Integrand size = 31, antiderivative size = 885 \[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\frac {(B d-A e) \left (c d^2-a e^2\right ) x \sqrt {a-c x^4}}{5 d e^3 \left (d+e x^2\right )^{5/2}}-\frac {\left (13 B c d^3-8 A c d^2 e-a B d e^2-4 a A e^3\right ) x \sqrt {a-c x^4}}{15 d^2 e^3 \left (d+e x^2\right )^{3/2}}+\frac {\left (45 B c^2 d^5-15 A c^2 d^4 e-31 a B c d^3 e^2+11 a A c d^2 e^3-2 a^2 B d e^4-8 a^2 A e^5\right ) x \sqrt {a-c x^4}}{15 d^3 e^3 \left (c d^2-a e^2\right ) \sqrt {d+e x^2}}-\frac {\left (105 B c^2 d^5-30 A c^2 d^4 e-77 a B c d^3 e^2+22 a A c d^2 e^3-4 a^2 B d e^4-16 a^2 A e^5\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{30 d^3 e^4 \left (c d^2-a e^2\right ) x}+\frac {\sqrt {c} \left (2 A e \left (15 c^2 d^4-11 a c d^2 e^2+8 a^2 e^4\right )-B \left (105 c^2 d^5-77 a c d^3 e^2-4 a^2 d e^4\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{30 d^3 e^4 \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\sqrt {a} \sqrt {c} \left (35 B c d^3-10 A c d^2 e+4 a B d e^2+16 a A e^3\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{30 d^3 e^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {c^2 (7 B d-2 A e) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 e^4 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
1/5*(-A*e+B*d)*(-a*e^2+c*d^2)*x*(-c*x^4+a)^(1/2)/d/e^3/(e*x^2+d)^(5/2)-1/1 5*(-4*A*a*e^3-8*A*c*d^2*e-B*a*d*e^2+13*B*c*d^3)*x*(-c*x^4+a)^(1/2)/d^2/e^3 /(e*x^2+d)^(3/2)+1/15*(-8*A*a^2*e^5+11*A*a*c*d^2*e^3-15*A*c^2*d^4*e-2*B*a^ 2*d*e^4-31*B*a*c*d^3*e^2+45*B*c^2*d^5)*x*(-c*x^4+a)^(1/2)/d^3/e^3/(-a*e^2+ c*d^2)/(e*x^2+d)^(1/2)-1/30*(-16*A*a^2*e^5+22*A*a*c*d^2*e^3-30*A*c^2*d^4*e -4*B*a^2*d*e^4-77*B*a*c*d^3*e^2+105*B*c^2*d^5)*(e*x^2+d)^(1/2)*(-c*x^4+a)^ (1/2)/d^3/e^4/(-a*e^2+c*d^2)/x+1/30*c^(1/2)*(2*A*e*(8*a^2*e^4-11*a*c*d^2*e ^2+15*c^2*d^4)-B*(-4*a^2*d*e^4-77*a*c*d^3*e^2+105*c^2*d^5))*(1-a/c/x^4)^(1 /2)*x^3*(a^(1/2)*(e*x^2+d)/(c^(1/2)*d+a^(1/2)*e)/x^2)^(1/2)*EllipticE(1/2* (1-a^(1/2)/c^(1/2)/x^2)^(1/2)*2^(1/2),2^(1/2)*(d/(d+a^(1/2)*e/c^(1/2)))^(1 /2))/d^3/e^4/(c^(1/2)*d-a^(1/2)*e)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+1/30*a ^(1/2)*c^(1/2)*(16*A*a*e^3-10*A*c*d^2*e+4*B*a*d*e^2+35*B*c*d^3)*(1-a/c/x^4 )^(1/2)*x^3*(a^(1/2)*(e*x^2+d)/(c^(1/2)*d+a^(1/2)*e)/x^2)^(1/2)*EllipticF( 1/2*(1-a^(1/2)/c^(1/2)/x^2)^(1/2)*2^(1/2),2^(1/2)*(d/(d+a^(1/2)*e/c^(1/2)) )^(1/2))/d^3/e^3/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)-1/2*c^2*(-2*A*e+7*B*d)*( 1-a/c/x^4)^(1/2)*x^3*(a^(1/2)*(e*x^2+d)/(c^(1/2)*d+a^(1/2)*e)/x^2)^(1/2)*E llipticPi(1/2*(1-a^(1/2)/c^(1/2)/x^2)^(1/2)*2^(1/2),2,2^(1/2)*(d/(d+a^(1/2 )*e/c^(1/2)))^(1/2))/e^4/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
\[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx \] Input:
Integrate[((A + B*x^2)*(a - c*x^4)^(3/2))/(d + e*x^2)^(7/2),x]
Output:
Integrate[((A + B*x^2)*(a - c*x^4)^(3/2))/(d + e*x^2)^(7/2), 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.
