Integrand size = 25, antiderivative size = 389 \[ \int \frac {x^3}{(c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=-\frac {2 c^3 \sqrt {a-b x^2}}{d^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}-\frac {2 \sqrt {c+d x} \sqrt {a-b x^2}}{3 b d^2}+\frac {2 \sqrt {a} c \left (8 b c^2-5 a d^2\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 \sqrt {b} d^3 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \left (8 b c^2+a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 b^{3/2} d^3 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
-2*c^3*(-b*x^2+a)^(1/2)/d^2/(-a*d^2+b*c^2)/(d*x+c)^(1/2)-2/3*(d*x+c)^(1/2) *(-b*x^2+a)^(1/2)/b/d^2+2/3*a^(1/2)*c*(-5*a*d^2+8*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)/d^3/(-a*d^2+b*c^2)/(b^(1/2 )*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)/(-b*x^2+a)^(1/2)-2/3*a^(1/2)*(a*d^2 +8*b*c^2)*(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^(3/2)/d^3/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 23.15 (sec) , antiderivative size = 548, normalized size of antiderivative = 1.41 \[ \int \frac {x^3}{(c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=-\frac {2 \left (c d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (8 b c^2-5 a d^2\right ) \left (-a+b x^2\right )+d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (a-b x^2\right ) \left (-a d^2 (c+d x)+b c^2 (4 c+d x)\right )-i \sqrt {b} c \left (8 b^{3/2} c^3-8 \sqrt {a} b c^2 d-5 a \sqrt {b} c d^2+5 a^{3/2} d^3\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+i \sqrt {a} d \left (-8 b^{3/2} c^3+2 \sqrt {a} b c^2 d+5 a \sqrt {b} c d^2+a^{3/2} d^3\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )\right )}{3 b d^4 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}} \] Input:
Integrate[x^3/((c + d*x)^(3/2)*Sqrt[a - b*x^2]),x]
Output:
(-2*(c*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(8*b*c^2 - 5*a*d^2)*(-a + b*x^2) + d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(a - b*x^2)*(-(a*d^2*(c + d*x)) + b* c^2*(4*c + d*x)) - I*Sqrt[b]*c*(8*b^(3/2)*c^3 - 8*Sqrt[a]*b*c^2*d - 5*a*Sq rt[b]*c*d^2 + 5*a^(3/2)*d^3)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqr t[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticE[I*Ar cSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]* d)/(Sqrt[b]*c - Sqrt[a]*d)] + I*Sqrt[a]*d*(-8*b^(3/2)*c^3 + 2*Sqrt[a]*b*c^ 2*d + 5*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)*Elli pticF[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)]))/(3*b*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqr t[b]]*(b*c^2 - a*d^2)*Sqrt[c + d*x]*Sqrt[a - b*x^2])
Time = 0.82 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {603, 27, 2185, 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.
