Integrand size = 24, antiderivative size = 103 \[ \int \frac {\left (a+b x^2\right )^p \left (c+d x^2\right )}{(e x)^{7/2}} \, dx=-\frac {2 c \left (a+b x^2\right )^{1+p}}{5 a e (e x)^{5/2}}-\frac {2 (5 a d-b c (1-4 p)) \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},-p,\frac {3}{4},-\frac {b x^2}{a}\right )}{5 a e^3 \sqrt {e x}} \] Output:
-2/5*c*(b*x^2+a)^(p+1)/a/e/(e*x)^(5/2)-2/5*(5*a*d-b*c*(1-4*p))*(b*x^2+a)^p *hypergeom([-1/4, -p],[3/4],-b*x^2/a)/a/e^3/(e*x)^(1/2)/((1+b*x^2/a)^p)
Time = 0.21 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^2\right )^p \left (c+d x^2\right )}{(e x)^{7/2}} \, dx=-\frac {2 x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (c \left (a+b x^2\right ) \left (1+\frac {b x^2}{a}\right )^p+(5 a d+b c (-1+4 p)) x^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},-p,\frac {3}{4},-\frac {b x^2}{a}\right )\right )}{5 a (e x)^{7/2}} \] Input:
Integrate[((a + b*x^2)^p*(c + d*x^2))/(e*x)^(7/2),x]
Output:
(-2*x*(a + b*x^2)^p*(c*(a + b*x^2)*(1 + (b*x^2)/a)^p + (5*a*d + b*c*(-1 + 4*p))*x^2*Hypergeometric2F1[-1/4, -p, 3/4, -((b*x^2)/a)]))/(5*a*(e*x)^(7/2 )*(1 + (b*x^2)/a)^p)
Time = 0.22 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {359, 279, 278}
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.
| \(\displaystyle \int \frac {\left (c+d x^2\right ) \left (a+b x^2\right )^p}{(e x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle \frac {(5 a d-b c (1-4 p)) \int \frac {\left (b x^2+a\right )^p}{(e x)^{3/2}}dx}{5 a e^2}-\frac {2 c \left (a+b x^2\right )^{p+1}}{5 a e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 279 |
| \(\displaystyle \frac {\left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} (5 a d-b c (1-4 p)) \int \frac {\left (\frac {b x^2}{a}+1\right )^p}{(e x)^{3/2}}dx}{5 a e^2}-\frac {2 c \left (a+b x^2\right )^{p+1}}{5 a e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle -\frac {2 \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} (5 a d-b c (1-4 p)) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},-p,\frac {3}{4},-\frac {b x^2}{a}\right )}{5 a e^3 \sqrt {e x}}-\frac {2 c \left (a+b x^2\right )^{p+1}}{5 a e (e x)^{5/2}}\) |
Input:
Int[((a + b*x^2)^p*(c + d*x^2))/(e*x)^(7/2),x]
Output:
(-2*c*(a + b*x^2)^(1 + p))/(5*a*e*(e*x)^(5/2)) - (2*(5*a*d - b*c*(1 - 4*p) )*(a + b*x^2)^p*Hypergeometric2F1[-1/4, -p, 3/4, -((b*x^2)/a)])/(5*a*e^3*S qrt[e*x]*(1 + (b*x^2)/a)^p)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p]) Int[(c*x)^m* (1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && !(ILtQ[p, 0] || GtQ[a, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
\[\int \frac {\left (b \,x^{2}+a \right )^{p} \left (x^{2} d +c \right )}{\left (e x \right )^{\frac {7}{2}}}d x\]
Input:
int((b*x^2+a)^p*(d*x^2+c)/(e*x)^(7/2),x)
Output:
int((b*x^2+a)^p*(d*x^2+c)/(e*x)^(7/2),x)
\[ \int \frac {\left (a+b x^2\right )^p \left (c+d x^2\right )}{(e x)^{7/2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )} {\left (b x^{2} + a\right )}^{p}}{\left (e x\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^p*(d*x^2+c)/(e*x)^(7/2),x, algorithm="fricas")
Output:
integral((d*x^2 + c)*sqrt(e*x)*(b*x^2 + a)^p/(e^4*x^4), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^p \left (c+d x^2\right )}{(e x)^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((b*x**2+a)**p*(d*x**2+c)/(e*x)**(7/2),x)
Output:
Timed out
\[ \int \frac {\left (a+b x^2\right )^p \left (c+d x^2\right )}{(e x)^{7/2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )} {\left (b x^{2} + a\right )}^{p}}{\left (e x\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^p*(d*x^2+c)/(e*x)^(7/2),x, algorithm="maxima")
Output:
integrate((d*x^2 + c)*(b*x^2 + a)^p/(e*x)^(7/2), x)
