Integrand size = 23, antiderivative size = 391 \[ \int x \sqrt {c+d x} \sqrt {a-b x^2} \, dx=-\frac {4}{105} \left (\frac {5 a}{b}-\frac {4 c^2}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}-\frac {2 (4 c-5 d x) (c+d x)^{3/2} \sqrt {a-b x^2}}{35 d^2}+\frac {16 \sqrt {a} c \left (b c^2-2 a d^2\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{105 \sqrt {b} d^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {4 \sqrt {a} \left (4 b^2 c^4-9 a b c^2 d^2+5 a^2 d^4\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{105 b^{3/2} d^3 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
-4/105*(5*a/b-4*c^2/d^2)*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)-2/35*(-5*d*x+4*c)* (d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/d^2+16/105*a^(1/2)*c*(-2*a*d^2+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/(b^(1/2)*( d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)/(-b*x^2+a)^(1/2)-4/105*a^(1/2)*(5*a^2* d^4-9*a*b*c^2*d^2+4*b^2*c^4)*(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 = 11.27 (sec) , antiderivative size = 533, normalized size of antiderivative = 1.36 \[ \int x \sqrt {c+d x} \sqrt {a-b x^2} \, dx=\frac {\sqrt {a-b x^2} \left (\frac {2 (c+d x) \left (-10 a d^2+b \left (-4 c^2+3 c d x+15 d^2 x^2\right )\right )}{b d^2}+\frac {4 \left (4 c d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (b c^2-2 a d^2\right ) \left (-a+b x^2\right )-4 i \sqrt {b} c \left (b^{3/2} c^3-\sqrt {a} b c^2 d-2 a \sqrt {b} c d^2+2 a^{3/2} d^3\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-i \sqrt {a} d \left (4 b^{3/2} c^3-\sqrt {a} b c^2 d-8 a \sqrt {b} c d^2+5 a^{3/2} d^3\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )\right )}{b d^4 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{105 \sqrt {c+d x}} \] Input:
Integrate[x*Sqrt[c + d*x]*Sqrt[a - b*x^2],x]
Output:
(Sqrt[a - b*x^2]*((2*(c + d*x)*(-10*a*d^2 + b*(-4*c^2 + 3*c*d*x + 15*d^2*x ^2)))/(b*d^2) + (4*(4*c*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(b*c^2 - 2*a*d^ 2)*(-a + b*x^2) - (4*I)*Sqrt[b]*c*(b^(3/2)*c^3 - Sqrt[a]*b*c^2*d - 2*a*Sqr t[b]*c*d^2 + 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)*EllipticE[I*Arc Sinh[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*(4*b^(3/2)*c^3 - Sqrt[a]*b*c^2*d - 8*a*Sqrt[b]*c*d^2 + 5*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)*Ellipt icF[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)]))/(b*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b] ]*(-a + b*x^2))))/(105*Sqrt[c + d*x])
Time = 0.60 (sec) , antiderivative size = 388, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {596, 682, 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.
| \(\displaystyle \int x \sqrt {a-b x^2} \sqrt {c+d x} \, dx\) |
| \(\Big \downarrow \) 596 |
| \(\displaystyle \frac {\int \frac {(a d+b c x) \sqrt {a-b x^2}}{\sqrt {c+d x}}dx}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}{7 b}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {-\frac {4 \int \frac {b \left (a d \left (b c^2-5 a d^2\right )+4 b c \left (b c^2-2 a d^2\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 b d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (-5 a d^2+4 b c^2-3 b c d x\right )}{15 d^2}}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}{7 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {2 \int \frac {a d \left (b c^2-5 a d^2\right )+4 b c \left (b c^2-2 a d^2\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (-5 a d^2+4 b c^2-3 b c d x\right )}{15 d^2}}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}{7 b}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {-\frac {2 \left (\frac {4 b c \left (b c^2-2 a d^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (4 b c^2-5 a d^2\right ) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (-5 a d^2+4 b c^2-3 b c d x\right )}{15 d^2}}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}{7 b}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {-\frac {2 \left (\frac {4 b c \sqrt {1-\frac {b x^2}{a}} \left (b c^2-2 a d^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (4 b c^2-5 a d^2\right ) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (-5 a d^2+4 b c^2-3 b c d x\right )}{15 d^2}}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}{7 b}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {-\frac {2 \left (-\frac {\left (4 b c^2-5 a d^2\right ) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {8 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (b c^2-2 a