Integrand size = 25, antiderivative size = 337 \[ \int \frac {x^2 (c+d x)^{3/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\frac {x (c+d x)^{3/2}}{b \sqrt {a-b x^2}}+\frac {5 d \sqrt {c+d x} \sqrt {a-b x^2}}{3 b^2}+\frac {11 \sqrt {a} c \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 b^{3/2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {5 \sqrt {a} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 b^{5/2} \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
x*(d*x+c)^(3/2)/b/(-b*x^2+a)^(1/2)+5/3*d*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/b^ 2+11/3*a^(1/2)*c*(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^(3/2)/(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)/(-b*x^2+a)^(1/2)-5/3 *a^(1/2)*(-a*d^2+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^(5/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1 /2)
Result contains complex when optimal does not.
Time = 22.95 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.38 \[ \int \frac {x^2 (c+d x)^{3/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {a-b x^2} \left (-\frac {(c+d x) (5 a d+b x (3 c-2 d x))}{b^2 \left (-a+b x^2\right )}-\frac {11 c d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (a-b x^2\right )+11 i \sqrt {b} c \left (\sqrt {b} c-\sqrt {a} d\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-i \left (6 b c^2-11 \sqrt {a} \sqrt {b} c d+5 a d^2\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{b^2 d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{3 \sqrt {c+d x}} \] Input:
Integrate[(x^2*(c + d*x)^(3/2))/(a - b*x^2)^(3/2),x]
Output:
(Sqrt[a - b*x^2]*(-(((c + d*x)*(5*a*d + b*x*(3*c - 2*d*x)))/(b^2*(-a + b*x ^2))) - (11*c*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(a - b*x^2) + (11*I)*Sqrt [b]*c*(Sqrt[b]*c - Sqrt[a]*d)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sq rt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticE[I*A rcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a] *d)/(Sqrt[b]*c - Sqrt[a]*d)] - I*(6*b*c^2 - 11*Sqrt[a]*Sqrt[b]*c*d + 5*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 )])/(b^2*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(-a + b*x^2))))/(3*Sqrt[c + d*x] )
Time = 0.66 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.23, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {602, 27, 687, 27, 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.
| \(\displaystyle \int \frac {x^2 (c+d x)^{3/2}}{\left (a-b x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 602 |
| \(\displaystyle \frac {\int -\frac {a (c+d x)^{3/2} \left (2 c^2+5 d x c-\frac {5 a d^2}{b}\right )}{2 \sqrt {a-b x^2}}dx}{a \left (b c^2-a d^2\right )}-\frac {(c+d x)^{5/2} (a d-b c x)}{b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {(c+d x)^{3/2} \left (2 c^2+5 d x c-\frac {5 a d^2}{b}\right )}{\sqrt {a-b x^2}}dx}{2 \left (b c^2-a d^2\right )}-\frac {(c+d x)^{5/2} (a d-b c x)}{b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle -\frac {-\frac {2 \int -\frac {5 \sqrt {c+d x} \left (2 c \left (b c^2-a d^2\right )+5 d x \left (b c^2-a d^2\right )\right )}{2 \sqrt {a-b x^2}}dx}{5 b}-\frac {2 c d \sqrt {a-b x^2} (c+d x)^{3/2}}{b}}{2 \left (b c^2-a d^2\right )}-\frac {(c+d x)^{5/2} (a d-b c x)}{b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\int \frac {\left (b c^2-a d^2\right ) \sqrt {c+d x} (2 c+5 d x)}{\sqrt {a-b x^2}}dx}{b}-\frac {2 c d \sqrt {a-b x^2} (c+d x)^{3/2}}{b}}{2 \left (b c^2-a d^2\right )}-\frac {(c+d x)^{5/2} (a d-b c x)}{b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\left (b c^2-a d^2\right ) \int \frac {\sqrt {c+d x} (2 c+5 d x)}{\sqrt {a-b x^2}}dx}{b}-\frac {2 c d \sqrt {a-b x^2} (c+d x)^{3/2}}{b}}{2 \left (b c^2-a d^2\right )}-\frac {(c+d x)^{5/2} (a d-b c x)}{b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle -\frac {\frac {\left (b c^2-a d^2\right ) \left (-\frac {2 \int -\frac {6 b c^2+11 b d x c+5 a d^2}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b}-\frac {10 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}\right )}{b}-\frac {2 c d \sqrt {a-b x^2} (c+d x)^{3/2}}{b}}{2 \left (b c^2-a d^2\right )}-\frac {(c+d x)^{5/2} (a d-b c x)}{b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\left (b c^2-a d^2\right ) \left (\frac {\int \frac {6 b c^2+11 b d x c+5 a d^2}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b}-\frac {10 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}\right )}{b}-\frac {2 c d \sqrt {a-b x^2} (c+d x)^{3/2}}{b}}{2 \left (b c^2-a d^2\right )}-\frac {(c+d x)^{5/2} (a d-b c x)}{b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle -\frac {\frac {\left (b c^2-a d^2\right ) \left (\frac {11 b c \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx-5 \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b}-\frac {10 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}\right )}{b}-\frac {2 c d \sqrt {a-b x^2} (c+d x)^{3/2}}{b}}{2 \left (b c^2-a d^2\right )}-\frac {(c+d x)^{5/2} (a d-b c x)}{b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle -\frac {\frac {\left (b c^2-a d^2\right ) \left (\frac {\frac {11 b c \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}-5 \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b}-\frac {10 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}\right )}{b}-\frac {2 c d \sqrt {a-b x^2} (c+d x)^{3/2}}{b}}{2 \left (b c^2-a d^2\right )}-\frac {(c+d x)^{5/2} (a d-b c x)}{b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle -\frac {\frac {\left (b c^2-a d^2\right ) \left (\frac {-5 \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {22 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}-\frac {10 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}\right )}{b}-\frac {2 c d \sqrt {a-b x^2} (c+d x)^{3/2}}{b}}{2 \left (b c^2-a d^2\right )}-\frac {(c+d x)^{5/2} (a d-b c x)}{b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {\frac {\left (b c^2-a d^2\right ) \left (\frac {-5 \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {22 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}-\frac {10 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}\right )}{b}-\frac {2 c d \sqrt {a-b x^2} (c+d x)^{3/2}}{b}}{2 \left (b c^2-a d^2\right )}-\frac {(c+d x)^{5/2} (a d-b c x)}{b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle -\frac {\frac {\left (b c^2-a d^2\right ) \left (\frac {-\frac {5 \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}-\frac {22 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}-\frac {10 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}\right )}{b}-\frac {2 c d \sqrt {a-b x^2} (c+d x)^{3/2}}{b}}{2 \left (b c^2-a d^2\right )}-\frac {(c+d x)^{5/2} (a d-b c x)}{b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle -\frac {\frac {\left (b c^2-a d^2\right ) \left (\frac {\frac {10 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {22 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}-\frac {10 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}\right )}{b}-\frac {2 c d \sqrt {a-b x^2} (c+d x)^{3/2}}{b}}{2 \left (b c^2-a d^2\right )}-\frac {(c+d x)^{5/2} (a d-b c x)}{b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {\frac {\left (b c^2-a d^2\right ) \left (\frac {\frac {10 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {22 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}-\frac {10 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}\right )}{b}-\frac {2 c d \sqrt {a-b x^2} (c+d x)^{3/2}}{b}}{2 \left (b c^2-a d^2\right )}-\frac {(c+d x)^{5/2} (a d-b c x)}{b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
