Optimal. Leaf size=61 \[ \frac {(b c-a d)^2 \log \left (a+b x^2\right )}{2 b^3}+\frac {d x^2 (b c-a d)}{2 b^2}+\frac {\left (c+d x^2\right )^2}{4 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {444, 43} \[ \frac {d x^2 (b c-a d)}{2 b^2}+\frac {(b c-a d)^2 \log \left (a+b x^2\right )}{2 b^3}+\frac {\left (c+d x^2\right )^2}{4 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 444
Rubi steps
\begin {align*} \int \frac {x \left (c+d x^2\right )^2}{a+b x^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(c+d x)^2}{a+b x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {d (b c-a d)}{b^2}+\frac {(b c-a d)^2}{b^2 (a+b x)}+\frac {d (c+d x)}{b}\right ) \, dx,x,x^2\right )\\ &=\frac {d (b c-a d) x^2}{2 b^2}+\frac {\left (c+d x^2\right )^2}{4 b}+\frac {(b c-a d)^2 \log \left (a+b x^2\right )}{2 b^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 49, normalized size = 0.80 \[ \frac {b d x^2 \left (-2 a d+4 b c+b d x^2\right )+2 (b c-a d)^2 \log \left (a+b x^2\right )}{4 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.48, size = 67, normalized size = 1.10 \[ \frac {b^{2} d^{2} x^{4} + 2 \, {\left (2 \, b^{2} c d - a b d^{2}\right )} x^{2} + 2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x^{2} + a\right )}{4 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.38, size = 67, normalized size = 1.10 \[ \frac {b d^{2} x^{4} + 4 \, b c d x^{2} - 2 \, a d^{2} x^{2}}{4 \, b^{2}} + \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 85, normalized size = 1.39 \[ \frac {d^{2} x^{4}}{4 b}-\frac {a \,d^{2} x^{2}}{2 b^{2}}+\frac {c d \,x^{2}}{b}+\frac {a^{2} d^{2} \ln \left (b \,x^{2}+a \right )}{2 b^{3}}-\frac {a c d \ln \left (b \,x^{2}+a \right )}{b^{2}}+\frac {c^{2} \ln \left (b \,x^{2}+a \right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.12, size = 66, normalized size = 1.08 \[ \frac {b d^{2} x^{4} + 2 \, {\left (2 \, b c d - a d^{2}\right )} x^{2}}{4 \, b^{2}} + \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.06, size = 68, normalized size = 1.11 \[ \frac {d^2\,x^4}{4\,b}-x^2\,\left (\frac {a\,d^2}{2\,b^2}-\frac {c\,d}{b}\right )+\frac {\ln \left (b\,x^2+a\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{2\,b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.37, size = 49, normalized size = 0.80 \[ x^{2} \left (- \frac {a d^{2}}{2 b^{2}} + \frac {c d}{b}\right ) + \frac {d^{2} x^{4}}{4 b} + \frac {\left (a d - b c\right )^{2} \log {\left (a + b x^{2} \right )}}{2 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________