Integrand size = 23, antiderivative size = 159 \[ \int \frac {x (c+d x)^{3/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\frac {(c+d x)^{3/2}}{b \sqrt {a-b x^2}}+\frac {3 \sqrt {a} d \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{b^{3/2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}} \] Output:
(d*x+c)^(3/2)/b/(-b*x^2+a)^(1/2)+3*a^(1/2)*d*(d*x+c)^(1/2)*(1-b*x^2/a)^(1/ 2)*EllipticE(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b ^(1/2)*c+a^(1/2)*d))^(1/2))/b^(3/2)/(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d) )^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 11.25 (sec) , antiderivative size = 435, normalized size of antiderivative = 2.74 \[ \int \frac {x (c+d x)^{3/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {c+d x} \left (\sqrt {b} \sqrt {\frac {d \left (\sqrt {a}+\sqrt {b} x\right )}{-\sqrt {b} c+\sqrt {a} d}} (c+d x) \sqrt {-\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}}+3 i d \sqrt {\frac {d \left (\sqrt {a}-\sqrt {b} x\right )}{\sqrt {b} c+\sqrt {a} d}} \left (\sqrt {a}+\sqrt {b} x\right ) E\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-3 i d \sqrt {\frac {d \left (\sqrt {a}-\sqrt {b} x\right )}{\sqrt {b} c+\sqrt {a} d}} \left (\sqrt {a}+\sqrt {b} x\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )\right )}{b^{3/2} \sqrt {\frac {d \left (\sqrt {a}+\sqrt {b} x\right )}{-\sqrt {b} c+\sqrt {a} d}} \sqrt {-\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}} \] Input:
Integrate[(x*(c + d*x)^(3/2))/(a - b*x^2)^(3/2),x]
Output:
(Sqrt[c + d*x]*(Sqrt[b]*Sqrt[(d*(Sqrt[a] + Sqrt[b]*x))/(-(Sqrt[b]*c) + Sqr t[a]*d)]*(c + d*x)*Sqrt[-((Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d))] + (3*I)*d*Sqrt[(d*(Sqrt[a] - Sqrt[b]*x))/(Sqrt[b]*c + Sqrt[a]*d)]*(Sqrt[a] + Sqrt[b]*x)*EllipticE[I*ArcSinh[Sqrt[-((Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sq rt[a]*d))]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] - (3*I)*d*Sq rt[(d*(Sqrt[a] - Sqrt[b]*x))/(Sqrt[b]*c + Sqrt[a]*d)]*(Sqrt[a] + Sqrt[b]*x )*EllipticF[I*ArcSinh[Sqrt[-((Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d))] ], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)]))/(b^(3/2)*Sqrt[(d*(Sq rt[a] + Sqrt[b]*x))/(-(Sqrt[b]*c) + Sqrt[a]*d)]*Sqrt[-((Sqrt[b]*(c + d*x)) /(Sqrt[b]*c + Sqrt[a]*d))]*Sqrt[a - b*x^2])
Time = 0.27 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {592, 509, 508, 327}
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.
| \(\displaystyle \int \frac {x (c+d x)^{3/2}}{\left (a-b x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 592 |
| \(\displaystyle \frac {(c+d x)^{3/2}}{b \sqrt {a-b x^2}}-\frac {3 d \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{2 b}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {(c+d x)^{3/2}}{b \sqrt {a-b x^2}}-\frac {3 d \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{2 b \sqrt {a-b x^2}}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {3 \sqrt {a} d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{b^{3/2} \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}+\frac {(c+d x)^{3/2}}{b \sqrt {a-b x^2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {3 \sqrt {a} d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{b^{3/2} \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}+\frac {(c+d x)^{3/2}}{b \sqrt {a-b x^2}}\) |
Input:
Int[(x*(c + d*x)^(3/2))/(a - b*x^2)^(3/2),x]
Output:
(c + d*x)^(3/2)/(b*Sqrt[a - b*x^2]) + (3*Sqrt[a]*d*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/ ((Sqrt[b]*c)/Sqrt[a] + d)])/(b^(3/2)*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c *q))])) Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[(x_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c + d*x)^n*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[d*(n/(2*b* (p + 1))) Int[(c + d*x)^(n - 1)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d}, x] && LtQ[p, -1] && GtQ[n, 0] && (IntegerQ[n] || IntegerQ[p] || I ntegersQ[2*n, 2*p])
Leaf count of result is larger than twice the leaf count of optimal. \(596\) vs. \(2(128)=256\).
