Optimal. Leaf size=51 \[ \frac {2 x^{15/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {x^{17/2}}{7 a \left (a x+b x^3\right )^{7/2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2015, 2014} \begin {gather*} \frac {2 x^{15/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {x^{17/2}}{7 a \left (a x+b x^3\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2014
Rule 2015
Rubi steps
\begin {align*} \int \frac {x^{17/2}}{\left (a x+b x^3\right )^{9/2}} \, dx &=\frac {x^{17/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {2 \int \frac {x^{15/2}}{\left (a x+b x^3\right )^{7/2}} \, dx}{7 a}\\ &=\frac {x^{17/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {2 x^{15/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 44, normalized size = 0.86 \begin {gather*} \frac {x^{9/2} \sqrt {x \left (a+b x^2\right )} \left (7 a+2 b x^2\right )}{35 a^2 \left (a+b x^2\right )^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.11, size = 42, normalized size = 0.82 \begin {gather*} \frac {x^{19/2} \left (a+b x^2\right ) \left (7 a+2 b x^2\right )}{35 a^2 \left (x \left (a+b x^2\right )\right )^{9/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 76, normalized size = 1.49 \begin {gather*} \frac {{\left (2 \, b x^{6} + 7 \, a x^{4}\right )} \sqrt {b x^{3} + a x} \sqrt {x}}{35 \, {\left (a^{2} b^{4} x^{8} + 4 \, a^{3} b^{3} x^{6} + 6 \, a^{4} b^{2} x^{4} + 4 \, a^{5} b x^{2} + a^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.29, size = 29, normalized size = 0.57 \begin {gather*} \frac {x^{5} {\left (\frac {2 \, b x^{2}}{a^{2}} + \frac {7}{a}\right )}}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 37, normalized size = 0.73 \begin {gather*} \frac {\left (b \,x^{2}+a \right ) \left (2 b \,x^{2}+7 a \right ) x^{\frac {19}{2}}}{35 \left (b \,x^{3}+a x \right )^{\frac {9}{2}} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{\frac {17}{2}}}{{\left (b x^{3} + a x\right )}^{\frac {9}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^{17/2}}{{\left (b\,x^3+a\,x\right )}^{9/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [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]
________________________________________________________________________________________