Integrand size = 25, antiderivative size = 407 \[ \int \frac {x^2}{(c+d x)^{3/2} \left (a-b x^2\right )^{3/2}} \, dx=-\frac {a d-b c x}{b \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {d \left (3 b c^2+a d^2\right ) \sqrt {a-b x^2}}{b \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {\sqrt {a} \left (3 b c^2+a d^2\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {b} \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {a} c \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {b} \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
-(-b*c*x+a*d)/b/(-a*d^2+b*c^2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)-d*(a*d^2+3*b *c^2)*(-b*x^2+a)^(1/2)/b/(-a*d^2+b*c^2)^2/(d*x+c)^(1/2)+a^(1/2)*(a*d^2+3*b *c^2)*(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^(1/2)/(-a *d^2+b*c^2)^2/(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)/(-b*x^2+a)^(1/ 2)-a^(1/2)*c*(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)*(1-b*x^2/a)^(1/ 2)*EllipticF(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^(1/2)/(-a*d^2+b*c^2)/(d*x+c)^(1/2)/(-b*x^2+a )^(1/2)
Result contains complex when optimal does not.
Time = 23.14 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.21 \[ \int \frac {x^2}{(c+d x)^{3/2} \left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {a-b x^2} \left (c^2 d+\frac {a d^3}{b}-\frac {(c+d x) \left (b c^2 x+a d (-2 c+d x)\right )}{-a+b x^2}+\frac {i \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (3 b c^2+a d^2\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d \left (-a+b x^2\right )}+\frac {i \left (2 b^{3/2} c^3-3 \sqrt {a} b c^2 d+2 a \sqrt {b} c d^2-a^{3/2} d^3\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {b} d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{\left (b c^2-a d^2\right )^2 \sqrt {c+d x}} \] Input:
Integrate[x^2/((c + d*x)^(3/2)*(a - b*x^2)^(3/2)),x]
Output:
(Sqrt[a - b*x^2]*(c^2*d + (a*d^3)/b - ((c + d*x)*(b*c^2*x + a*d*(-2*c + d* x)))/(-a + b*x^2) + (I*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(3*b*c^2 + a*d^2)*Sq rt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x) /(c + d*x))]*(c + d*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqr t[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/(d *(-a + b*x^2)) + (I*(2*b^(3/2)*c^3 - 3*Sqrt[a]*b*c^2*d + 2*a*Sqrt[b]*c*d^2 - a^(3/2)*d^3)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a] *d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/(Sqrt[b]*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(-a + b*x^2))))/ ((b*c^2 - a*d^2)^2*Sqrt[c + d*x])
Time = 0.65 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {602, 27, 688, 27, 600, 509, 508, 327, 512, 511, 321}
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.
| \(\displaystyle \int \frac {x^2}{\left (a-b x^2\right )^{3/2} (c+d x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 602 |
| \(\displaystyle \frac {\int -\frac {a \left (2 b c^2-b d x c+a d^2\right )}{2 b (c+d x)^{3/2} \sqrt {a-b x^2}}dx}{a \left (b c^2-a d^2\right )}-\frac {a d-b c x}{b \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {2 b c^2-b d x c+a d^2}{(c+d x)^{3/2} \sqrt {a-b x^2}}dx}{2 b \left (b c^2-a d^2\right )}-\frac {a d-b c x}{b \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle -\frac {\frac {2 \int \frac {b \left (2 c \left (b c^2+a d^2\right )+d \left (3 b c^2+a d^2\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{b c^2-a d^2}+\frac {2 d \sqrt {a-b x^2} \left (a d^2+3 b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{2 b \left (b c^2-a d^2\right )}-\frac {a d-b c x}{b \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {b \int \frac {2 c \left (b c^2+a d^2\right )+d \left (3 b c^2+a d^2\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{b c^2-a d^2}+\frac {2 d \sqrt {a-b x^2} \left (a d^2+3 b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{2 b \left (b c^2-a d^2\right )}-\frac {a d-b c x}{b \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle -\frac {\frac {b \left (\left (a d^2+3 b c^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx-c \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx\right )}{b c^2-a d^2}+\frac {2 d \sqrt {a-b x^2} \left (a d^2+3 b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{2 b \left (b c^2-a d^2\right )}-\frac {a d-b c x}{b \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle -\frac {\frac {b \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (a d^2+3 b c^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}-c \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx\right )}{b c^2-a d^2}+\frac {2 d \sqrt {a-b x^2} \left (a d^2+3 b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{2 b \left (b c^2-a d^2\right )}-\frac {a d-b c x}{b \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle -\frac {\frac {b \left (-c \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (a d^2+3 b c^2\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{b c^2-a d^2}+\frac {2 d \sqrt {a-b x^2} \left (a d^2+3 b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{2 b \left (b c^2-a d^2\right )}-\frac {a d-b c x}{b \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {\frac {b \left (-c \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (a d^2+3 b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{b c^2-a d^2}+\frac {2 d \sqrt {a-b x^2} \left (a d^2+3 b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{2 b \left (b c^2-a d^2\right )}-\frac {a d-b c x}{b \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle -\frac {\frac {b \left (-\frac {c \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (a d^2+3 b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{b c^2-a d^2}+\frac {2 d \sqrt {a-b x^2} \left (a d^2+3 b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{2 b \left (b c^2-a d^2\right )}-\frac {a d-b c x}{b \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle -\frac {\frac {b \left (\frac {2 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (a d^2+3 b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{b c^2-a d^2}+\frac {2 d \sqrt {a-b x^2} \left (a d^2+3 b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{2 b \left (b c^2-a d^2\right )}-\frac {a d-b c x}{b \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {\frac {b \left (\frac {2 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (a d^2+3 b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{b c^2-a d^2}+\frac {2 d \sqrt {a-b x^2} \left (a d^2+3 b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{2 b \left (b c^2-a d^2\right )}-\frac {a d-b c x}{b \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
Input:
Int[x^2/((c + d*x)^(3/2)*(a - b*x^2)^(3/2)),x]
Output:
-((a*d - b*c*x)/(b*(b*c^2 - a*d^2)*Sqrt[c + d*x]*Sqrt[a - b*x^2])) - ((2*d *(3*b*c^2 + a*d^2)*Sqrt[a - b*x^2])/((b*c^2 - a*d^2)*Sqrt[c + d*x]) + (b*( (-2*Sqrt[a]*(3*b*c^2 + a*d^2)*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)])/(Sqrt[b]*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) + (2*Sqrt[a]*c*(b*c^2 - a*d^2)*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]* c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/ Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*Sqrt[c + d*x ]*Sqrt[a - b*x^2])))/(b*c^2 - a*d^2))/(2*b*(b*c^2 - a*d^2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[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] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
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[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt [c + d*x])) Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] , x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(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[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[B/d Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp [(B*c - A*d)/d Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, b, c, d, A, B}, x] && NegQ[b/a]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m, a + b*x^2, x], e = Coeff[Polynomia lRemainder[x^m, a + b*x^2, x], x, 0], f = Coeff[PolynomialRemainder[x^m, a + b*x^2, x], x, 1]}, Simp[(-(c + d*x)^(n + 1))*(a + b*x^2)^(p + 1)*((a*(d*e - c*f) + (b*c*e + a*d*f)*x)/(2*a*(p + 1)*(b*c^2 + a*d^2))), x] + Simp[1/(2 *a*(p + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*ExpandToS um[2*a*(p + 1)*(b*c^2 + a*d^2)*Qx + e*(b*c^2*(2*p + 3) + a*d^2*(n + 2*p + 3 )) - a*c*d*f*n + d*(b*c*e + a*d*f)*(n + 2*p + 4)*x, x], x], x]] /; FreeQ[{a , b, c, d, n}, x] && IGtQ[m, 1] && LtQ[p, -1] && NeQ[b*c^2 + a*d^2, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( (m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Leaf count of result is larger than twice the leaf count of optimal. \(774\) vs. \(2(345)=690\).
