Optimal. Leaf size=120 \[ -\frac{x^{m+1} (b c-a d) (a d (1-m)+b c (m+3)) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{2 c^2 d^2 (m+1)}+\frac{x^{m+1} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac{b^2 x^{m+1}}{d^2 (m+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.103223, antiderivative size = 120, normalized size of antiderivative = 1., 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 = {463, 459, 364} \[ -\frac{x^{m+1} (b c-a d) (a d (1-m)+b c (m+3)) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{2 c^2 d^2 (m+1)}+\frac{x^{m+1} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac{b^2 x^{m+1}}{d^2 (m+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 463
Rule 459
Rule 364
Rubi steps
\begin{align*} \int \frac{x^m \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx &=\frac{(b c-a d)^2 x^{1+m}}{2 c d^2 \left (c+d x^2\right )}-\frac{\int \frac{x^m \left (-2 a^2 d^2+(b c-a d)^2 (1+m)-2 b^2 c d x^2\right )}{c+d x^2} \, dx}{2 c d^2}\\ &=\frac{b^2 x^{1+m}}{d^2 (1+m)}+\frac{(b c-a d)^2 x^{1+m}}{2 c d^2 \left (c+d x^2\right )}-\frac{((b c-a d) (a d (1-m)+b c (3+m))) \int \frac{x^m}{c+d x^2} \, dx}{2 c d^2}\\ &=\frac{b^2 x^{1+m}}{d^2 (1+m)}+\frac{(b c-a d)^2 x^{1+m}}{2 c d^2 \left (c+d x^2\right )}-\frac{(b c-a d) (a d (1-m)+b c (3+m)) x^{1+m} \, _2F_1\left (1,\frac{1+m}{2};\frac{3+m}{2};-\frac{d x^2}{c}\right )}{2 c^2 d^2 (1+m)}\\ \end{align*}
Mathematica [A] time = 0.108257, size = 118, normalized size = 0.98 \[ \frac{x^{m+1} \left (\frac{a^2 \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{m+1}+b x^2 \left (\frac{2 a \, _2F_1\left (2,\frac{m+3}{2};\frac{m+5}{2};-\frac{d x^2}{c}\right )}{m+3}+\frac{b x^2 \, _2F_1\left (2,\frac{m+5}{2};\frac{m+7}{2};-\frac{d x^2}{c}\right )}{m+5}\right )\right )}{c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.048, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( b{x}^{2}+a \right ) ^{2}{x}^{m}}{ \left ( d{x}^{2}+c \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}^{2} x^{m}}{{\left (d x^{2} + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} x^{m}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m} \left (a + b x^{2}\right )^{2}}{\left (c + d x^{2}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}^{2} x^{m}}{{\left (d x^{2} + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]