Integrand size = 35, antiderivative size = 837 \[ \int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{5/2}} \, dx=-\frac {(B d-A e) x \sqrt {a+b x^2+c x^4}}{3 d e \left (d+e x^2\right )^{3/2}}+\frac {\left (A e^2 (b d-2 a e)-B d \left (3 c d^2-e (2 b d-a e)\right )\right ) \sqrt {a+b x^2+c x^4}}{3 d e^2 \left (c d^2-b d e+a e^2\right ) x \sqrt {d+e x^2}}-\frac {\sqrt {b^2-4 a c} \left (A e^2 (b d-2 a e)-B d \left (3 c d^2-e (2 b d-a e)\right )\right ) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} x \sqrt {d+e x^2} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{3 \sqrt {2} d^2 e^2 \left (c d^2-b d e+a e^2\right ) \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} (B d+2 A e) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} x^3 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{3 d^2 e \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}+\frac {2 \sqrt {2} B c \sqrt {b^2-4 a c} \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} x^3 \operatorname {EllipticPi}\left (\frac {2 \sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}},\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) e^2 \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \] Output:
-1/3*(-A*e+B*d)*x*(c*x^4+b*x^2+a)^(1/2)/d/e/(e*x^2+d)^(3/2)+1/3*(A*e^2*(-2 *a*e+b*d)-B*d*(3*c*d^2-e*(-a*e+2*b*d)))*(c*x^4+b*x^2+a)^(1/2)/d/e^2/(a*e^2 -b*d*e+c*d^2)/x/(e*x^2+d)^(1/2)-1/6*(-4*a*c+b^2)^(1/2)*(A*e^2*(-2*a*e+b*d) -B*d*(3*c*d^2-e*(-a*e+2*b*d)))*(-a*(c+a/x^4+b/x^2)/(-4*a*c+b^2))^(1/2)*x*( e*x^2+d)^(1/2)*EllipticE(1/2*(1+(b+2*a/x^2)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1 /2),2^(1/2)*((-4*a*c+b^2)^(1/2)*d/(b*d+(-4*a*c+b^2)^(1/2)*d-2*a*e))^(1/2)) *2^(1/2)/d^2/e^2/(a*e^2-b*d*e+c*d^2)/(-a*(e+d/x^2)/((b+(-4*a*c+b^2)^(1/2)) *d-2*a*e))^(1/2)/(c*x^4+b*x^2+a)^(1/2)-1/3*2^(1/2)*(-4*a*c+b^2)^(1/2)*(2*A *e+B*d)*(-a*(c+a/x^4+b/x^2)/(-4*a*c+b^2))^(1/2)*(-a*(e+d/x^2)/((b+(-4*a*c+ b^2)^(1/2))*d-2*a*e))^(1/2)*x^3*EllipticF(1/2*(1+(b+2*a/x^2)/(-4*a*c+b^2)^ (1/2))^(1/2)*2^(1/2),2^(1/2)*((-4*a*c+b^2)^(1/2)*d/(b*d+(-4*a*c+b^2)^(1/2) *d-2*a*e))^(1/2))/d^2/e/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2)+2*2^(1/2)*B* c*(-4*a*c+b^2)^(1/2)*(-a*(c+a/x^4+b/x^2)/(-4*a*c+b^2))^(1/2)*(-a*(e+d/x^2) /((b+(-4*a*c+b^2)^(1/2))*d-2*a*e))^(1/2)*x^3*EllipticPi(1/2*(1+(b+2*a/x^2) /(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2*(-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^2)^(1 /2)),2^(1/2)*((-4*a*c+b^2)^(1/2)*d/(b*d+(-4*a*c+b^2)^(1/2)*d-2*a*e))^(1/2) )/(b+(-4*a*c+b^2)^(1/2))/e^2/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{5/2}} \, dx \] Input:
Integrate[((A + B*x^2)*Sqrt[a + b*x^2 + c*x^4])/(d + e*x^2)^(5/2),x]
Output:
Integrate[((A + B*x^2)*Sqrt[a + b*x^2 + c*x^4])/(d + e*x^2)^(5/2), x]
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.
