Optimal. Leaf size=140 \[ \frac {256 b^5 \left (a+b x^2\right )^{11/2}}{969969 a^6 x^{11}}-\frac {128 b^4 \left (a+b x^2\right )^{11/2}}{88179 a^5 x^{13}}+\frac {32 b^3 \left (a+b x^2\right )^{11/2}}{6783 a^4 x^{15}}-\frac {80 b^2 \left (a+b x^2\right )^{11/2}}{6783 a^3 x^{17}}+\frac {10 b \left (a+b x^2\right )^{11/2}}{399 a^2 x^{19}}-\frac {\left (a+b x^2\right )^{11/2}}{21 a x^{21}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac {256 b^5 \left (a+b x^2\right )^{11/2}}{969969 a^6 x^{11}}-\frac {128 b^4 \left (a+b x^2\right )^{11/2}}{88179 a^5 x^{13}}+\frac {32 b^3 \left (a+b x^2\right )^{11/2}}{6783 a^4 x^{15}}-\frac {80 b^2 \left (a+b x^2\right )^{11/2}}{6783 a^3 x^{17}}+\frac {10 b \left (a+b x^2\right )^{11/2}}{399 a^2 x^{19}}-\frac {\left (a+b x^2\right )^{11/2}}{21 a x^{21}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 264
Rule 271
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{9/2}}{x^{22}} \, dx &=-\frac {\left (a+b x^2\right )^{11/2}}{21 a x^{21}}-\frac {(10 b) \int \frac {\left (a+b x^2\right )^{9/2}}{x^{20}} \, dx}{21 a}\\ &=-\frac {\left (a+b x^2\right )^{11/2}}{21 a x^{21}}+\frac {10 b \left (a+b x^2\right )^{11/2}}{399 a^2 x^{19}}+\frac {\left (80 b^2\right ) \int \frac {\left (a+b x^2\right )^{9/2}}{x^{18}} \, dx}{399 a^2}\\ &=-\frac {\left (a+b x^2\right )^{11/2}}{21 a x^{21}}+\frac {10 b \left (a+b x^2\right )^{11/2}}{399 a^2 x^{19}}-\frac {80 b^2 \left (a+b x^2\right )^{11/2}}{6783 a^3 x^{17}}-\frac {\left (160 b^3\right ) \int \frac {\left (a+b x^2\right )^{9/2}}{x^{16}} \, dx}{2261 a^3}\\ &=-\frac {\left (a+b x^2\right )^{11/2}}{21 a x^{21}}+\frac {10 b \left (a+b x^2\right )^{11/2}}{399 a^2 x^{19}}-\frac {80 b^2 \left (a+b x^2\right )^{11/2}}{6783 a^3 x^{17}}+\frac {32 b^3 \left (a+b x^2\right )^{11/2}}{6783 a^4 x^{15}}+\frac {\left (128 b^4\right ) \int \frac {\left (a+b x^2\right )^{9/2}}{x^{14}} \, dx}{6783 a^4}\\ &=-\frac {\left (a+b x^2\right )^{11/2}}{21 a x^{21}}+\frac {10 b \left (a+b x^2\right )^{11/2}}{399 a^2 x^{19}}-\frac {80 b^2 \left (a+b x^2\right )^{11/2}}{6783 a^3 x^{17}}+\frac {32 b^3 \left (a+b x^2\right )^{11/2}}{6783 a^4 x^{15}}-\frac {128 b^4 \left (a+b x^2\right )^{11/2}}{88179 a^5 x^{13}}-\frac {\left (256 b^5\right ) \int \frac {\left (a+b x^2\right )^{9/2}}{x^{12}} \, dx}{88179 a^5}\\ &=-\frac {\left (a+b x^2\right )^{11/2}}{21 a x^{21}}+\frac {10 b \left (a+b x^2\right )^{11/2}}{399 a^2 x^{19}}-\frac {80 b^2 \left (a+b x^2\right )^{11/2}}{6783 a^3 x^{17}}+\frac {32 b^3 \left (a+b x^2\right )^{11/2}}{6783 a^4 x^{15}}-\frac {128 b^4 \left (a+b x^2\right )^{11/2}}{88179 a^5 x^{13}}+\frac {256 b^5 \left (a+b x^2\right )^{11/2}}{969969 a^6 x^{11}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 75, normalized size = 0.54 \[ \frac {\left (a+b x^2\right )^{11/2} \left (-46189 a^5+24310 a^4 b x^2-11440 a^3 b^2 x^4+4576 a^2 b^3 x^6-1408 a b^4 x^8+256 b^5 x^{10}\right )}{969969 a^6 x^{21}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.81, size = 126, normalized size = 0.90 \[ \frac {{\left (256 \, b^{10} x^{20} - 128 \, a b^{9} x^{18} + 96 \, a^{2} b^{8} x^{16} - 80 \, a^{3} b^{7} x^{14} + 70 \, a^{4} b^{6} x^{12} - 63 \, a^{5} b^{5} x^{10} - 80773 \, a^{6} b^{4} x^{8} - 271414 \, a^{7} b^{3} x^{6} - 351780 \, a^{8} b^{2} x^{4} - 206635 \, a^{9} b x^{2} - 46189 \, a^{10}\right )} \sqrt {b x^{2} + a}}{969969 \, a^{6} x^{21}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.25, size = 436, normalized size = 3.11 \[ \frac {512 \, {\left (646646 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{30} b^{\frac {21}{2}} + 4157010 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{28} a b^{\frac {21}{2}} + 14549535 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{26} a^{2} b^{\frac {21}{2}} + 30715685 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{24} a^{3} b^{\frac {21}{2}} + 44618574 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{22} a^{4} b^{\frac {21}{2}} + 44265858 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{20} a^{5} b^{\frac {21}{2}} + 31009615 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{18} a^{6} b^{\frac {21}{2}} + 14346045 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} a^{7} b^{\frac {21}{2}} + 4273290 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} a^{8} b^{\frac {21}{2}} + 592382 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{9} b^{\frac {21}{2}} + 20349 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{10} b^{\frac {21}{2}} - 5985 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{11} b^{\frac {21}{2}} + 1330 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{12} b^{\frac {21}{2}} - 210 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{13} b^{\frac {21}{2}} + 21 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{14} b^{\frac {21}{2}} - a^{15} b^{\frac {21}{2}}\right )}}{969969 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{21}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 72, normalized size = 0.51 \[ -\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} \left (-256 b^{5} x^{10}+1408 a \,b^{4} x^{8}-4576 a^{2} b^{3} x^{6}+11440 a^{3} b^{2} x^{4}-24310 a^{4} b \,x^{2}+46189 a^{5}\right )}{969969 a^{6} x^{21}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.53, size = 116, normalized size = 0.83 \[ \frac {256 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{5}}{969969 \, a^{6} x^{11}} - \frac {128 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{4}}{88179 \, a^{5} x^{13}} + \frac {32 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{3}}{6783 \, a^{4} x^{15}} - \frac {80 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{2}}{6783 \, a^{3} x^{17}} + \frac {10 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b}{399 \, a^{2} x^{19}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}}}{21 \, a x^{21}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 9.63, size = 211, normalized size = 1.51 \[ \frac {10\,b^6\,\sqrt {b\,x^2+a}}{138567\,a^2\,x^9}-\frac {1049\,b^4\,\sqrt {b\,x^2+a}}{12597\,x^{13}}-\frac {1898\,a\,b^3\,\sqrt {b\,x^2+a}}{6783\,x^{15}}-\frac {85\,a^3\,b\,\sqrt {b\,x^2+a}}{399\,x^{19}}-\frac {3\,b^5\,\sqrt {b\,x^2+a}}{46189\,a\,x^{11}}-\frac {a^4\,\sqrt {b\,x^2+a}}{21\,x^{21}}-\frac {80\,b^7\,\sqrt {b\,x^2+a}}{969969\,a^3\,x^7}+\frac {32\,b^8\,\sqrt {b\,x^2+a}}{323323\,a^4\,x^5}-\frac {128\,b^9\,\sqrt {b\,x^2+a}}{969969\,a^5\,x^3}+\frac {256\,b^{10}\,\sqrt {b\,x^2+a}}{969969\,a^6\,x}-\frac {820\,a^2\,b^2\,\sqrt {b\,x^2+a}}{2261\,x^{17}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 7.80, size = 1540, normalized size = 11.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________