Integrand size = 23, antiderivative size = 257 \[ \int \frac {x^2}{(c+d x)^{3/2} \left (a-b x^2\right )^2} \, dx=-\frac {2 c^2 d}{\left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {\sqrt {c+d x} \left (b c^2 x-a d (2 c-d x)\right )}{2 \left (b c^2-a d^2\right )^2 \left (a-b x^2\right )}+\frac {\left (2 \sqrt {b} c+\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{4 \sqrt {a} b^{3/4} \left (\sqrt {b} c-\sqrt {a} d\right )^{5/2}}-\frac {\left (2 \sqrt {b} c-\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{4 \sqrt {a} b^{3/4} \left (\sqrt {b} c+\sqrt {a} d\right )^{5/2}} \] Output:
-2*c^2*d/(-a*d^2+b*c^2)^2/(d*x+c)^(1/2)+1/2*(d*x+c)^(1/2)*(b*c^2*x-a*d*(-d *x+2*c))/(-a*d^2+b*c^2)^2/(-b*x^2+a)+1/4*(2*b^(1/2)*c+a^(1/2)*d)*arctanh(b ^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c-a^(1/2)*d)^(1/2))/a^(1/2)/b^(3/4)/(b^(1/2) *c-a^(1/2)*d)^(5/2)-1/4*(2*b^(1/2)*c-a^(1/2)*d)*arctanh(b^(1/4)*(d*x+c)^(1 /2)/(b^(1/2)*c+a^(1/2)*d)^(1/2))/a^(1/2)/b^(3/4)/(b^(1/2)*c+a^(1/2)*d)^(5/ 2)
Time = 1.75 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.18 \[ \int \frac {x^2}{(c+d x)^{3/2} \left (a-b x^2\right )^2} \, dx=\frac {1}{4} \left (-\frac {2 \left (b c^2 x (c+5 d x)+a d \left (-6 c^2-c d x+d^2 x^2\right )\right )}{\left (b c^2-a d^2\right )^2 \sqrt {c+d x} \left (-a+b x^2\right )}+\frac {\left (2 \sqrt {b} c-\sqrt {a} d\right ) \sqrt {-b c-\sqrt {a} \sqrt {b} d} \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a} b \left (\sqrt {b} c+\sqrt {a} d\right )^3}+\frac {\left (2 \sqrt {b} c+\sqrt {a} d\right ) \sqrt {-b c+\sqrt {a} \sqrt {b} d} \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {a} b \left (-\sqrt {b} c+\sqrt {a} d\right )^3}\right ) \] Input:
Integrate[x^2/((c + d*x)^(3/2)*(a - b*x^2)^2),x]
Output:
((-2*(b*c^2*x*(c + 5*d*x) + a*d*(-6*c^2 - c*d*x + d^2*x^2)))/((b*c^2 - a*d ^2)^2*Sqrt[c + d*x]*(-a + b*x^2)) + ((2*Sqrt[b]*c - Sqrt[a]*d)*Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d]*ArcTan[(Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d* x])/(Sqrt[b]*c + Sqrt[a]*d)])/(Sqrt[a]*b*(Sqrt[b]*c + Sqrt[a]*d)^3) + ((2* Sqrt[b]*c + Sqrt[a]*d)*Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]*ArcTan[(Sqrt[-(b*c ) + Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c - Sqrt[a]*d)])/(Sqrt[a]*b *(-(Sqrt[b]*c) + Sqrt[a]*d)^3))/4
Time = 2.12 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.45, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {561, 27, 1673, 27, 2195, 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.
