Integrand size = 33, antiderivative size = 398 \[ \int \frac {x \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\frac {2 (7 c C-5 B d) \sqrt {c+d x} \sqrt {a-b x^2}}{15 b d^2}-\frac {2 C (c+d x)^{3/2} \sqrt {a-b x^2}}{5 b d^2}-\frac {2 \sqrt {a} \left (15 A b+9 a C+\frac {2 b c (4 c C-5 B d)}{d^2}\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{15 b^{3/2} d \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \left (a d^2 (7 c C-5 B d)+b c \left (8 c^2 C-10 B c d+15 A d^2\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{15 b^{3/2} d^3 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
2/15*(-5*B*d+7*C*c)*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/b/d^2-2/5*C*(d*x+c)^(3/ 2)*(-b*x^2+a)^(1/2)/b/d^2-2/15*a^(1/2)*(15*A*b+9*a*C+2*b*c*(-5*B*d+4*C*c)/ d^2)*(d*x+c)^(1/2)*((-b*x^2+a)/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)/d /((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)/(-b*x^2+a)^(1/2)+2/15*a^(1/2)*(a*d^ 2*(-5*B*d+7*C*c)+b*c*(15*A*d^2-10*B*c*d+8*C*c^2))*((d*x+c)/(c+a^(1/2)*d/b^ (1/2)))^(1/2)*((-b*x^2+a)/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 = 25.61 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.35 \[ \int \frac {x \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\frac {2 \sqrt {a-b x^2} \left (-9 a C d^2-b \left (8 c^2 C-10 B c d+15 A d^2\right )+b (c+d x) (4 c C-5 B d-3 C d x)+\frac {i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (9 a C d^2+b \left (8 c^2 C-10 B c d+15 A d^2\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}+\frac {i \sqrt {a} \sqrt {b} \left (9 a C d^2-\sqrt {a} \sqrt {b} d (2 c C+5 B d)+b \left (8 c^2 C-10 B c d+15 A d^2\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{15 b^2 d^2 \sqrt {c+d x}} \] Input:
Integrate[(x*(A + B*x + C*x^2))/(Sqrt[c + d*x]*Sqrt[a - b*x^2]),x]
Output:
(2*Sqrt[a - b*x^2]*(-9*a*C*d^2 - b*(8*c^2*C - 10*B*c*d + 15*A*d^2) + b*(c + d*x)*(4*c*C - 5*B*d - 3*C*d*x) + (I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(9*a *C*d^2 + b*(8*c^2*C - 10*B*c*d + 15*A*d^2))*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*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[ b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/(d^2*Sqrt[-c + (Sqrt[a]*d)/Sqr t[b]]*(-a + b*x^2)) + (I*Sqrt[a]*Sqrt[b]*(9*a*C*d^2 - Sqrt[a]*Sqrt[b]*d*(2 *c*C + 5*B*d) + b*(8*c^2*C - 10*B*c*d + 15*A*d^2))*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)])/(d*Sqrt[-c + (Sqrt[a]*d )/Sqrt[b]]*(-a + b*x^2))))/(15*b^2*d^2*Sqrt[c + d*x])
Time = 1.53 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2185, 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.