Input:
Int[(x^2*(c + d*x)^(3/2))/(a - b*x^2)^(3/2),x]
Output:
-(((a*d - b*c*x)*(c + d*x)^(5/2))/(b*(b*c^2 - a*d^2)*Sqrt[a - b*x^2])) - ( (-2*c*d*(c + d*x)^(3/2)*Sqrt[a - b*x^2])/b + ((b*c^2 - a*d^2)*((-10*d*Sqrt [c + d*x]*Sqrt[a - b*x^2])/(3*b) + ((-22*Sqrt[a]*Sqrt[b]*c*Sqrt[c + d*x]*S qrt[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[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) + (10*Sqrt[a]*(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[S qrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/( Sqrt[b]*Sqrt[c + d*x]*Sqrt[a - b*x^2]))/(3*b)))/b)/(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_Symbo l] :> With[{Qx = PolynomialQuotient[x^m, a + b*x^2, x], e = Coeff[Polynomia lRemainder[x^m, a + b*x^2, x], x, 0], f = Coeff[PolynomialRemainder[x^m, a + b*x^2, x], x, 1]}, Simp[(-(c + d*x)^(n + 1))*(a + b*x^2)^(p + 1)*((a*(d*e - c*f) + (b*c*e + a*d*f)*x)/(2*a*(p + 1)*(b*c^2 + a*d^2))), x] + Simp[1/(2 *a*(p + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*ExpandToS um[2*a*(p + 1)*(b*c^2 + a*d^2)*Qx + e*(b*c^2*(2*p + 3) + a*d^2*(n + 2*p + 3 )) - a*c*d*f*n + d*(b*c*e + a*d*f)*(n + 2*p + 4)*x, x], x], x]] /; FreeQ[{a , b, c, d, n}, x] && IGtQ[m, 1] && LtQ[p, -1] && NeQ[b*c^2 + a*d^2, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[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])
Leaf count of result is larger than twice the leaf count of optimal. \(639\) vs. \(2(269)=538\).
Time = 4.78 (sec) , antiderivative size = 640, normalized size of antiderivative = 1.90
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {2 \left (-b d x -b c \right ) \left (\frac {c x}{2 b^{2}}+\frac {d a}{2 b^{3}}\right )}{\sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d x -b c \right )}}+\frac {2 d \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 b^{2}}+\frac {2 \left (-\frac {5 d^{2} a}{6 b^{2}}-\frac {c^{2}}{b}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {11 c d \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{3 b \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(640\) |
| default | \(\frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, \left (5 \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) \sqrt {a b}\, a \,d^{3}-5 \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) \sqrt {a b}\, b \,c^{2} d +6 \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) a b c \,d^{2}-6 \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) b^{2} c^{3}-11 \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) a b c \,d^{2}+11 \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) b^{2} c^{3}-2 b^{2} d^{3} x^{3}+b^{2} c \,d^{2} x^{2}+5 a b \,d^{3} x +3 b^{2} c^{2} d x +5 a b c \,d^{2}\right )}{3 d \left (-b d \,x^{3}-b c \,x^{2}+a d x +a c \right ) b^{3}}\) | \(946\) |
| risch | \(\text {Expression too large to display}\) | \(1126\) |
Input:
int(x^2*(d*x+c)^(3/2)/(-b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
((d*x+c)*(-b*x^2+a))^(1/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*(-2*(-b*d*x-b*c) *(1/2/b^2*c*x+1/2*d/b^3*a)/((x^2-a/b)*(-b*d*x-b*c))^(1/2)+2/3*d/b^2*(-b*d* x^3-b*c*x^2+a*d*x+a*c)^(1/2)+2*(-5/6*d^2/b^2*a-c^2/b)*(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))-11/3*c*d/b*(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))*Ellipt icE(((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 = 283, normalized size of antiderivative = 0.84 \[ \int \frac {x^2 (c+d x)^{3/2}}{\left (a-b x^2\right )^{3/2}} \, dx=-\frac {{\left (7 \, a b c^{2} + 15 \, a^{2} d^{2} - {\left (7 \, b^{2} c^{2} + 15 \, a b d^{2}\right )} x^{2}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 33 \, {\left (b^{2} c d x^{2} - a b c d\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) - 3 \, {\left (2 \, b^{2} d^{2} x^{2} - 3 \, b^{2} c d x - 5 \, a b d^{2}\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}}{9 \, {\left (b^{4} d x^{2} - a b^{3} d\right )}} \] Input:
integrate(x^2*(d*x+c)^(3/2)/(-b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
-1/9*((7*a*b*c^2 + 15*a^2*d^2 - (7*b^2*c^2 + 15*a*b*d^2)*x^2)*sqrt(-b*d)*w eierstrassPInverse(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) + 33*(b^2*c*d*x^2 - a*b*c*d)*sqrt(-b*d)*weie rstrassZeta(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*(2*b^2*d^2*x^2 - 3*b^2*c*d*x - 5*a *b*d^2)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(b^4*d*x^2 - a*b^3*d)
\[ \int \frac {x^2 (c+d x)^{3/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\int \frac {x^{2} \left (c + d x\right )^{\frac {3}{2}}}{\left (a - b x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(x**2*(d*x+c)**(3/2)/(-b*x**2+a)**(3/2),x)
Output:
Integral(x**2*(c + d*x)**(3/2)/(a - b*x**2)**(3/2), x)
\[ \int \frac {x^2 (c+d x)^{3/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{2}} x^{2}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^2*(d*x+c)^(3/2)/(-b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate((d*x + c)^(3/2)*x^2/(-b*x^2 + a)^(3/2), x)
\[ \int \frac {x^2 (c+d x)^{3/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{2}} x^{2}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^2*(d*x+c)^(3/2)/(-b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate((d*x + c)^(3/2)*x^2/(-b*x^2 + a)^(3/2), x)
Timed out. \[ \int \frac {x^2 (c+d x)^{3/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\int \frac {x^2\,{\left (c+d\,x\right )}^{3/2}}{{\left (a-b\,x^2\right )}^{3/2}} \,d x \] Input:
int((x^2*(c + d*x)^(3/2))/(a - b*x^2)^(3/2),x)
Output:
int((x^2*(c + d*x)^(3/2))/(a - b*x^2)^(3/2), x)
\[ \int \frac {x^2 (c+d x)^{3/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\text {too large to display} \] Input:
int(x^2*(d*x+c)^(3/2)/(-b*x^2+a)^(3/2),x)
Output:
( - 11*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a*c**2 - sqrt( a - b*x**2)*a*d**2*x**2 - sqrt(a - b*x**2)*b*c**2*x**2 + sqrt(a - b*x**2)* b*d**2*x**4),x)*a**4*c*d**4 + 11*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt( a - b*x**2)*a*c**2 - sqrt(a - b*x**2)*a*d**2*x**2 - sqrt(a - b*x**2)*b*c** 2*x**2 + sqrt(a - b*x**2)*b*d**2*x**4),x)*a**3*b*c**3*d**2 + 11*sqrt(a - b *x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a*c**2 - sqrt(a - b*x**2)*a*d** 2*x**2 - sqrt(a - b*x**2)*b*c**2*x**2 + sqrt(a - b*x**2)*b*d**2*x**4),x)*a **3*b*c*d**4*x**2 - 11*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2 )*a*c**2 - sqrt(a - b*x**2)*a*d**2*x**2 - sqrt(a - b*x**2)*b*c**2*x**2 + s qrt(a - b*x**2)*b*d**2*x**4),x)*a**2*b**2*c**3*d**2*x**2 - 2*sqrt(a - b*x* *2)*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**3)/(a**2*c + a**2*d*x - 2*a*b*c *x**2 - 2*a*b*d*x**3 + b**2*c*x**4 + b**2*d*x**5),x)*a**2*b**2*c*d**3 + 2* sqrt(a - b*x**2)*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**3)/(a**2*c + a**2* d*x - 2*a*b*c*x**2 - 2*a*b*d*x**3 + b**2*c*x**4 + b**2*d*x**5),x)*a*b**3*c **3*d + 2*sqrt(a - b*x**2)*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**3)/(a**2 *c + a**2*d*x - 2*a*b*c*x**2 - 2*a*b*d*x**3 + b**2*c*x**4 + b**2*d*x**5),x )*a*b**3*c*d**3*x**2 - 2*sqrt(a - b*x**2)*int((sqrt(c + d*x)*sqrt(a - b*x* *2)*x**3)/(a**2*c + a**2*d*x - 2*a*b*c*x**2 - 2*a*b*d*x**3 + b**2*c*x**4 + b**2*d*x**5),x)*b**4*c**3*d*x**2 - 6*sqrt(a - b*x**2)*int((sqrt(c + d*x)* sqrt(a - b*x**2)*x**2)/(a**2*c + a**2*d*x - 2*a*b*c*x**2 - 2*a*b*d*x**3...