Optimal. Leaf size=62 \[ -\frac {2 a^2}{5 x^{5/2}}+\frac {2}{3} x^{3/2} \left (2 a c+b^2\right )-\frac {4 a b}{\sqrt {x}}+\frac {4}{7} b c x^{7/2}+\frac {2}{11} c^2 x^{11/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1108} \[ -\frac {2 a^2}{5 x^{5/2}}+\frac {2}{3} x^{3/2} \left (2 a c+b^2\right )-\frac {4 a b}{\sqrt {x}}+\frac {4}{7} b c x^{7/2}+\frac {2}{11} c^2 x^{11/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1108
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2+c x^4\right )^2}{x^{7/2}} \, dx &=\int \left (\frac {a^2}{x^{7/2}}+\frac {2 a b}{x^{3/2}}+\left (b^2+2 a c\right ) \sqrt {x}+2 b c x^{5/2}+c^2 x^{9/2}\right ) \, dx\\ &=-\frac {2 a^2}{5 x^{5/2}}-\frac {4 a b}{\sqrt {x}}+\frac {2}{3} \left (b^2+2 a c\right ) x^{3/2}+\frac {4}{7} b c x^{7/2}+\frac {2}{11} c^2 x^{11/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 50, normalized size = 0.81 \[ \frac {2 \left (-231 a^2+385 x^4 \left (2 a c+b^2\right )-2310 a b x^2+330 b c x^6+105 c^2 x^8\right )}{1155 x^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.78, size = 46, normalized size = 0.74 \[ \frac {2 \, {\left (105 \, c^{2} x^{8} + 330 \, b c x^{6} + 385 \, {\left (b^{2} + 2 \, a c\right )} x^{4} - 2310 \, a b x^{2} - 231 \, a^{2}\right )}}{1155 \, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 47, normalized size = 0.76 \[ \frac {2}{11} \, c^{2} x^{\frac {11}{2}} + \frac {4}{7} \, b c x^{\frac {7}{2}} + \frac {2}{3} \, b^{2} x^{\frac {3}{2}} + \frac {4}{3} \, a c x^{\frac {3}{2}} - \frac {2 \, {\left (10 \, a b x^{2} + a^{2}\right )}}{5 \, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 49, normalized size = 0.79 \[ -\frac {2 \left (-105 c^{2} x^{8}-330 b c \,x^{6}-770 a c \,x^{4}-385 b^{2} x^{4}+2310 a b \,x^{2}+231 a^{2}\right )}{1155 x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.12, size = 45, normalized size = 0.73 \[ \frac {2}{11} \, c^{2} x^{\frac {11}{2}} + \frac {4}{7} \, b c x^{\frac {7}{2}} + \frac {2}{3} \, {\left (b^{2} + 2 \, a c\right )} x^{\frac {3}{2}} - \frac {2 \, {\left (10 \, a b x^{2} + a^{2}\right )}}{5 \, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 48, normalized size = 0.77 \[ x^{3/2}\,\left (\frac {2\,b^2}{3}+\frac {4\,a\,c}{3}\right )-\frac {\frac {2\,a^2}{5}+4\,b\,a\,x^2}{x^{5/2}}+\frac {2\,c^2\,x^{11/2}}{11}+\frac {4\,b\,c\,x^{7/2}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 9.11, size = 68, normalized size = 1.10 \[ - \frac {2 a^{2}}{5 x^{\frac {5}{2}}} - \frac {4 a b}{\sqrt {x}} + \frac {4 a c x^{\frac {3}{2}}}{3} + \frac {2 b^{2} x^{\frac {3}{2}}}{3} + \frac {4 b c x^{\frac {7}{2}}}{7} + \frac {2 c^{2} x^{\frac {11}{2}}}{11} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________