| \(\displaystyle \int \frac {x^3}{\sqrt {a-b x^2} (c+d x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 603 |
| \(\displaystyle \frac {2 \int -\frac {\frac {a c^2}{d}-\left (a-\frac {2 b c^2}{d^2}\right ) x c-\left (\frac {b c^2}{d}-a d\right ) x^2}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{b c^2-a d^2}-\frac {2 c^3 \sqrt {a-b x^2}}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\frac {a c^2}{d}-\left (a-\frac {2 b c^2}{d^2}\right ) x c-\left (\frac {b c^2}{d}-a d\right ) x^2}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{b c^2-a d^2}-\frac {2 c^3 \sqrt {a-b x^2}}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle -\frac {-\frac {2 \int -\frac {a d \left (2 b c^2+a d^2\right )+b c \left (8 b c^2-5 a d^2\right ) x}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b d^2}-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}-\frac {2 c^3 \sqrt {a-b x^2}}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\int \frac {a d \left (2 b c^2+a d^2\right )+b c \left (8 b c^2-5 a d^2\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b d^2}-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}-\frac {2 c^3 \sqrt {a-b x^2}}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle -\frac {\frac {\frac {b c \left (8 b c^2-5 a d^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (b c^2-a d^2\right ) \left (a d^2+8 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}}{3 b d^2}-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}-\frac {2 c^3 \sqrt {a-b x^2}}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle -\frac {\frac {\frac {b c \sqrt {1-\frac {b x^2}{a}} \left (8 b c^2-5 a d^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (a d^2+8 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}}{3 b d^2}-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}-\frac {2 c^3 \sqrt {a-b x^2}}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle -\frac {\frac {-\frac {\left (b c^2-a d^2\right ) \left (a d^2+8 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (8 b c^2-5 a d^2\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b d^2}-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}-\frac {2 c^3 \sqrt {a-b x^2}}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {\frac {-\frac {\left (b c^2-a d^2\right ) \left (a d^2+8 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (8 b c^2-5 a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b d^2}-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}-\frac {2 c^3 \sqrt {a-b x^2}}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle -\frac {\frac {-\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (a d^2+8 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (8 b c^2-5 a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b d^2}-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}-\frac {2 c^3 \sqrt {a-b x^2}}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle -\frac {\frac {\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (a d^2+8 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (8 b c^2-5 a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b d^2}-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}-\frac {2 c^3 \sqrt {a-b x^2}}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {\frac {\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (a d^2+8 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (8 b c^2-5 a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b d^2}-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}-\frac {2 c^3 \sqrt {a-b x^2}}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
Input:
Int[x^3/((c + d*x)^(3/2)*Sqrt[a - b*x^2]),x]
Output:
(-2*c^3*Sqrt[a - b*x^2])/(d^2*(b*c^2 - a*d^2)*Sqrt[c + d*x]) - ((-2*(a/b - c^2/d^2)*Sqrt[c + d*x]*Sqrt[a - b*x^2])/3 + ((-2*Sqrt[a]*Sqrt[b]*c*(8*b*c ^2 - 5*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)])/(d*Sqrt[( Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) + (2*Sqrt[a]* (b*c^2 - a*d^2)*(8*b*c^2 + a*d^2)*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sq rt[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]*d*Sqrt[c + d*x]*Sq rt[a - b*x^2]))/(3*b*d^2))/(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_Symbol ] :> With[{Qx = PolynomialQuotient[x^m, c + d*x, x], R = PolynomialRemainde r[x^m, c + d*x, x]}, Simp[d*R*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 1)*(b*c^2 + a*d^2))), x] + Simp[1/((n + 1)*(b*c^2 + a*d^2)) Int[(c + d*x) ^(n + 1)*(a + b*x^2)^p*ExpandToSum[(n + 1)*(b*c^2 + a*d^2)*Qx + b*c*R*(n + 1) - b*d*R*(n + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IGt Q[m, 1] && LtQ[n, -1] && NeQ[b*c^2 + a*d^2, 0]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) ^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si mp[1/(b*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x )^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p )*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d , e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && !(EqQ[d, 0] && True) && !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Leaf count of result is larger than twice the leaf count of optimal. \(690\) vs. \(2(321)=642\).