| \(\displaystyle \int \frac {x \left (A+B x+C x^2\right )}{\sqrt {a-b x^2} \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle -\frac {2 \int -\frac {-b (7 c C-5 B d) x^2 d^2+3 a c C d^2-\left (2 b C c^2-5 A b d^2-3 a C d^2\right ) x d}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{5 b d^3}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {-b (7 c C-5 B d) x^2 d^2+3 a c C d^2-\left (2 b C c^2-5 A b d^2-3 a C d^2\right ) x d}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{5 b d^3}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\frac {2}{3} d \sqrt {a-b x^2} \sqrt {c+d x} (7 c C-5 B d)-\frac {2 \int -\frac {b d^3 \left (a d (2 c C+5 B d)+\left (9 a C d^2+b \left (8 C c^2-10 B d c+15 A d^2\right )\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b d^2}}{5 b d^3}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} d \int \frac {a d (2 c C+5 B d)+\left (9 a C d^2+b \left (8 C c^2-10 B d c+15 A d^2\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx+\frac {2}{3} d \sqrt {a-b x^2} \sqrt {c+d x} (7 c C-5 B d)}{5 b d^3}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {\frac {1}{3} d \left (\frac {\left (9 a C d^2+b \left (15 A d^2-10 B c d+8 c^2 C\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (a d^2 (7 c C-5 B d)+b c \left (15 A d^2-10 B c d+8 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )+\frac {2}{3} d \sqrt {a-b x^2} \sqrt {c+d x} (7 c C-5 B d)}{5 b d^3}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\frac {1}{3} d \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (9 a C d^2+b \left (15 A d^2-10 B c d+8 c^2 C\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (a d^2 (7 c C-5 B d)+b c \left (15 A d^2-10 B c d+8 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )+\frac {2}{3} d \sqrt {a-b x^2} \sqrt {c+d x} (7 c C-5 B d)}{5 b d^3}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {\frac {1}{3} d \left (-\frac {\left (a d^2 (7 c C-5 B d)+b c \left (15 A d^2-10 B c d+8 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (9 a C d^2+b \left (15 A d^2-10 B c d+8 c^2 C\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )+\frac {2}{3} d \sqrt {a-b x^2} \sqrt {c+d x} (7 c C-5 B d)}{5 b d^3}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {1}{3} d \left (-\frac {\left (a d^2 (7 c C-5 B d)+b c \left (15 A d^2-10 B c d+8 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (9 a C d^2+b \left (15 A d^2-10 B c d+8 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )+\frac {2}{3} d \sqrt {a-b x^2} \sqrt {c+d x} (7 c C-5 B d)}{5 b d^3}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\frac {1}{3} d \left (-\frac {\sqrt {1-\frac {b x^2}{a}} \left (a d^2 (7 c C-5 B d)+b c \left (15 A d^2-10 B c d+8 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (9 a C d^2+b \left (15 A d^2-10 B c d+8 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )+\frac {2}{3} d \sqrt {a-b x^2} \sqrt {c+d x} (7 c C-5 B d)}{5 b d^3}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {\frac {1}{3} d \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (a d^2 (7 c C-5 B d)+b c \left (15 A d^2-10 B c d+8 c^2 C\right )\right ) \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (9 a C d^2+b \left (15 A d^2-10 B c d+8 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )+\frac {2}{3} d \sqrt {a-b x^2} \sqrt {c+d x} (7 c C-5 B d)}{5 b d^3}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\frac {1}{3} d \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (a d^2 (7 c C-5 B d)+b c \left (15 A d^2-10 B c d+8 c^2 C\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (9 a C d^2+b \left (15 A d^2-10 B c d+8 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )+\frac {2}{3} d \sqrt {a-b x^2} \sqrt {c+d x} (7 c C-5 B d)}{5 b d^3}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\) |
Input:
Int[(x*(A + B*x + C*x^2))/(Sqrt[c + d*x]*Sqrt[a - b*x^2]),x]
Output:
(-2*C*(c + d*x)^(3/2)*Sqrt[a - b*x^2])/(5*b*d^2) + ((2*d*(7*c*C - 5*B*d)*S qrt[c + d*x]*Sqrt[a - b*x^2])/3 + (d*((-2*Sqrt[a]*(9*a*C*d^2 + b*(8*c^2*C - 10*B*c*d + 15*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]*d*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x ^2]) + (2*Sqrt[a]*(a*d^2*(7*c*C - 5*B*d) + b*c*(8*c^2*C - 10*B*c*d + 15*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])))/3)/(5*b*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[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[(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]))
Time = 3.62 (sec) , antiderivative size = 663, normalized size of antiderivative = 1.67
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (d x +c \right )}\, \left (-\frac {2 C x \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{5 b d}-\frac {2 \left (B -\frac {4 c C}{5 d}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 b d}+\frac {2 \left (\frac {2 C a c}{5 b d}+\frac {\left (B -\frac {4 c C}{5 d}\right ) a}{3 b}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 \left (A +\frac {3 a C}{5 b}-\frac {2 \left (B -\frac {4 c C}{5 d}\right ) c}{3 d}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {d x +c}}\) | \(663\) |