d^2\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (-5 a d^2+4 b c^2-3 b c d x\right )}{15 d^2}}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}{7 b}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {-\frac {2 \left (-\frac {\left (4 b c^2-5 a d^2\right ) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {8 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (b c^2-2 a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (-5 a d^2+4 b c^2-3 b c d x\right )}{15 d^2}}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}{7 b}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {-\frac {2 \left (-\frac {\sqrt {1-\frac {b x^2}{a}} \left (4 b c^2-5 a d^2\right ) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {8 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (b c^2-2 a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (-5 a d^2+4 b c^2-3 b c d x\right )}{15 d^2}}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}{7 b}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {-\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (4 b c^2-5 a d^2\right ) \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {8 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (b c^2-2 a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (-5 a d^2+4 b c^2-3 b c d x\right )}{15 d^2}}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}{7 b}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {-\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (4 b c^2-5 a d^2\right ) \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {8 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (b c^2-2 a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (-5 a d^2+4 b c^2-3 b c d x\right )}{15 d^2}}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}{7 b}\) |
Input:
Int[x*Sqrt[c + d*x]*Sqrt[a - b*x^2],x]
Output:
(-2*Sqrt[c + d*x]*(a - b*x^2)^(3/2))/(7*b) + ((-2*Sqrt[c + d*x]*(4*b*c^2 - 5*a*d^2 - 3*b*c*d*x)*Sqrt[a - b*x^2])/(15*d^2) - (2*((-8*Sqrt[a]*Sqrt[b]* c*(b*c^2 - 2*a*d^2)*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqr t[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*Sq rt[a]*(4*b*c^2 - 5*a*d^2)*(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]*d*Sqrt[c + d*x]*Sqrt[a - b*x^2])))/(15*d^2))/(7*b)
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[(x_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c + d*x)^n*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 2))), x] - Simp[n/(b*(n + 2*p + 2)) Int[(c + d*x)^(n - 1)*(a + b*x^2)^p*(a*d - b*c*x), x], x] /; FreeQ[{a, b, c, d, p}, x] && GtQ[n, 0] && NeQ[n + 2*p + 2, 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[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))) Int[(d + e*x) ^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x ], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || ! RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Leaf count of result is larger than twice the leaf count of optimal. \(684\) vs. \(2(321)=642\).
Time = 1.21 (sec) , antiderivative size = 685, normalized size of antiderivative = 1.75
| method | result | size |
| risch | \(-\frac {2 \left (-15 b \,x^{2} d^{2}-3 b c d x +10 a \,d^{2}+4 b \,c^{2}\right ) \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{105 b \,d^{2}}+\frac {2 \left (\frac {4 c \left (2 a \,d^{2}-b \,c^{2}\right ) \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \left (\left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )-\frac {c \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{d}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {5 a^{2} d^{3} \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{b \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {a \,c^{2} d \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right ) \sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}}{105 b \,d^{2} \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(685\) |
| elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (\frac {2 x^{2} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{7}+\frac {2 c x \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{35 d}-\frac {2 \left (\frac {2 a d}{7}+\frac {4 c^{2} b}{35 d}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 b d}+\frac {2 \left (-\frac {2 c^{2} a}{35 d}+\frac {\left (\frac {2 a d}{7}+\frac {4 c^{2} b}{35 d}\right ) a}{3 b}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 \left (\frac {12 a c}{35}-\frac {2 \left (\frac {2 a d}{7}+\frac {4 c^{2} b}{35 d}\right ) c}{3 d}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(698\) |