Time = 4.84 (sec) , antiderivative size = 691, normalized size of antiderivative = 1.78
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (\frac {2 \left (-b d \,x^{2}+a d \right ) c^{3}}{d^{3} \left (a \,d^{2}-b \,c^{2}\right ) \sqrt {\left (x +\frac {c}{d}\right ) \left (-b d \,x^{2}+a d \right )}}-\frac {2 \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 d^{2} b}+\frac {2 \left (\frac {c^{2}}{d^{3}}+\frac {b \,c^{4}}{d^{3} \left (a \,d^{2}-b \,c^{2}\right )}+\frac {a}{3 b d}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 \left (-\frac {5 c}{3 d^{2}}+\frac {b \,c^{3}}{d^{2} \left (a \,d^{2}-b \,c^{2}\right )}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(691\) |
| risch | \(\text {Expression too large to display}\) | \(1134\) |
| default | \(\text {Expression too large to display}\) | \(1310\) |
Input:
int(x^3/(d*x+c)^(3/2)/(-b*x^2+a)^(1/2),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^2+a*d )/d^3/(a*d^2-b*c^2)*c^3/((x+c/d)*(-b*d*x^2+a*d))^(1/2)-2/3/d^2/b*(-b*d*x^3 -b*c*x^2+a*d*x+a*c)^(1/2)+2*(c^2/d^3+b*c^4/d^3/(a*d^2-b*c^2)+1/3*a/b/d)*(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) )+2*(-5/3*c/d^2+b/d^2*c^3/(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)*((-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 = 0.10 (sec) , antiderivative size = 370, normalized size of antiderivative = 0.95 \[ \int \frac {x^3}{(c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=-\frac {2 \, {\left ({\left (8 \, b^{2} c^{5} - 11 \, a b c^{3} d^{2} - 3 \, a^{2} c d^{4} + {\left (8 \, b^{2} c^{4} d - 11 \, a b c^{2} d^{3} - 3 \, a^{2} d^{5}\right )} x\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 3 \, {\left (8 \, b^{2} c^{4} d - 5 \, a b c^{2} d^{3} + {\left (8 \, b^{2} c^{3} d^{2} - 5 \, a b c d^{4}\right )} x\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) + 3 \, {\left (4 \, b^{2} c^{3} d^{2} - a b c d^{4} + {\left (b^{2} c^{2} d^{3} - a b d^{5}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}\right )}}{9 \, {\left (b^{3} c^{3} d^{4} - a b^{2} c d^{6} + {\left (b^{3} c^{2} d^{5} - a b^{2} d^{7}\right )} x\right )}} \] Input:
integrate(x^3/(d*x+c)^(3/2)/(-b*x^2+a)^(1/2),x, algorithm="fricas")
Output:
-2/9*((8*b^2*c^5 - 11*a*b*c^3*d^2 - 3*a^2*c*d^4 + (8*b^2*c^4*d - 11*a*b*c^ 2*d^3 - 3*a^2*d^5)*x)*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*(8*b^2 *c^4*d - 5*a*b*c^2*d^3 + (8*b^2*c^3*d^2 - 5*a*b*c*d^4)*x)*sqrt(-b*d)*weier strassZeta(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*b^2*c^3*d^2 - a*b*c*d^4 + (b^2*c ^2*d^3 - a*b*d^5)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(b^3*c^3*d^4 - a*b^2* c*d^6 + (b^3*c^2*d^5 - a*b^2*d^7)*x)
\[ \int \frac {x^3}{(c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\int \frac {x^{3}}{\sqrt {a - b x^{2}} \left (c + d x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(x**3/(d*x+c)**(3/2)/(-b*x**2+a)**(1/2),x)
Output:
Integral(x**3/(sqrt(a - b*x**2)*(c + d*x)**(3/2)), x)
\[ \int \frac {x^3}{(c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\int { \frac {x^{3}}{\sqrt {-b x^{2} + a} {\left (d x + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^3/(d*x+c)^(3/2)/(-b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate(x^3/(sqrt(-b*x^2 + a)*(d*x + c)^(3/2)), x)