| risch | \(-\frac {2 \left (3 C x d +5 B d -4 C c \right ) \sqrt {-b \,x^{2}+a}\, \sqrt {d x +c}}{15 b \,d^{2}}+\frac {\left (\frac {\left (15 A b \,d^{2}-10 B b c d +9 a C \,d^{2}+8 C 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 )}{b \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {5 a B \,d^{2} \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 {2 C a c 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 )}{b \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right ) \sqrt {\left (-b \,x^{2}+a \right ) \left (d x +c \right )}}{15 b \,d^{2} \sqrt {-b \,x^{2}+a}\, \sqrt {d x +c}}\) | \(688\) |
| default | \(\text {Expression too large to display}\) | \(2448\) |
Input:
int(x*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x,method=_RETURNVERBOSE )
Output:
((-b*x^2+a)*(d*x+c))^(1/2)/(-b*x^2+a)^(1/2)/(d*x+c)^(1/2)*(-2/5*C/b/d*x*(- b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/3*(B-4/5*c*C/d)/b/d*(-b*d*x^3-b*c*x^2+a *d*x+a*c)^(1/2)+2*(2/5*C/b/d*a*c+1/3*(B-4/5*c*C/d)/b*a)*(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*(A+3/5*a*C/b -2/3*(B-4/5*c*C/d)/d*c)*(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)*Ellipt icF(((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 = 283, normalized size of antiderivative = 0.71 \[ \int \frac {x \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\frac {2 \, {\left ({\left (8 \, C b c^{3} - 10 \, B b c^{2} d - 15 \, B a d^{3} + 3 \, {\left (C a + 5 \, A b\right )} c d^{2}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 3 \, {\left (8 \, C b c^{2} d - 10 \, B b c d^{2} + 3 \, {\left (3 \, C a + 5 \, A b\right )} d^{3}\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) - 3 \, {\left (3 \, C b d^{3} x - 4 \, C b c d^{2} + 5 \, B b d^{3}\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}\right )}}{45 \, b^{2} d^{4}} \] Input:
integrate(x*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x, algorithm="fri cas")
Output:
2/45*((8*C*b*c^3 - 10*B*b*c^2*d - 15*B*a*d^3 + 3*(C*a + 5*A*b)*c*d^2)*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*C*b*c^2*d - 10*B*b*c*d^2 + 3* (3*C*a + 5*A*b)*d^3)*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*(3*C*b*d^3*x - 4*C*b*c*d^2 + 5*B*b*d^3)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/ (b^2*d^4)
\[ \int \frac {x \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int \frac {x \left (A + B x + C x^{2}\right )}{\sqrt {a - b x^{2}} \sqrt {c + d x}}\, dx \] Input:
integrate(x*(C*x**2+B*x+A)/(d*x+c)**(1/2)/(-b*x**2+a)**(1/2),x)
Output:
Integral(x*(A + B*x + C*x**2)/(sqrt(a - b*x**2)*sqrt(c + d*x)), x)
\[ \int \frac {x \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} x}{\sqrt {-b x^{2} + a} \sqrt {d x + c}} \,d x } \] Input:
integrate(x*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x, algorithm="max ima")
Output:
integrate((C*x^2 + B*x + A)*x/(sqrt(-b*x^2 + a)*sqrt(d*x + c)), x)
\[ \int \frac {x \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} x}{\sqrt {-b x^{2} + a} \sqrt {d x + c}} \,d x } \] Input:
integrate(x*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x, algorithm="gia c")
Output:
integrate((C*x^2 + B*x + A)*x/(sqrt(-b*x^2 + a)*sqrt(d*x + c)), x)
Timed out. \[ \int \frac {x \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int \frac {x\,\left (C\,x^2+B\,x+A\right )}{\sqrt {a-b\,x^2}\,\sqrt {c+d\,x}} \,d x \] Input:
int((x*(A + B*x + C*x^2))/((a - b*x^2)^(1/2)*(c + d*x)^(1/2)),x)
Output:
int((x*(A + B*x + C*x^2))/((a - b*x^2)^(1/2)*(c + d*x)^(1/2)), x)
\[ \int \frac {x \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\frac {-10 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a b d -6 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a c d -4 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b \,c^{2} x -15 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a \,b^{2} d^{2}-9 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a b c \,d^{2}+10 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) b^{3} c d -8 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) b^{2} c^{3}+5 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{2} b \,d^{2}+3 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{2} c \,d^{2}+4 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a b \,c^{3}}{10 b^{2} c d} \] Input:
int(x*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x)
Output:
( - 10*sqrt(c + d*x)*sqrt(a - b*x**2)*a*b*d - 6*sqrt(c + d*x)*sqrt(a - b*x **2)*a*c*d - 4*sqrt(c + d*x)*sqrt(a - b*x**2)*b*c**2*x - 15*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**2* d**2 - 9*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*c*d**2 + 10*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**3*c*d - 8*int((sqrt(c + d*x)*sqr t(a - b*x**2)*x**2)/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*b**2*c**3 + 5*i nt((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*b*d**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*c*d**2 + 4*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**3)/(10*b**2*c*d)