Optimal. Leaf size=120 \[ \frac {(e x)^{m+1} (b c-a d) (B c-A d) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {d x^2}{c}\right )}{c d^2 e (m+1)}-\frac {(e x)^{m+1} (-a B d-A b d+b B c)}{d^2 e (m+1)}+\frac {b B (e x)^{m+3}}{d e^3 (m+3)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {570, 364} \[ \frac {(e x)^{m+1} (b c-a d) (B c-A d) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {d x^2}{c}\right )}{c d^2 e (m+1)}-\frac {(e x)^{m+1} (-a B d-A b d+b B c)}{d^2 e (m+1)}+\frac {b B (e x)^{m+3}}{d e^3 (m+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 364
Rule 570
Rubi steps
\begin {align*} \int \frac {(e x)^m \left (a+b x^2\right ) \left (A+B x^2\right )}{c+d x^2} \, dx &=\int \left (-\frac {(b B c-A b d-a B d) (e x)^m}{d^2}+\frac {b B (e x)^{2+m}}{d e^2}+\frac {\left (b B c^2-A b c d-a B c d+a A d^2\right ) (e x)^m}{d^2 \left (c+d x^2\right )}\right ) \, dx\\ &=-\frac {(b B c-A b d-a B d) (e x)^{1+m}}{d^2 e (1+m)}+\frac {b B (e x)^{3+m}}{d e^3 (3+m)}+\frac {((b c-a d) (B c-A d)) \int \frac {(e x)^m}{c+d x^2} \, dx}{d^2}\\ &=-\frac {(b B c-A b d-a B d) (e x)^{1+m}}{d^2 e (1+m)}+\frac {b B (e x)^{3+m}}{d e^3 (3+m)}+\frac {(b c-a d) (B c-A d) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {d x^2}{c}\right )}{c d^2 e (1+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 93, normalized size = 0.78 \[ \frac {x (e x)^m \left (\frac {(b c-a d) (B c-A d) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {d x^2}{c}\right )}{c (m+1)}+\frac {a B d+A b d-b B c}{m+1}+\frac {b B d x^2}{m+3}\right )}{d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B b x^{4} + {\left (B a + A b\right )} x^{2} + A a\right )} \left (e x\right )^{m}}{d x^{2} + c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x^{2} + A\right )} {\left (b x^{2} + a\right )} \left (e x\right )^{m}}{d x^{2} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{2}+a \right ) \left (B \,x^{2}+A \right ) \left (e x \right )^{m}}{d \,x^{2}+c}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x^{2} + A\right )} {\left (b x^{2} + a\right )} \left (e x\right )^{m}}{d x^{2} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (B\,x^2+A\right )\,{\left (e\,x\right )}^m\,\left (b\,x^2+a\right )}{d\,x^2+c} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 13.66, size = 428, normalized size = 3.57 \[ \frac {A a e^{m} m x x^{m} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{4 c \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {A a e^{m} x x^{m} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{4 c \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {A b e^{m} m x^{3} x^{m} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{4 c \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {3 A b e^{m} x^{3} x^{m} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{4 c \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {B a e^{m} m x^{3} x^{m} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{4 c \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {3 B a e^{m} x^{3} x^{m} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{4 c \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {B b e^{m} m x^{5} x^{m} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {5}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )}{4 c \Gamma \left (\frac {m}{2} + \frac {7}{2}\right )} + \frac {5 B b e^{m} x^{5} x^{m} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {5}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )}{4 c \Gamma \left (\frac {m}{2} + \frac {7}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________