Time = 2.56 (sec) , antiderivative size = 597, normalized size of antiderivative = 3.75
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {2 \left (-b d x -b c \right ) \left (\frac {c}{2 b^{2}}+\frac {d x}{2 b^{2}}\right )}{\sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d x -b c \right )}}-\frac {3 c d \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {3 d^{2} \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{b \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(597\) |
| default | \(\frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, \left (3 a \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) d^{2}-3 \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) b \,c^{2}-3 \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) a \,d^{2}+3 \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) b \,c^{2}+b \,x^{2} d^{2}+2 b c d x +b \,c^{2}\right )}{b^{2} \left (-b d \,x^{3}-b c \,x^{2}+a d x +a c \right )}\) | \(618\) |
Input:
int(x*(d*x+c)^(3/2)/(-b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
((d*x+c)*(-b*x^2+a))^(1/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*(-2*(-b*d*x-b*c) *(1/2*c/b^2+1/2*d/b^2*x)/((x^2-a/b)*(-b*d*x-b*c))^(1/2)-3*c*d/b*(c/d-1/b*( a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c /d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1 /2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*EllipticF(((x+c/d)/(c/d-1/b*(a*b)^( 1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))-3*d^2/ b*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b )^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b )^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*((-c/d-1/b*(a*b)^(1/2)) *EllipticE(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/( -c/d-1/b*(a*b)^(1/2)))^(1/2))+1/b*(a*b)^(1/2)*EllipticF(((x+c/d)/(c/d-1/b* (a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)) ))
Time = 0.09 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.45 \[ \int \frac {x (c+d x)^{3/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (b c x^{2} - a c\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) - 3 \, {\left (b d x^{2} - a d\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) - {\left (b d x + b c\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}}{b^{3} x^{2} - a b^{2}} \] Input:
integrate(x*(d*x+c)^(3/2)/(-b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
(2*(b*c*x^2 - a*c)*sqrt(-b*d)*weierstrassPInverse(4/3*(b*c^2 + 3*a*d^2)/(b *d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d) - 3*(b*d*x^2 - a*d)*sqrt(-b*d)*weierstrassZeta(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b* c^3 - 9*a*c*d^2)/(b*d^3), weierstrassPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2 ), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d)) - (b*d*x + b*c)* sqrt(-b*x^2 + a)*sqrt(d*x + c))/(b^3*x^2 - a*b^2)
\[ \int \frac {x (c+d x)^{3/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\int \frac {x \left (c + d x\right )^{\frac {3}{2}}}{\left (a - b x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(x*(d*x+c)**(3/2)/(-b*x**2+a)**(3/2),x)
Output:
Integral(x*(c + d*x)**(3/2)/(a - b*x**2)**(3/2), x)
\[ \int \frac {x (c+d x)^{3/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{2}} x}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x*(d*x+c)^(3/2)/(-b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate((d*x + c)^(3/2)*x/(-b*x^2 + a)^(3/2), x)
\[ \int \frac {x (c+d x)^{3/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{2}} x}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x*(d*x+c)^(3/2)/(-b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate((d*x + c)^(3/2)*x/(-b*x^2 + a)^(3/2), x)
Timed out. \[ \int \frac {x (c+d x)^{3/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\int \frac {x\,{\left (c+d\,x\right )}^{3/2}}{{\left (a-b\,x^2\right )}^{3/2}} \,d x \] Input:
int((x*(c + d*x)^(3/2))/(a - b*x^2)^(3/2),x)
Output:
int((x*(c + d*x)^(3/2))/(a - b*x^2)^(3/2), x)
\[ \int \frac {x (c+d x)^{3/2}}{\left (a-b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int(x*(d*x+c)^(3/2)/(-b*x^2+a)^(3/2),x)
Output:
( - 3*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a*c**2 - sqrt(a - b*x**2)*a*d**2*x**2 - sqrt(a - b*x**2)*b*c**2*x**2 + sqrt(a - b*x**2)*b *d**2*x**4),x)*a**3*c*d**4 + 3*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a*c**2 - sqrt(a - b*x**2)*a*d**2*x**2 - sqrt(a - b*x**2)*b*c**2* x**2 + sqrt(a - b*x**2)*b*d**2*x**4),x)*a**2*b*c**3*d**2 - sqrt(a - b*x**2 )*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**3)/(a**2*c + a**2*d*x - 2*a*b*c*x **2 - 2*a*b*d*x**3 + b**2*c*x**4 + b**2*d*x**5),x)*a*b**2*c*d**3 + sqrt(a - b*x**2)*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**3)/(a**2*c + a**2*d*x - 2 *a*b*c*x**2 - 2*a*b*d*x**3 + b**2*c*x**4 + b**2*d*x**5),x)*b**3*c**3*d - 3 *sqrt(a - b*x**2)*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a**2*c + a**2 *d*x - 2*a*b*c*x**2 - 2*a*b*d*x**3 + b**2*c*x**4 + b**2*d*x**5),x)*a**2*b* d**4 + 3*sqrt(a - b*x**2)*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a**2* c + a**2*d*x - 2*a*b*c*x**2 - 2*a*b*d*x**3 + b**2*c*x**4 + b**2*d*x**5),x) *a*b**2*c**2*d**2 + 2*sqrt(a - b*x**2)*int((sqrt(c + d*x)*sqrt(a - b*x**2) )/(a**2*c + a**2*d*x - 2*a*b*c*x**2 - 2*a*b*d*x**3 + b**2*c*x**4 + b**2*d* x**5),x)*a**2*b*c**2*d**2 - 2*sqrt(a - b*x**2)*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a**2*c + a**2*d*x - 2*a*b*c*x**2 - 2*a*b*d*x**3 + b**2*c*x**4 + b**2*d*x**5),x)*a*b**2*c**4 + 3*sqrt(a - b*x**2)*int((sqrt(c + d*x)*x)/(s qrt(a - b*x**2)*a*c**2 - sqrt(a - b*x**2)*a*d**2*x**2 - sqrt(a - b*x**2)*b *c**2*x**2 + sqrt(a - b*x**2)*b*d**2*x**4),x)*a**3*d**5 - 3*sqrt(a - b*...