Optimal. Leaf size=64 \[ \frac {(b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b}}+\frac {c (b c-2 a d)}{a^2 x}-\frac {c^2}{3 a x^3} \]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {461, 205} \begin {gather*} \frac {c (b c-2 a d)}{a^2 x}+\frac {(b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b}}-\frac {c^2}{3 a x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 461
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^2}{x^4 \left (a+b x^2\right )} \, dx &=\int \left (\frac {c^2}{a x^4}+\frac {c (-b c+2 a d)}{a^2 x^2}+\frac {(-b c+a d)^2}{a^2 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac {c^2}{3 a x^3}+\frac {c (b c-2 a d)}{a^2 x}+\frac {(b c-a d)^2 \int \frac {1}{a+b x^2} \, dx}{a^2}\\ &=-\frac {c^2}{3 a x^3}+\frac {c (b c-2 a d)}{a^2 x}+\frac {(b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 66, normalized size = 1.03 \begin {gather*} \frac {(a d-b c)^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b}}-\frac {c (2 a d-b c)}{a^2 x}-\frac {c^2}{3 a x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c+d x^2\right )^2}{x^4 \left (a+b x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.89, size = 190, normalized size = 2.97 \begin {gather*} \left [-\frac {3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-a b} x^{3} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 2 \, a^{2} b c^{2} - 6 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d\right )} x^{2}}{6 \, a^{3} b x^{3}}, \frac {3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {a b} x^{3} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) - a^{2} b c^{2} + 3 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d\right )} x^{2}}{3 \, a^{3} b x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.33, size = 72, normalized size = 1.12 \begin {gather*} \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {3 \, b c^{2} x^{2} - 6 \, a c d x^{2} - a c^{2}}{3 \, a^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 98, normalized size = 1.53 \begin {gather*} -\frac {2 b c d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a}+\frac {b^{2} c^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{2}}+\frac {d^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}-\frac {2 c d}{a x}+\frac {b \,c^{2}}{a^{2} x}-\frac {c^{2}}{3 a \,x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.46, size = 70, normalized size = 1.09 \begin {gather*} \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} - \frac {a c^{2} - 3 \, {\left (b c^{2} - 2 \, a c d\right )} x^{2}}{3 \, a^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.13, size = 90, normalized size = 1.41 \begin {gather*} \frac {b\,c^2}{a^2\,x}-\frac {c^2}{3\,a\,x^3}+\frac {b^{3/2}\,c^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{a^{5/2}}+\frac {d^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {b}}-\frac {2\,c\,d}{a\,x}-\frac {2\,\sqrt {b}\,c\,d\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{a^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.67, size = 172, normalized size = 2.69 \begin {gather*} - \frac {\sqrt {- \frac {1}{a^{5} b}} \left (a d - b c\right )^{2} \log {\left (- \frac {a^{3} \sqrt {- \frac {1}{a^{5} b}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{a^{5} b}} \left (a d - b c\right )^{2} \log {\left (\frac {a^{3} \sqrt {- \frac {1}{a^{5} b}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} + \frac {- a c^{2} + x^{2} \left (- 6 a c d + 3 b c^{2}\right )}{3 a^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________