Integrand size = 31, antiderivative size = 480 \[ \int \frac {(e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^3}{\left (a+b x^2\right )^3} \, dx=-\frac {d \left (A b \left (2 b^2 c^2 \left (3+2 m-m^2\right )+3 a b c d \left (3+4 m+m^2\right )-a^2 d^2 \left (15+8 m+m^2\right )\right )+a B \left (2 b^2 c^2 (1+m)^2-3 a b c d \left (15+8 m+m^2\right )+a^2 d^2 \left (35+12 m+m^2\right )\right )\right ) (e x)^{1+m}}{8 a^2 b^4 e (1+m)}-\frac {d^2 \left (A b (3+m) (b c (3-m)+a d (5+m))+a B \left (b c \left (3+4 m+m^2\right )-a d \left (35+12 m+m^2\right )\right )\right ) (e x)^{3+m}}{8 a^2 b^3 e^3 (3+m)}+\frac {(A b (b c (3-m)+a d (3+m))+a B (b c (1+m)-a d (7+m))) (e x)^{1+m} \left (c+d x^2\right )^2}{8 a^2 b^2 e \left (a+b x^2\right )}+\frac {(A b-a B) (e x)^{1+m} \left (c+d x^2\right )^3}{4 a b e \left (a+b x^2\right )^2}+\frac {(b c-a d) \left (A b \left (2 a b c d \left (3-2 m-m^2\right )+b^2 c^2 \left (3-4 m+m^2\right )+a^2 d^2 \left (15+8 m+m^2\right )\right )+a B \left (b^2 c^2 \left (1-m^2\right )+2 a b c d \left (5+6 m+m^2\right )-a^2 d^2 \left (35+12 m+m^2\right )\right )\right ) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{8 a^3 b^4 e (1+m)} \]
[Out]
Time = 0.67 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {591, 584, 371} \[ \int \frac {(e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^3}{\left (a+b x^2\right )^3} \, dx=-\frac {d^2 (e x)^{m+3} \left (A b (m+3) (a d (m+5)+b c (3-m))+a B \left (b c \left (m^2+4 m+3\right )-a d \left (m^2+12 m+35\right )\right )\right )}{8 a^2 b^3 e^3 (m+3)}+\frac {\left (c+d x^2\right )^2 (e x)^{m+1} (A b (a d (m+3)+b c (3-m))+a B (b c (m+1)-a d (m+7)))}{8 a^2 b^2 e \left (a+b x^2\right )}-\frac {d (e x)^{m+1} \left (A b \left (-a^2 d^2 \left (m^2+8 m+15\right )+3 a b c d \left (m^2+4 m+3\right )+2 b^2 c^2 \left (-m^2+2 m+3\right )\right )+a B \left (a^2 d^2 \left (m^2+12 m+35\right )-3 a b c d \left (m^2+8 m+15\right )+2 b^2 c^2 (m+1)^2\right )\right )}{8 a^2 b^4 e (m+1)}+\frac {(e x)^{m+1} (b c-a d) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\frac {b x^2}{a}\right ) \left (A b \left (a^2 d^2 \left (m^2+8 m+15\right )+2 a b c d \left (-m^2-2 m+3\right )+b^2 c^2 \left (m^2-4 m+3\right )\right )+a B \left (-a^2 d^2 \left (m^2+12 m+35\right )+2 a b c d \left (m^2+6 m+5\right )+b^2 c^2 \left (1-m^2\right )\right )\right )}{8 a^3 b^4 e (m+1)}+\frac {\left (c+d x^2\right )^3 (e x)^{m+1} (A b-a B)}{4 a b e \left (a+b x^2\right )^2} \]
[In]
[Out]
Rule 371
Rule 584
Rule 591