\[ \int \frac {x^3}{(c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\int { \frac {x^{3}}{\sqrt {-b x^{2} + a} {\left (d x + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^3/(d*x+c)^(3/2)/(-b*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate(x^3/(sqrt(-b*x^2 + a)*(d*x + c)^(3/2)), x)
Timed out. \[ \int \frac {x^3}{(c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\int \frac {x^3}{\sqrt {a-b\,x^2}\,{\left (c+d\,x\right )}^{3/2}} \,d x \] Input:
int(x^3/((a - b*x^2)^(1/2)*(c + d*x)^(3/2)),x)
Output:
int(x^3/((a - b*x^2)^(1/2)*(c + d*x)^(3/2)), x)
\[ \int \frac {x^3}{(c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\frac {-2 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a d -4 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b c x -\left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b \,d^{2} x^{4}-2 b c d \,x^{3}+a \,d^{2} x^{2}-b \,c^{2} x^{2}+2 a c d x +a \,c^{2}}d x \right ) a b c \,d^{2}-\left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b \,d^{2} x^{4}-2 b c d \,x^{3}+a \,d^{2} x^{2}-b \,c^{2} x^{2}+2 a c d x +a \,c^{2}}d x \right ) a b \,d^{3} x -8 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b \,d^{2} x^{4}-2 b c d \,x^{3}+a \,d^{2} x^{2}-b \,c^{2} x^{2}+2 a c d x +a \,c^{2}}d x \right ) b^{2} c^{3}-8 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b \,d^{2} x^{4}-2 b c d \,x^{3}+a \,d^{2} x^{2}-b \,c^{2} x^{2}+2 a c d x +a \,c^{2}}d x \right ) b^{2} c^{2} d x -\left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b \,d^{2} x^{4}-2 b c d \,x^{3}+a \,d^{2} x^{2}-b \,c^{2} x^{2}+2 a c d x +a \,c^{2}}d x \right ) a^{2} c \,d^{2}-\left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b \,d^{2} x^{4}-2 b c d \,x^{3}+a \,d^{2} x^{2}-b \,c^{2} x^{2}+2 a c d x +a \,c^{2}}d x \right ) a^{2} d^{3} x +4 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b \,d^{2} x^{4}-2 b c d \,x^{3}+a \,d^{2} x^{2}-b \,c^{2} x^{2}+2 a c d x +a \,c^{2}}d x \right ) a b \,c^{3}+4 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b \,d^{2} x^{4}-2 b c d \,x^{3}+a \,d^{2} x^{2}-b \,c^{2} x^{2}+2 a c d x +a \,c^{2}}d x \right ) a b \,c^{2} d x}{6 b^{2} c d \left (d x +c \right )} \] Input:
int(x^3/(d*x+c)^(3/2)/(-b*x^2+a)^(1/2),x)
Output:
( - 2*sqrt(c + d*x)*sqrt(a - b*x**2)*a*d - 4*sqrt(c + d*x)*sqrt(a - b*x**2 )*b*c*x - int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c**2 + 2*a*c*d*x + a*d**2*x**2 - b*c**2*x**2 - 2*b*c*d*x**3 - b*d**2*x**4),x)*a*b*c*d**2 - in t((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c**2 + 2*a*c*d*x + a*d**2*x**2 - b*c**2*x**2 - 2*b*c*d*x**3 - b*d**2*x**4),x)*a*b*d**3*x - 8*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c**2 + 2*a*c*d*x + a*d**2*x**2 - b*c**2*x **2 - 2*b*c*d*x**3 - b*d**2*x**4),x)*b**2*c**3 - 8*int((sqrt(c + d*x)*sqrt (a - b*x**2)*x**2)/(a*c**2 + 2*a*c*d*x + a*d**2*x**2 - b*c**2*x**2 - 2*b*c *d*x**3 - b*d**2*x**4),x)*b**2*c**2*d*x - int((sqrt(c + d*x)*sqrt(a - b*x* *2))/(a*c**2 + 2*a*c*d*x + a*d**2*x**2 - b*c**2*x**2 - 2*b*c*d*x**3 - b*d* *2*x**4),x)*a**2*c*d**2 - int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c**2 + 2 *a*c*d*x + a*d**2*x**2 - b*c**2*x**2 - 2*b*c*d*x**3 - b*d**2*x**4),x)*a**2 *d**3*x + 4*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c**2 + 2*a*c*d*x + a*d **2*x**2 - b*c**2*x**2 - 2*b*c*d*x**3 - b*d**2*x**4),x)*a*b*c**3 + 4*int(( sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c**2 + 2*a*c*d*x + a*d**2*x**2 - b*c**2 *x**2 - 2*b*c*d*x**3 - b*d**2*x**4),x)*a*b*c**2*d*x)/(6*b**2*c*d*(c + d*x) )