Time = 6.60 (sec) , antiderivative size = 775, normalized size of antiderivative = 1.90
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (\frac {2 b d \left (\frac {\left (a \,d^{2}+3 b \,c^{2}\right ) x^{2}}{2 \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) b}-\frac {c x}{2 d b \left (a \,d^{2}-b \,c^{2}\right )}-\frac {2 c^{2} a}{\left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) b}\right )}{\sqrt {-\left (x^{3}+\frac {c \,x^{2}}{d}-\frac {a x}{b}-\frac {a c}{b d}\right ) b d}}+\frac {2 \left (-\frac {2 c \,d^{2} a}{a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}}+\frac {c}{a \,d^{2}-b \,c^{2}}\right ) \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 )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {d \left (a \,d^{2}+3 b \,c^{2}\right ) \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 )}{\left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(775\) |
| default | \(\text {Expression too large to display}\) | \(1244\) |
Input:
int(x^2/(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*(1/2*(a*d ^2+3*b*c^2)/(a^2*d^4-2*a*b*c^2*d^2+b^2*c^4)/b*x^2-1/2*c/d/b/(a*d^2-b*c^2)* x-2*c^2/(a^2*d^4-2*a*b*c^2*d^2+b^2*c^4)*a/b)/(-(x^3+c/d*x^2-a*x/b-a/b*c/d) *b*d)^(1/2)+2*(-2*c*d^2/(a^2*d^4-2*a*b*c^2*d^2+b^2*c^4)*a+c/(a*d^2-b*c^2)) *(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))-d*(a*d^2+3*b*c^2)/(a^2*d^4-2*a*b*c^2*d^2+b^2*c^4)*(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 = 553, normalized size of antiderivative = 1.36 \[ \int \frac {x^2}{(c+d x)^{3/2} \left (a-b x^2\right )^{3/2}} \, dx=\frac {{\left (3 \, a b c^{4} + 5 \, a^{2} c^{2} d^{2} - {\left (3 \, b^{2} c^{3} d + 5 \, a b c d^{3}\right )} x^{3} - {\left (3 \, b^{2} c^{4} + 5 \, a b c^{2} d^{2}\right )} x^{2} + {\left (3 \, a b c^{3} d + 5 \, a^{2} c d^{3}\right )} x\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) - 3 \, {\left (3 \, a b c^{3} d + a^{2} c d^{3} - {\left (3 \, b^{2} c^{2} d^{2} + a b d^{4}\right )} x^{3} - {\left (3 \, b^{2} c^{3} d + a b c d^{3}\right )} x^{2} + {\left (3 \, a b c^{2} d^{2} + a^{2} d^{4}\right )} x\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) - 3 \, {\left (4 \, a b c^{2} d^{2} - {\left (3 \, b^{2} c^{2} d^{2} + a b d^{4}\right )} x^{2} - {\left (b^{2} c^{3} d - a b c d^{3}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}}{3 \, {\left (a b^{3} c^{5} d - 2 \, a^{2} b^{2} c^{3} d^{3} + a^{3} b c d^{5} - {\left (b^{4} c^{4} d^{2} - 2 \, a b^{3} c^{2} d^{4} + a^{2} b^{2} d^{6}\right )} x^{3} - {\left (b^{4} c^{5} d - 2 \, a b^{3} c^{3} d^{3} + a^{2} b^{2} c d^{5}\right )} x^{2} + {\left (a b^{3} c^{4} d^{2} - 2 \, a^{2} b^{2} c^{2} d^{4} + a^{3} b d^{6}\right )} x\right )}} \] Input:
integrate(x^2/(d*x+c)^(3/2)/(-b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
1/3*((3*a*b*c^4 + 5*a^2*c^2*d^2 - (3*b^2*c^3*d + 5*a*b*c*d^3)*x^3 - (3*b^2 *c^4 + 5*a*b*c^2*d^2)*x^2 + (3*a*b*c^3*d + 5*a^2*c*d^3)*x)*sqrt(-b*d)*weie rstrassPInverse(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*(3*a*b*c^3*d + a^2*c*d^3 - (3*b^2*c^2*d^2 + a*b*d^4)*x^3 - (3*b^2*c^3*d + a*b*c*d^3)*x^2 + (3*a*b*c^2*d^2 + a^2*d^4)* x)*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)) - 3*(4*a*b*c^2*d^2 - (3*b^2*c^2*d^2 + a*b*d^4)*x^2 - (b^2*c^3*d - a*b*c*d^3)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(a*b^3*c^5*d - 2*a^2*b^2*c^3*d^3 + a^3*b*c*d^5 - (b^4*c^ 4*d^2 - 2*a*b^3*c^2*d^4 + a^2*b^2*d^6)*x^3 - (b^4*c^5*d - 2*a*b^3*c^3*d^3 + a^2*b^2*c*d^5)*x^2 + (a*b^3*c^4*d^2 - 2*a^2*b^2*c^2*d^4 + a^3*b*d^6)*x)
\[ \int \frac {x^2}{(c+d x)^{3/2} \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {x^{2}}{\left (a - b x^{2}\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(x**2/(d*x+c)**(3/2)/(-b*x**2+a)**(3/2),x)
Output:
Integral(x**2/((a - b*x**2)**(3/2)*(c + d*x)**(3/2)), x)
\[ \int \frac {x^2}{(c+d x)^{3/2} \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {x^{2}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^2/(d*x+c)^(3/2)/(-b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate(x^2/((-b*x^2 + a)^(3/2)*(d*x + c)^(3/2)), x)
\[ \int \frac {x^2}{(c+d x)^{3/2} \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {x^{2}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^2/(d*x+c)^(3/2)/(-b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate(x^2/((-b*x^2 + a)^(3/2)*(d*x + c)^(3/2)), x)
Timed out. \[ \int \frac {x^2}{(c+d x)^{3/2} \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {x^2}{{\left (a-b\,x^2\right )}^{3/2}\,{\left (c+d\,x\right )}^{3/2}} \,d x \] Input:
int(x^2/((a - b*x^2)^(3/2)*(c + d*x)^(3/2)),x)
Output:
int(x^2/((a - b*x^2)^(3/2)*(c + d*x)^(3/2)), x)
\[ \int \frac {x^2}{(c+d x)^{3/2} \left (a-b x^2\right )^{3/2}} \, dx=\frac {-\sqrt {-b \,x^{2}+a}\, \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{3}}{b^{2} d^{2} x^{6}+2 b^{2} c d \,x^{5}-2 a b \,d^{2} x^{4}+b^{2} c^{2} x^{4}-4 a b c d \,x^{3}+a^{2} d^{2} x^{2}-2 a b \,c^{2} x^{2}+2 a^{2} c d x +a^{2} c^{2}}d x \right ) b c \,d^{2}-\sqrt {-b \,x^{2}+a}\, \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{3}}{b^{2} d^{2} x^{6}+2 b^{2} c d \,x^{5}-2 a b \,d^{2} x^{4}+b^{2} c^{2} x^{4}-4 a b c d \,x^{3}+a^{2} d^{2} x^{2}-2 a b \,c^{2} x^{2}+2 a^{2} c d x +a^{2} c^{2}}d x \right ) b \,d^{3} x -\sqrt {-b \,x^{2}+a}\, \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x}{b^{2} d^{2} x^{6}+2 