Optimal. Leaf size=84 \[ -\frac {2 d x^5 \left (c+\frac {d}{x^2}\right )^{5/2} (9 b c-4 a d)}{315 c^3}+\frac {x^7 \left (c+\frac {d}{x^2}\right )^{5/2} (9 b c-4 a d)}{63 c^2}+\frac {a x^9 \left (c+\frac {d}{x^2}\right )^{5/2}}{9 c} \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {453, 271, 264} \begin {gather*} \frac {x^7 \left (c+\frac {d}{x^2}\right )^{5/2} (9 b c-4 a d)}{63 c^2}-\frac {2 d x^5 \left (c+\frac {d}{x^2}\right )^{5/2} (9 b c-4 a d)}{315 c^3}+\frac {a x^9 \left (c+\frac {d}{x^2}\right )^{5/2}}{9 c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 264
Rule 271
Rule 453
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^8 \, dx &=\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^9}{9 c}+\frac {(9 b c-4 a d) \int \left (c+\frac {d}{x^2}\right )^{3/2} x^6 \, dx}{9 c}\\ &=\frac {(9 b c-4 a d) \left (c+\frac {d}{x^2}\right )^{5/2} x^7}{63 c^2}+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^9}{9 c}-\frac {(2 d (9 b c-4 a d)) \int \left (c+\frac {d}{x^2}\right )^{3/2} x^4 \, dx}{63 c^2}\\ &=-\frac {2 d (9 b c-4 a d) \left (c+\frac {d}{x^2}\right )^{5/2} x^5}{315 c^3}+\frac {(9 b c-4 a d) \left (c+\frac {d}{x^2}\right )^{5/2} x^7}{63 c^2}+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^9}{9 c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 66, normalized size = 0.79 \begin {gather*} \frac {x \sqrt {c+\frac {d}{x^2}} \left (c x^2+d\right )^2 \left (a \left (35 c^2 x^4-20 c d x^2+8 d^2\right )+9 b c \left (5 c x^2-2 d\right )\right )}{315 c^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.09, size = 66, normalized size = 0.79 \begin {gather*} \frac {x \sqrt {c+\frac {d}{x^2}} \left (c x^2+d\right )^2 \left (35 a c^2 x^4-20 a c d x^2+8 a d^2+45 b c^2 x^2-18 b c d\right )}{315 c^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 106, normalized size = 1.26 \begin {gather*} \frac {{\left (35 \, a c^{4} x^{9} + 5 \, {\left (9 \, b c^{4} + 10 \, a c^{3} d\right )} x^{7} + 3 \, {\left (24 \, b c^{3} d + a c^{2} d^{2}\right )} x^{5} + {\left (9 \, b c^{2} d^{2} - 4 \, a c d^{3}\right )} x^{3} - 2 \, {\left (9 \, b c d^{3} - 4 \, a d^{4}\right )} x\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{315 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 105, normalized size = 1.25 \begin {gather*} \frac {2 \, {\left (9 \, b c d^{\frac {7}{2}} - 4 \, a d^{\frac {9}{2}}\right )} \mathrm {sgn}\relax (x)}{315 \, c^{3}} + \frac {35 \, {\left (c x^{2} + d\right )}^{\frac {9}{2}} a \mathrm {sgn}\relax (x) + 45 \, {\left (c x^{2} + d\right )}^{\frac {7}{2}} b c \mathrm {sgn}\relax (x) - 90 \, {\left (c x^{2} + d\right )}^{\frac {7}{2}} a d \mathrm {sgn}\relax (x) - 63 \, {\left (c x^{2} + d\right )}^{\frac {5}{2}} b c d \mathrm {sgn}\relax (x) + 63 \, {\left (c x^{2} + d\right )}^{\frac {5}{2}} a d^{2} \mathrm {sgn}\relax (x)}{315 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 67, normalized size = 0.80 \begin {gather*} \frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} \left (35 a \,x^{4} c^{2}-20 a c d \,x^{2}+45 b \,c^{2} x^{2}+8 a \,d^{2}-18 b c d \right ) \left (c \,x^{2}+d \right ) x^{3}}{315 c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.76, size = 90, normalized size = 1.07 \begin {gather*} \frac {{\left (5 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}} x^{7} - 7 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} d x^{5}\right )} b}{35 \, c^{2}} + \frac {{\left (35 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {9}{2}} x^{9} - 90 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}} d x^{7} + 63 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} d^{2} x^{5}\right )} a}{315 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.55, size = 97, normalized size = 1.15 \begin {gather*} \sqrt {c+\frac {d}{x^2}}\,\left (\frac {x\,\left (8\,a\,d^4-18\,b\,c\,d^3\right )}{315\,c^3}+\frac {x^7\,\left (45\,b\,c^4+50\,a\,d\,c^3\right )}{315\,c^3}+\frac {a\,c\,x^9}{9}+\frac {d\,x^5\,\left (a\,d+24\,b\,c\right )}{105\,c}-\frac {d^2\,x^3\,\left (4\,a\,d-9\,b\,c\right )}{315\,c^2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 7.64, size = 1340, normalized size = 15.95
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________