Integrand size = 25, antiderivative size = 423 \[ \int \frac {x^3 \sqrt {c+d x}}{\sqrt {a-b x^2}} \, dx=-\frac {2 \left (8 b c^2+25 a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{105 b^2 d^2}+\frac {18 c (c+d x)^{3/2} \sqrt {a-b x^2}}{35 b d^2}-\frac {2 (c+d x)^{5/2} \sqrt {a-b x^2}}{7 b d^2}-\frac {2 \sqrt {a} c \left (8 b c^2+19 a d^2\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{105 b^{3/2} d^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \left (8 b^2 c^4+17 a b c^2 d^2-25 a^2 d^4\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{105 b^{5/2} d^3 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
-2/105*(25*a*d^2+8*b*c^2)*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/b^2/d^2+18/35*c*( d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/b/d^2-2/7*(d*x+c)^(5/2)*(-b*x^2+a)^(1/2)/b/d ^2-2/105*a^(1/2)*c*(19*a*d^2+8*b*c^2)*(d*x+c)^(1/2)*(1-b*x^2/a)^(1/2)*Elli pticE(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/(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^( 1/2)/(-b*x^2+a)^(1/2)+2/105*a^(1/2)*(-25*a^2*d^4+17*a*b*c^2*d^2+8*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^(5/2)/d^3/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 23.39 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.16 \[ \int \frac {x^3 \sqrt {c+d x}}{\sqrt {a-b x^2}} \, dx=\frac {2 \sqrt {a-b x^2} \left (-8 b c^3-19 a c d^2+(c+d x) \left (-25 a d^2+b \left (4 c^2-3 c d x-15 d^2 x^2\right )\right )-\frac {i b c \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (8 b c^2+19 a d^2\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d^2 \left (-a+b x^2\right )}+\frac {i \sqrt {a} \left (8 b^{3/2} c^3-2 \sqrt {a} b c^2 d+19 a \sqrt {b} c d^2-25 a^{3/2} d^3\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{105 b^2 d^2 \sqrt {c+d x}} \] Input:
Integrate[(x^3*Sqrt[c + d*x])/Sqrt[a - b*x^2],x]
Output:
(2*Sqrt[a - b*x^2]*(-8*b*c^3 - 19*a*c*d^2 + (c + d*x)*(-25*a*d^2 + b*(4*c^ 2 - 3*c*d*x - 15*d^2*x^2)) - (I*b*c*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(8*b*c^ 2 + 19*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*(-a + b*x^2)) + (I*Sqrt[a]*(8*b^(3/2)*c^3 - 2*Sqrt[a]*b *c^2*d + 19*a*Sqrt[b]*c*d^2 - 25*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]], (Sqr t[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/(d*Sqrt[-c + (Sqrt[a]*d)/Sqr t[b]]*(-a + b*x^2))))/(105*b^2*d^2*Sqrt[c + d*x])
Time = 0.80 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {604, 27, 2185, 27, 687, 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.
| \(\displaystyle \int \frac {x^3 \sqrt {c+d x}}{\sqrt {a-b x^2}} \, dx\) |
| \(\Big \downarrow \) 604 |
| \(\displaystyle -\frac {2 \int -\frac {\sqrt {c+d x} \left (-9 b c x^2 d^2+5 a c d^2-\left (2 b c^2-5 a d^2\right ) x d\right )}{2 \sqrt {a-b x^2}}dx}{7 b d^3}-\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {c+d x} \left (-9 b c x^2 d^2+5 a c d^2-\left (2 b c^2-5 a d^2\right ) x d\right )}{\sqrt {a-b x^2}}dx}{7 b d^3}-\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^2}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\frac {18}{5} c d \sqrt {a-b x^2} (c+d x)^{3/2}-\frac {2 \int \frac {b d^3 \sqrt {c+d x} \left (2 a c d-\left (8 b c^2+25 a d^2\right ) x\right )}{2 \sqrt {a-b x^2}}dx}{5 b d^2}}{7 b d^3}-\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {18}{5} c d \sqrt {a-b x^2} (c+d x)^{3/2}-\frac {1}{5} d \int \frac {\sqrt {c+d x} \left (2 a c d-\left (8 b c^2+25 a d^2\right ) x\right )}{\sqrt {a-b x^2}}dx}{7 b d^3}-\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^2}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {\frac {18}{5} c d \sqrt {a-b x^2} (c+d x)^{3/2}-\frac {1}{5} d \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a d^2+8 b c^2\right )}{3 b}-\frac {2 \int \frac {a d \left (2 b c^2+25 a d^2\right )+b c \left (8 b c^2+19 a d^2\right ) x}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b}\right )}{7 b d^3}-\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {18}{5} c d \sqrt {a-b x^2} (c+d x)^{3/2}-\frac {1}{5} d \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a d^2+8 b c^2\right )}{3 b}-\frac {\int \frac {a d \left (2 b c^2+25 a d^2\right )+b c \left (8 b c^2+19 a d^2\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b}\right )}{7 b d^3}-\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^2}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {\frac {18}{5} c d \sqrt {a-b x^2} (c+d x)^{3/2}-\frac {1}{5} d \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a d^2+8 b c^2\right )}{3 b}-\frac {\frac {b c \left (19 a d^2+8 b c^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (b c^2-a d^2\right ) \left (25 a d^2+8 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}}{3 b}\right )}{7 b d^3}-\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^2}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\frac {18}{5} c d \sqrt {a-b x^2} (c+d x)^{3/2}-\frac {1}{5} d \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a d^2+8 b c^2\right )}{3 b}-\frac {\frac {b c \sqrt {1-\frac {b x^2}{a}} \left (19 a d^2+8 b c^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (25 a d^2+8 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}}{3 b}\right )}{7 b d^3}-\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^2}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {\frac {18}{5} c d \sqrt {a-b x^2} (c+d x)^{3/2}-\frac {1}{5} d \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a d^2+8 b c^2\right )}{3 b}-\frac {-\frac {\left (b c^2-a d^2\right ) \left (25 a d^2+8 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (19 a d^2+8 b c^2\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}\right )}{7 b d^3}-\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {18}{5} c d \sqrt {a-b x^2} (c+d x)^{3/2}-\frac {1}{5} d \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a d^2+8 b c^2\right )}{3 b}-\frac {-\frac {\left (b c^2-a d^2\right ) \left (25 a d^2+8 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (19 a d^2+8 b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}\right )}{7 b d^3}-\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^2}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\frac {18}{5} c d \sqrt {a-b x^2} (c+d x)^{3/2}-\frac {1}{5} d \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a d^2+8 b c^2\right )}{3 b}-\frac {-\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (25 a d^2+8 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (19 a d^2+8 b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}\right )}{7 b d^3}-\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^2}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {\frac {18}{5} c d \sqrt {a-b x^2} (c+d x)^{3/2}-\frac {1}{5} d \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a d^2+8 b c^2\right )}{3 b}-\frac {\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (25 a d^2+8 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (19 a d^2+8 b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}\right )}{7 b d^3}-\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\frac {18}{5} c d \sqrt {a-b x^2} (c+d x)^{3/2}-\frac {1}{5} d \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a d^2+8 b c^2\right )}{3 b}-\frac {\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (25 a d^2+8 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (19 a d^2+8 b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}\right )}{7 b d^3}-\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^2}\) |
Input:
Int[(x^3*Sqrt[c + d*x])/Sqrt[a - b*x^2],x]
Output:
(-2*(c + d*x)^(5/2)*Sqrt[a - b*x^2])/(7*b*d^2) + ((18*c*d*(c + d*x)^(3/2)* Sqrt[a - b*x^2])/5 - (d*((2*(8*b*c^2 + 25*a*d^2)*Sqrt[c + d*x]*Sqrt[a - b* x^2])/(3*b) - ((-2*Sqrt[a]*Sqrt[b]*c*(8*b*c^2 + 19*a*d^2)*Sqrt[c + d*x]*Sq rt[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 + 25* 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*b)))/5)/ (7*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[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[(c + d*x)^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*d^(m - 1)*(m + n + 2*p + 1))), x] + Simp[1/(b*d^m*(m + n + 2*p + 1)) Int[(c + d*x)^n*(a + b* x^2)^p*ExpandToSum[b*d^m*(m + n + 2*p + 1)*x^m - b*(m + n + 2*p + 1)*(c + d *x)^m - (c + d*x)^(m - 2)*(a*d^2*(m + n - 1) - b*c^2*(m + n + 2*p + 1) - 2* b*c*d*(m + n + p)*x), x], x], x] /; FreeQ[{a, b, c, d, n, p}, x] && IGtQ[m, 1] && NeQ[m + n + 2*p + 1, 0] && IntegerQ[2*p]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) ), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp [c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x ] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && Eq Q[f, 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]))
Time = 2.01 (sec) , antiderivative size = 684, normalized size of antiderivative = 1.62
| method | result | size |
