Integrand size = 29, antiderivative size = 274 \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{c+d x^2} \, dx=\frac {B x \sqrt {a-c x^4}}{3 d}+\frac {a^{3/4} \sqrt [4]{c} (B c-A d) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{d^2 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \left (3 B c^3-3 A c^2 d-2 a B d^2+3 \sqrt {a} \sqrt {c} d (B c-A d)\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{3 \sqrt [4]{c} d^3 \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} (B c-A d) \left (c^3-a d^2\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{5/4} d^3 \sqrt {a-c x^4}} \] Output:
1/3*B*x*(-c*x^4+a)^(1/2)/d+a^(3/4)*c^(1/4)*(-A*d+B*c)*(1-c*x^4/a)^(1/2)*El lipticE(c^(1/4)*x/a^(1/4),I)/d^2/(-c*x^4+a)^(1/2)-1/3*a^(1/4)*(3*B*c^3-3*A *c^2*d-2*B*a*d^2+3*a^(1/2)*c^(1/2)*d*(-A*d+B*c))*(1-c*x^4/a)^(1/2)*Ellipti cF(c^(1/4)*x/a^(1/4),I)/c^(1/4)/d^3/(-c*x^4+a)^(1/2)+a^(1/4)*(-A*d+B*c)*(- a*d^2+c^3)*(1-c*x^4/a)^(1/2)*EllipticPi(c^(1/4)*x/a^(1/4),-a^(1/2)*d/c^(3/ 2),I)/c^(5/4)/d^3/(-c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 11.09 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.81 \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{c+d x^2} \, dx=\frac {a B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c d^2 x-B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^2 d^2 x^5-3 i \sqrt {a} c^{3/2} d (B c-A d) \sqrt {1-\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )-i c \left (3 A \sqrt {c} d \left (c^{3/2}+\sqrt {a} d\right )+B \left (-3 c^3-3 \sqrt {a} c^{3/2} d+2 a d^2\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-3 i B c^4 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+3 i A c^3 d \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+3 i a B c d^2 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-3 i a A d^3 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )}{3 \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c d^3 \sqrt {a-c x^4}} \] Input:
Integrate[((A + B*x^2)*Sqrt[a - c*x^4])/(c + d*x^2),x]
Output:
(a*B*Sqrt[-(Sqrt[c]/Sqrt[a])]*c*d^2*x - B*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^2*d^2 *x^5 - (3*I)*Sqrt[a]*c^(3/2)*d*(B*c - A*d)*Sqrt[1 - (c*x^4)/a]*EllipticE[I *ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - I*c*(3*A*Sqrt[c]*d*(c^(3/2) + Sqrt[a]*d) + B*(-3*c^3 - 3*Sqrt[a]*c^(3/2)*d + 2*a*d^2))*Sqrt[1 - (c*x^4)/ a]*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - (3*I)*B*c^4*Sqrt [1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*d)/c^(3/2)), I*ArcSinh[Sqrt[-(Sqrt[c ]/Sqrt[a])]*x], -1] + (3*I)*A*c^3*d*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt [a]*d)/c^(3/2)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + (3*I)*a*B*c* d^2*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*d)/c^(3/2)), I*ArcSinh[Sqrt[ -(Sqrt[c]/Sqrt[a])]*x], -1] - (3*I)*a*A*d^3*Sqrt[1 - (c*x^4)/a]*EllipticPi [-((Sqrt[a]*d)/c^(3/2)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1])/(3*Sq rt[-(Sqrt[c]/Sqrt[a])]*c*d^3*Sqrt[a - c*x^4])
Time = 0.56 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.37, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2259, 2009}
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.
| \(\displaystyle \int \frac {\sqrt {a-c x^4} \left (A+B x^2\right )}{c+d x^2} \, dx\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \int \left (\frac {a A d^3-a B c d^2-A c^3 d+B c^4}{d^3 \sqrt {a-c x^4} \left (c+d x^2\right )}-\frac {-a B d^2-A c^2 d+B c^3}{d^3 \sqrt {a-c x^4}}+\frac {c x^2 (B c-A d)}{d^2 \sqrt {a-c x^4}}-\frac {B c x^4}{d \sqrt {a-c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^{3/4} \sqrt [4]{c} \sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{d^2 \sqrt {a-c x^4}}+\frac {a^{3/4} \sqrt [4]{c} \sqrt {1-\frac {c x^4}{a}} (B c-A d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{d^2 \sqrt {a-c x^4}}-\frac {a^{5/4} B \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{3 \sqrt [4]{c} d \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (c^3-a d^2\right ) \sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{5/4} d^3 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (-a B d^2-A c^2 d+B c^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} d^3 \sqrt {a-c x^4}}+\frac {B x \sqrt {a-c x^4}}{3 d}\) |
Input:
Int[((A + B*x^2)*Sqrt[a - c*x^4])/(c + d*x^2),x]
Output:
(B*x*Sqrt[a - c*x^4])/(3*d) + (a^(3/4)*c^(1/4)*(B*c - A*d)*Sqrt[1 - (c*x^4 )/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(d^2*Sqrt[a - c*x^4]) - ( a^(5/4)*B*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/ (3*c^(1/4)*d*Sqrt[a - c*x^4]) - (a^(3/4)*c^(1/4)*(B*c - A*d)*Sqrt[1 - (c*x ^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(d^2*Sqrt[a - c*x^4]) - (a^(1/4)*(B*c^3 - A*c^2*d - a*B*d^2)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin [(c^(1/4)*x)/a^(1/4)], -1])/(c^(1/4)*d^3*Sqrt[a - c*x^4]) + (a^(1/4)*(B*c - A*d)*(c^3 - a*d^2)*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*d)/c^(3/2)) , ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(5/4)*d^3*Sqrt[a - c*x^4])
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a + c*x^4], Px*(d + e*x^2)^q*(a + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x] && IntegerQ[p + 1/ 2] && IntegerQ[q]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 455 vs. \(2 (224 ) = 448\).