b^{2} c d \,x^{5}-2 a b \,d^{2} x^{4}+b^{2} c^{2} x^{4}-4 a b c d \,x^{3}+a^{2} d^{2} x^{2}-2 a b \,c^{2} x^{2}+2 a^{2} c d x +a^{2} c^{2}}d x \right ) a c \,d^{2}-\sqrt {-b \,x^{2}+a}\, \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x}{b^{2} d^{2} x^{6}+2 b^{2} c d \,x^{5}-2 a b \,d^{2} x^{4}+b^{2} c^{2} x^{4}-4 a b c d \,x^{3}+a^{2} d^{2} x^{2}-2 a b \,c^{2} x^{2}+2 a^{2} c d x +a^{2} c^{2}}d x \right ) a \,d^{3} x -4 \sqrt {-b \,x^{2}+a}\, \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x}{b^{2} d^{2} x^{6}+2 b^{2} c d \,x^{5}-2 a b \,d^{2} x^{4}+b^{2} c^{2} x^{4}-4 a b c d \,x^{3}+a^{2} d^{2} x^{2}-2 a b \,c^{2} x^{2}+2 a^{2} c d x +a^{2} c^{2}}d x \right ) b \,c^{3}-4 \sqrt {-b \,x^{2}+a}\, \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x}{b^{2} d^{2} x^{6}+2 b^{2} c d \,x^{5}-2 a b \,d^{2} x^{4}+b^{2} c^{2} x^{4}-4 a b c d \,x^{3}+a^{2} d^{2} x^{2}-2 a b \,c^{2} x^{2}+2 a^{2} c d x +a^{2} c^{2}}d x \right ) b \,c^{2} d x +4 \sqrt {d x +c}\, c +2 \sqrt {d x +c}\, d x}{6 \sqrt {-b \,x^{2}+a}\, b c d \left (d x +c \right )} \] Input:
int(x^2/(d*x+c)^(3/2)/(-b*x^2+a)^(3/2),x)
Output:
( - sqrt(a - b*x**2)*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**3)/(a**2*c**2 + 2*a**2*c*d*x + a**2*d**2*x**2 - 2*a*b*c**2*x**2 - 4*a*b*c*d*x**3 - 2*a*b *d**2*x**4 + b**2*c**2*x**4 + 2*b**2*c*d*x**5 + b**2*d**2*x**6),x)*b*c*d** 2 - sqrt(a - b*x**2)*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**3)/(a**2*c**2 + 2*a**2*c*d*x + a**2*d**2*x**2 - 2*a*b*c**2*x**2 - 4*a*b*c*d*x**3 - 2*a*b *d**2*x**4 + b**2*c**2*x**4 + 2*b**2*c*d*x**5 + b**2*d**2*x**6),x)*b*d**3* x - sqrt(a - b*x**2)*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x)/(a**2*c**2 + 2 *a**2*c*d*x + a**2*d**2*x**2 - 2*a*b*c**2*x**2 - 4*a*b*c*d*x**3 - 2*a*b*d* *2*x**4 + b**2*c**2*x**4 + 2*b**2*c*d*x**5 + b**2*d**2*x**6),x)*a*c*d**2 - sqrt(a - b*x**2)*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x)/(a**2*c**2 + 2*a* *2*c*d*x + a**2*d**2*x**2 - 2*a*b*c**2*x**2 - 4*a*b*c*d*x**3 - 2*a*b*d**2* x**4 + b**2*c**2*x**4 + 2*b**2*c*d*x**5 + b**2*d**2*x**6),x)*a*d**3*x - 4* sqrt(a - b*x**2)*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x)/(a**2*c**2 + 2*a** 2*c*d*x + a**2*d**2*x**2 - 2*a*b*c**2*x**2 - 4*a*b*c*d*x**3 - 2*a*b*d**2*x **4 + b**2*c**2*x**4 + 2*b**2*c*d*x**5 + b**2*d**2*x**6),x)*b*c**3 - 4*sqr t(a - b*x**2)*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x)/(a**2*c**2 + 2*a**2*c *d*x + a**2*d**2*x**2 - 2*a*b*c**2*x**2 - 4*a*b*c*d*x**3 - 2*a*b*d**2*x**4 + b**2*c**2*x**4 + 2*b**2*c*d*x**5 + b**2*d**2*x**6),x)*b*c**2*d*x + 4*sq rt(c + d*x)*c + 2*sqrt(c + d*x)*d*x)/(6*sqrt(a - b*x**2)*b*c*d*(c + d*x))