Optimal. Leaf size=133 \[ \frac{d x^{m+1} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3 (m+1)}+\frac{d^2 x^{m+3} (3 b c-a d)}{b^2 (m+3)}+\frac{x^{m+1} (b c-a d)^3 \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a b^3 (m+1)}+\frac{d^3 x^{m+5}}{b (m+5)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0860909, antiderivative size = 133, 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} \[ \frac{d x^{m+1} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3 (m+1)}+\frac{d^2 x^{m+3} (3 b c-a d)}{b^2 (m+3)}+\frac{x^{m+1} (b c-a d)^3 \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a b^3 (m+1)}+\frac{d^3 x^{m+5}}{b (m+5)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 461
Rule 364
Rubi steps
\begin{align*} \int \frac{x^m \left (c+d x^2\right )^3}{a+b x^2} \, dx &=\int \left (\frac{d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^m}{b^3}+\frac{d^2 (3 b c-a d) x^{2+m}}{b^2}+\frac{d^3 x^{4+m}}{b}+\frac{\left (b^3 c^3-3 a b^2 c^2 d+3 a^2 b c d^2-a^3 d^3\right ) x^m}{b^3 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac{d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{1+m}}{b^3 (1+m)}+\frac{d^2 (3 b c-a d) x^{3+m}}{b^2 (3+m)}+\frac{d^3 x^{5+m}}{b (5+m)}+\frac{(b c-a d)^3 \int \frac{x^m}{a+b x^2} \, dx}{b^3}\\ &=\frac{d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{1+m}}{b^3 (1+m)}+\frac{d^2 (3 b c-a d) x^{3+m}}{b^2 (3+m)}+\frac{d^3 x^{5+m}}{b (5+m)}+\frac{(b c-a d)^3 x^{1+m} \, _2F_1\left (1,\frac{1+m}{2};\frac{3+m}{2};-\frac{b x^2}{a}\right )}{a b^3 (1+m)}\\ \end{align*}
Mathematica [C] time = 1.36008, size = 114, normalized size = 0.86 \[ \frac{x^{m+1} \left (d x^2 \left (3 c^2 \Phi \left (-\frac{b x^2}{a},1,\frac{m+3}{2}\right )+d x^2 \left (3 c \Phi \left (-\frac{b x^2}{a},1,\frac{m+5}{2}\right )+d x^2 \Phi \left (-\frac{b x^2}{a},1,\frac{m+7}{2}\right )\right )\right )+c^3 \Phi \left (-\frac{b x^2}{a},1,\frac{m+1}{2}\right )\right )}{2 a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( d{x}^{2}+c \right ) ^{3}{x}^{m}}{b{x}^{2}+a}}\, 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 (d x^{2} + c\right )}^{3} x^{m}}{b x^{2} + a}\,{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 (d^{3} x^{6} + 3 \, c d^{2} x^{4} + 3 \, c^{2} d x^{2} + c^{3}\right )} x^{m}}{b x^{2} + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 16.9858, size = 411, normalized size = 3.09 \begin{align*} \frac{c^{3} m x x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{c^{3} x x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{3 c^{2} d m x^{3} x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )} + \frac{9 c^{2} d x^{3} x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )} + \frac{3 c d^{2} m x^{5} x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{5}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{7}{2}\right )} + \frac{15 c d^{2} x^{5} x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{5}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{7}{2}\right )} + \frac{d^{3} m x^{7} x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{7}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{7}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{9}{2}\right )} + \frac{7 d^{3} x^{7} x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{7}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{7}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{9}{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 (d x^{2} + c\right )}^{3} x^{m}}{b x^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]