Time = 2.34 (sec) , antiderivative size = 456, normalized size of antiderivative = 1.66
| method | result | size |
| risch | \(\frac {B x \sqrt {-c \,x^{4}+a}}{3 d}-\frac {\frac {-\frac {3 \sqrt {c}\, d \left (A d -B c \right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {3 B \,c^{3} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {3 A \,c^{2} d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {2 B a \,d^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}}{d^{2}}-\frac {3 \left (A a \,d^{3}-A \,c^{3} d -B a c \,d^{2}+B \,c^{4}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{d^{2} c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}}{3 d}\) | \(456\) |
| default | \(\frac {B \left (\frac {x \sqrt {-c \,x^{4}+a}}{3}+\frac {2 a \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{d}+\frac {\left (A d -B c \right ) \left (\frac {c^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {\sqrt {c}\, \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\sqrt {c}\, \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) a}{c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {c^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{d}\) | \(504\) |
| elliptic | \(\text {Expression too large to display}\) | \(909\) |
Input:
int((B*x^2+A)*(-c*x^4+a)^(1/2)/(d*x^2+c),x,method=_RETURNVERBOSE)
Output:
1/3*B*x*(-c*x^4+a)^(1/2)/d-1/3/d*(1/d^2*(-3*c^(1/2)*d*(A*d-B*c)*a^(1/2)/(c ^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^(1/2)*x^2/a^(1/2) )^(1/2)/(-c*x^4+a)^(1/2)*(EllipticF(x*(c^(1/2)/a^(1/2))^(1/2),I)-EllipticE (x*(c^(1/2)/a^(1/2))^(1/2),I))+3*B*c^3/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)* x^2/a^(1/2))^(1/2)*(1+c^(1/2)*x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)*Elliptic F(x*(c^(1/2)/a^(1/2))^(1/2),I)-3*A*c^2*d/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2 )*x^2/a^(1/2))^(1/2)*(1+c^(1/2)*x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)*Ellipt icF(x*(c^(1/2)/a^(1/2))^(1/2),I)-2*B*a*d^2/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1 /2)*x^2/a^(1/2))^(1/2)*(1+c^(1/2)*x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)*Elli pticF(x*(c^(1/2)/a^(1/2))^(1/2),I))-3*(A*a*d^3-A*c^3*d-B*a*c*d^2+B*c^4)/d^ 2/c/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^(1/2)*x^2/a ^(1/2))^(1/2)/(-c*x^4+a)^(1/2)*EllipticPi(x*(c^(1/2)/a^(1/2))^(1/2),-a^(1/ 2)*d/c^(3/2),(-c^(1/2)/a^(1/2))^(1/2)/(c^(1/2)/a^(1/2))^(1/2)))
Timed out. \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{c+d x^2} \, dx=\text {Timed out} \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/(d*x^2+c),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{c+d x^2} \, dx=\int \frac {\left (A + B x^{2}\right ) \sqrt {a - c x^{4}}}{c + d x^{2}}\, dx \] Input:
integrate((B*x**2+A)*(-c*x**4+a)**(1/2)/(d*x**2+c),x)
Output:
Integral((A + B*x**2)*sqrt(a - c*x**4)/(c + d*x**2), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{c+d x^2} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )}}{d x^{2} + c} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/(d*x^2+c),x, algorithm="maxima")
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)/(d*x^2 + c), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{c+d x^2} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )}}{d x^{2} + c} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/(d*x^2+c),x, algorithm="giac")
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)/(d*x^2 + c), x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{c+d x^2} \, dx=\int \frac {\left (B\,x^2+A\right )\,\sqrt {a-c\,x^4}}{d\,x^2+c} \,d x \] Input:
int(((A + B*x^2)*(a - c*x^4)^(1/2))/(c + d*x^2),x)
Output:
int(((A + B*x^2)*(a - c*x^4)^(1/2))/(c + d*x^2), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{c+d x^2} \, dx=\frac {\sqrt {-c \,x^{4}+a}\, b x +3 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a^{2} d -\left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a b c -3 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{4}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a c d +3 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{4}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) b \,c^{2}+2 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a b d}{3 d} \] Input:
int((B*x^2+A)*(-c*x^4+a)^(1/2)/(d*x^2+c),x)
Output:
(sqrt(a - c*x**4)*b*x + 3*int(sqrt(a - c*x**4)/(a*c + a*d*x**2 - c**2*x**4 - c*d*x**6),x)*a**2*d - int(sqrt(a - c*x**4)/(a*c + a*d*x**2 - c**2*x**4 - c*d*x**6),x)*a*b*c - 3*int((sqrt(a - c*x**4)*x**4)/(a*c + a*d*x**2 - c** 2*x**4 - c*d*x**6),x)*a*c*d + 3*int((sqrt(a - c*x**4)*x**4)/(a*c + a*d*x** 2 - c**2*x**4 - c*d*x**6),x)*b*c**2 + 2*int((sqrt(a - c*x**4)*x**2)/(a*c + a*d*x**2 - c**2*x**4 - c*d*x**6),x)*a*b*d)/(3*d)