| \(\displaystyle \int \frac {x^2}{\left (a-b x^2\right )^2 (c+d x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 561 |
| \(\displaystyle \frac {2 \int \frac {x^2}{(c+d x) \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \int \frac {d^2 x^2}{(c+d x) \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{d^3}\) |
| \(\Big \downarrow \) 1673 |
| \(\displaystyle \frac {2 \left (\frac {d^4 \int -\frac {2 \left (\frac {4 a b c^2}{d^2}-\frac {a b \left (b c^2-5 a d^2\right ) (c+d x) c}{d^2 \left (b c^2-a d^2\right )}-\frac {a b \left (b c^2+a d^2\right ) (c+d x)^2}{d^2 \left (b c^2-a d^2\right )}\right )}{(c+d x) \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}}{8 a b \left (b c^2-a d^2\right )}-\frac {d^2 \sqrt {c+d x} \left (c \left (3 a d^2+b c^2\right )-(c+d x) \left (a d^2+b c^2\right )\right )}{4 \left (b c^2-a d^2\right )^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{d^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (-\frac {d^4 \int \frac {\frac {4 a b c^2}{d^2}-\frac {a b \left (b c^2-5 a d^2\right ) (c+d x) c}{d^2 \left (b c^2-a d^2\right )}-\frac {a b \left (b c^2+a d^2\right ) (c+d x)^2}{d^2 \left (b c^2-a d^2\right )}}{(c+d x) \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}}{4 a b \left (b c^2-a d^2\right )}-\frac {d^2 \sqrt {c+d x} \left (c \left (3 a d^2+b c^2\right )-(c+d x) \left (a d^2+b c^2\right )\right )}{4 \left (b c^2-a d^2\right )^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{d^3}\) |
| \(\Big \downarrow \) 2195 |
| \(\displaystyle \frac {2 \left (-\frac {d^4 \int \left (\frac {a b \left (\left (5 b c^2+a d^2\right ) (c+d x)-c \left (7 b c^2+5 a d^2\right )\right )}{\left (b c^2-a d^2\right ) \left (b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2\right )}-\frac {4 a b c^2}{\left (b c^2-a d^2\right ) (c+d x)}\right )d\sqrt {c+d x}}{4 a b \left (b c^2-a d^2\right )}-\frac {d^2 \sqrt {c+d x} \left (c \left (3 a d^2+b c^2\right )-(c+d x) \left (a d^2+b c^2\right )\right )}{4 \left (b c^2-a d^2\right )^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{d^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (-\frac {d^4 \left (-\frac {\sqrt {a} \sqrt [4]{b} \left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} d+2 \sqrt {b} c\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 d \left (\sqrt {b} c-\sqrt {a} d\right )^{3/2}}+\frac {\sqrt {a} \sqrt [4]{b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (2 \sqrt {b} c-\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 d \left (\sqrt {a} d+\sqrt {b} c\right )^{3/2}}+\frac {4 a b c^2}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{4 a b \left (b c^2-a d^2\right )}-\frac {d^2 \sqrt {c+d x} \left (c \left (3 a d^2+b c^2\right )-(c+d x) \left (a d^2+b c^2\right )\right )}{4 \left (b c^2-a d^2\right )^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{d^3}\) |
Input:
Int[x^2/((c + d*x)^(3/2)*(a - b*x^2)^2),x]