| default | \(\text {Expression too large to display}\) | \(1321\) |
Input:
int(x*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/105*(-15*b*d^2*x^2-3*b*c*d*x+10*a*d^2+4*b*c^2)/b*(d*x+c)^(1/2)/d^2*(-b* x^2+a)^(1/2)+2/105/b/d^2*(4*c*(2*a*d^2-b*c^2)*(a*b)^(1/2)*2^(1/2)*((x+1/b* (a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*(- 2*(x-1/b*(a*b)^(1/2))*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(1/2*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a *b)^(1/2))^(1/2),(-2/b*(a*b)^(1/2)/(c/d-1/b*(a*b)^(1/2)))^(1/2))-c/d*Ellip ticF(1/2*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2),(-2/b*(a*b)^(1/ 2)/(c/d-1/b*(a*b)^(1/2)))^(1/2)))+5*a^2*d^3/b*(a*b)^(1/2)*2^(1/2)*((x+1/b* (a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*(- 2*(x-1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1 /2)*EllipticF(1/2*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2),(-2/b* (a*b)^(1/2)/(c/d-1/b*(a*b)^(1/2)))^(1/2))-a*c^2*d*(a*b)^(1/2)*2^(1/2)*((x+ 1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2 )*(-2*(x-1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c )^(1/2)*EllipticF(1/2*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2),(- 2/b*(a*b)^(1/2)/(c/d-1/b*(a*b)^(1/2)))^(1/2)))*((d*x+c)*(-b*x^2+a))^(1/2)/ (d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Time = 0.10 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.71 \[ \int x \sqrt {c+d x} \sqrt {a-b x^2} \, dx=-\frac {2 \, {\left (2 \, {\left (4 \, b^{2} c^{4} - 11 \, a b c^{2} d^{2} + 15 \, a^{2} d^{4}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 24 \, {\left (b^{2} c^{3} d - 2 \, a b c d^{3}\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) - 3 \, {\left (15 \, b^{2} d^{4} x^{2} + 3 \, b^{2} c d^{3} x - 4 \, b^{2} c^{2} d^{2} - 10 \, a b d^{4}\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}\right )}}{315 \, b^{2} d^{4}} \] Input:
integrate(x*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2),x, algorithm="fricas")
Output:
-2/315*(2*(4*b^2*c^4 - 11*a*b*c^2*d^2 + 15*a^2*d^4)*sqrt(-b*d)*weierstrass PInverse(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) + 24*(b^2*c^3*d - 2*a*b*c*d^3)*sqrt(-b*d)*weierstrassZ eta(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), 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*(15*b^2*d^4*x^2 + 3*b^2*c*d^3*x - 4*b^2*c^ 2*d^2 - 10*a*b*d^4)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(b^2*d^4)
\[ \int x \sqrt {c+d x} \sqrt {a-b x^2} \, dx=\int x \sqrt {a - b x^{2}} \sqrt {c + d x}\, dx \] Input:
integrate(x*(d*x+c)**(1/2)*(-b*x**2+a)**(1/2),x)
Output:
Integral(x*sqrt(a - b*x**2)*sqrt(c + d*x), x)
\[ \int x \sqrt {c+d x} \sqrt {a-b x^2} \, dx=\int { \sqrt {-b x^{2} + a} \sqrt {d x + c} x \,d x } \] Input:
integrate(x*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(-b*x^2 + a)*sqrt(d*x + c)*x, x)
\[ \int x \sqrt {c+d x} \sqrt {a-b x^2} \, dx=\int { \sqrt {-b x^{2} + a} \sqrt {d x + c} x \,d x } \] Input:
integrate(x*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(-b*x^2 + a)*sqrt(d*x + c)*x, x)
Timed out. \[ \int x \sqrt {c+d x} \sqrt {a-b x^2} \, dx=\int x\,\sqrt {a-b\,x^2}\,\sqrt {c+d\,x} \,d x \] Input:
int(x*(a - b*x^2)^(1/2)*(c + d*x)^(1/2),x)
Output:
int(x*(a - b*x^2)^(1/2)*(c + d*x)^(1/2), x)
\[ \int x \sqrt {c+d x} \sqrt {a-b x^2} \, dx=\frac {-\frac {12 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a d}{35}+\frac {2 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b c x}{35}+\frac {2 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b d \,x^{2}}{7}-\frac {8 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a b \,d^{2}}{35}+\frac {4 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) b^{2} c^{2}}{35}+\frac {6 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{2} d^{2}}{35}-\frac {2 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a b \,c^{2}}{35}}{b d} \] Input:
int(x*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2),x)
Output:
(2*( - 6*sqrt(c + d*x)*sqrt(a - b*x**2)*a*d + sqrt(c + d*x)*sqrt(a - b*x** 2)*b*c*x + 5*sqrt(c + d*x)*sqrt(a - b*x**2)*b*d*x**2 - 4*int((sqrt(c + d*x )*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a*b*d**2 + 2*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x - b*c*x**2 - b*d *x**3),x)*b**2*c**2 + 3*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a**2*d**2 - int((sqrt(c + d*x)*sqrt(a - b*x**2)) /(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a*b*c**2))/(35*b*d)