Rubi steps \begin{align*} \text {integral}& = \frac {(A b-a B) (e x)^{1+m} \left (c+d x^2\right )^3}{4 a b e \left (a+b x^2\right )^2}-\frac {\int \frac {(e x)^m \left (c+d x^2\right )^2 \left (-c (A b (3-m)+a B (1+m))+d (A b (3+m)-a B (7+m)) x^2\right )}{\left (a+b x^2\right )^2} \, dx}{4 a b} \\ & = \frac {(A b (b c (3-m)+a d (3+m))+a B (b c (1+m)-a d (7+m))) (e x)^{1+m} \left (c+d x^2\right )^2}{8 a^2 b^2 e \left (a+b x^2\right )}+\frac {(A b-a B) (e x)^{1+m} \left (c+d x^2\right )^3}{4 a b e \left (a+b x^2\right )^2}+\frac {\int \frac {(e x)^m \left (c+d x^2\right ) \left (c \left (a B (1+m) (a d (7+m)+b (c-c m))+A b \left (b c \left (3-4 m+m^2\right )-a d \left (3+4 m+m^2\right )\right )\right )-d \left (A b (3+m) (b c (3-m)+a d (5+m))+a B \left (b c \left (3+4 m+m^2\right )-a d \left (35+12 m+m^2\right )\right )\right ) x^2\right )}{a+b x^2} \, dx}{8 a^2 b^2} \\ & = \frac {(A b (b c (3-m)+a d (3+m))+a B (b c (1+m)-a d (7+m))) (e x)^{1+m} \left (c+d x^2\right )^2}{8 a^2 b^2 e \left (a+b x^2\right )}+\frac {(A b-a B) (e x)^{1+m} \left (c+d x^2\right )^3}{4 a b e \left (a+b x^2\right )^2}+\frac {\int \left (-\frac {d \left (A b \left (2 b^2 c^2 \left (3+2 m-m^2\right )+3 a b c d \left (3+4 m+m^2\right )-a^2 d^2 \left (15+8 m+m^2\right )\right )+a B \left (2 b^2 c^2 (1+m)^2-3 a b c d \left (15+8 m+m^2\right )+a^2 d^2 \left (35+12 m+m^2\right )\right )\right ) (e x)^m}{b^2}-\frac {d^2 \left (A b (3+m) (b c (3-m)+a d (5+m))+a B \left (b c \left (3+4 m+m^2\right )-a d \left (35+12 m+m^2\right )\right )\right ) (e x)^{2+m}}{b e^2}+\frac {\left (3 A b^4 c^3+a b^3 B c^3+3 a A b^3 c^2 d+9 a^2 b^2 B c^2 d+9 a^2 A b^2 c d^2-45 a^3 b B c d^2-15 a^3 A b d^3+35 a^4 B d^3-4 A b^4 c^3 m+12 a^2 b^2 B c^2 d m+12 a^2 A b^2 c d^2 m-24 a^3 b B c d^2 m-8 a^3 A b d^3 m+12 a^4 B d^3 m+A b^4 c^3 m^2-a b^3 B c^3 m^2-3 a A b^3 c^2 d m^2+3 a^2 b^2 B c^2 d m^2+3 a^2 A b^2 c d^2 m^2-3 a^3 b B c d^2 m^2-a^3 A b d^3 m^2+a^4 B d^3 m^2\right ) (e x)^m}{b^2 \left (a+b x^2\right )}\right ) \, dx}{8 a^2 b^2} \\ & = -\frac {d \left (A b \left (2 b^2 c^2 \left (3+2 m-m^2\right )+3 a b c d \left (3+4 m+m^2\right )-a^2 d^2 \left (15+8 m+m^2\right )\right )+a B \left (2 b^2 c^2 (1+m)^2-3 a b c d \left (15+8 m+m^2\right )+a^2 d^2 \left (35+12 m+m^2\right )\right )\right ) (e x)^{1+m}}{8 a^2 b^4 e (1+m)}-\frac {d^2 \left (A b (3+m) (b c (3-m)+a d (5+m))+a B \left (b c \left (3+4 m+m^2\right )-a d \left (35+12 m+m^2\right )\right )\right ) (e x)^{3+m}}{8 a^2 b^3 e^3 (3+m)}+\frac {(A b (b c (3-m)+a d (3+m))+a B (b c (1+m)-a d (7+m))) (e x)^{1+m} \left (c+d x^2\right )^2}{8 a^2 b^2 e \left (a+b x^2\right )}+\frac {(A b-a B) (e x)^{1+m} \left (c+d x^2\right )^3}{4 a b e \left (a+b x^2\right )^2}+\frac {\left ((b c-a d) \left (A b \left (2 a b c d \left (3-2 m-m^2\right )+b^2 c^2 \left (3-4 m+m^2\right )+a^2 d^2 \left (15+8 m+m^2\right )\right )+a B \left (b^2 c^2 \left (1-m^2\right )+2 a b c d \left (5+6 m+m^2\right )-a^2 d^2 \left (35+12 m+m^2\right )\right )\right )\right ) \int \frac {(e x)^m}{a+b x^2} \, dx}{8 a^2 b^4} \\ & = -\frac {d \left (A b \left (2 b^2 c^2 \left (3+2 m-m^2\right )+3 a b c d \left (3+4 m+m^2\right )-a^2 d^2 \left (15+8 