Optimal. Leaf size=94 \[ \frac{x^{m+1} (b c-a d)^2 \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{c d^2 (m+1)}-\frac{b x^{m+1} (b c-2 a d)}{d^2 (m+1)}+\frac{b^2 x^{m+3}}{d (m+3)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0597319, antiderivative size = 94, normalized size of antiderivative = 1., 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, 364} \[ \frac{x^{m+1} (b c-a d)^2 \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{c d^2 (m+1)}-\frac{b x^{m+1} (b c-2 a d)}{d^2 (m+1)}+\frac{b^2 x^{m+3}}{d (m+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 461
Rule 364
Rubi steps
\begin{align*} \int \frac{x^m \left (a+b x^2\right )^2}{c+d x^2} \, dx &=\int \left (-\frac{b (b c-2 a d) x^m}{d^2}+\frac{b^2 x^{2+m}}{d}+\frac{\left (b^2 c^2-2 a b c d+a^2 d^2\right ) x^m}{d^2 \left (c+d x^2\right )}\right ) \, dx\\ &=-\frac{b (b c-2 a d) x^{1+m}}{d^2 (1+m)}+\frac{b^2 x^{3+m}}{d (3+m)}+\frac{(b c-a d)^2 \int \frac{x^m}{c+d x^2} \, dx}{d^2}\\ &=-\frac{b (b c-2 a d) x^{1+m}}{d^2 (1+m)}+\frac{b^2 x^{3+m}}{d (3+m)}+\frac{(b c-a d)^2 x^{1+m} \, _2F_1\left (1,\frac{1+m}{2};\frac{3+m}{2};-\frac{d x^2}{c}\right )}{c d^2 (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0971034, size = 118, normalized size = 1.26 \[ \frac{x^{m+1} \left (\frac{a^2 \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{m+1}+b x^2 \left (\frac{2 a \, _2F_1\left (1,\frac{m+3}{2};\frac{m+5}{2};-\frac{d x^2}{c}\right )}{m+3}+\frac{b x^2 \, _2F_1\left (1,\frac{m+5}{2};\frac{m+7}{2};-\frac{d x^2}{c}\right )}{m+5}\right )\right )}{c} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( b{x}^{2}+a \right ) ^{2}{x}^{m}}{d{x}^{2}+c}}\, 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}}{d x^{2} + c}\,{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 x^{2} + c}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 9.36321, size = 299, normalized size = 3.18 \begin{align*} \frac{a^{2} m x x^{m} \Phi \left (\frac{d x^{2} e^{i \pi }}{c}, 1, \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )}{4 c \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{a^{2} x x^{m} \Phi \left (\frac{d x^{2} e^{i \pi }}{c}, 1, \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )}{4 c \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{a b m x^{3} x^{m} \Phi \left (\frac{d x^{2} e^{i \pi }}{c}, 1, \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )}{2 c \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )} + \frac{3 a b x^{3} x^{m} \Phi \left (\frac{d x^{2} e^{i \pi }}{c}, 1, \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )}{2 c \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )} + \frac{b^{2} m x^{5} x^{m} \Phi \left (\frac{d x^{2} e^{i \pi }}{c}, 1, \frac{m}{2} + \frac{5}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )}{4 c \Gamma \left (\frac{m}{2} + \frac{7}{2}\right )} + \frac{5 b^{2} x^{5} x^{m} \Phi \left (\frac{d x^{2} e^{i \pi }}{c}, 1, \frac{m}{2} + \frac{5}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )}{4 c \Gamma \left (\frac{m}{2} + \frac{7}{2}\right )} \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}}{d x^{2} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]