Integrand size = 22, antiderivative size = 146 \[ \int x^3 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\frac {\left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^{1+q}}{2 b d (2+p+q)}-\frac {(b c (1+p)+a d (1+q)) \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^q \left (\frac {b \left (c+d x^2\right )}{b c-a d}\right )^{-q} \operatorname {Hypergeometric2F1}\left (1+p,-q,2+p,-\frac {d \left (a+b x^2\right )}{b c-a d}\right )}{2 b^2 d (1+p) (2+p+q)} \]
1/2*(b*x^2+a)^(p+1)*(d*x^2+c)^(1+q)/b/d/(2+p+q)-1/2*(b*c*(p+1)+a*d*(1+q))* (b*x^2+a)^(p+1)*(d*x^2+c)^q*hypergeom([-q, p+1],[2+p],-d*(b*x^2+a)/(-a*d+b *c))/b^2/d/(p+1)/(2+p+q)/((b*(d*x^2+c)/(-a*d+b*c))^q)
Time = 0.13 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.81 \[ \int x^3 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\frac {\left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^q \left (b \left (c+d x^2\right )-\frac {(b c (1+p)+a d (1+q)) \left (\frac {b \left (c+d x^2\right )}{b c-a d}\right )^{-q} \operatorname {Hypergeometric2F1}\left (1+p,-q,2+p,\frac {d \left (a+b x^2\right )}{-b c+a d}\right )}{1+p}\right )}{2 b^2 d (2+p+q)} \]
((a + b*x^2)^(1 + p)*(c + d*x^2)^q*(b*(c + d*x^2) - ((b*c*(1 + p) + a*d*(1 + q))*Hypergeometric2F1[1 + p, -q, 2 + p, (d*(a + b*x^2))/(-(b*c) + a*d)] )/((1 + p)*((b*(c + d*x^2))/(b*c - a*d))^q)))/(2*b^2*d*(2 + p + q))
Time = 0.24 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {354, 90, 80, 79}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^3 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {1}{2} \int x^2 \left (b x^2+a\right )^p \left (d x^2+c\right )^qdx^2\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {1}{2} \left (\frac {\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{b d (p+q+2)}-\frac {(a d (q+1)+b c (p+1)) \int \left (b x^2+a\right )^p \left (d x^2+c\right )^qdx^2}{b d (p+q+2)}\right )\) |
\(\Big \downarrow \) 80 |
\(\displaystyle \frac {1}{2} \left (\frac {\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{b d (p+q+2)}-\frac {\left (c+d x^2\right )^q (a d (q+1)+b c (p+1)) \left (\frac {b \left (c+d x^2\right )}{b c-a d}\right )^{-q} \int \left (b x^2+a\right )^p \left (\frac {b d x^2}{b c-a d}+\frac {b c}{b c-a d}\right )^qdx^2}{b d (p+q+2)}\right )\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {1}{2} \left (\frac {\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{b d (p+q+2)}-\frac {\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^q (a d (q+1)+b c (p+1)) \left (\frac {b \left (c+d x^2\right )}{b c-a d}\right )^{-q} \operatorname {Hypergeometric2F1}\left (p+1,-q,p+2,-\frac {d \left (b x^2+a\right )}{b c-a d}\right )}{b^2 d (p+1) (p+q+2)}\right )\) |
(((a + b*x^2)^(1 + p)*(c + d*x^2)^(1 + q))/(b*d*(2 + p + q)) - ((b*c*(1 + p) + a*d*(1 + q))*(a + b*x^2)^(1 + p)*(c + d*x^2)^q*Hypergeometric2F1[1 + p, -q, 2 + p, -((d*(a + b*x^2))/(b*c - a*d))])/(b^2*d*(1 + p)*(2 + p + q)* ((b*(c + d*x^2))/(b*c - a*d))^q))/2
3.12.46.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) ^FracPart[n]) Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) ), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !Integ erQ[n] && (RationalQ[m] || !SimplerQ[n + 1, m + 1])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
\[\int x^{3} \left (b \,x^{2}+a \right )^{p} \left (d \,x^{2}+c \right )^{q}d x\]
\[ \int x^3 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\int { {\left (b x^{2} + a\right )}^{p} {\left (d x^{2} + c\right )}^{q} x^{3} \,d x } \]
Timed out. \[ \int x^3 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\text {Timed out} \]
\[ \int x^3 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\int { {\left (b x^{2} + a\right )}^{p} {\left (d x^{2} + c\right )}^{q} x^{3} \,d x } \]
\[ \int x^3 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\int { {\left (b x^{2} + a\right )}^{p} {\left (d x^{2} + c\right )}^{q} x^{3} \,d x } \]
Timed out. \[ \int x^3 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\int x^3\,{\left (b\,x^2+a\right )}^p\,{\left (d\,x^2+c\right )}^q \,d x \]