m+m^2\right )\right )+a B \left (2 b^2 c^2 (1+m)^2-3 a b c d \left (15+8 m+m^2\right )+a^2 d^2 \left (35+12 m+m^2\right )\right )\right ) (e x)^{1+m}}{8 a^2 b^4 e (1+m)}-\frac {d^2 \left (A b (3+m) (b c (3-m)+a d (5+m))+a B \left (b c \left (3+4 m+m^2\right )-a d \left (35+12 m+m^2\right )\right )\right ) (e x)^{3+m}}{8 a^2 b^3 e^3 (3+m)}+\frac {(A b (b c (3-m)+a d (3+m))+a B (b c (1+m)-a d (7+m))) (e x)^{1+m} \left (c+d x^2\right )^2}{8 a^2 b^2 e \left (a+b x^2\right )}+\frac {(A b-a B) (e x)^{1+m} \left (c+d x^2\right )^3}{4 a b e \left (a+b x^2\right )^2}+\frac {(b c-a d) \left (A b \left (2 a b c d \left (3-2 m-m^2\right )+b^2 c^2 \left (3-4 m+m^2\right )+a^2 d^2 \left (15+8 m+m^2\right )\right )+a B \left (b^2 c^2 \left (1-m^2\right )+2 a b c d \left (5+6 m+m^2\right )-a^2 d^2 \left (35+12 m+m^2\right )\right )\right ) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{8 a^3 b^4 e (1+m)} \\ \end{align*}
Time = 0.79 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.45 \[ \int \frac {(e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^3}{\left (a+b x^2\right )^3} \, dx=\frac {x (e x)^m \left (\frac {d^2 (3 b B c+A b d-3 a B d)}{1+m}+\frac {b B d^3 x^2}{3+m}+\frac {3 d (b c-a d) (b B c+A b d-2 a B d) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{a (1+m)}+\frac {(b c-a d)^2 (b B c+3 A b d-4 a B d) \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{a^2 (1+m)}+\frac {(-A b+a B) (-b c+a d)^3 \operatorname {Hypergeometric2F1}\left (3,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{a^3 (1+m)}\right )}{b^4} \]
[In]
[Out]
\[\int \frac {\left (e x \right )^{m} \left (x^{2} B +A \right ) \left (d \,x^{2}+c \right )^{3}}{\left (b \,x^{2}+a \right )^{3}}d x\]
[In]
[Out]
\[ \int \frac {(e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^3}{\left (a+b x^2\right )^3} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (d x^{2} + c\right )}^{3} \left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {(e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^3}{\left (a+b x^2\right )^3} \, dx=\int \frac {\left (e x\right )^{m} \left (A + B x^{2}\right ) \left (c + d x^{2}\right )^{3}}{\left (a + b x^{2}\right )^{3}}\, dx \]
[In]
[Out]
\[ \int \frac {(e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^3}{\left (a+b x^2\right )^3} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (d x^{2} + c\right )}^{3} \left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {(e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^3}{\left (a+b x^2\right )^3} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (d x^{2} + c\right )}^{3} \left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^3}{\left (a+b x^2\right )^3} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (e\,x\right )}^m\,{\left (d\,x^2+c\right )}^3}{{\left (b\,x^2+a\right )}^3} \,d x \]
[In]
[Out]