\[ \int \frac {\left (a+b x^2\right )^p \left (c+d x^2\right )}{(e x)^{7/2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )} {\left (b x^{2} + a\right )}^{p}}{\left (e x\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^p*(d*x^2+c)/(e*x)^(7/2),x, algorithm="giac")
Output:
integrate((d*x^2 + c)*(b*x^2 + a)^p/(e*x)^(7/2), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^p \left (c+d x^2\right )}{(e x)^{7/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^p\,\left (d\,x^2+c\right )}{{\left (e\,x\right )}^{7/2}} \,d x \] Input:
int(((a + b*x^2)^p*(c + d*x^2))/(e*x)^(7/2),x)
Output:
int(((a + b*x^2)^p*(c + d*x^2))/(e*x)^(7/2), x)
\[ \int \frac {\left (a+b x^2\right )^p \left (c+d x^2\right )}{(e x)^{7/2}} \, dx=\frac {2 \sqrt {e}\, \left (4 \left (b \,x^{2}+a \right )^{p} a d p +4 \left (b \,x^{2}+a \right )^{p} b c p -\left (b \,x^{2}+a \right )^{p} b c +4 \left (b \,x^{2}+a \right )^{p} b d p \,x^{2}-5 \left (b \,x^{2}+a \right )^{p} b d \,x^{2}+160 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \left (b \,x^{2}+a \right )^{p}}{16 b \,p^{2} x^{6}-24 b p \,x^{6}+16 a \,p^{2} x^{4}+5 b \,x^{6}-24 a p \,x^{4}+5 a \,x^{4}}d x \right ) a^{2} d \,p^{3} x^{2}-240 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \left (b \,x^{2}+a \right )^{p}}{16 b \,p^{2} x^{6}-24 b p \,x^{6}+16 a \,p^{2} x^{4}+5 b \,x^{6}-24 a p \,x^{4}+5 a \,x^{4}}d x \right ) a^{2} d \,p^{2} x^{2}+50 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \left (b \,x^{2}+a \right )^{p}}{16 b \,p^{2} x^{6}-24 b p \,x^{6}+16 a \,p^{2} x^{4}+5 b \,x^{6}-24 a p \,x^{4}+5 a \,x^{4}}d x \right ) a^{2} d p \,x^{2}+128 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \left (b \,x^{2}+a \right )^{p}}{16 b \,p^{2} x^{6}-24 b p \,x^{6}+16 a \,p^{2} x^{4}+5 b \,x^{6}-24 a p \,x^{4}+5 a \,x^{4}}d x \right ) a b c \,p^{4} x^{2}-224 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \left (b \,x^{2}+a \right )^{p}}{16 b \,p^{2} x^{6}-24 b p \,x^{6}+16 a \,p^{2} x^{4}+5 b \,x^{6}-24 a p \,x^{4}+5 a \,x^{4}}d x \right ) a b c \,p^{3} x^{2}+88 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \left (b \,x^{2}+a \right )^{p}}{16 b \,p^{2} x^{6}-24 b p \,x^{6}+16 a \,p^{2} x^{4}+5 b \,x^{6}-24 a p \,x^{4}+5 a \,x^{4}}d x \right ) a b c \,p^{2} x^{2}-10 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \left (b \,x^{2}+a \right )^{p}}{16 b \,p^{2} x^{6}-24 b p \,x^{6}+16 a \,p^{2} x^{4}+5 b \,x^{6}-24 a p \,x^{4}+5 a \,x^{4}}d x \right ) a b c p \,x^{2}\right )}{\sqrt {x}\, b \,e^{4} x^{2} \left (16 p^{2}-24 p +5\right )} \] Input:
int((b*x^2+a)^p*(d*x^2+c)/(e*x)^(7/2),x)
Output:
(2*sqrt(e)*(4*(a + b*x**2)**p*a*d*p + 4*(a + b*x**2)**p*b*c*p - (a + b*x** 2)**p*b*c + 4*(a + b*x**2)**p*b*d*p*x**2 - 5*(a + b*x**2)**p*b*d*x**2 + 16 0*sqrt(x)*int((sqrt(x)*(a + b*x**2)**p)/(16*a*p**2*x**4 - 24*a*p*x**4 + 5* a*x**4 + 16*b*p**2*x**6 - 24*b*p*x**6 + 5*b*x**6),x)*a**2*d*p**3*x**2 - 24 0*sqrt(x)*int((sqrt(x)*(a + b*x**2)**p)/(16*a*p**2*x**4 - 24*a*p*x**4 + 5* a*x**4 + 16*b*p**2*x**6 - 24*b*p*x**6 + 5*b*x**6),x)*a**2*d*p**2*x**2 + 50 *sqrt(x)*int((sqrt(x)*(a + b*x**2)**p)/(16*a*p**2*x**4 - 24*a*p*x**4 + 5*a *x**4 + 16*b*p**2*x**6 - 24*b*p*x**6 + 5*b*x**6),x)*a**2*d*p*x**2 + 128*sq rt(x)*int((sqrt(x)*(a + b*x**2)**p)/(16*a*p**2*x**4 - 24*a*p*x**4 + 5*a*x* *4 + 16*b*p**2*x**6 - 24*b*p*x**6 + 5*b*x**6),x)*a*b*c*p**4*x**2 - 224*sqr t(x)*int((sqrt(x)*(a + b*x**2)**p)/(16*a*p**2*x**4 - 24*a*p*x**4 + 5*a*x** 4 + 16*b*p**2*x**6 - 24*b*p*x**6 + 5*b*x**6),x)*a*b*c*p**3*x**2 + 88*sqrt( x)*int((sqrt(x)*(a + b*x**2)**p)/(16*a*p**2*x**4 - 24*a*p*x**4 + 5*a*x**4 + 16*b*p**2*x**6 - 24*b*p*x**6 + 5*b*x**6),x)*a*b*c*p**2*x**2 - 10*sqrt(x) *int((sqrt(x)*(a + b*x**2)**p)/(16*a*p**2*x**4 - 24*a*p*x**4 + 5*a*x**4 + 16*b*p**2*x**6 - 24*b*p*x**6 + 5*b*x**6),x)*a*b*c*p*x**2))/(sqrt(x)*b*e**4 *x**2*(16*p**2 - 24*p + 5))