Output:
(2*(-1/4*(d^2*Sqrt[c + d*x]*(c*(b*c^2 + 3*a*d^2) - (b*c^2 + a*d^2)*(c + d* x)))/((b*c^2 - a*d^2)^2*(a - (b*c^2)/d^2 + (2*b*c*(c + d*x))/d^2 - (b*(c + d*x)^2)/d^2)) - (d^4*((4*a*b*c^2)/((b*c^2 - a*d^2)*Sqrt[c + d*x]) - (Sqrt [a]*b^(1/4)*(Sqrt[b]*c + Sqrt[a]*d)*(2*Sqrt[b]*c + Sqrt[a]*d)*ArcTanh[(b^( 1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d]])/(2*d*(Sqrt[b]*c - Sqrt[a ]*d)^(3/2)) + (Sqrt[a]*b^(1/4)*(Sqrt[b]*c - Sqrt[a]*d)*(2*Sqrt[b]*c - Sqrt [a]*d)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]*d]])/(2*d* (Sqrt[b]*c + Sqrt[a]*d)^(3/2))))/(4*a*b*(b*c^2 - a*d^2))))/d^3
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{k = Denominator[n]}, Simp[k/d Subst[Int[x^(k*(n + 1) - 1)*(-c /d + x^k/d)^m*Simp[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^k/d^2) + b*(x^(2*k)/d^2), x]^p, x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, m, p}, x] && Frac tionQ[n] && IntegerQ[p] && IntegerQ[m]
Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^ 4)^(p_), x_Symbol] :> With[{f = Coeff[PolynomialRemainder[x^m*(d + e*x^2)^q , a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[x^m*(d + e*x^ 2)^q, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a *b*g - f*(b^2 - 2*a*c) - c*(b*f - 2*a*g)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[x^m*(a + b*x^2 + c*x^4)^(p + 1)*Simp[ExpandToSum[(2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[x^m*(d + e*x^2)^q, a + b*x^2 + c*x^4, x])/x^m + (b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5 ) - a*b*g)/x^m + c*(4*p + 7)*(b*f - 2*a*g)*x^(2 - m), x], x], x], x]] /; Fr eeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IGtQ[q, 1] && ILtQ[m/2, 0]
Int[(Pq_)*((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d*x)^m*Pq*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && PolyQ[Pq, x^2] && IGtQ[p, -2]
Time = 0.55 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.28
| method | result | size |
| derivativedivides | \(2 d \left (-\frac {c^{2}}{\left (a \,d^{2}-b \,c^{2}\right )^{2} \sqrt {d x +c}}-\frac {\frac {\left (-\frac {a \,d^{2}}{4}-\frac {b \,c^{2}}{4}\right ) \left (d x +c \right )^{\frac {3}{2}}+\left (\frac {3}{4} a \,d^{2} c +\frac {1}{4} b \,c^{3}\right ) \sqrt {d x +c}}{-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}}+\frac {b \left (-\frac {\left (-4 a b c \,d^{2}-2 c^{3} b^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+5 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (4 a b c \,d^{2}+2 c^{3} b^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+5 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4}}{\left (a \,d^{2}-b \,c^{2}\right )^{2}}\right )\) | \(330\) |