| risch | \(-\frac {2 \left (15 b \,x^{2} d^{2}+3 b c d x +25 a \,d^{2}-4 b \,c^{2}\right ) \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{105 b^{2} d^{2}}+\frac {\left (\frac {c \left (19 a \,d^{2}+8 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 {25 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 {2 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^{2} d^{2} \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(684\) |
| elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {2 x^{2} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{7 b}-\frac {2 c x \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{35 d b}-\frac {2 \left (-\frac {4 c^{2}}{35 d}+\frac {5 a d}{7 b}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 b d}+\frac {2 \left (\frac {2 a \,c^{2}}{35 b d}+\frac {a \left (-\frac {4 c^{2}}{35 d}+\frac {5 a d}{7 b}\right )}{3 b}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 \left (\frac {23 a c}{35 b}-\frac {2 c \left (-\frac {4 c^{2}}{35 d}+\frac {5 a d}{7 b}\right )}{3 d}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(716\) |
| default | \(\text {Expression too large to display}\) | \(1321\) |
Input:
int(x^3*(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+25*a*d^2-4*b*c^2)/b^2*(d*x+c)^(1/2)/d^2*(-b *x^2+a)^(1/2)+1/105/b^2/d^2*(c*(19*a*d^2+8*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*E llipticF(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)))+25*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))+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 = 0.08 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.65 \[ \int \frac {x^3 \sqrt {c+d x}}{\sqrt {a-b x^2}} \, dx=\frac {2 \, {\left ({\left (8 \, b^{2} c^{4} + 13 \, a b c^{2} d^{2} - 75 \, a^{2} d^{4}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 3 \, {\left (8 \, b^{2} c^{3} d + 19 \, a b c d^{3}\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) - 3 \, {\left (15 \, b^{2} d^{4} x^{2} + 3 \, b^{2} c d^{3} x - 4 \, b^{2} c^{2} d^{2} + 25 \, a b d^{4}\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}\right )}}{315 \, b^{3} d^{4}} \] Input:
integrate(x^3*(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x, algorithm="fricas")
Output:
2/315*((8*b^2*c^4 + 13*a*b*c^2*d^2 - 75*a^2*d^4)*sqrt(-b*d)*weierstrassPIn verse(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^3*d + 19*a*b*c*d^3)*sqrt(-b*d)*weierstrassZe ta(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), weier strassPInverse(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 + 25*a*b*d^4)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(b^3*d^4)
\[ \int \frac {x^3 \sqrt {c+d x}}{\sqrt {a-b x^2}} \, dx=\int \frac {x^{3} \sqrt {c + d x}}{\sqrt {a - b x^{2}}}\, dx \] Input:
integrate(x**3*(d*x+c)**(1/2)/(-b*x**2+a)**(1/2),x)
Output:
Integral(x**3*sqrt(c + d*x)/sqrt(a - b*x**2), x)
\[ \int \frac {x^3 \sqrt {c+d x}}{\sqrt {a-b x^2}} \, dx=\int { \frac {\sqrt {d x + c} x^{3}}{\sqrt {-b x^{2} + a}} \,d x } \] Input:
integrate(x^3*(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(d*x + c)*x^3/sqrt(-b*x^2 + a), x)
\[ \int \frac {x^3 \sqrt {c+d x}}{\sqrt {a-b x^2}} \, dx=\int { \frac {\sqrt {d x + c} x^{3}}{\sqrt {-b x^{2} + a}} \,d x } \] Input:
integrate(x^3*(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(d*x + c)*x^3/sqrt(-b*x^2 + a), x)
Timed out. \[ \int \frac {x^3 \sqrt {c+d x}}{\sqrt {a-b x^2}} \, dx=\int \frac {x^3\,\sqrt {c+d\,x}}{\sqrt {a-b\,x^2}} \,d x \] Input:
int((x^3*(c + d*x)^(1/2))/(a - b*x^2)^(1/2),x)
Output:
int((x^3*(c + d*x)^(1/2))/(a - b*x^2)^(1/2), x)
\[ \int \frac {x^3 \sqrt {c+d x}}{\sqrt {a-b x^2}} \, dx=\frac {-46 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a d -4 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b c x -20 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b d \,x^{2}-19 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a b \,d^{2}-8 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) b^{2} c^{2}+23 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{2} d^{2}+4 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a b \,c^{2}}{70 b^{2} d} \] Input:
int(x^3*(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x)
Output:
( - 46*sqrt(c + d*x)*sqrt(a - b*x**2)*a*d - 4*sqrt(c + d*x)*sqrt(a - b*x** 2)*b*c*x - 20*sqrt(c + d*x)*sqrt(a - b*x**2)*b*d*x**2 - 19*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 - 8*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 + 23*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 + 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**2)/(70*b**2*d)