Integrand size = 22, antiderivative size = 242 \[ \int x^5 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=-\frac {(b c (2+p)+a d (2+q)) \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^{1+q}}{2 b^2 d^2 (2+p+q) (3+p+q)}+\frac {x^2 \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^{1+q}}{2 b d (3+p+q)}+\frac {\left (b^2 c^2 \left (2+3 p+p^2\right )+2 a b c d (1+p) (1+q)+a^2 d^2 \left (2+3 q+q^2\right )\right ) \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^q \left (\frac {b \left (c+d x^2\right )}{b c-a d}\right )^{-q} \operatorname {Hypergeometric2F1}\left (1+p,-q,2+p,-\frac {d \left (a+b x^2\right )}{b c-a d}\right )}{2 b^3 d^2 (1+p) (2+p+q) (3+p+q)} \]
[Out]
Time = 0.21 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {457, 92, 81, 72, 71} \[ \int x^5 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\frac {\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^q \left (a^2 d^2 \left (q^2+3 q+2\right )+2 a b c d (p+1) (q+1)+b^2 c^2 \left (p^2+3 p+2\right )\right ) \left (\frac {b \left (c+d x^2\right )}{b c-a d}\right )^{-q} \operatorname {Hypergeometric2F1}\left (p+1,-q,p+2,-\frac {d \left (b x^2+a\right )}{b c-a d}\right )}{2 b^3 d^2 (p+1) (p+q+2) (p+q+3)}-\frac {\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (a d (q+2)+b c (p+2))}{2 b^2 d^2 (p+q+2) (p+q+3)}+\frac {x^2 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{2 b d (p+q+3)} \]
[In]
[Out]
Rule 71
Rule 72
Rule 81
Rule 92
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int x^2 (a+b x)^p (c+d x)^q \, dx,x,x^2\right ) \\ & = \frac {x^2 \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^{1+q}}{2 b d (3+p+q)}+\frac {\text {Subst}\left (\int (a+b x)^p (c+d x)^q (-a c-(b c (2+p)+a d (2+q)) x) \, dx,x,x^2\right )}{2 b d (3+p+q)} \\ & = -\frac {(b c (2+p)+a d (2+q)) \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^{1+q}}{2 b^2 d^2 (2+p+q) (3+p+q)}+\frac {x^2 \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^{1+q}}{2 b d (3+p+q)}+\frac {\left (b^2 c^2 \left (2+3 p+p^2\right )+2 a b c d (1+p) (1+q)+a^2 d^2 \left (2+3 q+q^2\right )\right ) \text {Subst}\left (\int (a+b x)^p (c+d x)^q \, dx,x,x^2\right )}{2 b^2 d^2 (2+p+q) (3+p+q)} \\ & = -\frac {(b c (2+p)+a d (2+q)) \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^{1+q}}{2 b^2 d^2 (2+p+q) (3+p+q)}+\frac {x^2 \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^{1+q}}{2 b d (3+p+q)}+\frac {\left (\left (b^2 c^2 \left (2+3 p+p^2\right )+2 a b c d (1+p) (1+q)+a^2 d^2 \left (2+3 q+q^2\right )\right ) \left (c+d x^2\right )^q \left (\frac {b \left (c+d x^2\right )}{b c-a d}\right )^{-q}\right ) \text {Subst}\left (\int (a+b x)^p \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^q \, dx,x,x^2\right )}{2 b^2 d^2 (2+p+q) (3+p+q)} \\ & = -\frac {(b c (2+p)+a d (2+q)) \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^{1+q}}{2 b^2 d^2 (2+p+q) (3+p+q)}+\frac {x^2 \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^{1+q}}{2 b d (3+p+q)}+\frac {\left (b^2 c^2 \left (2+3 p+p^2\right )+2 a b c d (1+p) (1+q)+a^2 d^2 \left (2+3 q+q^2\right )\right ) \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^q \left (\frac {b \left (c+d x^2\right )}{b c-a d}\right )^{-q} \, _2F_1\left (1+p,-q;2+p;-\frac {d \left (a+b x^2\right )}{b c-a d}\right )}{2 b^3 d^2 (1+p) (2+p+q) (3+p+q)} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.81 \[ \int x^5 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\frac {\left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^q \left (-\frac {(b c (2+p)+a d (2+q)) \left (c+d x^2\right )}{b d (2+p+q)}+x^2 \left (c+d x^2\right )+\frac {\left (b^2 c^2 \left (2+3 p+p^2\right )+2 a b c d (1+p) (1+q)+a^2 d^2 \left (2+3 q+q^2\right )\right ) \left (\frac {b \left (c+d x^2\right )}{b c-a d}\right )^{-q} \operatorname {Hypergeometric2F1}\left (1+p,-q,2+p,\frac {d \left (a+b x^2\right )}{-b c+a d}\right )}{b^2 d (1+p) (2+p+q)}\right )}{2 b d (3+p+q)} \]
[In]
[Out]
\[\int x^{5} \left (b \,x^{2}+a \right )^{p} \left (d \,x^{2}+c \right )^{q}d x\]
[In]
[Out]
\[ \int x^5 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\int { {\left (b x^{2} + a\right )}^{p} {\left (d x^{2} + c\right )}^{q} x^{5} \,d x } \]
[In]
[Out]
Timed out. \[ \int x^5 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int x^5 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\int { {\left (b x^{2} + a\right )}^{p} {\left (d x^{2} + c\right )}^{q} x^{5} \,d x } \]
[In]
[Out]
\[ \int x^5 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\int { {\left (b x^{2} + a\right )}^{p} {\left (d x^{2} + c\right )}^{q} x^{5} \,d x } \]
[In]
[Out]
Timed out. \[ \int x^5 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\int x^5\,{\left (b\,x^2+a\right )}^p\,{\left (d\,x^2+c\right )}^q \,d x \]
[In]
[Out]