| default | \(2 d \left (-\frac {c^{2}}{\left (a \,d^{2}-b \,c^{2}\right )^{2} \sqrt {d x +c}}-\frac {\frac {\left (-\frac {a \,d^{2}}{4}-\frac {b \,c^{2}}{4}\right ) \left (d x +c \right )^{\frac {3}{2}}+\left (\frac {3}{4} a \,d^{2} c +\frac {1}{4} b \,c^{3}\right ) \sqrt {d x +c}}{-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}}+\frac {b \left (-\frac {\left (-4 a b c \,d^{2}-2 c^{3} b^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+5 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (4 a b c \,d^{2}+2 c^{3} b^{2}+\sqrt {a b \,d^{2}}\, a \,d^{2}+5 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4}}{\left (a \,d^{2}-b \,c^{2}\right )^{2}}\right )\) | \(330\) |
| pseudoelliptic | \(-\frac {d \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (\frac {\left (a \,d^{2}+5 b \,c^{2}\right ) \sqrt {a b \,d^{2}}}{4}+b c \left (a \,d^{2}+\frac {b \,c^{2}}{2}\right )\right ) \sqrt {d x +c}\, \left (-b \,x^{2}+a \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\left (d \sqrt {d x +c}\, \left (-b \,x^{2}+a \right ) \left (\frac {\left (-a \,d^{2}-5 b \,c^{2}\right ) \sqrt {a b \,d^{2}}}{4}+b c \left (a \,d^{2}+\frac {b \,c^{2}}{2}\right )\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+3 \left (-\frac {a \,d^{3} x^{2}}{6}+\frac {a c \,d^{2} x}{6}+c^{2} \left (-\frac {5 b \,x^{2}}{6}+a \right ) d -\frac {b \,c^{3} x}{6}\right ) \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {a b \,d^{2}}\, \sqrt {d x +c}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (-b \,x^{2}+a \right ) \left (a \,d^{2}-b \,c^{2}\right )^{2}}\) | \(339\) |
Input:
int(x^2/(d*x+c)^(3/2)/(-b*x^2+a)^2,x,method=_RETURNVERBOSE)
Output:
2*d*(-c^2/(a*d^2-b*c^2)^2/(d*x+c)^(1/2)-1/(a*d^2-b*c^2)^2*(((-1/4*a*d^2-1/ 4*b*c^2)*(d*x+c)^(3/2)+(3/4*a*d^2*c+1/4*b*c^3)*(d*x+c)^(1/2))/(-b*(d*x+c)^ 2+2*b*c*(d*x+c)+a*d^2-b*c^2)+1/4*b*(-1/2*(-4*a*b*c*d^2-2*c^3*b^2+(a*b*d^2) ^(1/2)*a*d^2+5*(a*b*d^2)^(1/2)*b*c^2)/b/(a*b*d^2)^(1/2)/((b*c+(a*b*d^2)^(1 /2))*b)^(1/2)*arctanh(b*(d*x+c)^(1/2)/((b*c+(a*b*d^2)^(1/2))*b)^(1/2))+1/2 *(4*a*b*c*d^2+2*c^3*b^2+(a*b*d^2)^(1/2)*a*d^2+5*(a*b*d^2)^(1/2)*b*c^2)/b/( a*b*d^2)^(1/2)/((-b*c+(a*b*d^2)^(1/2))*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((- b*c+(a*b*d^2)^(1/2))*b)^(1/2)))))
Leaf count of result is larger than twice the leaf count of optimal. 6054 vs. \(2 (198) = 396\).
Time = 1.82 (sec) , antiderivative size = 6054, normalized size of antiderivative = 23.56 \[ \int \frac {x^2}{(c+d x)^{3/2} \left (a-b x^2\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(x^2/(d*x+c)^(3/2)/(-b*x^2+a)^2,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {x^2}{(c+d x)^{3/2} \left (a-b x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate(x**2/(d*x+c)**(3/2)/(-b*x**2+a)**2,x)
Output:
Timed out
\[ \int \frac {x^2}{(c+d x)^{3/2} \left (a-b x^2\right )^2} \, dx=\int { \frac {x^{2}}{{\left (b x^{2} - a\right )}^{2} {\left (d x + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^2/(d*x+c)^(3/2)/(-b*x^2+a)^2,x, algorithm="maxima")
Output:
integrate(x^2/((b*x^2 - a)^2*(d*x + c)^(3/2)), x)
Leaf count of result is larger than twice the leaf count of optimal. 1409 vs. \(2 (198) = 396\).