| \(\displaystyle \int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2260 |
| \(\displaystyle \int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{5/2}}dx\) |
Input:
Int[((A + B*x^2)*Sqrt[a + b*x^2 + c*x^4])/(d + e*x^2)^(5/2),x]
Output:
$Aborted
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^ (p_.), x_Symbol] :> Unintegrable[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x] /; FreeQ[{a, b, c, d, e, p, q}, x] && PolyQ[Px, x]
\[\int \frac {\left (B \,x^{2}+A \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]
Input:
int((B*x^2+A)*(c*x^4+b*x^2+a)^(1/2)/(e*x^2+d)^(5/2),x)
Output:
int((B*x^2+A)*(c*x^4+b*x^2+a)^(1/2)/(e*x^2+d)^(5/2),x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((B*x^2+A)*(c*x^4+b*x^2+a)^(1/2)/(e*x^2+d)^(5/2),x, algorithm="fr icas")
Output:
Timed out
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {\left (A + B x^{2}\right ) \sqrt {a + b x^{2} + c x^{4}}}{\left (d + e x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((B*x**2+A)*(c*x**4+b*x**2+a)**(1/2)/(e*x**2+d)**(5/2),x)
Output:
Integral((A + B*x**2)*sqrt(a + b*x**2 + c*x**4)/(d + e*x**2)**(5/2), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{2} + a} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(c*x^4+b*x^2+a)^(1/2)/(e*x^2+d)^(5/2),x, algorithm="ma xima")
Output:
integrate(sqrt(c*x^4 + b*x^2 + a)*(B*x^2 + A)/(e*x^2 + d)^(5/2), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{2} + a} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(c*x^4+b*x^2+a)^(1/2)/(e*x^2+d)^(5/2),x, algorithm="gi ac")
Output:
integrate(sqrt(c*x^4 + b*x^2 + a)*(B*x^2 + A)/(e*x^2 + d)^(5/2), x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,\sqrt {c\,x^4+b\,x^2+a}}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \] Input:
int(((A + B*x^2)*(a + b*x^2 + c*x^4)^(1/2))/(d + e*x^2)^(5/2),x)
Output:
int(((A + B*x^2)*(a + b*x^2 + c*x^4)^(1/2))/(d + e*x^2)^(5/2), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{5/2}} \, dx=\text {too large to display} \] Input:
int((B*x^2+A)*(c*x^4+b*x^2+a)^(1/2)/(e*x^2+d)^(5/2),x)
Output:
( - sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*a*c*x - sqrt(d + e*x**2)*sq rt(a + b*x**2 + c*x**4)*b**2*x + int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c *x**4)*x**6)/(a*b*d**3*e + 3*a*b*d**2*e**2*x**2 + 3*a*b*d*e**3*x**4 + a*b* e**4*x**6 - 3*a*c*d**4 - 9*a*c*d**3*e*x**2 - 9*a*c*d**2*e**2*x**4 - 3*a*c* d*e**3*x**6 + b**2*d**3*e*x**2 + 3*b**2*d**2*e**2*x**4 + 3*b**2*d*e**3*x** 6 + b**2*e**4*x**8 - 3*b*c*d**4*x**2 - 8*b*c*d**3*e*x**4 - 6*b*c*d**2*e**2 *x**6 + b*c*e**4*x**10 - 3*c**2*d**4*x**4 - 9*c**2*d**3*e*x**6 - 9*c**2*d* *2*e**2*x**8 - 3*c**2*d*e**3*x**10),x)*b**3*c*d**2*e**2 + 2*int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*x**6)/(a*b*d**3*e + 3*a*b*d**2*e**2*x**2 + 3*a*b*d*e**3*x**4 + a*b*e**4*x**6 - 3*a*c*d**4 - 9*a*c*d**3*e*x**2 - 9* a*c*d**2*e**2*x**4 - 3*a*c*d*e**3*x**6 + b**2*d**3*e*x**2 + 3*b**2*d**2*e* *2*x**4 + 3*b**2*d*e**3*x**6 + b**2*e**4*x**8 - 3*b*c*d**4*x**2 - 8*b*c*d* *3*e*x**4 - 6*b*c*d**2*e**2*x**6 + b*c*e**4*x**10 - 3*c**2*d**4*x**4 - 9*c **2*d**3*e*x**6 - 9*c**2*d**2*e**2*x**8 - 3*c**2*d*e**3*x**10),x)*b**3*c*d *e**3*x**2 + int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*x**6)/(a*b*d* *3*e + 3*a*b*d**2*e**2*x**2 + 3*a*b*d*e**3*x**4 + a*b*e**4*x**6 - 3*a*c*d* *4 - 9*a*c*d**3*e*x**2 - 9*a*c*d**2*e**2*x**4 - 3*a*c*d*e**3*x**6 + b**2*d **3*e*x**2 + 3*b**2*d**2*e**2*x**4 + 3*b**2*d*e**3*x**6 + b**2*e**4*x**8 - 3*b*c*d**4*x**2 - 8*b*c*d**3*e*x**4 - 6*b*c*d**2*e**2*x**6 + b*c*e**4*x** 10 - 3*c**2*d**4*x**4 - 9*c**2*d**3*e*x**6 - 9*c**2*d**2*e**2*x**8 - 3*...