| \(\displaystyle \int \frac {\left (a-c x^4\right )^{3/2} \left (A+B x^2\right )}{\left (d+e x^2\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 2261 |
| \(\displaystyle \int \frac {\left (a-c x^4\right )^{3/2} \left (A+B x^2\right )}{\left (d+e x^2\right )^{7/2}}dx\) |
Input:
Int[((A + B*x^2)*(a - c*x^4)^(3/2))/(d + e*x^2)^(7/2),x]
Output:
$Aborted
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol ] :> Unintegrable[Px*(d + e*x^2)^q*(a + c*x^4)^p, x] /; FreeQ[{a, c, d, e, p, q}, x] && PolyQ[Px, x]
\[\int \frac {\left (B \,x^{2}+A \right ) \left (-c \,x^{4}+a \right )^{\frac {3}{2}}}{\left (e \,x^{2}+d \right )^{\frac {7}{2}}}d x\]
Input:
int((B*x^2+A)*(-c*x^4+a)^(3/2)/(e*x^2+d)^(7/2),x)
Output:
int((B*x^2+A)*(-c*x^4+a)^(3/2)/(e*x^2+d)^(7/2),x)
\[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\int { \frac {{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(3/2)/(e*x^2+d)^(7/2),x, algorithm="fricas" )
Output:
integral(-(B*c*x^6 + A*c*x^4 - B*a*x^2 - A*a)*sqrt(-c*x^4 + a)*sqrt(e*x^2 + d)/(e^4*x^8 + 4*d*e^3*x^6 + 6*d^2*e^2*x^4 + 4*d^3*e*x^2 + d^4), x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((B*x**2+A)*(-c*x**4+a)**(3/2)/(e*x**2+d)**(7/2),x)
Output:
Timed out
\[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\int { \frac {{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(3/2)/(e*x^2+d)^(7/2),x, algorithm="maxima" )
Output:
integrate((-c*x^4 + a)^(3/2)*(B*x^2 + A)/(e*x^2 + d)^(7/2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\int { \frac {{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(3/2)/(e*x^2+d)^(7/2),x, algorithm="giac")
Output:
integrate((-c*x^4 + a)^(3/2)*(B*x^2 + A)/(e*x^2 + d)^(7/2), x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (a-c\,x^4\right )}^{3/2}}{{\left (e\,x^2+d\right )}^{7/2}} \,d x \] Input:
int(((A + B*x^2)*(a - c*x^4)^(3/2))/(d + e*x^2)^(7/2),x)
Output:
int(((A + B*x^2)*(a - c*x^4)^(3/2))/(d + e*x^2)^(7/2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\text {too large to display} \] Input:
int((B*x^2+A)*(-c*x^4+a)^(3/2)/(e*x^2+d)^(7/2),x)
Output:
( - 28*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*b*d*e*x + 45*sqrt(d + e*x**2 )*sqrt(a - c*x**4)*a**2*c*d**2*x + 18*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a* *2*c*d*e*x**3 - 4*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*c*e**2*x**5 + 42* sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*c*d**2*x**3 - 15*sqrt(d + e*x**2)*sq rt(a - c*x**4)*a*c**2*d**2*x**5 - 32*int((sqrt(d + e*x**2)*sqrt(a - c*x**4 )*x**10)/(4*a**2*d**4*e**2 + 16*a**2*d**3*e**3*x**2 + 24*a**2*d**2*e**4*x* *4 + 16*a**2*d*e**5*x**6 + 4*a**2*e**6*x**8 + 15*a*c*d**6 + 60*a*c*d**5*e* x**2 + 86*a*c*d**4*e**2*x**4 + 44*a*c*d**3*e**3*x**6 - 9*a*c*d**2*e**4*x** 8 - 16*a*c*d*e**5*x**10 - 4*a*c*e**6*x**12 - 15*c**2*d**6*x**4 - 60*c**2*d **5*e*x**6 - 90*c**2*d**4*e**2*x**8 - 60*c**2*d**3*e**3*x**10 - 15*c**2*d* *2*e**4*x**12),x)*a**3*c**2*d**3*e**5 - 96*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**10)/(4*a**2*d**4*e**2 + 16*a**2*d**3*e**3*x**2 + 24*a**2*d**2*e **4*x**4 + 16*a**2*d*e**5*x**6 + 4*a**2*e**6*x**8 + 15*a*c*d**6 + 60*a*c*d **5*e*x**2 + 86*a*c*d**4*e**2*x**4 + 44*a*c*d**3*e**3*x**6 - 9*a*c*d**2*e* *4*x**8 - 16*a*c*d*e**5*x**10 - 4*a*c*e**6*x**12 - 15*c**2*d**6*x**4 - 60* c**2*d**5*e*x**6 - 90*c**2*d**4*e**2*x**8 - 60*c**2*d**3*e**3*x**10 - 15*c **2*d**2*e**4*x**12),x)*a**3*c**2*d**2*e**6*x**2 - 96*int((sqrt(d + e*x**2 )*sqrt(a - c*x**4)*x**10)/(4*a**2*d**4*e**2 + 16*a**2*d**3*e**3*x**2 + 24* a**2*d**2*e**4*x**4 + 16*a**2*d*e**5*x**6 + 4*a**2*e**6*x**8 + 15*a*c*d**6 + 60*a*c*d**5*e*x**2 + 86*a*c*d**4*e**2*x**4 + 44*a*c*d**3*e**3*x**6 -...