Time = 0.32 (sec) , antiderivative size = 1409, normalized size of antiderivative = 5.48 \[ \int \frac {x^2}{(c+d x)^{3/2} \left (a-b x^2\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(x^2/(d*x+c)^(3/2)/(-b*x^2+a)^2,x, algorithm="giac")
Output:
1/4*(((5*sqrt(a*b)*a*b*c^2*d^3 + sqrt(a*b)*a^2*d^5)*(b^2*c^4*d - 2*a*b*c^2 *d^3 + a^2*d^5)^2*abs(b) - (7*a*b^4*c^7*d^3 - 9*a^2*b^3*c^5*d^5 - 3*a^3*b^ 2*c^3*d^7 + 5*a^4*b*c*d^9)*abs(b^2*c^4*d - 2*a*b*c^2*d^3 + a^2*d^5)*abs(b) + 2*(sqrt(a*b)*b^6*c^12*d^3 - 2*sqrt(a*b)*a*b^5*c^10*d^5 - 2*sqrt(a*b)*a^ 2*b^4*c^8*d^7 + 8*sqrt(a*b)*a^3*b^3*c^6*d^9 - 7*sqrt(a*b)*a^4*b^2*c^4*d^11 + 2*sqrt(a*b)*a^5*b*c^2*d^13)*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(b^3*c^5 - 2*a*b^2*c^3*d^2 + a^2*b*c*d^4 + sqrt((b^3*c^5 - 2*a*b^2*c^3*d^2 + a^2*b *c*d^4)^2 - (b^3*c^6 - 3*a*b^2*c^4*d^2 + 3*a^2*b*c^2*d^4 - a^3*d^6)*(b^3*c ^4 - 2*a*b^2*c^2*d^2 + a^2*b*d^4)))/(b^3*c^4 - 2*a*b^2*c^2*d^2 + a^2*b*d^4 )))/((a*b^6*c^9 - 4*a^2*b^5*c^7*d^2 + 6*a^3*b^4*c^5*d^4 - 4*a^4*b^3*c^3*d^ 6 + a^5*b^2*c*d^8 - sqrt(a*b)*a*b^5*c^8*d + 4*sqrt(a*b)*a^2*b^4*c^6*d^3 - 6*sqrt(a*b)*a^3*b^3*c^4*d^5 + 4*sqrt(a*b)*a^4*b^2*c^2*d^7 - sqrt(a*b)*a^5* b*d^9)*sqrt(-b^2*c - sqrt(a*b)*b*d)*abs(b^2*c^4*d - 2*a*b*c^2*d^3 + a^2*d^ 5)) - ((5*sqrt(a*b)*a*b*c^2*d^3 + sqrt(a*b)*a^2*d^5)*(b^2*c^4*d - 2*a*b*c^ 2*d^3 + a^2*d^5)^2*abs(b) + (7*a*b^4*c^7*d^3 - 9*a^2*b^3*c^5*d^5 - 3*a^3*b ^2*c^3*d^7 + 5*a^4*b*c*d^9)*abs(b^2*c^4*d - 2*a*b*c^2*d^3 + a^2*d^5)*abs(b ) + 2*(sqrt(a*b)*b^6*c^12*d^3 - 2*sqrt(a*b)*a*b^5*c^10*d^5 - 2*sqrt(a*b)*a ^2*b^4*c^8*d^7 + 8*sqrt(a*b)*a^3*b^3*c^6*d^9 - 7*sqrt(a*b)*a^4*b^2*c^4*d^1 1 + 2*sqrt(a*b)*a^5*b*c^2*d^13)*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(b^3*c^ 5 - 2*a*b^2*c^3*d^2 + a^2*b*c*d^4 - sqrt((b^3*c^5 - 2*a*b^2*c^3*d^2 + a...
Time = 11.90 (sec) , antiderivative size = 9402, normalized size of antiderivative = 36.58 \[ \int \frac {x^2}{(c+d x)^{3/2} \left (a-b x^2\right )^2} \, dx=\text {Too large to display} \] Input:
int(x^2/((a - b*x^2)^2*(c + d*x)^(3/2)),x)
Output:
atan(-(((-(a^3*d^7*(a^3*b^3)^(1/2) - 4*a*b^5*c^7 - 9*a^4*b^2*c*d^6 - 61*a^ 2*b^4*c^5*d^2 - 70*a^3*b^3*c^3*d^4 + 24*b^3*c^6*d*(a^3*b^3)^(1/2) + 85*a*b ^2*c^4*d^3*(a^3*b^3)^(1/2) + 34*a^2*b*c^2*d^5*(a^3*b^3)^(1/2))/(64*(a^2*b^ 8*c^10 - a^7*b^3*d^10 - 5*a^3*b^7*c^8*d^2 + 10*a^4*b^6*c^6*d^4 - 10*a^5*b^ 5*c^4*d^6 + 5*a^6*b^4*c^2*d^8)))^(1/2)*((c + d*x)^(1/2)*(-(a^3*d^7*(a^3*b^ 3)^(1/2) - 4*a*b^5*c^7 - 9*a^4*b^2*c*d^6 - 61*a^2*b^4*c^5*d^2 - 70*a^3*b^3 *c^3*d^4 + 24*b^3*c^6*d*(a^3*b^3)^(1/2) + 85*a*b^2*c^4*d^3*(a^3*b^3)^(1/2) + 