} x + a b^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 92.14, size = 131, normalized size = 1.36 \begin {gather*} \frac {{\left (2 \, \sqrt {{\left (b x + a\right )} b - a b} \sqrt {b x + a} {\left (\frac {2 \, {\left (b x + a\right )}}{b^{3}} - \frac {9 \, a}{b^{3}}\right )} - \frac {32 \, a^{3}}{{\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} b^{\frac {3}{2}}} - \frac {15 \, a^{2} \log \left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{b^{\frac {5}{2}}}\right )} {\left | b \right |}}{8 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 119, normalized size = 1.24 \begin {gather*} \frac {\left (\frac {15 a^{2} \ln \left (\frac {b x +\frac {a}{2}}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {7}{2}}}-\frac {2 \sqrt {-\left (x +\frac {a}{b}\right ) a +\left (x +\frac {a}{b}\right )^{2} b}\, a^{2}}{\left (x +\frac {a}{b}\right ) b^{4}}\right ) \sqrt {\left (b x +a \right ) x}}{\sqrt {b x +a}\, \sqrt {x}}-\frac {\left (-2 b x +7 a \right ) \sqrt {b x +a}\, \sqrt {x}}{4 b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.92, size = 131, normalized size = 1.36 \begin {gather*} -\frac {8 \, a^{2} b^{2} - \frac {25 \, {\left (b x + a\right )} a^{2} b}{x} + \frac {15 \, {\left (b x + a\right )}^{2} a^{2}}{x^{2}}}{4 \, {\left (\frac {\sqrt {b x + a} b^{5}}{\sqrt {x}} - \frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{4}}{x^{\frac {3}{2}}} + \frac {{\left (b x + a\right )}^{\frac {5}{2}} b^{3}}{x^{\frac {5}{2}}}\right )}} - \frac {15 \, a^{2} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right )}{8 \, b^{\frac {7}{2}}} \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 {x^{5/2}}{{\left (a+b\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 8.14, size = 105, normalized size = 1.09 \begin {gather*} - \frac {15 a^{\frac {3}{2}} \sqrt {x}}{4 b^{3} \sqrt {1 + \frac {b x}{a}}} - \frac {5 \sqrt {a} x^{\frac {3}{2}}}{4 b^{2} \sqrt {1 + \frac {b x}{a}}} + \frac {15 a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{4 b^{\frac {7}{2}}} + \frac {x^{\frac {5}{2}}}{2 \sqrt {a} b \sqrt {1 + \frac {b x}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________