34*a^2*b*c^2*d^5*(a^3*b^3)^(1/2))/(64*(a^2*b^8*c^10 - a^7*b^3*d^10 - 5* a^3*b^7*c^8*d^2 + 10*a^4*b^6*c^6*d^4 - 10*a^5*b^5*c^4*d^6 + 5*a^6*b^4*c^2* d^8)))^(1/2)*(2048*a*b^14*c^21*d^2 + 2048*a^11*b^4*c*d^22 - 20480*a^2*b^13 *c^19*d^4 + 92160*a^3*b^12*c^17*d^6 - 245760*a^4*b^11*c^15*d^8 + 430080*a^ 5*b^10*c^13*d^10 - 516096*a^6*b^9*c^11*d^12 + 430080*a^7*b^8*c^9*d^14 - 24 5760*a^8*b^7*c^7*d^16 + 92160*a^9*b^6*c^5*d^18 - 20480*a^10*b^5*c^3*d^20) - 1792*a*b^12*c^19*d^3 - 1280*a^10*b^3*c*d^21 + 13056*a^2*b^11*c^17*d^5 - 39936*a^3*b^10*c^15*d^7 + 64512*a^4*b^9*c^13*d^9 - 53760*a^5*b^8*c^11*d^11 + 10752*a^6*b^7*c^9*d^13 + 21504*a^7*b^6*c^7*d^15 - 21504*a^8*b^5*c^5*d^1 7 + 8448*a^9*b^4*c^3*d^19) - (c + d*x)^(1/2)*(32*a^9*b^2*d^20 + 128*b^11*c ^18*d^2 + 544*a*b^10*c^16*d^4 - 5120*a^2*b^9*c^14*d^6 + 12160*a^3*b^8*c^12 *d^8 - 12032*a^4*b^7*c^10*d^10 + 2752*a^5*b^6*c^8*d^12 + 4096*a^6*b^5*c^6* d^14 - 3200*a^7*b^4*c^4*d^16 + 640*a^8*b^3*c^2*d^18))*(-(a^3*d^7*(a^3*b...
Time = 0.61 (sec) , antiderivative size = 1862, normalized size of antiderivative = 7.25 \[ \int \frac {x^2}{(c+d x)^{3/2} \left (a-b x^2\right )^2} \, dx =\text {Too large to display} \] Input:
int(x^2/(d*x+c)^(3/2)/(-b*x^2+a)^2,x)
Output:
( - 2*sqrt(a)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d *x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**3*d**4 - 18*sqrt(a)*sqr t(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*s qrt(sqrt(b)*sqrt(a)*d - b*c)))*a**2*b*c**2*d**2 + 2*sqrt(a)*sqrt(c + d*x)* sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b) *sqrt(a)*d - b*c)))*a**2*b*d**4*x**2 - 4*sqrt(a)*sqrt(c + d*x)*sqrt(sqrt(b )*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a*b**2*c**4 + 18*sqrt(a)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b *c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a*b**2 *c**2*d**2*x**2 + 4*sqrt(a)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*at an((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*b**3*c**4*x* *2 - 10*sqrt(b)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**3*c*d**3 - 14*sqrt(b) *sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt( b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**2*b*c**3*d + 10*sqrt(b)*sqrt(c + d*x )*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt( b)*sqrt(a)*d - b*c)))*a**2*b*c*d**3*x**2 + 14*sqrt(b)*sqrt(c + d*x)*sqrt(s qrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt( a)*d - b*c)))*a*b**2*c**3*d*x**2 - sqrt(a)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt (a)*d + b*c)*log( - sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*...